SSTRING THEORY, GRAVITY AND EXPERIMENT
Thibault Damour and Marc Lilley Institut des Hautes Etudes Scientifiques, 35 route de Chartres, F-91440 Bures-sur-Yvette, France Institut d’Astrophysique de Paris, 98, bis Blvd. Arago, F-75014 Paris, France a r X i v : . [ h e p - t h ] F e b ontents
1. Introduction 32. Classical black holes as dissipative branes 42.1. Global properties of black holes 52.2. Black hole electrodynamics 132.3. Black hole viscosity 192.4. Irreversible thermodynamics of black holes 232.5. Hawking Radiation 253. Experimental tests of gravity 323.1. Universal coupling of matter to gravity 333.2. Experimental tests of the coupling of matter to gravity 363.2.1. How constant are the constants? 363.2.2. Tests of local Lorentz invariance 373.2.3. Universality of free fall 383.2.4. Universality of the gravitational redshift 393.3. Tests of the dynamics of the gravitational field 393.3.1. Brief review of the theoretical background 393.3.2. Experimental tests in the solar system 453.3.3. Objects with strong self-gravity: binary pulsars 473.3.4. Tests of gravity on very large scales 514. String-inspired phenomenology of the gravitational sector 524.1. Overview 524.2. Long range modifications of gravity 544.2.1. The cosmological attractor mechanism 554.2.2. Observable consequences of the Cosmological Attractor Mechanism 575. String-related signals in cosmology 625.1. Alternatives to slow-roll inflation 625.2. Cosmic superstrings 635.2.1. Phenomenological origin 635.2.2. Observational signatures 655.2.3. String dynamics 675.2.4. Gravitational waves from a cosmological string network 716. Conclusion 72Bibliography 72 bstract The aim of these lectures is to give an introduction to several topics which lieat the intersection of string theory, gravity theory and gravity phenomenology. Onesuccessively reviews: (i) the “membrane” approach to the dissipative dynamics ofclassical black holes, (ii) the current experimental tests of gravity, and their theoret-ical interpretation, (iii) some aspects of the string-inspired phenomenology of thegravitational sector, and (iv) some possibilities for observing string-related signalsin cosmology (including a discussion of gravitational wave signals from cosmicsuperstrings).
1. Introduction
The common theme of these lectures is gravity , and their aim is to discuss a fewcases where string theory might have an interesting interplay either with gravitytheory, or with gravity phenomenology. We shall discuss the following topics: • Classical black holes as dissipative branes.
The idea here is to review the“classic” work on black holes of the seventies which led to the picture of blackholes as being analog to dissipative branes endowed with finite electrical resis-tivity, and finite surface viscosity. In particular, we shall review the derivationof the (classical) surface viscosity of black holes, which has recently acquired anew (quantum) interest in view of AdS/CFT duality. • Hawking radiation from black holes.
To complete our classical account ofirreversible properties of black holes, we shall also give a direct derivation of thephenomenon of Hawking radiation, because of its crucial importance in fixingthe coefficient between the area of the horizon and black hole “entropy”. • Experimental tests of gravity.
Before discussing possible phenomenologicalconsequences of string theory in the gravitational sector, we find useful to sum-marize the present status of experimental tests of gravity, as well as the theoreticalframeworks used to interpret them. In particular, we emphasize that binary pulsarexperiments have already given us accurate tests of some aspects of strong-field(and radiative) relativistic gravity. • String-inspired phenomenology of the gravitational sector.
In this section weshall discuss (without any attempt at completeness) some of the ideas that havebeen suggested about observable signals possibly connected to string theory. In3
T. Damour and M. Lilley particular, we shall discuss the cosmological attractor mechanism which leadsto a rather rich gravitational phenomenology that will be probed soon by variousgravitational experiments. • String-related signals in cosmology.
After discussing a few alternatives toslow-roll inflation (and the possible relaxation of the Lyth bound when usingnon-linear kinetic terms for the inflaton), one discusses in some detail cosmicsuperstrings . We explain, in particular, how one computes the gravitational waveburst signal emitted by the cusps that periodically form during the dynamicalevolution of generic string loops.A final warning: by lack of time (and energy), no attempt has been made togive exhaustive and fair references to original and/or relevant work. The givenreferences are indicative, and should be viewed as entry points into the relevantliterature. With the modern, web-based, easy access to the scientific literature itis hoped that the reader will have no difficulty in using the few given referencesas starting points for an instructive navigation on the vast sea of the physics liter-ature.
2. Classical black holes as dissipative branes
Early work on (Schwarzschild, Reissner-Nordström, or Kerr-Newman) blackholes (BHs) in the 1950’s and 1960’s treated them as passive objects , i.e., asgiven geometrical backgrounds (and potential wells). This viewpoint changed inthe early 1970’s when the study of the dynamics of BHs was initiated by Pen-rose [1], Christodoulou and Ruffini [2, 3], Hawking [4], and Bardeen, Carter andHawking [5]. In the works [1–5], only the global dynamics of BHs was consid-ered, i.e., their total mass, their total angular momentum, their total irreduciblemass, and the variation of these quantities. This viewpoint further evolved in theworks of Hartle and Hawking [6], Hanni and Ruffini [7], Damour [8–10], andZnajek [11], which studied the local dynamics of BH horizons . In this new ap-proach (which was later called the “membrane paradigm” [12]) a BH horizon isinterpreted as a brane with dissipative properties, such as, for instance, an elec-trical resistivity ρ , equal to Ohms [8, 11] independently of the type of BH,and a surface (shear) viscosity, equal to η = π [9, 10]. When divided by theentropy density found by Hawking ( S/A = ), the latter shear viscosity yieldsthe ratio π , a result which has recently raised a renewed interest in connectionwith AdS/CFT, through the work of Kovtun, Son, and Starinets [13, 14]. tring theory, gravity and experiment Let us start by reviewing the study of the global dynamics of BHs. Initially, BHswere thought of as given geometrical backgrounds. In the case of a sphericallysymmetric object of mass M without any additional attribute, Schwarzschild de-rived the first exact solution of Einstein’s equations only a few weeks after Ein-stein had obtained the final form of the field equations. Schwarzschild’s solutionis as follows. In dimensions, setting G = c = 1 , the metric for a sphericallysymmetric background can be written in the form d s = − A ( r )d T + B ( r )d r + r (cid:0) d θ + sin θ d ϕ (cid:1) (2.1)where T denotes the usual Schwarzschild-type time coordinate, and where thecoefficients A ( r ) and B ( r ) read A ( r ) = 1 − GMr ,B ( r ) = A ( r ) . (2.2)This result was generalized in independent works by Reissner, and by Nordström(1918) for electrically charged spherically symmetric objects, in which case A ( r ) and B ( r ) are given by A ( r ) = 1 − Mr + Q r ,B ( r ) = A ( r ) . (2.3)We shall not review here the long historical path which led to interpreting theabove solutions, as well as their later generalizations due to Kerr (who added tothe mass M the spin J ), and Newman et al. (mass, spin and charge), as BHs. Upto the 1960’s BHs were viewed only as passive gravitational wells. For instance,one could think of adiabatically lowering a small mass m at the end of a stringuntil it disappears within the BH, thereby converting its mass-energy mc intowork. More realistically, one was thinking of matter orbiting a BH and radiatingaway its potential energy (up to a maximum, given by the binding energy of thelast stable circular orbit around a BH). This viewpoint changed in the 1970’s,when BHs started being considered as dynamical objects, able to exchange mass,angular momentum and charge with the external world. Whereas in the simplestcase above, one uses the attractive potential well created by the mass M withoutextracting energy from the BH, Penrose [1], showed that energy could in princi-ple be extracted from a BH itself by means of what is now called a (gedanken)“Penrose process” (see FIG. 1). Namely, if one considers a time-independent Note that in the case of a spinning BH, one often introduces the useful quantity a = J/M , i.e., the ratio of the total angular momentum to the mass of the BH, which has the dimension of length. T. Damour and M. Lilley
M, Q, J z E , p , e ϕ E , p , e ϕ E , p , e ϕ Figure 1. In this figure, we schematically illustrate the “Penrose process”, i.e., the splitting of aningoing particle into one that falls into the BH and another that exits at infinity. background and a BH that is more complicated than Schwarzschild’s, say a KerrBH, one may extract energy using a test particle coming in from infinity withenergy E , angular momentum p ϕ , and electric charge e . By Noether’s theo-rem, the time-translation, axial and U (1) gauge symmetries of the backgroundguarantee the conservation of E , p ϕ and e during the “fall” of the test particle.Moreover, if, in a quantum process, the test particle splits, near the BH, intotwo particles and , with E , p ϕ , e , and E , p ϕ , e respectively, then, undercertain conditions, one finds that particle 3 can be absorbed by the BH, and thatparticle 2 may come out at infinity with more energy than the incoming particle1 . A detailed analysis of the efficiency of such gedanken Penrose processes byChristodoulou and Ruffini [2, 3] then led to the understanding of the existence ofa fundamental irreversibility in BH dynamics, and to the discovery of the BH mass formula . Let us explain these results.The basic idea is to explore the physics of BHs through a sequence of infinites-imal changes of their state obtained by injecting in them some test particles. Onestarts by writing that the total mass-energy, spin and charge of the BH change, byabsorption of particle 3, as δM = E = E − E ,δJ = J = J − J ,δQ = e = e − e . (2.4) tring theory, gravity and experiment This preliminary result can be further exploited by making use of the Hamilton-Jacobi equation. Considering an on-shell particle of mass µ , and adopting the ( − + ++) signature, the Hamilton-Jacobi equation reads g µν ( p µ − eA µ ) ( p ν − eA ν ) = − µ , (2.5)in which p µ = ∂S/∂x µ , S is the action and the partial derivatives are takenw.r.t. the coordinates x µ . The details of the splitting process will be irrelevant,as only particle 3 matters in the calculation. In an axially symmetric and time-independent background, S can be taken as a linear function of T and ϕ , S = − ET + p ϕ ϕ + S ( r, θ ) . (2.6)where E = − p T = − p is the conserved energy, p ϕ is the conserved ϕ -component of angular momentum and the last term is the contribution from termsthat depend on the angle θ and on the radial distance r . Let us consider the caseof a Reissner-Nordström BH, where calculations are easier: the inverse metric iseasily computed and (2.5) can then be written explicitely as − A ( r ) ( p − eA ) + A ( r ) p r + 1 r (cid:18) p θ + 1sin θ p ϕ (cid:19) = − µ (2.7)which we re-write as ( p − eA ) = A ( r ) p r + A ( r ) (cid:18) µ + L r (cid:19) (2.8)The electric potential is − A = + V = + Q/r . The above expression is quadraticin E (it is the generalization of the famous flat-spacetime E = µ + p ) and onefinds two possible solutions for the energy as a function of momenta and charge(see FIG. 2): E = eQr ± (cid:115) A ( r ) p r + A ( r ) (cid:18) µ + L r (cid:19) . (2.9)In flat space, A ( r ) = 1 , so that, if we ignore charge, we recover the usual Diracdichotomy on the choice of the + or − sign between particle and antiparticle: E = ± (cid:112) µ + p . This shows that one should take the plus sign in the equationabove. We remind the reader that for a charged BH, there exists a regular horizononly if Q < M (which can be interpreted as a BPS bound). [We have set G = 1 ].Remembering that A ( r ) = 1 − M/r + Q /r , there exists both an outer andan inner horizon defined by r ± = M ± (cid:112) M − Q (which are the two roots T. Damour and M. Lilley − µµ +r=r + E r
TUNNELING
Figure 2. This figure depicts the classically allowed energy levels (shaded region) as a functionof radius, for test particles in the neighborhood of a BH. There exist positive- and negative-energysolutions, corresponding (after second quantization) to particles and anti-particles. Classically (as inthe Penrose process) one should consider only the “positive-square-root” energy levels, located in theupper shaded region. The white region is classically forbidden. Note the possibility of tunneling (thiscorresponds to particle creation via the “super-radiant”, non-thermal mechanism briefly mentionedbelow). of A ( r ) = 0 ). The horizon of relevance for BH physics is the outer one r + = M + (cid:112) M − Q (it gives the usual result M when Q = 0 ). As particle 3is absorbed by the BH, we can compute its (conserved) energy when it crossesthe horizon, i.e., in the limit where the radial coordinate r is equal to r + . Thissimplifies the expression of E to E = e Qr + + | p r | , (2.10)where we have introduced the contravariant component p r = g rr p r = A ( r ) p r ,which has a finite limit on the horizon. Note the presence of the absolute value of p r (coming from the limit of a positive square-root). The change in the mass ofthe BH is equal to the energy E of the particle absorbed, i.e., particle . Using e = δQ , this yields δM = QδQr + ( M, Q ) + | p r | . (2.11)From the positivity of | p r | we deduce that δM ≥ QδQr + ( M, Q ) . (2.12) tring theory, gravity and experiment We have derived an inequality and have thereby demonstrated (by followingChristodoulou and Ruffini) the irreversibility property of BH energetics. Thereexist two types of processes, the reversible ones with an ‘ = ’ sign in (2.12),and the irreversible ones with an ‘ > ’ sign. The former ones are reversible be-cause if a BH first absorbs a particle of charge + e with vanishing | p r | (therebychanging its mass by δ (cid:48) M = eQ/r + ( M, Q ) and its charge by δ (cid:48) Q = e ), andthen a particle of charge − e with vanishing | p r | (thereby changing its mass by δ (cid:48)(cid:48) M = − eQ/r + ( M, Q ) and its charge by δ (cid:48)(cid:48) Q = − e ), it will be left, at theend, in the same state as the original one (with mass M + δ (cid:48) M + δ (cid:48)(cid:48) M = M and charge Q + δ (cid:48) Q + δ (cid:48)(cid:48) Q = Q ). Evidently, such reversible transformationsare delicate to perform, and one expects that irreversibility will occur in most BHprocesses. The situation here is clearly similar to the relation between reversibleand irreversible processes in thermodynamics.The same computation as for the Reissner-Nordström BH can be performedfor the Kerr-Newman BH. One obtains in that case, by a slightly more compli-cated calculation, δM − aδJ + r + QδQr + a = r + a cos θr + a | p r | . (2.13)in which r + ( M, J, Q ) = M + (cid:112) M − Q − a . We recall that a = J/M , andthat one has the bound Q + ( J/M ) ≤ M .The idea now is to consider an infinite sequence of infinitesimal reversiblechanges ( i.e. , p r → ), and to study the BH states which are reversibly connectedto some initial BH state with given mass M , angular momentum J and charge Q . This leads to a partial differential equation for δM , δM = aδJ + r + QδQr + a , (2.14)which is found to be integrable. Integrating it, one finds the Christodoulou-Ruffini mass formula [3] M = (cid:18) M irr + Q M irr (cid:19) + J M . (2.15)Here the irreducible mass M irr = (cid:113) r + a appears as an integration con-stant. The mass squared thus appears as a function of three contributions, withone term containing the square of the sum of the irreducible mass and of theCoulomb energy, and the other one containing the rotational energy. Insertingthis expression into Eq. (2.13), one finds δM irr ≥ (2.16) T. Damour and M. Lilley with δM irr = 0 under reversible transformations and δM irr > under irre-versible transformations. The irreducible mass M irr can only increase or stayconstant. This behaviour is certainly reminiscent of the second law of thermody-namics. The free energy of a BH is therefore M − M irr , i.e., this is the maximumextractable energy. In this view, BHs are no longer passive geometrical back-grounds but contain stored energy that can be extracted. Actually, the stored en-ergy can be enormous because a BH can store up to 29 % of its mass as rotationalenergy, and up to 50 % as Coulomb energy!The irreducible mass is related to the area of the horizon of the BH, by A =16 πM so that in a reversible process δA = 0 , while in an irreversible one δA > . Hawking showed [4] that this irreversible evolution of the area of thehorizon was a general consequence of Einstein’s equations, when assuming theweak energy condition. He also showed that in the merging of two BHs of area A and A , the total final area satisfied A tot ≥ A + A .Such results evidently evoque the second law of thermodynamics. The analogof the first law [ d E ( S, extensive parameters) = d W + d Q , where the work d W is linked to the variation of extensive parameters (volume, etc.) and where d Q = T d S is the heat exchange] reads, for BH processes, d M ( Q, J, A ) = V d Q + Ωd J + g π d A. (2.17)Comparing this result with expression (2.13), one has V = Qr + r + a , Ω = ar + a , (2.18)and g = 12 r + − r − r + a , (2.19)which, in the Kerr-Newman case, is given by g = (cid:112) M − a − Q r + a . (2.20) V is interpreted as the electric potential of the BH, and Ω as its angular velocity.Expression (2.17) resembles the usual form of the first law of thermodynamics inwhich the area term has to be interpreted as some kind of entropy. The parameter g is called the “surface gravity”. [In the Schwarschild case, it reduces to M/r (in G = 1 units), i.e., the usual formula for the surface gravitational acceleration g = GM/R .]. In the Les Houches Summer School of 1972, a more general tring theory, gravity and experiment version of the first law was derived, that included the presence of matter aroundthe BH, and energy exchange [5]. An analog of the zeroth law was also derived[15], in the sense that the surface gravity g (which is analog to the temperature)was found to be uniform on the surface of a BH in equilibrium.In 1974, Bekenstein went further in taking seriously (and no longer as a simpleanalogy) the thermodynamics of BHs. First, note that one can write the formalBH “heat exchange” term in various ways d Q = T d S = g d A π = 4 gM irr d M irr . (2.21)In light of this, is the appropriate physical analog of the entropy the irreduciblemass or the area of a BH? Is the analog of temperature proportional to the sur-face gravity g or to the product M irr g ? Can one give a physical meaning to thetemperature and entropy of a BH ? To address such questions, Bekenstein usedseveral different approaches.In particular, he used Carnot-cycle-type arguments. For instance, one may ex-tract work from a BH by slowly lowering into it a box of radiation of infinitesimalsize. In fact, in this ideal case, one can theoretically convert all the energy of thebox of radiation, mc , into work. The efficiency of Carnot cycles is defined interms of both a hot and a cold source as η = 1 − T cold T hot . (2.22)From what we just said, it would seem that the efficiency of classical BHs asthermodynamic engines is 100%, η = 1 . This would then correspond to a BHtemperature (= the cold source) T BH = T cold = 0 . The point made by Beken-stein was that this classical result will be modified by quantum effects. Indeed,one expects (because of the uncertainty principle) that a box of thermal radiationat temperature T (made of typical wavelengths λ ∼ /T ) cannot be made in-finitesimally small, but will have a minimum finite size ∼ λ . From this limit onthe size of the box, Bekenstein then deduced an upper bound on the efficiency η ,and therefore a lower bound on the BH temperature T BH (cid:54) = 0 .Let us indicate another reasoning (of Bekenstein) which suggests that the ab-sorption of a single particle by a BH augments its surface by a finite amountproportional to (cid:126) . As we said above the change of BH energy as it absorbs aparticle is (when a = 0 , for simplicity) E = eQr + + lim r → r + | p r | (2.23)We also showed that the transformation will be reversible ( i.e., will not increasethe surface area of the BH) only if lim r → r + | p r | = 0 . However, for this to be T. Damour and M. Lilley true both the (radial) position and the (radial) momentum of the particle must beexactly fixed: namely, r = r + and p r = 0 . This would clearly be in contradictionwith the Heisenberg uncertainty principle. Technically, we must consider theconjugate momentum to the position r which is the covariant component p r ofthe radial momentum (instead of the contravariant component p r used in theequation above). The uncertainty relation therefore reads δrδp r ≥ (cid:126) . (2.24)Near the horizon (i.e. when δr ≡ r − r + is small), the contravariant radialmomentum reads (using g rr = 1 /g rr = A ( r ) ) p r = A ( r ) p r = ( r − r + ) ( r − r − ) r p r (cid:39) δr ( r + − r − ) r p r (cid:39) (cid:18) ∂A∂r (cid:19) r + δrp r , (2.25)so that Heisenberg’s uncertainty relation yields a lower bound for p r . We canreexpress this lower bound in terms of the BH surface gravity g introduced aboveby noting that the partial derivative of A w.r.t. r , (cid:0) ∂A∂r (cid:1) r + , entering the last equa-tion, is proportional to g : (cid:18) ∂A∂r (cid:19) r + = 2 g. (2.26)This then gives p r (cid:39) gδrδp r ≥ g (cid:126) (2.27)>From the relation δM = QδQr + ( M,Q ) + | p r | r + , we finally obtain δM − QδQr + = | p r | ≥ g (cid:126) , (2.28)which can be rewritten as δA ≥ π (cid:126) . (2.29)In other words, quantum mechanics tells us that when one lets a particle fall intoa BH, one cannot do so in a perfectly reversible way. The area must increaseby a quantity of order (cid:126) . If (still following Bekenstein) one considers that theirreversible absorption of a particle by a BH corresponds to the loss of one bitof information (for the outside world), we are led to the idea of attributing to tring theory, gravity and experiment a BH an entropy (in the sense of “negentropy”) equal (after re-introducing theconstants c and G ) to [16] S BH = ˆ α c (cid:126) G A, (2.30)with a dimensionless numerical coefficient equal to ˆ α = ln 2 / π according tothe reasoning just made. More generally, Bekenstein suggested that the aboveformula should hold with a dimensionless coefficient ˆ α ≈ O (1) , without beingable to fix in a unique, and convincing, manner the value of ˆ α . This result in turnimplies (by applying the law of thermodynamics) that one should attribute to aBH a temperature equal to T BH = 18 π ˆ α (cid:126) c g. (2.31)This attribution of a finite temperature to a BH looked rather strange in view ofthe definition of a BH has being “black”, i.e., as allowing no radiation to come outof it. In particular, Stephen Hawking resisted this idea, and tried to prove it wrongby studying quantum field theory in a BH background. However, much to his ownsurprise, he so discovered (in 1974) the phenomenon of quantum radiation fromBH horizons (see below) which remarkably vindicated the physical correctnessof Bekenstein’s suggestion. Hawking’s calculation also unambiguously fixed thenumerical value of ˆ α to be ˆ α = [17]. [We shall give below a simple derivation(from Ref. [18]) of Hawking’s radiation.]Summarizing so far: The results on BH dynamics and thermodynamics of theearly 1970’s modified the early view of BHs as passive potential wells by endow-ing them with global dynamical and thermodynamical quantities, such as mass,charge, irreducible mass, entropy, and temperature. In the following section, weshall review the further changes in viewpoint brought by work in the mid andlate 1970’s ( [6, 8–11]) which attributed local dynamical and thermodynamicalquantities to BHs, and led to considering BH horizons as some kind of dissi-pative branes . Note that, in the following section, we shall no longer consideronly Kerr-Newman BHs ( i.e., stationary BHs in equilibrium, which are not dis-torted by sources at infinity). We shall consider more general non-stationary BHsdistorted by outside forces. The description of BHs we give from here on is essentially “holographic” innature since it will consist of excising the interior of a BH, and replacing thedescription of the interior BH physics by quantities and phenomena taking placeentirely on the “surface of the BH” ( i.e., the horizon). The surface of the BHis defined as being a null hypersurface, i.e., a surface everywhere tangent to thelightcone, separating the region inside the BH from the region outside. As just T. Damour and M. Lilley said, we ignore the region inside, including the spacetime singularity, and con-sider the physics in the outside region, completing it with suitable “boundaryeffects” on the horizon. These boundary effects are fictitious, and do not reallyexist on the BH surface but play the role of representing, in a holographic sense,the physics that goes on inside. In the end, we shall have a horizon, a set ofsurface quantities on the horizon and a set of bulk properties outside the horizon.We first consider Maxwell’s equations, namely F µν = ∂ µ A ν − ∂ ν A µ , and ∇ ν F µν = 4 πJ µ , ∇ µ J µ = 0 . (2.32)A priori, the electromagnetic field F µν permeates the full space time, existingboth inside and outside the horizon, and the current, i.e., the source term of F µν that carries charge, is also distributed both outside and inside the BH. In order toreplace the internal electrodynamics of the BH by surface effects, we replace thereal F µν ( x ) by F µν ( x )Θ H , where Θ H is a Heaviside-like step function, equal to outside the BH and inside. Then we consider what equations are satisfied bythis Θ H -modified electromagnetic field. The corresponding modified Maxwellequations contain two types of source terms, ∇ ν ( F µν Θ) = ( ∇ ν F µν ) Θ + F µν ∇ ν Θ= 4 π ( J µ Θ + j µν ) , (2.33)where we have introduced a BH surface current j µH as j µH = 14 π F µν ∇ ν Θ . (2.34)This surface current contains a Dirac δ -function which restricts it to the horizon.Indeed, let us consider a scalar function ϕ ( x ) such that ϕ ( x ) = 0 on the horizon,with ϕ ( x ) < inside the BH, and ϕ ( x ) > outside it. The BH Θ -function in-troduced above is simply equal to Θ H = θ ( ϕ ( x )) , where θ denotes the standardstep function of one real variable. Therefore, the gradient of Θ H reads ∂ µ Θ H = ∂ µ θ ( ϕ ( x )) = δ ( ϕ ( x )) ∂ µ ϕ, (2.35)where δ is the (one dimensional) usual Dirac delta, so that δ ( ϕ ( x )) is a deltafunction with support on the horizon. Morally, the gradient ∂ µ ϕ yields a vector“normal to the horizon”. In the case of a BH (by contrast to the usual case ofa hypersurface in Euclidean space), there exists an extra subtlety in the exactdefinition of the normal to the horizon. The horizon is a null hypersurface whichby definition is normal to a null covariant vector (cid:96) µ satisfying both (cid:96) µ (cid:96) µ = 0 and (cid:96) µ d x µ for any infinitesimal displacement d x µ within the hypersurface. Since (cid:96) µ tring theory, gravity and experiment is null, it cannot be normalized in the same way as in Euclidean space. Thisleads to an ambiguity in the physical observables related to (cid:96) µ . In stationary-axisymmetric spacetimes, one uniquely normalizes (cid:96) µ by demanding that thecorresponding directional gradient (cid:96) µ ∂ µ be of the form ∂/∂t + Ω ∂/∂φ (witha coefficient one in front of the time-derivative term). We shall assume (in thegeneral non-stationary case) that (cid:96) µ is normalized so that its normalization iscompatible with the usual normalization when considering the limiting case ofstationary-axisymmetric spacetimes. Anyway, given any normalization, thereexists a scalar ω such that (cid:96) µ = ω∂ µ ϕ, (2.36)and we can then define an “horizon δ -function” δ H = 1 ω δ ( ϕ ) , (2.37)such that ∂ µ Θ H = (cid:96) µ δ H . (2.38)This leads to defining a “BH surface current density” K µ = 14 π F µν (cid:96) ν . (2.39)With this definition, the BH current j µH reads j µH = K µ δ H , (2.40)and satisfies ∇ µ (Θ H J µ + K µ δ H ) = 0 , (2.41)which is a conservation law for the sum of the outside bulk current Θ H J µ andof the boundary current K µ δ H . In picturesque terms, the surface current K µ δ H effectively “closes” the external current lines penetrating the BH (analogously tothe case of external currents being injected in a perfect conductor and leading tocurrents flowing on its surface). In addition, Eq. (2.39) shows that this surfacecurrent is linked to the electromagnetic fields which are on the horizon. We havethus endowed the horizon with surface quantities, defined uniquely and locallyon the horizon.Before we proceed, we introduce a convenient coordinate system to describethe physics on the horizon of a general BH. We assume some regular “slicing” ofthe horizon and its neighbourhood by some (advanced) Eddington-Finkelstein-like time coordinate t = x . Then we assume that the first coordinate x issuch that it is equal to zero on the horizon (like r − r + in the Kerr-Newman T. Damour and M. Lilley case). Finally x A for A = 2 , denote some angular-like coordinates on the two-dimensional spatial slice S t ( x = t ) of the horizon. In this coordinate system,we normalize (cid:96) µ such that (cid:96) µ ∂ µ = ∂∂t + v A ∂∂x A . (2.42)Here, we have used the fact that the “normal” vector (cid:96) µ , being null, is also tan-gent to the horizon, so that (cid:96) µ ∂ µ is a general combination of ∂/∂t and ∂/∂x A but has no component along the “radial” (or “transverse”) coordinate x . Be-cause (cid:96) µ is a vector tangent to the hypersurface, we can consider its integral lines (cid:96) µ = dx µ /dt , which lie within the horizon. These integral curves are called the generators of the horizon. They are null geodesics curves, lying entirely withinthe horizon.Expression (2.42) for the directional gradient along (cid:96) µ suggests that v A beinterpreted as the velocity of some “fluid particles” on the horizon, which arethe “constituents” of a null membrane. Similarly to the usual description of themotion of a fluid, one has to keep track of the changes in the distance between twofluid particles as the fluid expands and shears. For a usual fluid, one considers thegradient of the velocity field, splitting it into its symmetric and anti-symmetricparts. The antisymmetric part is simply a local rotation which has no incidenceon the physics and can be ignored. The symmetric part is further split into itstrace and tracefree parts, namely
12 ( ∂ i v j + ∂ j v i ) = σ ij + 1 d ∂ · vδ ij (2.43)where d is the spatial dimension of the considered fluid (which will be d = 2 inour case). Here the first term describes the shear, and the second describes therate of expansion. We will see later how the BH analogs of these quantities aredefined. For the moment let us consider the distances on the horizon. They aremeasured by considering the restriction to the horizon of the spacetime metric(which is assumed to satisfy Einstein’s equations). As we are considering a nullhypersurface, we have d s | x =0 = γ AB (cid:0) t, x C (cid:1) (cid:0) d x A − v A d t (cid:1) (cid:0) d x B − v B d t (cid:1) (2.44)where v A = d x A d t . Note that d s is a degenerate metric: indeed, on a (three-dimensional) null hypersurface, there is no real time direction ( d s vanishesalong the generators). One has only two positive-definite space dimensions along, e.g., the spatial slices S t . This metric describes the geometry on the horizon fromwhich one can compute the area element of the spatial sections S t d A = (cid:112) det γ AB d x ∧ d x . (2.45) tring theory, gravity and experiment One can decompose the current density K µ into a time component σ H = K ,and two spatial components K A tangent to the spatial slices S t ( t = const.) ofthe horizon, K µ ∂ µ = σ H ∂ t + K A ∂ A (2.46)in which ∂ t = (cid:96) µ ∂ µ − v A ∂ A so that K µ ∂ µ = σ H (cid:96) µ + ( K A − σ H v A ) ∂ A (2.47)The total electric charge of the spacetime is defined by a surface integral at ∞ ,say Q tot = 14 π (cid:73) S ∞ F µν d S µν . (2.48)This result can be re-written as the sum of a surface integral on the horizon anda volume integral in between the horizon and ∞ . The volume integral is simplythe usual charge contained in space, so that we can define the BH charge Q H as Q H = 14 π (cid:73) H F µν d S µν , (2.49)where the tensorial horizon surface element reads d S µν = ε µνρσ d x ρ ∧ d x σ =( n µ (cid:96) ν − n ν (cid:96) µ ) d A . Here, n µ is a second null vector, which is transverse to thehorizon, and which is orthogonal to the spatial sections S t . It is normalized suchthat n µ (cid:96) µ = +1 . Using the definitions above of the BH surface current, oneeasily finds that the total BH charge can be rewritten as Q H = (cid:73) H σ H dA, (2.50)where σ H is the time component of the BH surface current introduced above.Though it is a priori only the integrated BH charge which has a clear physicalmeaning, it is natural to consider the density σ H appearing in the above surfaceintegral as defining a charge distribution on the horizon. Then the link σ H = K µ n µ = 14 π F µν n µ l ν (2.51)can be thought of as being analog to the result σ = π E i n i giving the electriccharge distribution on a metallic object. This can again be viewed as part of aholographic approach in which the interior of the BH is replaced by boundaryeffects. This analogy extends to the (spatial) currents flowing along the surfaceof the BH. Indeed, using the conservation law ∇ µ (Θ H J µ + K µ δ H ) = 0 , whichis just a Bianchi identity, one has √ γ ∂∂t ( √ γσ H ) + 1 √ γ ∂∂x A (cid:0) √ γK A (cid:1) = − J µ (cid:96) µ . (2.52) T. Damour and M. Lilley
This shows, in a mathematically precise way, how an external current injected“normally” to the horizon “closes” onto a combination of currents flowing alongthe horizon, and/or of an increase in the local horizon charge density. One canalso introduce the electromagnetic 2-form and restrict it on the horizon. It thendefines the electric and magnetic fields on the horizon according to F µν d x µ ∧ d x ν | H = E A d x A ∧ d t + B ⊥ d A. (2.53)Taking the exterior derivative of the left-hand-side then gives ∇ × (cid:126)E = − √ γ ∂ t ( √ γB ⊥ ) . (2.54)which relates the electric and magnetic fields on the horizon.>From the various formal definitions above, one also gets the following rela-tion E A + (cid:15) AB B ⊥ v B = 4 πγ AB (cid:0) K B − σ H v B (cid:1) , (2.55)or (cid:126)E + (cid:126)v × (cid:126)B ⊥ = 4 π (cid:16) (cid:126)K − σ H (cid:126)v (cid:17) . (2.56)We recognize here a BH analog of the usual Ohm’s law relating the electric fieldto the current (especially in the case where v → , i.e., in the absence of thevarious “convection effects” linked to the horizon “velocity” (cid:126)v ). From this formof Ohm’s law, we can read off that BHs have a surface electric resistivity equalto ρ = 4 π = 377 Ohm [8, 11].Let us give an example in which this BH Ohm’s law can be “applied” to aspecific system. We consider for simplicity the case of a Schwarzschild BH andset up an electric circuit “on the surface of the BH” by injecting on the North pole(through an electrode penetrating the horizon under a polar angle θ , with, say, θ (cid:28) ) an electric current I , and letting it escape from the South pole (via anelectrode penetrating the horizon under a polar angle θ , with, say, π − θ (cid:28) ).When viewing the BH as a membrane with surface resistivity ρ , this set up willgive rise to a fictitious current flow on the horizon, closing the circuit betweenthe North and the South poles. Associated to the current flow on the horizon,there will be a potential drop V between the poles. This potential drop is simplygiven by the usual Ohm’s law, V = RI , i.e., the product of the current I by a“resistance” R : V = − A ( θ ) + A ( θ ) = RI. (2.57) Actually, as (classical) charges cannot escape from a BH, we need to inject in the South electrodea flow of negative charges (while injecting a flow of positive charges down the North pole). tring theory, gravity and experiment The BH resistance R can be computed in two different ways, either by solvingMaxwell’s equations in a Schwarzschild background, or by computing, in usualEuclidean space, the total resistance of a spherical metallic shell with a uniformsurface resistivity ρ = 4 π (by decomposing the problem in many elementaryresistances, some being in parallel, and others in series). Both methods give thesame answer, namely R = 2 ln tan θ tan θ , (2.58)expressed in units of
30 Ω . This result is saying that the typical total resistivityof a BH is of the order of
30 Ω . In addition, if one considers a rotating BH placedin a magnetic field out of alignment with its axis of rotation (a field uniform at ∞ ,but distorted on the horizon), one expects to find eddy currents on the horizon,currents which dissipate the energy. These currents exist, can be computed anddo indeed brake the rotation of the BH. In such a situation, one also finds a torquewhich acts to restore the alignment of the BH with the field [8]. In the previous section, we introduced the electromagnetic dissipative propertiesof a BH, using a holographic approach which kept the physics outside up to in-finity, and replaced the physics inside the BH by defining suitable quantities onthe horizon, and then showed that they satisfied equations similar to well-knownones (such as Ohm’s law). We now turn to the viscous properties of BHs andshow how suitably defined “surface hydrodynamical” quantities satisfy a sort ofNavier-Stokes equation. Technically, we would like to do, for the gravitationalsurface properties, something similar to what we did for the electrodynamic prop-erties. Namely, we would like to replace the spacetime connection, say ω , bysome sort of “screened connection” Θ H ω , and see what kind of quantities andphysics will be so induced on the surface of the BH. However, Einstein’s equa-tions being nonlinear, one cannot simply use a BH step function Θ H as was donefor BH electrodynamics. We shall therefore motivate the definition of suitable“surface quantities” related to ω in a slightly different way and then study theevolution of these surface quantities and their connection to the physics outsidethe horizon, up to ∞ . Our presentation will be sketchy; for technical details,see [9, 10, 19, 20].Let us start by considering an axisymmetric spacetime. Then there exists aKilling vector (cid:126)m = m µ ∂/∂x µ = ∂/∂ϕ , to which, by Noether’s theorem, one Indeed, in CGS-Gaussian units (as used, say, in the treatise of Landau and Lifshitz) ohms isequal to the velocity of light (or its inverse, depending on whether one uses esu or emu). Then, whenusing (as we do here) units where c = 1 ,
30Ω = 1 . T. Damour and M. Lilley can associate a conserved total angular momentum, which can be written as asurface integral at ∞ . The total angular momentum J z w.r.t. ϕ reads J ∞ = − π (cid:90) S ∞ ∇ ν m µ d S µν , (2.59)where d S µν = ε µνρσ d x ρ ∧ d x σ , ∇ ν denotes a covariant derivative, and thesurface integral is performed over the 2-sphere, S ∞ . This starting point is theanalog of the surface-integral expression for the total electric charge used aboveto motivate the definition of a BH surface charge distribution.In a way similar to what was done in the electromagnetic case, we can useGauss’theorem to rewrite this integral as the sum of two contributions: (i) a vol-ume integral (over the 3-volume contained between the horizon and infinity) mea-suring the angular momentum of the matter present outside the horizon, and (ii)a surface integral over a (topological) 2-sphere S H at the horizon, representingwhat we can call the BH angular momentum J H , i.e., J = J matter + J H , (2.60)where J H is given by the same surface-integral formula as J ∞ , except for thereplacement of S ∞ by S H as integration domain.The horizon being tangent to the lightcone, one defines on the horizon, asabove, a null vector (cid:96) µ both normal and tangent to it. (cid:96) µ can in turn be comple-mented by another null vector n µ such that (cid:96) µ n µ = 1 and such that the surfaceelement dS µν can then be re-expressed as ( n µ (cid:96) ν − n ν (cid:96) µ ) d A . Remembering thatthe Killing symmetry preserves the generators of the horizon i.e., the commuta-tor [ (cid:126)(cid:96), (cid:126)m ] = 0 , one has (cid:96) ν ∇ ν m µ = m ν ∇ ν (cid:96) µ , so that we can re-express the BHangular momentum J H as the following surface integral J H = − π (cid:90) S H n µ m ν ∇ ν (cid:96) µ d A. (2.61)This result involves the directional (covariant) derivative of the horizon normalvector (cid:126)(cid:96) along a vector (cid:126)m which is tangent to the horizon. The crucial point nowis to realize that, very generally, given any hypersurface, the parallel transportalong some tangent direction, say (cid:126)t , of the (normalized) vector (cid:126)(cid:96) normal to thehypersurface yields another tangent vector . The technical proof of this fact con-sists of starting from the fact that (cid:126)(cid:96) · (cid:126)(cid:96) = (cid:15) , where (cid:15) is a constant which is equalto ± in the case of a time-like or spacelike hypersurface, and to in the case (ofinterest here) of a null hypersurface. Then, taking the directional gradient of thisstarting equality along an arbitrary tangent vector (cid:126)t yields ( ∇ (cid:126)t (cid:126)(cid:96) ) .(cid:126)(cid:96) = 0 . Fromthis result, one deduces that the vector ( ∇ (cid:126)t (cid:126)(cid:96) ) must be tangent to the hypersur-face. Therefore, there exists a certain linear map K , acting in the tangent plane tring theory, gravity and experiment to the hypersurface, such that ∇ (cid:126)t (cid:126)(cid:96) = K ( (cid:126)t ) . For a usual (time-like or space-like)hypersurface, the linear map K is called the “Weingarten map” and is simply themixed-component K ij version of the extrinsic curvature of the hypersurface (usu-ally thought of a being a symmetric covariant tensor K ij ). On the other hand, inthe case of a null hypersurface, there is no unique way to define the analog ofthe covariant tensor K ij (where the indices i, j are “tangent” to the hypersur-face), but it is natural, and useful, to consider the mixed-component tensor K ij ,intrinsically defined as the Weingarten map K in ∇ (cid:126)t (cid:126)(cid:96) = K ( (cid:126)t ) .To explicitly write out the various components of the linear map K (actingon the hypersurface tangent plane), we need to define a basis of vectors tangentto the horizon. This basis contains the null vector (cid:126)(cid:96) (which is both normal andtangent to the horizon), and two spacelike vectors. Using a coordinate system x , x , x A ( A = 2 , ) of the type already introduced (with the horizon beinglocated at x = 0 ), we can choose, as two spacelike horizon tangent vectors, thevectors (cid:126)e A = ∂ A . Then one finds that the Weingarten map K is fully describedby the set of equations ∇ (cid:126)(cid:96) (cid:126)(cid:96) = g (cid:126)(cid:96), ∇ A (cid:126)(cid:96) = Ω A (cid:126)(cid:96) + D BA (cid:126)e B . (2.62)The first equation follows from the fact that (cid:126)(cid:96) is tangent to a null geodesic ly-ing within the null hypersurface. [In turn, this follows from the fact that (cid:126)(cid:96) isproportional to the gradient of some scalar, say ϕ (satisfying the eikonal equa-tion ( ∇ ϕ ) = 0 ).] The coefficient g entering the first equation defines (in themost general manner) the surface gravity of the BH. We see that it representsone component of the Weingarten map K . The other components are the two-vector Ω A , and the mixed two-tensor D BA . One can show that the components D BA are the mixed components of a symmetric two-tensor D AB , which mea-sures the “deformation”, in time, of the geometry of the horizon. We remindthe reader of the expression of the horizon metric, introduced above, d s | H = γ AB ( t, (cid:126)x ) (cid:0) d x A − v A d t (cid:1) (cid:0) d x B − v B d t (cid:1) . Here, γ AB ( t, (cid:126)x ) is a symmetric rank2 tensor i.e., a time-dependent 2-metric such that the horizon may by viewed as a2-dimensional brane. In addition, we have the generators, which are the vectorstangent to (cid:126)(cid:96) . When decomposing (cid:126)(cid:96) = ∂ t + v A ∂ A w.r.t. our coordinate system,they appear to have a “velocity” v A which can also be viewed as the velocity ofa fluid particle on the horizon. D AB is then defined as the deformation tensor ofthe horizon geometry, namely D AB = γ BC D CA = Dγ AB d t , where D/dt denotes T. Damour and M. Lilley the
Lie derivative along (cid:126)(cid:96) = ∂ t + v A ∂ A . It is explicitly given by D AB = 12 (cid:0) ∂ t γ AB + v C ∂ C γ AB + ∂ A v C γ CB + ∂ B v C γ AC (cid:1) = 12 (cid:0) ∂ t γ AB + v A | B + v B | A (cid:1) (2.63)where ‘ | ’ denotes a covariant derivative w.r.t. the Christoffel symbols of the 2-geometry γ AB . Note the contribution from the ordinary time derivative of γ AB ,and that from the variation of the generators of velocity v A along the horizon.It is then convenient to split the deformation tensor D AB into a tracefree partand a trace, i.e., D AB = σ AB + θγ AB , where the tracefree part σ AB is the“shear tensor” and the trace, θ = D AA = γ AB ∂ t γ AB + v A | A , the “expansion”.The remaining component of the Weingarten map, namely the 2-vector Ω A , isdefined as Ω A = (cid:126)n. ∇ A (cid:126)(cid:96) with (cid:126)(cid:96).(cid:126)n = 1 . Its physical meaning can be seen fromlooking at the BH angular momentum J H .Indeed, from the definition above of J H , one finds that the total BH angu-lar momentum is the projection of Ω A on the direction of the rotational Killingvector (cid:126)m = ∂ ϕ introduced at the beginning of this section, so that we have J H = − π (cid:73) S m A Ω A d A, (2.64)where m A Ω A is the ϕ -component of Ω A . It is therefore natural to define, for aBH, a “surface density of linear momentum” as π A = − π Ω A = − π (cid:126)n · ∇ A (cid:126)(cid:96) .With this definition, one has J H = (cid:90) S π ϕ d A, (2.65)which is similar to the result above giving the BH electric charge as the surfaceintegral of the “charge surface density” σ H .Having so defined some (fictitious) “hydrodynamical” quantities on the sur-face of a BH (fluid velocity, linear momentum density, shear tensor, expansionrate, etc.), let us now see what evolution equations they satisfy as a consequenceof Einstein’s equations. By contracting Einstein’s equations with the normal tothe horizon, we can relate the quantities just defined to the flux of the energy-momentum tensor T µν into the horizon. For instance, by projecting Einstein’sequations along (cid:96) µ e νA , one finds Dπ A d t = − ∂∂x A (cid:16) g π (cid:17) + 18 π σ BA | B − π ∂ A θ − (cid:96) µ T µA (2.66) tring theory, gravity and experiment where Dπ A d t = ( ∂ t + θ ) π A + v B π A | B + v B | A π B ,σ AB = 12 (cid:0) ∂ t γ AB + v A | B + v B | A (cid:1) − θγ AB ,θ = ∂ t √ γ √ γ + v A | A (2.67)correspond to a convective derivative, a shear and an expansion rate respectively.Let us recall that the Navier-Stokes equation for a viscous fluid reads ( ∂ t + θ ) π i + v k π i,k = − ∂∂x i p + 2 ησ ki ,k + ζθ ,i + f i , (2.68)where π i is the momentum density, p the pressure, η the shear viscosity, σ ij = ( v i,j + v j,i ) − Trace , the shear tensor, ζ the bulk viscosity, θ = v i,i the expan-sion rate, and f i the external force density. The two equations are remarkablysimilar. This suggests that a BH can be viewed as a (non-relativistic ) branewith (positive ) surface pressure p = + g π , external force-density f A = − (cid:96) µ T µA which corresponds to the flow of external linear momentum, surface shear vis-cosity η = + π , and surface bulk viscosity ζ = − π (in units where G = 1 ).Note, finally, that both the surface shear viscosity and the surface bulk viscosityapply to any type of deformed non-stationary BH. In previous sections, we have introduced some electrodynamic and fluid dynami-cal quantities associated to a kind of dissipative dynamics of BH horizons. In ad-dition, following Bekenstein, we would like to endow a BH with a surface densityof entropy equal to a dimensionless constant ˆ α (in units where (cid:126) = G = c = 1 ).Any dissipative system verifying Ohm’s law and the Navier-Stokes equationis also expected to satisfy corresponding thermodynamic dissipative equations,namely Joule’s law and the usual expression of the viscous heat rate proportionalto the sum of the squares of the shear tensor and of the expansion rate. More pre-cisely, we would expect to have a “heat production rate” in each surface element d A of the form ˙ q = d A (cid:20) ησ AB σ AB + ζθ + ρ (cid:16) (cid:126) K − σ H (cid:126)v (cid:17) (cid:21) , (2.69) The non-relativistic character of the BH hydrodynamical-like equations may seem surprising inview of the “ultra-relativistic” nature of a BH. This non-relativistic-looking character is due to ouruse of an adapted “light-cone frame” ( (cid:96), n, e A ) . It is well-known that light-cone-gauge results havea distinct “non-relativistic” flavour. This is consistent with the idea that the BH surface pressure must counteract the self-gravity. T. Damour and M. Lilley where ρ is the surface resistivity, and η and ζ the shear and bulk viscosities. Inaddition, one expects that this heat production rate should be associated with acorresponding local increase of the entropy s = ˆ α d A contained in any localsurface element of the form d s d t = ˙ qT . (2.70)with a local temperature T expected to be equal to T = g π ˆ α .Remarkably, the “scalar” ( (cid:96) µ (cid:96) ν ) projection of Einstein’s equations, ( i.e., , theRaychauduri equation) yields an evolution law for the entropy s = ˆ α d A of alocal surface element which is very analogous to what one would expect. Indeed,it yields d s d t − τ d s d t = d AT (cid:20) ησ AB σ AB + ζθ + ρ (cid:16) (cid:126) K − σ H (cid:126)v (cid:17) (cid:21) , (2.71)where T = g π ˆ α , where ησ AB σ AB + ζθ are exactly the expected viscouscontributions, and where ρ (cid:16) (cid:126) K − σ H (cid:126)v (cid:17) is Joule’s law.The only unexpected term in this result is the second term on the l.h.s.; this termgoes beyond usual near-equilibrium thermodynamics (which involves only thefirst order time derivative of the entropy), and is proportional to the second timederivative of the entropy density and to a time scale τ = g . It is interesting to notethat, for the value ˆ α = 1 / , corresponding to the Bekenstein-Hawking entropydensity, τ is equal to πT , i.e., the inverse of the lowest “Matsubara frequency”associated to the temperature T (one also notes that τ = D = 2 D , where D , D are the diffusion constants of [14]). The minus sign in front of this new termis also a particularity of BH physics. In the approximation of a constant τ , andin solving for the non-equilibrium second law of thermodynamics, one finds thatthe rate of increase of entropy is given by d s d t = (cid:90) ∞ t d t (cid:48) τ e − ( t (cid:48)− t ) τ (cid:18) ˙ qT (cid:19) ( t (cid:48) ) , (2.72) i.e., it is defined not as the value of the heat dissipated instantaneously, nor asan integral over the past heat dissipation, but as an integral over the future . Thishighlights the acausal nature of BHs, i.e., a BH is defined as a null hypersurfacewhich will become stationary in the far future . As such, it has to anticipate anyexternal perturbation. Failing this, the null hypersurface would generically tendeither to collapse, or blow up toward ∞ .We also note that the ratio of the shear viscosity η = 1 / (16 π ) to the entropy tring theory, gravity and experiment density ˆ s = s/dA = ˆ α is given by η ˆ s = 1ˆ α π = 14 π (2.73)where, in the second equality, we have used the Bekenstein-Hawking value for ˆ s = s/dA = ˆ α = . It is interesting to note that the result η ˆ s = π is indeed theratio found by Kovtun, Son and Starinets in the gravity duals of strongly coupledgauge theories [13, 14].Finally, let us note another remarkable agreement between BH dissipative dy-namics and a rather general property of ordinary (near-equilibrium) irreversiblethermodynamics. This agreement concerns what Prigogine has called the “mini-mum entropy production principle”. Let us consider the total “dissipation func-tion” D = (cid:73) S ˙ q (2.74)as a functional of the velocity field v A (cid:0) x , x (cid:1) and of the electric potential φ (cid:0) x , x (cid:1) in the presence of given external influences such as an external mag-netic field or tidal forces acting on the rotating BH. Then, D [ φ ] or D [ v A ] (impos-ing v A | A = 0 as a constraint), reach a minimum when (and only when) the lowestorder (Einstein-Maxwell) dynamical equations for φ or v A are satisfied. In this section we discuss the phenomenon of Hawking radiation, first obtained inRef. [17] (we shall follow here the derivation of Ref. [18]). For simplicity, we willconsider a 3+1 dimensional spherically symmetric BH. We remind the reader thatthe coefficient A ( r ) , associated to the time coordinate (here denoted as T ), goesto zero on the horizon, so that the horizon is an infinite redshift surface . It is also a Killing horizon, i.e., the (suitably normalized) normal vector (cid:126)(cid:96) = ∂/∂t + Ω ∂/∂φ is a Killing vector. These points will be crucial in the following. Since thecoefficient of the radial coordinate is defined by B ( r ) = A ( r ) (see Section 2.1),it is singular on the horizon. To get a good coordinate system on the horizon, wefirst factorize A ( r ) , d s = − A ( r ) d T + d r A ( r ) + r dΩ = − A ( r ) (cid:32) d T − (cid:18) d rA ( r ) (cid:19) (cid:33) + r dΩ , (2.75) T. Damour and M. Lilley and then introduce a new radial coordinate, the so-called tortoise coordinate,defined by r ∗ = (cid:90) d rA ( r ) . (2.76)Note that as r → r + , A ( r ) (cid:39) (cid:0) ∂A∂r (cid:1) r + ( r − r + ) , where (cid:0) ∂A∂r (cid:1) r + is (as mentionedabove) equal to twice the surface gravity g . This implies r ∗ (cid:39) (cid:90) d r ( r − r + ) (cid:0) ∂A∂r (cid:1) r + (cid:39) ln ( r − r + )2 g (2.77)such that as r → + ∞ , r ∗ (cid:39) r + 2 M ln r and as r → r + , r ∗ (cid:39) ln( r − r +)2 g . Theline element can thus be re-written as d s = − A (d T − d r ∗ ) (d T + d r ∗ ) + r dΩ . (2.78)We now switch to the so-called Eddington-Finkenstein coordinates, ( t, r, θ, ϕ ) ,where the combination t = T + r ∗ of T and r ∗ remains regular across the (future)horizon and define t = T + r ∗ . The time translation Killing vector ∂/∂t coïncides with the usual one ∂/∂T . In terms of these new coordinates the metricreads d s = − A ( r ) dt + 2 dtdr + r dΩ . (2.79)This metric is now regular (with a non-vanishing determinant) as the radial coor-dinate r penetrates within the horizon, i.e., becomes smaller than r + .Having defined a regular coordinate system, we now consider a masslessscalar field, the dynamics of which is given by the massless Klein Gordon equa-tion (cid:3) g ϕ = 1 √ g ∂ µ ( √ gg µν ∂ ν ϕ ) (2.80)The solutions to this equation in a spherically symmetric and time-independentbackground are given by mode functions which are themselves given simply byproducts of a Fourier decomposition into frequencies, spherical harmonics and aradial dependence and thus read ϕ ω(cid:96)m ( T, r, θ, ϕ ) = e − iωT (cid:112) π | ω | u ω(cid:96)m ( r ) r Y (cid:96)m ( θ, ϕ ) . (2.81) Given that the horizon is an infinite redshift surface, it takes an infinite time T to fall into it, while r ∗ (cid:39) ln ( r − r + ) / g goes to −∞ at the horizon.The sum of the two remains, however, finite. tring theory, gravity and experiment The problem is then reduced to solving a radial equation with the radial coordi-nate r ∗ , ∂ u∂r ∗ + (cid:2) ω − V (cid:96) ( r ( r ∗ )) (cid:3) u = 0 . (2.82)In the case of the Schwarzschild metric ( i.e., when A ( r ) = 1 − M/r ) the effec-tive radial potential V (cid:96) ( r ) is given by V (cid:96) ( r ) = (cid:18) − Mr (cid:19) (cid:18) (cid:96) ( (cid:96) + 1) r (cid:19) . (2.83)An essential point to note is that the effective potential vanishes both at ∞ (likea massless centrifugal potential (cid:96) ( (cid:96) + 1) /r ), and at the horizon (where it isproportional to A ( r ) ). Therefore, in these two regimes (which correspond to r ∗ → ±∞ ), the solution of the wave equation behaves essentially as in flat space(see FIG. 3). The effect of the coupling to curvature is non-negligible only inan intermediate region, where the combined effect of curvature and centrifugaleffects yield a positive potential barrier. In turn, this potential barrier yields a greybody factor which diminishes the amplitude of the quantum modes consideredbelow, i.e., those generated near the horizon and which must penetrate throughthe potential barrier on their way towards ∞ . Far from the potential barrier, thegeneral solution for ϕ is ϕ ω(cid:96)m ∼ e − iω ( T ± r ∗ ) (cid:112) π | ω | r Y (cid:96)m ( θ, ϕ ) . (2.84)The quantification of the scalar field ϕ is rather standard (and similar to what onedoes when studying the amplification of quantum fluctuations during cosmologi-cal inflation). The quantum operator for the scalar field is decomposed into modefunctions, i.e., the eigenfunctions of the Klein Gordon equation, with coefficientsgiven by creation and annihilation operators. The subtlety lies, however, in thedefinition of positive and negative frequencies.Let us start by formally considering the simpler case of a quantum scalar field ˆ ϕ ( x ) in a background spacetime which becomes stationary both in the infinitepast, and in the infinite future. In that case, one can define positive and negativefrequencies in the usual way, in the two asymptotic regions t → ±∞ . The coeffi-cient of the positive frequency modes, say p ( x ) , then defines an annihilation op-erator ˆ a . However, there are two sorts of positive-frequency ( p ( x ) ), and negative-frequency ( n ( x ) ) modes. The in ones p ini , n ini (defined in the asymptotic region t → −∞ , and then extended everywhere by solving the Klein-Gordon equa-tion), and the out ones p outi , n outi (defined in the asymptotic region t → + ∞ ).The operator-valued coefficients of these modes define some corresponding in T. Damour and M. Lilley and out annihilation or creation operators, so that we can write the field operator ˆ ϕ ( x ) as (here i is a label that runs over a basis of modes) ˆ ϕ ( x ) = (cid:88) i ˆ a ini p ini ( x ) + (cid:0) ˆ a ini (cid:1) † n ini ( x ) , = (cid:88) i ˆ a outi p outi ( x ) + (cid:0) ˆ a outi (cid:1) † n outi ( x ) . (2.85)Here, the modes are normalized as ( p i , p j ) = δ ij and ( n i , n j ) = − δ ij , where‘ ( , ) ’ is the Klein-Gordon scalar product, i.e., ∼ i (cid:82) dσ µ ( ϕ ∗ ∂ µ ϕ − ∂ µ ϕ ∗ ϕ ) .The operators a i and a † j are correspondingly normalized in the usual way as (cid:104) a i , a † j (cid:105) = δ ij . One defines both an in vacuum | in (cid:105) and an out vacuum | out (cid:105) asthe states that are respectively annihilated by a ini or a outi . Then, the phenomenonof particle creation corresponds to the fact that the out vacuum differs from the in one. More quantitatively, the expectation value of the number of out particles,in the mode labelled by i , which will be observed when the quantum field is inthe in vacuum state is given by (cid:104) N i (cid:105) = (cid:104) in | (cid:0) a outi (cid:1) † a outi | in (cid:105) = (cid:88) j | T ij | (2.86)where we have introduced the transition amplitude T ij = (cid:0) p outi , n inj (cid:1) from an ini-tial negative frequency mode n inj into a final positive frequency one p outi . Thesetransition amplitudes (also called Bogoliubov coefficients) enter the calculationbecause, as is easily deduced from the double expansion of the field ˆ ϕ ( x ) above,they give the part of a outi which is proportional to ( a inj ) † . The application of theprevious general formalism to the BH case is delicate since a BH backgroundis not asymptotically stationary in the infinite future (because of the BH interiorwhere the Killing vector ∂/∂t is spacelike), and is asymptotically stationary inthe infinite past only if we do consider explicitly the collapse leading to the for-mation of a BH from an initially stationary star. However, Hawking showed howto essentially bypass these difficulties by focussing on two types of modes: • The high-frequency modes coming from the infinite past, which reach thehorizon with practically no changes (because of their high-frequency nature) and, • the outgoing modes, viewed in the asymptotically flat region and in the farfuture.Concerning the outgoing modes, they can be unambiguously decomposed inpositive- and negative-frequency parts, because, as explained above, their asymp-totic behaviour is given by a sum of essentially flat-spacetime modes, (2.84). One tring theory, gravity and experiment r * V HORIZON
Figure 3. This figure is a schematic representation of the effective gravitational potential in theneighborhood of a BH. Note that as far as the particles are concerned, the spacetime is essentiallyflat both at infinity and near the horizon. The tidal-centrifugal barrier that separates the horizon frominfinity gives rise to the grey body factor. then defines the outgoing p outi ’s as being proportional to e − iω ( T − r ∗ ) with a pos-itive ω .Let us now focus on the definition of positive- and negative-frequency modesnear the horizon. We recall that, as mentioned at the beginning of this sec-tion, there is a physically infinite redshift between the surface of the horizonand asymptotically flat space at infinity. If one is interested in particle creationwith a finite given frequency, as observed at infinity, the corresponding wavepackets will have very high frequency near the horizon and can therefore be ap-proximated by very localized wave packets. Given that the spacetime geometryin the vicinity of the horizon is regular, with a finite radius of curvature, it canbe regarded as a piece of flat spacetime locally if one looks in a small enoughregion. In this approximation, the calculation can be performed in a single step.We wish to compute the average number of final outgoing particles seen inthe in vacuum. Then the average number of outgoing particles of type p outi is Note that the “out” label, in the general discussion of particle creation above, referred to “final”particles (as defined in the final, asymptotic, stationary spacetime background). In the case of aBH background, the “final” spacetime is made of two separate asymptotic regions: (i) the outgoingwave region at spatial infinity, and (ii) the vicinity of the (spacelike?) singularity within the BH.The definition of positive- and negative-frequency modes in the latter region is ill-defined. However,luckily, the calculation of the physically relevant flux of final, outgoing modes can be performedwithout worrying about the physics near the BH interior singularity. In other words, it is enough toconsider as “out” positive-frequency modes p outi only the ones outgoing at spatial infinity ( i.e., on“scri + ”), though they do not constitute a complete basis of final modes. T. Damour and M. Lilley given by (cid:80) j | T ij | , where T ij = (cid:0) p outi , n inj (cid:1) is the transition amplitude froman initial negative frequency mode n inj into a final outgoing positive frequencyone p outi (recorded at spatial infinity). To compute this transition amplitude, weneed to describe what is an initial negative frequency mode n inj . As said above,Hawking suggested that only high-frequency initial modes are important, and thatthey essentially look the same (some kind of WKB wave) in the real in region (inthe far past, before the formation of the BH) as in the vicinity of the horizon. Ourtechnical problem is then reduced to characterizing what is a negative frequencymode n inj as seen in a small neighborhood of the horizon, which looks like theMinkowski vacuum.To do this, it is convenient to have a technical criterion for characterizingpositive and negative frequency modes in (a local) Minkowski spacetime. Lo-cally, one can perform a Fourier decomposition of the wave packet and usethe mathematical fact that the Fourier space properties are mapped onto ana-lytic continuation properties in x -space. This relation can then be used to definepositive and negative frequency modes. This is easy to see. Consider a gen-eral negative frequency wave packet in Minkowski spacetime. It has the form ϕ − ( x ) = (cid:72) C − d k ˜ ϕ ( k ) e ik µ x µ where k µ is timelike-or-null and past-directed , i.e., k µ ∈ C − . We now perform a complex shift of the spacetime coordi-nate, x µ → x µ + iy µ , where y µ is timelike-or-null and future-directed ( i.e. , y µ ∈ C + ), then, the e ik µ x µ term will be suppressed by a e − k µ y µ term, where thescalar product k µ y µ is positive because it involves two timelike vectors that pointin opposite directions (we use the “mostly plus” signature). This ensures that anegative-frequency wavepacket can indeed be analytically continued to complexspacetime points of the form x µ + iy µ , with y µ ∈ C + .The strategy for applying this criterion to characterizing negative-frequencymodes n inj in the vicinity of the horizon is then the following. One starts froma wavepacket which is not purely a “negative-frequency” one near the hori-zon, but which has the property of evolving into an outgoing positive-frequencywave packet (so that it will have a non-zero transition amplitude to some p outi ).Then, one modifies the initial wavepacket so that it becomes a purely negative-frequency mode n inj near the horizon.When looking at a wavepacket of the form of ϕ outω ( t, r ) ∝ e − iω ( T − r ∗ ) (withpositive ω ) just outside the horizon, one must first switch to well-defined coordi-nates to examine its physical content. We therefore replace the Schwarzschild-type time coordinate T by the Eddington-Fikenstein time coordinate, t = T + r ∗ which is regular on the horizon. After rearranging terms according to T − r ∗ = tring theory, gravity and experiment ( T + r ∗ ) − r ∗ the previous (outgoing, positive-frequency) wave reads (cid:2) ϕ outω ( t, r ) (cid:3) r + ∝ e − iω ( T − r ∗ ) = e − iωt e iωr ∗ = e − iωt e i ωg ln( r − r + ) = e − iωt ( r − r + ) iωg . (2.87)This describes the behaviour, just outside the horizon, of a wavepacket whichwill become (modulo some grey-body factor) an outgoing positive frequencywavepacket at ∞ . However, locally on the horizon, it is neither a positive nora negative frequency wavepacket because, at this stage, it is defined only out-side the horizon, but not inside. Let us now show how one must continue thiswavepacket inside the horizon, so that it becomes a genuine negative-frequencywavepacket “straddling” the horizon. Using the criterion explained above, we can“continue” the wavepacket inside the horizon by a suitable analytic continuation .More precisely, we need an analytic continuation of the form x µ → x µ + iy µ ,where y µ belongs to the future lightcone to ensure that we shall then be dealingwith a local negative frequency wavepacket. It is easy to see, from a spacetimediagram of the lightcone on the horizon, that the vector ∂/∂r is everywhere nulland past-directed, such that r → r − ε , where ε > , is everywhere null and fu-ture directed. The analytic continuation of ϕ outω ( t, r ) to r → r − iε will thereforedefine for us a good local negative-frequency mode n inj = n inω(cid:96)m . One easily seesthat this analytic continuation in r generates a new component to the wavepacketwhich is located inside the BH ( i.e., for r < r + ). More precisely, a one linecalculation yields n inω(cid:96)m ( r, t ) = N ω ϕ out ω ( t, r − iε )= N ω (cid:104) θ ( r − r + ) ϕ out ω ( r − r + ) + e πωg θ ( r + − r ) ϕ out ω ( r + − r ) (cid:105) , (2.88)where the second term is the wavefunction inside the horizon that has acquiredan additional exponential factor due to the rotation e − iπ in the complex planefrom r > r + to r < r + . The overall factor N ω is a normalization factor (neededbecause we have extended the mode inside the BH), such that (cid:104) n inω(cid:96)m ( r, t ) n inω (cid:48) (cid:96) (cid:48) m (cid:48) ( r, t ) (cid:105) = δ ( ω − ω (cid:48) ) δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) , (2.89)from which we obtain (when remembering that ϕ out ω was correctly normalized) | N ω(cid:96)m | = 1 e πω/g − . (2.90) T. Damour and M. Lilley
The physical meaning of equation (2.88) is the description of the splitting of thein mode n inj = n inω(cid:96)m into a positive-frequency wave outgoing from the hori-zon and a wave falling from the horizon towards the singularity. One can readoff from it the needed transition amplitude T ij = (cid:0) p outi , n inj (cid:1) . It is essentiallygiven by the factor N ω , which must, however, be corrected by a grey-body factor (cid:112) Γ (cid:96) ( ω ) taking into account the attenuation of the outgoing wave e − iω ( T − r ∗ ) asit crosses the curvature + centrifugal potential barrier V (cid:96) ( r ) on its way from thehorizon to ∞ . Then (using Fermi’s golden rule), one easily finds that the generalresult (2.86) yields a rate of particle creation given by d (cid:104) N (cid:105) d t = (cid:88) (cid:96),m (cid:90) d ω π Γ (cid:96) ( ω ) e πωg − . (2.91)One recognizes here a thermal (Planck) spectrum (corrected by a grey-body fac-tor). From the Planck factor, one reads off the Hawking temperature, T = (cid:126) g π .This result fixes the dimensionless coefficient ˆ α in the Bekenstein entropy to thefamous result ˆ α = , i.e., S BH = A G (cid:126) . (2.92)Let us end by two final comments. First, the generalisation of the Hawkingradiation to a more general rotating and/or charged BH is given essentially byreplacing in the result above the frequency ω by ω − ω where ω = m Ω + eV exhibits the couplings of the created particles to the angular velocity Ω and theelectric potential V of the BH. Then, in astrophysically realistic conditions, the“Hawking” part of the particle creation ( i.e., the thermal aspect) is too small tobe relevant, while the combined effect of the grey body factor and of the zero-temperature limit of ( e π ( ω − ω g − − yield potentially relevant particle creationphenomena in Kerr-Newman BHs, associated to the “superradiance” of modeswith frequencies µ < ω < ω , where µ is the mass of the created particle (see, e.g., [18] for more details and references). We conclude by noting that the situ-ation just described is not only technically similar to the one in the inflationaryscenario for cosmological perturbations, but also physically similar in that, inboth cases, transplanckian frequency modes in the ultraviolet are redshifted to afinite, observable frequency.
3. Experimental tests of gravity
Before discussing various possibilities of string-inspired phenomenology (and ofpossible string-inspired deviations from Einstein’s theory of General Relativity)we give an overview of what is known experimentally about the gravitationalsector. tring theory, gravity and experiment The standard model of gravity is Einstein’s General Relativity (GR). In GR, allfields of the standard model of particle physics (SM) are universally coupled togravity by replacing the flat spacetime metric η µν by a curved spacetime one g µν .In “standard GR” one also assumes that gravity is the only long range coupling(apart from electromagnetism). We shall see below, how the presence of otherlong range interactions (coupled to bulk matter) modify the usual “pure GR”phenomenology. The action for the matter sector, S SM , has the structure S SM = (cid:90) d x (cid:104) − (cid:88) √ gg µα g νβ F aµν F aαβ − (cid:88) √ g ¯ ψγ µ D µ ψ − √ gg µν D µ HD ν H − √ gV ( H ) − (cid:88) λ √ g ¯ ψHψ − √ gρ vac (cid:105) , (3.1)where D denotes a (gauge and gravity) covariant derivative, while the dynamicsof g µν is described by the Einstein-Hilbert action, S EH , S EH = (cid:90) d x c πG √ gg µν R µν ( g ) . (3.2)The total action is therefore given by S = S EH [ g µν ] + S SM [ ψ, A µ , H ; g µν ] , (3.3)and its variation w.r.t. g µν yields the well-known Einstein field equations R µν − R g µν = 8 πGc T µν (3.4)where T µν = √ g δ L SM δg µν . The universal coupling of any type of particle to g µν is made manifest in S SM while S EH contains all the information on thepropagation of gravity. For instance, expanding S EH in powers of h µν (where g µν ≡ η µν + h µν ), one obtains, at quadratic order in h µν , the spin-two Pauli-Fierz Lagrangian. Higher orders in h µν contain an infinite series of nonlinearself-couplings of gravity: ∂∂hhh, ∂∂hhhh, etc. As we shall see, this nonlinearstructure has been verified experimentally to high accuracy (both in the weak-field regime, where the cubic vertex ∂∂hhh has been checked, and in the strong-field regime of binary pulsars, where the fully nonlinear GR dynamics has beenconfirmed). In the following, we discuss, successively, (i) the experimental testsof the coupling of matter to gravity, and (ii) the tests of the dynamics of the grav-itational field: kinetic terms (describing the propagation of gravity), and cubicand higher gravitational vertices. T. Damour and M. Lilley
The universal nature of matter’s coupling to gravity, i.e., the coupling of matterto a universal deformation of spacetime, has many experimental consequences.These experimental consequences can be derived by using a simple theorem byFermi and Cartan. Given any pseudo-Riemannian manifold, for instance a curvedspacetime endowed with a metric g µν , and given any worldline L in this space-time (not assumed to be a geodesic), there always exists a coordinate system suchthat, all along L , g µν (cid:0) x λ (cid:1) = η µν + O (cid:0) (cid:126)x (cid:1) , where (cid:126)x denotes the spatial devi-ation away from the central worldline L . It is important to note that there is nolinear term in (cid:126)x , but only (cid:126)x effects, i.e., tidal effects. There exists a very simpleand intuitive demonstration of this Fermi-Cartan theorem. Let us view the curvedmanifold as being some “brane” embedded within a flat ambient auxiliary man-ifold. For instance, consider an ordinary 2-surface Σ within a three-dimensionalflat euclidean space. Given any (smooth) curve L traced on Σ , we can take aflat sheet of paper and progressively “apply” (or “fit”) this sheet on Σ along thecurve L . The orthogonal projection of Σ onto this applied flat sheet defines amap from Σ to a coordinatized flat manifold which has the property enunciatedabove. Note that, in this “development” of the neighbourhood of L within Σ ontoa flat sheet, the shape (as seen on the flat sheet) of the “developped” curve L isgenerically not a straight line. It is only when L was a geodesic line on Σ , that itsdevelopment will be a straight line. This proof, and its consequences, are validin any dimension and signature.Here, we have in mind applying this result to the “center of mass” worldline L of an arbitrary body moving in a background spacetime, or more generally of anysufficiently small laboratory (containing several bodies, between which we canneglect gravitational effects). We assume that we can neglect the backreaction ofthe body (or bodies) on the spacetime. In the approximation where we can neglectthe tidal effects (linked to the O (cid:0) (cid:126)x (cid:1) terms in g µν in Fermi coordinates), we canconsider that we have a body, or a small lab, moving in a flat spacetime. In otherwords, the theorem of Fermi and Cartan tells us that we can essentially “efface”(modulo small, controllable tidal effects) the background gravitational field g µν all along the history of a small lab, or a body. This “effacement property” istelling us, for instance, that the physical properties we can measure in a small labwill be independent of where the lab is, and when the measurements are made. Inparticular, all the (dimensionless) coupling constants that enter the interpretationof local experiments (such as various mass ratios, the fine-structure constant, etc.)must be independent of where and when they are (locally) measured ( constancyof the constants ). A second consequence of this effacement property is that localphysics should be Lorentz SO (3 , invariant, because this is a symmetry of We assume here that the cutoff length scale (cid:15) = 1 / Λ of any low-energy effective QFT descrip-tion of the physics in a small lab is is fixed, when measured in units of ds = p g µν dx µ dx ν . tring theory, gravity and experiment the (approximate) flat spacetime appearing after one has effaced the tidal effects( local Lorentz invariance ).Moreover, in absence of coupling to other long-range fields (such as electro-magnetism for a charged body), the center of mass of an isolated body (viewedas moving in a flat spacetime) must follow a straight worldline (principle of iner-tia). We therefore conclude (by the theorem above) that L has to be a geodesic inthe original curved spacetime. This is true independently of the internal proper-ties of the object. One may thus conclude that isolated neutral bodies fall alonggeodesics independently of the internal properties of the object, since at no pointin the demonstration had we to rely on any internal properties of the object. Thisis therefore a proof of the weak equivalence principle , i.e., all bodies in a gravi-tational field fall with the same acceleration. Note, once again, that the absenceof other long range fields besides g µν that could influence the object consideredis crucial.Finally, another universality property, that of the gravitational redshift, maybe shown by a comparison of the GR formulation with the Newtonian one. Inlowest order approximation, the deviation of the g component of the metricfrom η = − is twice the Newtonian potential U ( x ) . Indeed, comparing theaction for a geodesic, S E = − m (cid:90) d t (cid:112) − g µν ˙ x µ ˙ x ν (3.5)with S Newton = (cid:90) d t (cid:20) m ˙ x + mU ( x ) (cid:21) , (3.6)one finds g = − c U ( x ) + O (cid:0) c (cid:1) where U = (cid:88) a Gm a | (cid:126)x − (cid:126)x a | .Experimentally, one may transfer electromagnetic signals from one clock toanother identical clock located in a gravitational field. If we are in a stationarysituation ( i.e., if there exists a coordinate system w.r.t. which the physics is in-dependent of time x = ct ), the time translation invariance of the backgroundshows that electromagnetic signals will take a constant coordinate time to prop-agate from clock 1 to clock 2. We can then use the link dτ i = (cid:112) − g ( (cid:126)x i ) dt i between the proper time (at the location of clock i ( i = 1 , )) and the correspond-ing coordinate time, as well as the (approximate) result above for g . Finally,we conclude that two identically constructed clocks located at two different po-sitions in a static external Newtonian potential exhibit, when intercompared byelectromagnetic signals, the (apparent) difference in clock rate τ τ = ν ν = 1 + 1 c [ U ( (cid:126)x ) − U ( (cid:126)x )] + O (cid:18) c (cid:19) . (3.7) T. Damour and M. Lilley
This gravitational redshift effect is proportional to the difference in the Newto-nian potential between the two locations, independently of the constitution of theclocks (say Hydrogen maser, or Cesium clock, etc.). This is a property known asthe universality of the gravitational redshift .The various consequences, discussed above, of the universal character of thecoupling of matter to gravity are usually summarized under the generic name of equivalence principle . In the next section, we discuss the experimental tests ofthe equivalence principle and their accuracy.
The best tests of the “constancy of the constants” concern the fine structure con-stant α = e / (cid:126) c (cid:39) / . , and the ratio of the electron mass to that of theproton m e m p (see Ref. [21] for a review). There exist several types of tests, based,for instance, on geological data ( e.g., measurements made on the nuclear decayproducts of old meteorites), or on measurements (of astronomical origin) of thefine structure of absorption and emission spectra of distant atoms, as, e.g., theabsorption lines of atoms on the line-of-sight of quasars at high redshift. Suchkinds of tests all depend on the value of α . There exist, in addition, several lab-oratory tests such as, for example, comparisons made between several differenthigh-stability clocks. However, the best measurement of the constancy of α todate is the Oklo phenomenon . It sets the following (conservative) limits on thevariation of α over a period of two billion years [22–24] − . × − < α Oklo − α today α today < . × − . (3.8)Converting this result into an average time variation, one finds − . × − yr − < ˙ αα < × − yr − . (3.9) The Oklo phenomenon was discovered by scientists at the
Commissariat à l’énergie atomique (CEA) in France. A study of the uranium ore in a Gabonese mine revealed an unusual depletion in U (used in fission reactors) w.r.t. the usual proportion. Uranium ore is a mix of two isotopes, with,in usual samples, . U and . U . By contrast, the Oklo ore had only (cid:28) . of U . It was realized that a natural fission process took place, prompted by the presence of groundwater, in Oklo some two billion years ago, and lasted for about two million years. Scientists analysedin detail the 2 billion year-old fission decay products. One can then infer from these measurementsthe scattering cross-sections of slow neutrons on various isotropes. Then, modulo some further as-sumptions about the dependence of various nuclear quantities on α , one could constrain the variationof α between the time of the fission reaction (roughly two billion years ago) and now. For detailsabout the analysis and interpretation of Oklo data see [22] and references therein. tring theory, gravity and experiment Note that this variation is a factor of ∼ smaller than the Hubble scale, whichis itself ∼ − yr − . Comparably stringent limits were obtained using theRhenium to Osmium ratio in meteorites [25] yielding an upper bound ∆ αα = (8 ± × − over . × years. Laboratory limits were also obtainedfrom the comparison, over time, of stable atomic clocks. More precisely, giventhat vc ∼ α for electrons in the first Bohr orbit, direct measurements of the vari-ation of α over time can be made by comparing the frequencies of atomic clocksthat rely on different atomic transitions. The upper bound on the variation of α using such methods is ˙ αα = ( − . ± . × − yr − [26]. It should be men-tioned that a few years ago claims were made concerning observational evidenceof non-zero time variations of α and m e m p from analyses of some astronomicalspectra (see Ref. [21]). Other recent astronomical data indicate no variabilityof these constants (see Ref. [21] and the chapter 18 of the Review of ParticlePhysics for references). We should first mention that the Michelson-Morley experiment has been re-peated (with high accuracy) and strong limits have been obtained on a possibleanisotropy of the propagation of light. In its modern realizations (Brillet andHall, 1979), it has been performed with laser technology on rotating platforms.This experiment is now viewed as a test of the isotropy of space on the movingEarth, and thereby as a test of local Lorentz invariance. There also exists anotheridea for testing the isotropy of space, and although its interpretation is not totallyclear, it is a conceptually interesting idea. This is why we choose to outline it inthese lectures.For simplicity, consider the hydrogen atom. Assuming the isotropy of space, i.e., the existence of a SO(3) symmetry, we know that there should exist a degen-eracy in the energy levels, given by the magnetic quantum number m . However,it is interesting to understand how the SO(3) symmetry comes about dynamically(and therefore, how it might be dynamically violated). The Hamiltonian for theelectron is given by ˆ H = − (cid:126) m (cid:52) − e r (3.10)where the first term is the kinetic term ( (cid:52) being the Laplacian), and the secondterm is the Coulomb potential. Note that in fact, (cid:52) = δ ij ∂ ij and r = δ ij x i x j ,such that both terms depend on the same spatial structure δ ij , the flat metric, Available on http://pdg.lbl.gov/ First performed as part of a series of experiments, beginning in Potsdam in 1881 (by Michelsonalone) and then in the US until 1887 (by both Michelson and Morley) to test the existence of the aether . T. Damour and M. Lilley thereby ensuring the SO(3) symmetry. However, both terms also come froman underlying field theoretic formulation: (i) the non-relativistic electron kineticenergy term ∝ (cid:52) = δ ij ∂ ij comes from the kinetic term in the Dirac action, ¯ ψγ µ ∂ µ ψ − m ¯ ψψ , with { γ µ , γ ν } = η µν , while (ii) the e /r term is the staticGreen’s function of the electromagnetic field, which comes from inverting thekinetic term of the photon η αµ η βν F αβ F µν , which manifestly depends, by as-sumption, on the same spacetime metric η µν . Einstein assumed that, in orderto take into account the coupling to gravity, it was sufficient to replace η µν bythe same g µν both for the electron and the photon. By contrast, let us considerthe possibility that electrons (“matter”) and photons (“electromagnetism”) havea different coupling to gravity, e.g., described by saying that they couple to twodifferent (spatial) metrics, say g matter ij = δ ij g em ij = δ ij + h ij , (3.11)Then, computing the new propagators for the electron and the photon in their re-spective metrics, one finds that the SO(3) symmetry would be violated by tensorterms, appearing in the Hamiltonian, of the form δH ∼ e h ij x i x j r . This is aviolation, at a deep level, of the universality discussed in the previous section.The usual SO(3) symmetry implies that all energy levels with magnetic quantumnumber m are degenerate. But if tensor terms violating SO(3) were to exist, then,observables effects would include potentially measurable quadrupole-type split-tings in the energy levels, which, applied to the atomic nucleus (whose energylevels are a more sensitive probe of anisotropy), are ∝ (cid:104) I M | ˆ Q ij | I M (cid:105) , where I and M are the nuclear spin quantum numbers, and where ˆ Q ij is a symmetrictracefree tensor operator that couples to the tracefree part of h ij . Such types ofmeasurements have been performed on the energy levels of nuclei with impres-sively high accuracy, the current upper bound being (cid:12)(cid:12)(cid:12)(cid:12) h ij − h kk δ ij (cid:12)(cid:12)(cid:12)(cid:12) ≤ − . (3.12)The universality of space is thus valid to one part in , showing how delicateEinstein’s postulate is. The most recent limits on the deviation from the universality of free fall havebeen obtained by Eric’s Adelberger’s group [27]. In particular, they compared theacceleration of a Beryllium mass and a Copper one in the Earth’s gravitationalfield and found (cid:18) ∆ aa (cid:19) Be − Cu = ( − . ± . × − , (3.13) tring theory, gravity and experiment where ∆ a = a Be − a Cu . Other limits exist, such as, for instance, the fractionaldifference in acceleration of earth-core-like ( ∼ iron) and moon-mantle-like (sil-ica) bodies, (cid:18) ∆ aa (cid:19) Earth − core − Moon − mantle = (3 . ± . × − . (3.14)There are also excellent limits concerning celestial bodies. In particular thepossible difference in the accelerations of the Earth and the Moon towards theSun have been measured using laser ranging (with 5 mm accuracy) with retro-reflectors (corner cubes) placed on the Moon, giving the result [28] (cid:18) ∆ aa (cid:19) Earth − Moon = ( − . ± . × − . (3.15)One should, however, remember that only a fraction ( ∼ / ) of the Earth massis made of iron, while the rest is mostly silica (which is the main material theMoon is made of). As, independently of the equivalence principle, silica mustfall like silica, one looses a factor 3, so that the resulting bound on a possibleviolation of the equivalence principle is only around the × − level, whichis comparable to laboratory bounds. We conclude the section on the tests of the coupling of matter to gravity by justmentioning that the universality of the gravitational redshift, namely the apparentchange in the frequencies of two similar clocks in a gravitational field, has beentested by comparing the frequencies of hydrogen masers at the Earth surface andin a rocket. Vessot and Levine (1979) in Ref. [29] verified that the fractionalchange in the measured frequencies is consistent with GR to the − level: ∆ νν = (cid:0) ± − (cid:1) ∆ Uc . (3.16)The universality of this redshift has also been verified by measurements involvingother types of clocks. Until now we have only considered the coupling between matter and gravity, andvarious tests of its universality. We now discuss the tests of the dynamics ofthe gravitational field , i.e., tests probing either the propagator of the gravitationalfield, or the cubic or higher order gravitational vertices (for more detailed reviews T. Damour and M. Lilley see Refs. [30, 31]). We first consider the weak field regime, regime in which wecan write g µν = η µν + h µν , where h µν is numerically much smaller than one.For instance, in the solar system, h µν ∼ − on the surface of the Sun, ∼ − on the Earth orbit around the Sun, or ∼ − on the Earth surface. With valuesso small, it is clear that the solar system will not allow one to test many nonlinearterms in the perturbative expansion of g µν .We start by considering the gravitational interaction between two particlesof masses m A and m B . At linear order in h µν , we will have an interactioncorresponding to the following (classical Feynman-like) graph (cid:1) To compute explicitly what the preceding graph means, we must start from thefull action describing two gravitationally interacting bodies A and B : S = − m A (cid:90) d s A − m B (cid:90) d s B + (cid:90) √ gR πG . (3.17)Expanding S in the deviations of g µν away from η µν , one obtains (denoting h ≡ η µν h µν ) S = − m A (cid:90) (cid:113) − η µν d x µA d x νA − m B (cid:90) (cid:113) − η µν d x µB d x νB +12 (cid:90) h µν T µνA + 12 (cid:90) h µν T µνB + (cid:90) πG h µν (cid:3) (cid:18) h µν − hη µν (cid:19) + O ( h T ) + O ( h ) , (3.18)where T µνA is the (flat-space limit) of the stress-energy tensor of particle A (givenby a δ -function localized on the worldline of A ), and where the kinetic term of h µν is the one corresponding to the harmonic gauge ( i.e. , ∂ ν (cid:0) √ gg µν (cid:1) = 0 ).Inverting this kinetic term yields for h µν the following lowest-order equation(corresponding to Einstein’s equations at linearized order) (cid:3) h µν = − πGc (cid:18) T µν − D − T η µν (cid:19) . (3.19)with T µν = η µα η νβ ( T αβA + T αβB ) . We can then “integrate out” h µν by solving thelatter field equation for h , and replacing the result in the original action. Modulo tring theory, gravity and experiment self-interaction terms ∝ T µνA (cid:3) − P µνρσ T ρσA , the action then splits into the sumof three terms, a term − m A (cid:82) (cid:112) − η µν d x µA d x νA , describing the free propagationof body A , a similar term for B , and an interaction term, S int = − πGc (cid:90) T µνA (cid:3) − (cid:18) T Bµν − D − T B η µν (cid:19) . (3.20)More explicitly, if we introduce the scalar Green’s function G ( x ) , such that (cid:3) G ( x ) = − πδ D ( x ) , this lowest-order interaction term reads S int = 2 G (cid:90) (cid:90) d s A d s B m A u µA u νA P ρσµν G ( x A ( s A ) − x B ( s B )) m B u Bρ u Bσ , (3.21)in which one easily identifies the usual structure of a Feynman graph (namelythe one depicted above), with the coupling constant G in front, and a gravitonpropagator (comprising the scalar Green’s function, together with the spin-twoprojection operator P ρσµν , which can be read off the previous explicit result) sand-wiched between two source terms.In the stationary approximation, the scalar Green’s function reduces to theusual Newtonian propagator /r . If one further neglects the relative velocityof the two worldlines one can replace the spacetime velocities u µA and u µB by (1 , , , , · · · ) . This yields the usual Newtonian interaction term G (cid:82) dtm A m B /r .However, the “one-graviton exchange” diagram above contains many Einsteinianeffects that go beyond the Newtonian approximation. To compute them explic-itly, we first need the explicit expression of the relativistic scalar Green’s function G ( x ) . As we are deriving here the part of the gravitational interaction which is“conservative” ( i.e., energy conserving), we must use the time-symmetric (half-advanced half-retarded) Green’s function. In four dimensions, it is given by δ (cid:0) ( x A − x B ) (cid:1) . It is the sum of two terms (a retarded and an advanced one), asdepicted in FIG. 4. Note in passing that this classical time-symmetric propagatorcorresponds to the real part of the Feynman propagator. Indeed, for a masslessscalar particle in D = 4 the Feynman propagator (in x space) is proportional to ix + iε = iP P x + πδ (cid:0) x (cid:1) , (3.22)where the first term on the r.h.s. is a distributional “principal part” (it is pureimaginary and “quantum”), while the second (real) term is the classical contribu-tion (classically the interaction propagates along the light cone, see FIG. 4). Notethat, contrary to the Newtonian picture where the interaction is instantaneous, wehave here an interaction which depends both on the future and on the past . T. Damour and M. Lilley (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1) h µν h µν + A B A B
Figure 4. Time-symmetric half-advanced half-retarded contributions to the gravitational interactionbetween particles A and B.
When considering the case (of most importance in applications) where A and B move slowly relative to c , the time-symmetric propagator can be expandedin powers of /c . The first term in this expansion yields the usual Newtonianinstantaneous interaction, while all the higher-order terms can be expressed interms of successive derivatives of the positions of A and B , (so that the acausal-ity formally disappears, and is replaced by a dependence of the Lagrangian onderivatives higher than the velocities, i.e., accelerations, derivatives of accelera-tions, and so forth. Actually, such higher-derivative terms start appearing only atthe so-called “second post-Newtonian” (2PN) order, i.e., the order O (cid:0) c (cid:1) . Suchhigher-order post-Newtonian (PN) contributions are important for some appli-cations (binary pulsars, coalescing black holes), and have been computed up tothe 3PN ( O (cid:0) c (cid:1) ), as well as 3.5 PN ( O (cid:0) c (cid:1) ) levels. Here we shall consideronly the first post-Newtonian, 1PN, level, i.e., O (cid:0) c (cid:1) . At this level, the actioncan be written entirely in terms of the velocities of A and B (taken at the sameinstant t in some Lorentz frame). By expanding the time-symmetric acausal one-graviton-exchange action written above to order /c one finds the followingexplicit 1PN Lagrangian (now considered for an N -body system made of masses This “acausal” behaviour is due to our considering the conservative (“Fokker”) action. If we werecomputing the “real” classical equations of motion of the two particles, we would use only a retarded
Green’s function. The equations of motion so obtained would then be “causal” and would automati-cally contain some (physically needed), time-asymmetric “radiation reaction” terms. The trick, usedhere, to employ an acausal time-symmetric Green’s function is a technical shortcut allowing one toderive the action yielding the conservative part of the equations of motion. tring theory, gravity and experiment labelled A, B = 1 , , · · · , N. ) L − body = 12 (cid:88) A (cid:54) = B Gm A m B r AB (cid:104) c (cid:16) (cid:126)v A + (cid:126)v B (cid:17) − c (cid:126)v A . (cid:126)v B − c ( (cid:126)n AB . (cid:126)v A ) ( (cid:126)n AB .(cid:126)v B ) + O (cid:18) c (cid:19) (cid:105) , (3.23)Note that the coefficients / , / , / , etc., arise from the spin 2 nature of thegraviton, i.e., they are uniquely fixed by Einstein’s propagator.When considering a gravitationally bound N -body system, we must remem-ber that there is a link v ∼ GMr , due to the “virial theorem”. This link saysthat the v c contributions in the one-graviton-exchange graph considered abovemust be completed by computing non-linear interaction graphs containing moregravitons, namely the ones of order G /c involving two powers of the couplingconstant G . These contributions correspond to the terms O ( h T ) or O ( h ) in the h -expansion of the exact Einstein action. In terms of Feynmam-like diagrams,this means the following graphs: (cid:1) (cid:2) (cid:3) (cid:4) Note, in particular, that these terms involve the graviton cubic vertex (whosestructure will therefore be probed by solar-system experiments). Note also thatsome of these terms involve only two bodies (being proportional, say, to m A m B ),while others can involve three distinct bodies ( ∝ m A m B m C ). The full G /c result (containing both two-body and three-body terms) is found to be equal to L − body = − (cid:88) B (cid:54) = A (cid:54) = C G m A m B m C r AB r AC c , (3.24)where the factor of / is a prediction from Einstein’s theory. Note that thesummation is restricted by B (cid:54) = A (cid:54) = C which allows for the two-body termswhere B = C .When looking at the nonlinear diagrams above one sees some “loops” madeby graviton propagator lines closing up on a matter worldline. This may seemparadoxical because we are considering here classical gravitational effects, andclassical theory is usually thought of as involving only tree diagrams. Indeed, if T. Damour and M. Lilley we replace all our “source worldlines” (drawn above as continuous worldlines)by separate external sources ( i.e., by replacing the line describing T µνA ( x ) byseparate “blobs” on which graviton propagators start or end), we see that all thediagrams above open up and become tree diagrams. However, the presence ofloops in the diagrams used here do correspond to essentially some of the samephysical effects that “quantum loops” describe. This is particularly clear for thediagrams below (which are included in the classical calculations) (cid:1) (cid:2) It is clear that these diagrams describe the self-gravity effects of a mass m A onitself. As such, they do describe the classical limit of quantum loops such as thesimplest one-loop diagram (cid:1) which describes the back action of the emission and reabsorption of a gravitonon a quantum particle. Another similarity beween “classical loops” and quantumones, is that, in practice, multi-loops are associated to the presence of multipleintegrals which are increasingly difficult to compute. In addition, the loop di-agrams depicted in the penultimate graph lead (like quantum loops) to formallydivergent integrals. The origin of these divergences is that we have been de-scribing the gravitationally interacting bodies as “pointlike”, i.e., mathematicallydescribed by a δ -function (on a worldline). There are several ways of dealingwith this technical problem. One can complete the formal perturbative calcu-lations done with point-like bodies by another approximation scheme in whicheach body is locally viewed (in its own rest frame) as a weakly perturbed iso-lated body. The development of such a dual perturbation method [32, 33] showsthat the non-point-like, internal structure of non-rotating compact bodies (neu-tron stars or black holes) will enter their translational dynamics only at the 5PN Note that classical diagrams must all be computed in x -space. An increasing number of loopssignals the presence of intermediate vertices in x -space on which one must integrate. tring theory, gravity and experiment level ( ∼ G /c ), corresponding to 5 loops ! Knowing this, one expects thatthe use of a gauge-invariant regularization method for treating gravitationally-interacting point masses should give a physically unique answer up to 5 loops(excluded). Using dimensional regularization , one finds that all self-gravity ef-fects are unambiguous and finite at 1PN and 2PN [33]. Recent work has pushedthe calculation to the 3PN ( i.e., δ -function source [35]. The 1PN-level results described above are accurate enough for describing thegravitational dynamics in the solar system. Testing the validity of GR’s descrip-tion of the gravitational field’s dynamics is then achieved by verifying the agree-ment of the coefficients introduced above with experimental measurements. Inthis section, we will see that these GR-predicted coefficients agree with theirexperimentally measured value to better than the − level for the coefficientsentering the one-graviton-exchange term, and to about the − level for the ad-ditional 1PN multi-graviton term. There are many observables that can be usedto test relativistic gravity in the solar system. One may use the advance of theperihelion of planets, the deflection, by the local curvature, of light reaching theEarth from distant stars, the additional time delays suffered by electromagneticsignals compared to their flat spacetime counterparts, or also, general relativis-tic corrections to the Moon’s motion using the laser ranging technique alreadymentioned in previous sections.When testing Einstein’s predictions it is convenient to embed GR within aclass of alternative gravity theories. For instance, one could consider not only theinteraction of matter with the usual Einstein (pure spin-2) graviton but also an in-teraction with a long-range scalar field ϕ , i.e., a spin-zero massless field, coupledto the trace of T µν with strength √ Gα ( ϕ ) . This leads to an additional attractiveforce, so that the effective gravitational constant measured in a Cavendish exper-iment is G eff = G (1 + α ) . This also modifies the v /c terms in the two-bodyaction introduced above by terms proportional to α . These modifications are of-ten summarized by writing the 1PN-level metric generated by an N -body systemin the form d s = − (cid:18) − Uc + 2 (cid:0) β (cid:1) U c (cid:19) c d t + (cid:18) γ ) Uc (cid:19) δ ij d x i d x j , where U = G eff (cid:80) A m A /r A is the (effective) Newtonian potential and where T. Damour and M. Lilley the two dimensionless coefficients ¯ β and ¯ γ encode the two possible (Lorentz-invariant) deviations from GR which enter the 1PN level. For an additionalcoupling to a scalar with ϕ -dependent coupling strength α ( ϕ ) , one finds thatthe “post-Einstein” parameter ¯ γ is given by ¯ γ = − α / (1 + α ) . As for theother “post-Einstein” parameter ¯ β it measures a possible modification of the(cubic-vertex related) three-body action L − body written above, and it is givenby ¯ β = + βα / (1 + α ) where β denotes the derivative of the scalar coupling α w.r.t. the field ϕ .The most accurate test of GR in the solar system is the one made using theCassini spacecraft by the authors of Ref. [36]. This test is (essentially) onlysensitive to the post-Einstein parameter ¯ γ ( i.e., it depends only on the gravitonpropagator and the coupling to matter but not on nonlinear terms). This experi-ment used electromagnetic signals sent from the Earth to the Cassini spacecraftand transponded back to Earth, and monitored the ratio between the electromag-netic frequency ν + ∆ ν recorded back on Earth to the initial frequency ν . Thisratio was used to probe the change in the geometry of spacetime in the vicinityof the Sun as the line of sight moved, especially when it was nearly grazing theSun. The theoretical prediction for the experimental quantity measured in thisexperiment is (cid:18) ∆ νν (cid:19) − way = − γ ) GM sun c b d b d t (3.25)where b is the impact parameter, i.e., the distance of closest approach of the sig-nal’s trajectory to the center of the Sun. The experimental data gave the followingresult for the parameter ¯ γ ¯ γ = (2 . ± . × − . (3.26)This confirms GR (namely ¯ γ GR = 0 ) to the O (1) × − level.The three-graviton vertex can be probed by considering a body, having a non-negligible gravitational self-binding energy, in an external gravitational field. In-deed, as emphasized by Nordtvedt [37, 38], the free fall acceleration of a self-gravitating body is, in most gravity theories (except GR) sensitive to its gravi-tational binding energy. For instance, the Earth and the Moon will have, in ageneral theory, a slightly different acceleration of free fall towards the Sun. Theeffect is proportional to the combination of post-Einstein parameters β − ¯ γ . Lu-nar laser ranging data have allowed one to put a stringent upper limit on such apossibility [28], namely β − ¯ γ = (4 . ± . × − . (3.27)Thus, to date, predictions of Einstein’s theory in the linear (one-graviton-exchange) tring theory, gravity and experiment approximation have been verified to the − level, while some of the cubicallynonlinear aspects have been verified to the − level.The tests discussed up to now concern the quasi-stationary, weak-field regime,as it can be probed in the solar system. We shall now discuss the tests obtainedin binary pulsar data, which have gone beyond the solar-system tests in probingpart of the strongly nonlinear regime of gravity. Binary pulsars were discovered by Hulse and Taylor in 1974. Such systems aremade of two objects going around each other in very elliptical orbits. Both ob-jects are neutron stars , of which one is a pulsar, i.e., a rotating, magnetizedobject that emits a beam of electromagnetic noise (which includes radio waves,as well as other parts of the electromagnetic spectrum). When one looks at thegeometry generated by a neutron star, and computes deviations from flat space ofthe metric components, one finds (on the surface of the star) g = − GMc R (cid:39) − . (3.28)for a star of (typical) mass . M (cid:12) , and radius R = 10 km. This is a 40%deviation from flat space. By contrast, we recall that, in the solar system, thelargest metric deviation from flat space occurs on the Sun’s surface and is oforder GM/ ( c R ) ∼ − . We should therefore a priori expect that such ob-jects might provide tests that go beyond the solar system ones in probing someof the strong-field aspects of relativistic gravity. In addition, in the solar sys-tem, the time-irreversible radiative aspects of gravity ( i.e., radiation reaction )are negligible (this is why we focussed above on time-symmetric interactions).Here, not only are strongly nonlinear effects relevant, but one must also takeinto account the time-dissymetric effects linked to using a retarded propagator ∝ δ (cid:16) t − | x A − x B | c (cid:17) / | x A − x B | . The corresponding time delay | x A − x B | c is typ-ically (cid:39) sec (since typical separations between the two objects are of order , km) and plays an essential role in the equations of motion of a binarypulsar. As a consequence, binary pulsars have given us firm experimental evi-dence for the reality of gravitational radiation, and for the fact that on-shell grav-itational radiation is described by two transverse tensorial degrees of freedomtravelling at the velocity of light.In practice, only a subset of the known binary pulsars can be used for testingthe strong nonlinear regime of GR, and/or its radiative regime. Among these, the Except in a few cases where the companion is another compact (though less compact) star rem-nant, a white dwarf. T. Damour and M. Lilley very best ones are PSR1913+16 (where the numbers 19h13m and +16 deg mea-sure angles on the sky), PSR1534+12, PSRJ1141-6545, and PSRJ0737-3039, thefirst double binary pulsar (made of two radio pulsars, simultaneously emitting to-ward the Earth).>From the theoretical point of view, methods have been developed to dealwith strongly self-gravitating objects, both in Einstein’s theory and in alternativetheories (see [39] for a recent review, and references). To adequately discuss theobservations of binary pulsars, one has had to push the post-Newtonian pertur-bative calculation to the 2.5 PN level, i.e., to order ( v/c ) . This odd power ofthe ratio v/c is linked to the time-dissymetric, retarded nature of the propagator(together with some nonlinear effects). It is the first PN level where radiationreaction effects arise. Any experimental test of the presence of such ( v/c ) termsin the equations of motion is a probe of the reality of gravitational radiation. Inaddition, as mentionned above, one must carefully treat, and disentangle, the var-ious strong-field effects that are linked to the self-gravity of each neutron star inthe system.The existing experimental tests are based on the timing of binary pulsars. Eachtime the beam of radio waves sweeps across the Earth, one observes a pulse ofelectromagnetic radiation. The data consists in recording the successive arrivaltimes, say t N (with N = 1 , , , · · · ), of these pulses. Were the pulsar fixed inspace, these arrival times would be equally spaced in time, i.e., t N = t + N P ,where P would be the (fixed) period of the pulsar. However, in a binary pulsar,the sequence of arrival times is a more complicated function of the integer N than such a simple linear dependence. Indeed, one must take into account manyeffects: the fact that the pulsar moves on an approximately elliptical orbit, thedeviation of this orbit from a usual Keplerian ellipse, the deviation of the or-bital velocity from the usual Kepler areal velocity law, the existence of variousadditional relativistic effects: gravitational redshift, second-order (relativistic)Doppler effect, time-delay when the electromagetnic pulse passes near the com-panion, radiation reaction effects in the orbital motion, etc. To compute all theseeffects, one needs to solve Einstein’s equations of motion with high ( ∼ ( v/c ) )accuracy.The final result of these theoretical calculations is to derive the so-called DDtiming formula , which gives the N th pulse arrival time t N as an explicit functionof various “Keplerian” ( p K ), and “post-Keplerian” ( p PK ) parameters, say t N − t = F (cid:2) N ; p K ; p PK (cid:3) . (3.29)Here, the Keplerian parameters ( p K ) comprise parameters that would exist in apurely Keplerian description of the timing: the orbital period P b , the eccentric-ity of the orbit e , the time of passage at some initial periastron T , and some tring theory, gravity and experiment corresponding angular position of the periastron ω , and, finally, the projectedsemi-major axis x = a sin ic , where a is the semi-major axis of the orbit of theobservable pulsar and i is the inclination angle w.r.t. the plane of the sky. Thepost-Keplerian parameters ( p PK ) then correspond to many relativistic effects thatgo beyond a Keplerian description, namely: a dimensionless parameter k mea-suring the progressive advance of the periastron k = (cid:104) ˙ ω (cid:105) P b / π , a parameter γ t measuring the combined second-order Doppler and gravitational redshift effects,possible secular variations in Keplerian parameters ˙ e , ˙ x , ˙ P b , two parameters r , s measuring the “range” and the “shape” of the additional time delay that ap-pears when the radio waves pass near the companion, and finally a parameter δ θ measuring the distortion of the orbit w.r.t an ellipse. By least-squares fitting theobserved arrival times t obs N to the above general theoretical timing formula onecan accurately determine the numerical values of all the Keplerian parameters,as well as some of the post-Keplerian ones. At this stage, the determination ofthese phenomenological parameters is (in great part) independent of the choiceof a theory of gravity. On the other hand, in any specific theory of gravity, eachpost-Keplerian parameter is predicted to be some well-defined function of theKeplerian parameters and of the two masses, m and m , of the pulsar and itscompanion. For instance, within GR the advance of the periastron is given by k GR (cid:0) p K , m , m (cid:1) = 3 c ( GM n ) / − e , (3.30)where n = 2 π/P b and M = m + m , while the secular variation of the orbitalperiod (caused by radiation reaction effects) is given by ˙ P GR b (cid:0) p K , m , m (cid:1) = − π c e + e (1 − e ) / ( GM n ) / m m M . (3.31)Note that while k is proportional to /c (1PN level), the secular variation ofthe orbital period is proportional to /c (and is indeed numerically of order ( v/c ) ). The GR-predicted value for ˙ P b is a direct reflection of the presence of O (cid:0) ( v/c ) (cid:1) time-asymmetric radiation damping terms in the equations of motion.Numerically, ˙ P b (which is dimensionless) is predicted to be of typical order ofmagnitude ˙ P b ∼ − , which seems very small, but happens to be large enoughto be measured with good accuracy in several binary pulsars.The crucial point to notice is that the GR predictions (of which two are givenhere as examples) for the link between the post-Keplerian parameters and themasses are specific to the structure of GR, and will be replaced, in other theoriesof gravity, by different functions k theory (cid:0) p K , m , m (cid:1) , ˙ P theory b (cid:0) p K , m , m (cid:1) , In general, only one of the two objects, here labelled as , is a pulsar. T. Damour and M. Lilley etc. In particular, it has been explicitly shown in various cases (and notably inthe case of generic tensor-scalar theories where gravity is mediated both by aspin-2 field and a spin-0 one) that the large self-gravity of neutron stars wouldgenerically enter these functions, and drastically modify the usual prediction ofGR, see [40].To see which theory of gravity is in agreement with pulsar timing data, onecan proceed as follows. Within each theory of gravity, the measurement of eachpost-Keplerian parameter defines a corresponding curve in the m , m plane.Therefore, in general, the measurement of two post-Keplerian parameters is suf-ficient to determine the (a priori unknown) numerical values of the two masses m and m (as the location where the two curves intersect). Then, the measure-ment of any additional post-Keplerian parameter yields a clear test of the validityof the theory considered: the corresponding third curve should pass preciselythrough the intersection point of the first two curves. If it does not, the theoryis invalidated by the binary pulsar data considered. By the same reasoning, themeasurement of n different post-Keplerian parameters yields n − tests of theunderlying theory of gravity. Many such stringent tests have been obtained inbinary pulsar observations (more precisely, nine different tests in all have beenobtained when considering the data from four binary pulsars). Remarkably, GRhas been found to be consistent with all these tests . Many alternative gravitytheories have fallen by the wayside, or their parameters have been constrainedso as to make the theory extremely close to GR in all circumstances (includingstrong-field ones).Let us just give two impressive examples of the beautiful agreement betweenGR and pulsar data. In the case of the original Hulse-Taylor pulsar PSR1913+16the ratio between the observed value of ˙ P b to that predicted by GR is given by (cid:34) ˙ P obs b − ˙ P gal b ˙ P GR b [ k obs , γ obs ] (cid:35) = 1 . ± . , (3.32)where ˙ P gal b is a Galactic correction. The fact that this ratio is close to one corre-sponds to a confirmation of the relativistic force law acting on the pulsar, of thesymbolic form F = GMr (cid:16) · · · + (cid:0) vc (cid:1) (cid:17) , where the crucial last term ∼ ( v/c ) ( i.e., an effect of order − ) has been verified with a fractional accuracy of or-der − . Note that this corresponds to an absolute accuracy of order − compared to the leading Newtonian term ∼ GM/r !The timing data from the recently discovered double binary pulsar PSRJ0737-3039 led to the following ratio between the observed, and GR-predicted, values tring theory, gravity and experiment of the post-Keplerian parameter s (cid:20) s obs s GR [ k obs , R obs ] (cid:21) = 0 . ± . . (3.33)The agreement for this parameter is at the × − level.Summarizing: binary pulsar timing data have led to accurate confirmations ofthe strong-field and radiative structure of GR. Roughly speaking, these confir-mations exclude any alternative theory containing long-range fields coupled tobulk (hadronic) matter. So far, we have mainly focussed on tests of GR on spatial scales of several as-tronomical units (the size of the solar system), and on scales of , km (thetypical separation between two neutron stars). We conclude this section on ex-perimental tests of gravity by mentioning the existence of tests made on verylarge spatial and temporal scales. Gravitational lensing effects by galaxy clus-ters allow one to probe some aspects of relativistic gravity on scales ∼ kpc.Here, one is talking of the effect of the curved spacetime metric generated bythe cluster on light emitted by very distant quasars and passing near a galaxycluster containing (in addition to visible galaxies) a lot of dark matter, as wellas some X-ray gas. Data on the temperature distribution of the X-ray gas allowsone to directly probe the Newtonian gravitational potential U ( x ) of the cluster(without having to assume much about the (dark) matter distribution). In turn,the potential U ( x ) determines the relativistic lensing of light, via the spacetimemetric predicted by Einstein’s theory, i.e., − g = 1 − Uc , g ij = (cid:0) Uc (cid:1) δ ij .According to Ref. [41], the agreement is of the order of 30%. This confirms thevalidity of GR on scales ∼ kpc.Primordial nucleosynthesis of light elements ( e.g., Helium, Lithium, Deu-terium) in the early universe depends on both the expansion rate and on the weak-interaction reaction rate for the conversion between neutrons and protons. Giventhat the Hubble parameter H ∝ Gρ ∝ GT , the creation of light elementsat early times (and high temperatures T ) depends on Newton’s constant. Thecomparison between theoretical predictions and observations of the abundanceof light elements typically constrains the value of G at the time of Big Bangnucleosynthesis, say G BB to differ by less than O (10%) from its current value G now (see e.g., Chapter 18 of the Review of Particle Physics, http://pdg.lbl.gov/). By which, one really means here fields with range larger than the distance between the twopulsars, i.e., ∼
300 000 km. T. Damour and M. Lilley
4. String-inspired phenomenology of the gravitational sector >From the previous sections, one can conclude that GR is a very well confirmedtheory so that one might be tempted to require of any future theory (and espe-cially string theory) that it lead to essentially no observable deviations from usual4-dimensional GR. For instance, one might require that all the a priori masslessscalar fields that abound in (tree level, compactified) string theory acquire largemasses. However, as there is yet no clear understanding of how to fit our worldwithin string theory, it is phenomenologically interesting to keep an open mindand explore whether there exist possibilities for deviations of GR that have natu-rally escaped detection so far.String theory predicts the existence of an extended mass spectrum ( g µν ( x ) , Φ ( x ) , B µν , moduli fields, etc.) from which there could result some long rangeor short range modification of gravity. The existence of branes and large extra di-mensions could also be sources of modified gravity ( e.g., KK gravity). Therecould exist short distance effects at scales of order the string scale (cid:96) s whichare observable in cosmology or in high energy astrophysics. We shall also con-sider possible gravitational wave signals from string-cosmology models. Finally,we refer the reader to the lectures by Juan Maldacena for a discussion of non-gaussianities in CMB data.A phenomenologically interesting idea (though it is not supported by precisetheoretical arguments) is a possible breakdown of Lorentz invariance, on largescale physics, linked to string-scale cutoff-related effects. An example of this isa modified dispersion relation of the type E = m + (cid:126)p + β E m P + β E m P + . . . (4.1)where m P denotes the Planck mass. One could think that because of the largevalue of the Planck mass, any such corrections to the usual dispersion relation areunobservable. However, there exist astrophysical phenomena, such as high en-ergy cosmic rays, for instance high energy γ -rays, for which such a small changein this relation could be observed. For example, by comparing the times-of-arrival of γ -rays of different energies, one has been able to place strong limits onthe parameter β . Such modifications of the dispersion relation have also beenused in the analysis of the CMB, since, in the standard inflationary model, initialquantum fluctuations (the seeds of today’s large scale structures) arise in the deepultraviolet i.e., at transplanckian scales. Note that there exist theoretical difficul-ties with the inclusion of the β E m P term (the one which is severely constrained Linked to its proportionality to /m P , while most theoretical models suggest a proportionality tring theory, gravity and experiment L C STD MODELBULK GRAVITY
Figure 5. The ends of open strings are attached to a brane, giving rise to SM particles, while closedstrings are free to propagate in the bulk. experimentally), while the more conventional fourth order term would be toosmall to be observed. Note that in the case of the photon, a modification on shortscales could imply a birefringence of the vacuum as ω ± = | k | (cid:16) ± β | k | m P (cid:17) . Forreferences on these issues see [42, 43]. Speaking of string-inspired astrophysicaleffects, let us mention the suggestion of Ref. [44] that string theory might imply aviolation of the usual Kerr bound on the spin of rotating black holes: J ≤ GM .Other possible predictions of string theory arise from the picture in whichone considers the existence of branes on which (open string) SM particles areconfined, while (closed string) gravitons are free to propagate in the bulk (FIG.5). The extra dimensions of the bulk can then be compactified, on a Calabi-Yau or simply on a torus (thereby “localizing” gravity around the SM brane).Constraints on the size of the compactified dimensions then come either fromthe gravitational phenomenology, or from effects on SM particles. This is the“large” extra dimensions idea [45] which could be tested at the LHC, and sois of interest today. Other realizations include models with “very large” extradimensions [46], but it is less clear how they are realized in string theory. Inthe Randall-Sundrum model [46], a brane can be like a defect in a bulk with a to /m P . T. Damour and M. Lilley negative cosmological constant, in which case the zero mode of bulk gravitationalwaves behaves as a surface wave localized on the brane due to the discontinuitylocated at the interface of the brane with the bulk. In the DGP model [47], theapproximate localization of bulk gravity on the SM brane is achieved through theinterplay of two dynamics for the gravitational sector: a 5D Einstein action, plus a4D “induced” Einstein action, with a different value of Newton’s constant, on thebrane. Combining the two inverse propagators, the global propagator drasticallymodifies gravity on large length scales r : r ≥ L = G G . (4.2)In addition, even on length scales r ≤ L there exist modifications of usual gravity.Indeed, the claim is that Newton’s potential is modified as [48] U (cid:39) GMr (cid:34) − L (cid:114) r c GM (cid:35) . (4.3)At the phenomenological level, it is interesting that ( Newtonian ) gravity be mod-ified in this way. Estimates indicate that effects are small enough to have escapeddetection so far, but could be seen in refined solar system experiments ( e.g.,
LunarLaser Ranging). Some authors have argued that such models may have acausalbehaviours, with, for instance, the appearance of closed timelike curves [49].Another conceptually interesting idea involves the possible existence of sev-eral (parallel) Randall-Sundrum branes. The confining mechanism of gravity inthe Randall-Sundrum model is such that the wavefunction of surface gravitonsis exponentially decaying away from the brane. If two branes are nearby, suchquasi-confined gravitational effects can tunnel from one brane to the other via ex-ponentially small effects. As a consequence, the effective Lagrangian would con-tain two metric tensors with two gravitons, one massless, the other massive [50].There are, however, theoretical difficulties with any massive gravity theory, inrelation with the van Dam-Veltman-Zakharov discontinuity (see, e.g., [51] andreferences therein).
It is well known that, at tree level in string theory, there exist many masslessscalar fields with gravitational strength coupling, the so-called moduli fields.Phenomenologically, one would expect that having massless scalar fields at lowenergies is undesirable (Would a theory containing such massless fields not im-mediately fail the GR tests discussed in the section above?). General argumentssuggest that such scalar fields should not be expected to remain massless after tring theory, gravity and experiment supersymmetry breaking [52]. Recently, a large “industry” has been devoted totry to construct explicit compactification models where all moduli are fixed, andactually acquire very heavy masses, which is needed if inflation is to happen inthe usual way. Here however, in the spirit of keeping an open mind, we willinstead assume that a scalar field remains massless in the low-energy effectivetheory, and discuss ways in which it might not disagree with existing tests ofgeneral relativity. In other words, we suppose there exists a flat or almost flatdirection in the total scalar potential V ( ϕ ) , such that there remains a masslessfield after supersymmetry breaking. Let us mention in this respect the idea sug-gested, in particular, by Eliezer Rabinovici [53] that the ultimate explanation forthe smallness of the cosmological constant might be a mechanism of spontaneousbreaking of an underlying scale invariance. In that case, we would expect to havean associated massless Goldstone boson (the “dilaton”, in the original sense ofthe word). Let us discuss here the idea of the least coupling principle , realized via a cos-mological attractor mechanism (see e.g.,
Refs [54, 55]), which can reconcile theexistence of a massless scalar field in the low energy world with existing tests ofGR (and with cosmological inflation). Note that, to date, it is not known whetherthis mechanism can be realized in string theory. We assume the existence of amassless scalar field Φ ( i.e., of a flat direction in the potential), with gravitational-strength coupling to matter. A priori , this looks phenomenologically forbiddenbut we are going to see that the cosmological attactor mechanism (CAM) tendsto drive Φ towards a value where its coupling to matter becomes naturally (cid:28) .In the string frame, we start with an effective action of the generic form S eff = (cid:90) d x (cid:112) ˆ g (cid:104) B g (Φ) ˆ Rα (cid:48) + B Φ (Φ) α (cid:48) (cid:16) (cid:3) Φ − ∇ Φ) (cid:17) − B F (Φ) k F µν − B Ψ (Φ) ΨD ¯Ψ − B χ (Φ) (cid:16) ˆ ∇ ˆ χ (cid:17) − m χ (Φ) χ (cid:105) , where Φ is the massless dilaton field, χ is the inflaton, to which has been asso-ciated a simple chaotic-inflation-type potential term, with the exception that here m χ is a function of Φ . In heterotic string theory for instance, B g , B Φ , B F , B Ψ and B χ are given by expansions in powers of the string coupling g s = e Φ , as B i = e − + c ( i )0 + c ( i )1 e + . . . (4.4)where the first term is the tree level term, followed by an infinite series of cor-rection terms involving positive powers of g s (or non-perturbative functions of T. Damour and M. Lilley g s ). Switching to the Einstein frame, and redefining ˆ g µν and the nonstandard Φ kinetic terms according to ˆ g µν → g µν = CB g (Φ) ˆ g µν ,ϕ = (cid:90) d Φ (cid:34) (cid:18) B (cid:48) g B g (cid:19) + 2 B (cid:48) Φ B g + 2 B Φ B g (cid:35) / , (4.5)(where a prime denotes d / dΦ ), the effective action turns into S eff = (cid:90) d x √ g (cid:104) ˜ m p R − ˜ m p ∇ ϕ ) − ˜ m p F ( ϕ ) ( ∇ χ ) − m ϕ ( χ ) χ (cid:105) + · · · , (4.6)with ˜ m P ≡ πG , and in which the χ terms are important during inflation whileadditional terms that include the gauge fields and ordinary matter such as − B F ( ϕ ) F µν − (cid:88) A (cid:90) m A [ B F ( ϕ ( x A ))] × (cid:113) − g µν ( x A ) d x µA d x νA − V vac ( ϕ ) (4.7)are relevant in the matter dominated era.As we shall see, the CAM leads to some generic predictions even withoutknowing the specific structure of the various coupling functions, such as e.g., m χ ( ϕ ) , m A ( B F ( ϕ )) , · · · . The basic assumption one has to make is that thestring-loop corrections are such that there exists a minimum in (some of) the func-tions m ( ϕ ) at some (finite or infinite) value, ϕ m . During inflation, the dynamicsis governed by a set of coupled differential equations for the scale factor, χ and ϕ . In particular, the equation of motion for ϕ contains a term ∝ − ∂∂ϕ m χ ( ϕ ) χ .During inflation ( i.e., when χ has a large vacuum expectation value, this couplingdrives ϕ towards the special point ϕ m where m χ ( ϕ ) reaches a minimum. Once ϕ has been so attracted near ϕ m , ϕ essentially (classically) decouples from χ (sothat inflation proceeds as if ϕ was not there). A similar attractor mechanism ex-ists during the other phases of cosmological evolution, and tends to decouple ϕ from the dominant cosmological matter. For this mechanism to efficiently decou-ple ϕ from all types of matter, one needs the special point ϕ m to approximatelyminimize all the important coupling functions. This can be naturally realized byassuming that ϕ m is a special point in field space: for instance it could be thefixed point of some Z symmetry of the T - or S -duality type (so that one couldsay that “symmetry is attractive”). An alternative way of having such a special tring theory, gravity and experiment point in field space is to assume that ϕ m = + ∞ is a limiting point where allcoupling functions have finite limits. This leads to the so-called runaway dilaton scenario [55]. In that case the mere assumption that B i (Φ) (cid:39) c i + O (cid:0) e − (cid:1) as Φ → + ∞ implies that ϕ m = + ∞ is an attractor where all couplings vanish. Before discussing the observational predictions of the CAM, let us remind thereader of a few facts that are relevant for studying the possible effects of a string-inspired modification of gravity. The main source of modification of gravitycomes from the fact that the “moduli” field ϕ will influence the values of themasses of the (low-energy) particles and nuclei. This means that the classicalaction of, say an atom A , will be − (cid:90) m A ( ϕ )d s A = − (cid:90) m A ( ϕ ) (cid:113) − g µν d x µA d x νA (4.8)where g µν is the Einstein-frame metric. Then, one finds that the scalar field ϕ will be coupled to the atom A with the strength α A √ G , where the dimensionlesscoupling strength α A (with the same normalization as the one discussed abovefor usual tensor-scalar theories ) is simply given by α A = ∂∂ϕ ln m A ( ϕ ) . (4.9)To see better the various ways in which ϕ might enter into m A , let us considerfor instance the various parts constituting the mass of an atom: m A ( ϕ ) = Zm p + N m n + Zm e + E nucleusSU3 + E nucleusU1 , (4.10)where Z is the atomic number, m p the mass of the proton, N the neutron number, m n the mass of the neutron, m e the mass of the electron, E nucleusSU3 and E nucleusU1 the nuclear and Coulomb interaction energies of the nucleus, respectively. Inaddition, one must note that the mass of the proton is given by m p ( ϕ ) = a Λ QCD (cid:0) g ( ϕ ) (cid:1) + b u m u ( ϕ ) + b d m d ( ϕ ) + c p Λ QCD α em ( ϕ ) . (4.11)The main scale that determines the mass of the proton is Λ QCD . It depends on allthe moduli including the massless field ϕ and is roughly of the form Λ QCD ( ϕ ) = C / g ( ϕ ) B − / g ( ϕ ) exp (cid:20) − πB F ( ϕ ) b (cid:21) ˜ M string . (4.12) This is viewed as a strong-(bare-)coupling limit, by contrast to the usual weak-coupling limit ϕ → −∞ and Φ → −∞ . In particular, the effective Newton constant for a Cavendish experiment between a body made ofatoms A and another one made of atoms B is G eff AB = G (1 + α A α B ) . T. Damour and M. Lilley
Here, C g is the conformal factor from the string to the Einstein frame. The mostimportant contribution to the ϕ dependence of Λ QCD is that given by the ϕ de-pendence of the exponential term. This dependence comes from the well-knownrunning (via the β -function of SU(3) ) of some (unified) gauge coupling constantbetween its value /g ∝ B F ( ϕ ) considered at a GUT-scale cut-off (here ap-proximately related to ˜ M string ), to a value of order unity at the confining scale Λ QCD . The other contributions to the mass of the proton are the quark masses,which are determined by the vev of the Higgs boson and by the Yukawa couplingconstants, which, again, are expected to be functions of ϕ at high energy. Therealso exists a contribution from the electromagnetic sector since part of the massof the proton is a function of the fine structure constant α em ( ϕ ) . Finally the nu-clear binding energy of a nucleus is quite important and must also be expressedas a function of basic scales. In an approximate form it reads E nucleusSU3 (cid:39) ( N + Z ) a + ( N + Z ) / b (4.13)where a (cid:39) a chiral limit3 + ∂a ∂m π m π ( ϕ ) . (4.14)In the chiral limit ( i.e. , taking the quark masses to zero) one gets a non-zerolimit a chiral limit3 to which must be added a term approximately proportional tothe squared pion mass. In turn, m π is proportional to the product of Λ QCD and m u + m d , both of which are expected to be functions of ϕ . Incidentally, let usnote that there exists a delicate balance between attractive and repulsive nuclearinteractions [56], which implies a strong sensitivity of the binding energy of nu-clei to the value of the quark masses [57]. A recent result shows that if the quarkmasses were to increase by 50% (at one σ , or 64% at σ ), all heavy nuclei wouldfall apart because there would be no nuclear binding [58].At leading order, the mass of any nucleus is a pure number times Λ QCD . Inthis approximation, m A would depend universally on ϕ (via Λ QCD ( ϕ ) ), and thescalar coupling strength α A would be independent of the atomic species A con-sidered. As a consequence, there would be no violation of the universality of freefall. This shows that the violations of the universality of free fall will depend onthe small fractional corrections in m A proportional to the ratios m u Λ QCD , m d Λ QCD , and α em . (4.15)When differentiating the mass of an atom w.r.t. ϕ , say m A ( ϕ ) = N Λ QCD (cid:18) ε σA m u + m d Λ QCD + ε δA m d − m u Λ QCD + ε em A α em (cid:19) , (4.16) tring theory, gravity and experiment where N is a pure number (which depends on N and Z ), one obtains for thescalar coupling strength α A ( ϕ ) = ∂∂ϕ ln m A ( ϕ ) an (approximate) expression ofthe form α A ( ϕ ) (cid:39) α had ( ϕ )+ ε σA ∂∂ϕ (cid:18) m u + m d Λ QCD (cid:19) + ε δA ∂∂ϕ (cid:18) m d − m u Λ QCD (cid:19) + ε em A ∂∂ϕ α em , (4.17)where α had ≡ ∂∂ϕ ln Λ QCD ( ϕ ) . When the CAM has attracted ϕ near a value ϕ m which minimizes all the separate coupling functions entering the variousingredients of m A ( ϕ ) , each term in the above expression for α A ( ϕ ) will be(approximately) proportional to the small difference ϕ − ϕ m . As a consequenceall the contributions to α A ( ϕ ) will be small, so that all the observable deviationsfrom GR will be naturally small.Let us describe more precisely the possible observable consequences of theCAM. In this mechanism, the couplings of the massless scalar field to the vari-ous physical sectors are not assumed to be initially small (they are given by thevarious coupling functions B i ( ϕ ) entering the Lagrangian, and these functionsare “of order unity”). However, via its coupling to cosmological evolution, thescalar field is driven towards a point where the couplings to matter become small,but not exactly zero. Indeed, one can analytically estimate the “efficiency” of thecosmological evolution in driving ϕ towards ϕ m , and one finds some expressionfor the difference δϕ ≡ ϕ − ϕ m [54, 55]. The deviations from GR are all pro-portional to the small quantity δϕ because the scalar coupling strengths α A , α B are proportional to δϕ , and all “post-Einstein” observables contain two scalarcouplings , say α A α B when talking about the scalar exchange between A and B (for instance the modified gravitational constant for a Cavendish experiment in-volving two bodies made of atoms A and B is G AB = G (1 + α A α B ) ). In addi-tion to predicting small values for the (approximately composition-independent)“post-Einstein” parameters ¯ γ and ¯ β this mechanism also predicts various (small)violations of the equivalence principle.For instance, the above expressions for the ingredients entering m A and α A lead to generic predictions about the type of violation of the universality of freefall that one might expect in string theory. Indeed, one finds that the fractionaldifference in the free fall acceleration of two bodies (made of atoms A and B ) When ϕ m is infinite, δϕ ≡ ϕ − ϕ m is replaced, e.g., by e − cϕ . T. Damour and M. Lilley takes the form a A − a B (cid:104) a (cid:105) (cid:39) × − α (cid:104) ∆ (cid:18) EM (cid:19) AB + c B ∆ (cid:18) N + ZM (cid:19) AB + c D ∆ (cid:18) N − ZM (cid:19) AB (cid:105) , (4.18)with EM = Z ( Z − N + Z ) / . (4.19)where (∆ Q ) AB ≡ Q A − Q B , and where the first, second and third terms inthe brackets are contributions from the Coulomb energy of the nucleus ( ∝ α em ),and from the ϕ -dependence of the sum and difference of the quarks masses, i.e., m u + m d and m u − m d .This mechanism also predicts (approximately composition-independent) val-ues for the post-Einstein parameters ¯ γ and ¯ β parametrizing 1PN-level deviationsfrom GR. They are of the form ¯ γ = − α α (cid:39) − α , (4.20)and ¯ β = 12 α ∂α had ∂ϕ (1 + α ) (cid:39) α ∂α had ∂ϕ . (4.21)In this model, one in fact violates all tests of GR. However, all these violationsare correlated. For instance, using the numerical value ∆ (cid:0) EM (cid:1) (cid:39) . (whichapplies both to the pair Cu–Be and to the pair Pt–Ti), one finds the following linkbetween equivalence-principle violations and solar-system deviations (cid:18) ∆ aa (cid:19) (cid:39) − . × − ¯ γ. (4.22)Given that present tests of the equivalence principle place a limit on the ratio ∆ a/a of the order of − , one finds | ¯ γ | ≤ × − . Note that the upper limitgiven on ¯ γ by the Cassini experiment was − , so that in this case the necessarysensitivity has not yet been reached to test the CAM.As another example, one can compute the evolution of the fine structure con-stant w.r.t. time. Given that it is a function of ϕ , and that ϕ evolves as a functionof cosmological evolution due to its coupling to matter, α em is indeed a functionof time, and its time derivative can be written as dd t ln α em ∼ ± − (cid:114) q − m (cid:114) ∆ aa yr − . (4.23) tring theory, gravity and experiment The first square root on the r.h.s. of this equation can also be written as Ω m α m + 4Ω v α v Ω m + 2Ω v . (4.24)where Ω m denotes the fraction of the cosmological closure density due to darkmatter, and α m the scalar coupling to dark matter, while Ω v and α v denote thecorresponding quantities for “dark energy” (or “vacuum energy”). For instance,if we assume α v ∼ (so that ϕ is a kind of “quintessence”) while α m (cid:28) , wesee from the result above that the current experimental limit ∆ aa < − , im-plies the following upper bound on a possible time variation of the fine-structureconstant: dd t (ln α em ) ≤ − yr − . This upper bound is below the currentlaboratory limits on ˙ α/α , but comparable to the Oklo limit mentioned above.When working out the generic predictions of the runaway dilaton version of thecosmological attractor mechanism, one finds that it naturally predicts (when as-suming an inflationary potential ∝ χ ) a level of deviation from GR of order − ¯ γ ∼ C × − , corresponding, for instance, to a violation of the equiva-lence principle at the level ∆ a/a ∼ C × − . Here, C is a combination ofmodel-dependent dimensionless parameters, which are generically expected tobe “of order unity”. This suggests (if C is smaller, but not much smaller than1) that the current sensitivity of equivalence principle experiments may be closeto what is needed to test the deviations from GR predicted by such a runawaydilaton. Let us note in this respect that ongoing improved lunar laser ranging ex-periments will probe ∆ a/a to better than the ∼ − level, and that the CNESsatellite mission MICROSCOPE (to be launched in the coming years) will reach ∆ a/a ∼ − . Another more ambitious satellite mission (which is not yetapproved), STEP (Satellite Test of the Equivalence Principle), plans to probeviolations of the equivalence principle down to the − level . In addition,post-Newtonian solar system experiments at the − level would be of interest.The approved micro-arcsecond global astrometry experiment GAIA will probe ¯ γ ∼ − , while the planned laser experiment LATOR might reach ¯ γ ∼ − .In addition, the comparison of cold-atom clocks might soon reach the interestinglevel dd t (ln α em ) ∼ − yr − .Finally, let us mention that one can combine the basic mechanism of the CAM(which consists in using the coupling of ϕ to matter, i.e., the presence of a termof the form a ( ϕ ) ρ matter in the action) with the presence of a “quintessence”-like potential V ( ϕ ) ∝ /ϕ p . This yields the “chameleon” mechanism [60] inwhich both the value ϕ m towards which ϕ is attracted, and the effective mass(or inverse range) of ϕ , depend on the local matter density ρ matter . Whatever be Let us also mention the suggestion [59] that atom interferometry might be used for testing theequivalence principle down to the ∆ a/a ∼ − level. T. Damour and M. Lilley one’s opinion concerning the a priori plausibility of having some nearly masslessmoduli field surviving in the low-energy physics of string theory, it is clear thatsuch experiments are important and could teach us something new about reality.
5. String-related signals in cosmology
In the usual inflationary scenario, the period of exponential expansion is basedon the slow roll mechanism, i.e., one has to assume a sufficiently flat potentialso that the scalar field, the inflaton, slowly rolls down to its minimum in such away that the approximate equality p ϕ (cid:39) − ρ ϕ lasts sufficiently long, say for aminimum of 60–70 e-folds. The simplest inflationary Lagrangian reads L = −
12 ( ∂ϕ ) − V ( ϕ ) (5.1)with a usual kinetic term and a potential V ( ϕ ) . This simple inflationary frame-work leads to specific predictions such as a relation between the ratio of tensorto scalar primordial perturbations and the “distance” in field space over which ϕ runs during inflation (the so-called Lyth bound, see the lectures by Juan Mal-dacena in these proceedings). Let us, however, emphasize that these predictions(which lead to constraints on the model) do depend on the assumption that infla-tion is realized by the simple action (5.1) with a slow-roll potential. There are,however, other ways of realizing inflation, in which these constraints might be re-laxed. Let us note in this respect that inflation can be realized even if the potential V ( ϕ ) in (5.1) is not of the slow-roll type [61]. Moreover, one may have inflationwithout a potential at all if the Lagrangian is a complicated enough function of X ≡ − ( ∂ϕ ) . Indeed, if one has an action of the type L = p ( X ) , one findsthat there can exist attractors toward a de Sitter expansion phase, correspondingto a line where the effective equation of state deduced from L = p ( X ) is p = − ρ ( e.g., k-inflation [62]; ghost inflation [63]). To have a “graceful exit” from this deSitter phase one needs, for instance, to introduce some additional ϕ dependencein L . It has been suggested in [64] that such a mechanism might be realized instring theory, via a Dirac-Born-Infeld-type action, say p ( X, ϕ ) = − ϕ λ (cid:32)(cid:115) − λ ˙ ϕ ϕ − (cid:33) − V ( ϕ ) . (5.2) For instance, if one considers it very unlikely that such a field can exist, these experiments areimportant because they can falsify string theory. By contrast, if one finds a violation of the equivalenceprinciple which is nicely consistent with the prediction (4.18) for the composition dependence of amoduli field, this might be viewed as a confirmation of string theory. tring theory, gravity and experiment In such a “DBI inflation”, the use of non-standard kinetic terms greatly relaxesthe restrictions imposed on the flatness of the potential V ( ϕ ) which must be im-posed in the usual case of (5.1). It also tends to produce larger non-gaussianitiesin the CMB [65]. Let us also point out that the use of a non-linear kinetic termmight significantly affect the Lyth bound. For instance, if one considers the ac-tion L = p ( X, ϕ ) = K ( X ) − V ( ϕ ) , (5.3)where K ( X ) is a non-linear function of the kinetic term X ≡ − ( ∂ϕ ) , one findsthe following modified form of the relation between the ratio r of tensor to scalarprimordial perturbations and the derivative of ϕ w.r.t. the number of efolds N : r (cid:18) d ϕ d N (cid:19) a (5.4)Here a is an additional amplification factor, which is given by the following ex-pression in terms of the kinetic function K ( X ) a = 2 K (cid:48) (cid:113) X K (cid:48)(cid:48) K (cid:48) = K = X , − √ − X, (cid:29) , e.g., − α (1 − X ) α , with α < . A large amplification factor a (cid:28) would (formally) correspond to a relaxedLyth bound on the excursion of ϕ , given a minimum number N of efolds. It isinteresting to note that a = 1 (unchanged Lyth bound) in the DBI-type squareroot model. However, we note that a more general power α , with α < / , wouldformally relax the Lyth bound. [A more detailed study is, however, necessary forseeing whether the bound is physically relaxed.]The present section presented only a very partial and sketchy picture. It wasonly intended as an illustration that folklore results and constraints on inflationarymodels do depend on using the standard slow-roll action (5.1), and that there existother mechanisms in which those results and constraints might be different andpossibly relaxed. The existence and detection of cosmic superstrings is an exciting possibility thatwas first suggested in Ref. [66], then kept alive for a number of years, and re-cently revived dramatically notably in Ref. [67] and in other papers [68–70].They arise in brane antibrane scenarios where the inflaton is the brane-antibrane T. Damour and M. Lilley
D D ϕ V ϕ Figure 6. Left: The brane-antibrane distance as a scalar field ϕ . Right: V ( ϕ ) behaves as c − c ϕ − for large brane-antibrane separations separation (see e.g., [71]). In such scenarios, for large enough brane-antibraneseparations, the potential behaves like c − c /ϕ such that it satisfies the slowroll conditions (FIG. 6). When the branes are near, some of the modes connect-ing the two branes become tachyonic, i.e., a complex field (with kinetic term − ∂T ∂ ¯ T ) having a potential V ( | T | ) with wrong-sign curvature near T = 0 .This instability can generate topological defects since the phase of the vev of T need not be uniformly the same all over space. Contrary to the situation inwhich strings are created at the beginning of inflation and then diluted away, thisscenario naturally produces strings at the end of inflation so that they are not di-luted by the expansion. For causality reasons, the value of the field’s vev in agiven Hubble patch should be uncorrelated to that in other Hubble patches. Thisis what creates a network of strings and one can then compute the initial den-sity and correlation length of the string network. The string tension µ in Planckunits, i.e., the dimensionless parameter G µ , where G is Newton’s constant, wasinitially thought to be high, of the order of − at best, because of the stringtheoretic origin of these objects and of the then expected relation between α (cid:48) andthe Planck length. However, in models with warping factors and large fluxes,the string tension can be lowered to much smaller values. In practice, the stringtension is tuned to fit current CMB data. Tye and collaborators [68] find a win-dow of the type − < G µ < − , while in the more detailed KKLMMTmodel [67] one finds Gµ ∼ − .In trying to gain insight into the observational predictions that can be madefrom cosmic superstring models, one must consider not only the stretching by thecosmological expansion of an initial network of cosmic strings with a correlationlength of the order of the Hubble scale , but also string interactions. A string canfor instance self-intersect or two strings can intersect and reconnect. The Hubbleexpansion tends to locally straighten out the strings while interconnections tendto produce loops and small-scale structure. Given an initial correlation length and tring theory, gravity and experiment reconnection probability p , working out the time evolution of a string network isessentially a classical problem. Two types of strings develop, long strings withcorrelation length of the order of the time scale t , and small loops that looseenergy by gravitational radiation [72–74].In order to define the typical size of string loops (at the time they are formed)we introduce a dimensionless parameter α such that (cid:96) loop ( t ) = α c t . It wasinitially thought that α = 50 G µ , an estimate linked to the idea that gravita-tional damping is the essential mechanism wich determines the lifetime of loops.More recently, is has been suggested that α might be significantly different from G µ . There is, however, no consensus on the “correct” value of α . Estimatesvary between α ∼ (50 G µ ) β , with β > (leading to “small loops”) and α ∼ . (leading to large loops). [For an introduction to this problem, and references,see e.g., the talk of Joe Polchinski at the 2007 String meeting in Madrid.] Hap-pily, some of the predictions we shall discuss below (notably those concerningthe observability of gravitational waves from a cosmic string network) are ratherinsensitive to the value of α .Several numerical simulations confirm the tendency of string networks to dis-play a scale-invariant behavior [75,76]. There have been recent attempts at refin-ing the theoretical description of string networks [77, 78]. However, there is, todate, no consensus among experts as to the typical size distribution of loops ( i.e., the dominant gravitational wave-emitting string type). In several simulations, thedistribution of the size of loops is bimodal, with one peak at α ∼ . and anotherpeak at the UV cutoff. It has been argued by Vilenkin and collaborators that onlythe “large loop” part, i.e., α ∼ . , will survive.In the following, assuming KKLMMT-type brane inflation and the stability ofstrings over cosmologically interesting time scales, we discuss the phenomeno-logical predictions made by treating p and α as free parameters and their possibleobservable signals. Partly for historical reasons, the phenomenology of cosmic strings has been stud-ied mostly in the context of CMB observations. Slow-roll inflation generates arandom δTT angular distribution on the sky that fits well the observations. Addinga random network of cosmic strings generates additional (non-Gaussian) fluctua-tions in the CMB which have less angular structure (the string has a lensing effectproportional to its velocity v over the sky, δT /T ∼ πGµvγ ). CMB observa-tions can then be used to place an upper bound on Gµ , of order × − . Muchsmaller values of the string tension will not lead to any observable signature inthe CMB. Let us also mention that cosmic (super)strings might be detected viatheir gravitational lensing of galaxies, or microlensing of stars.By contrast to the CMB (or lensing) observations, which are only sensitive to T. Damour and M. Lilley string tensions
Gµ > − , existing or planned gravitational wave interferom-eters could detect cosmic (super)strings with tensions in the much wider range − < G µ < − . Let us recall that a gravitational wave (GW) detector isactually measuring tidal forces, and more precisely a component of the Riemanntensor projected “along” a detector having a quadrupolar structure . In otherwords, a GW antenna measures the second time derivative ¨ h ( t ) of a projectionof the metric fluctuation h µν . Current detectors are sensitive down to the level h ∼ − , for frequencies f ∼ Hz.The GW signal from a string network is an incoherent background of GWsmade of the superposition of all GWs ever emitted by string loops (from zeroto very large redshifts). This signal is distributed over a very large spectrumof frequencies (including wavelengths of the size of the universe, as well as veryshort ones). In order to determine the frequency distribution, the number of loopsand how they evolve, one needs to know the evolution of the universe during theinflationary, radiation and matter dominated eras.Besides detecting the GWs from a string network in a man-made interferom-eter, another observational possibility lies in the timing of isolated millisecondpulsars. In a stationary spacetime, the pulses emitted by an isolated pulsar wouldbe observed on Earth (after correcting for the Earth motion) at very regular in-tervals. By contrast, in presence of a fluctuating background of GWs, the timesof arrival of successive pulses would fluctuate, and exhibit some red noise. Pul-sar timing over some time interval T (which is typically several years) is mostsensitive to the part of the GW frequency spectrum with frequencies f ∼ /T .Therefore, pulsar timing is most likely to detect long wavelength GWs (severallight years long).Along with LIGO-type ground based interferometers, a space-based one, theLaser Interferometer Space Antenna (LISA) has been conceived, with arm lengthsof the order of km instead of the 3 or km ones constructed on the ground.LISA can therefore explore much smaller frequencies. The best achievable sen-sitivity for LIGO-type instruments is reached for frequencies f ∼ Hz, i.e., rather fast events lasting ∼ − seconds, while space experiments may probeevents with periods ∼ secs, which are quite slow events (see FIG. 7). InRef. [74], the possible existence of sharp gravitational wave bursts above thebackground caused by string cusps was pointed out. For a typical oscillatingloop, there occurs a cusp once or twice per oscillation with the extremity of thecusp going at the velocity of light and emitting a strong gravitational wave sig-nal in the direction in which the cusp is moving. Statistically, these events arerandom. A GW burst will be detected if it happens to be emitted towards the This is the spin-2 analog of saying that electromagnetic antennas are sensitive to the projectionof the electric field along the direction of a dipolar antenna. tring theory, gravity and experiment ρ gwc ρ
10 Hz10 −3 Hz1/3 yr −1 f LIGOLISAPULSAR
Figure 7. Expected frequency distribution of the ratio Ω GW = ρ GW ρ c for the stochastic gravitationalwave background of cosmic strings. detector. Under some conditions, these cusps can create signals which standmuch above the quasi-Gaussian random mean square background “GW noise”.This raises the exciting possibility that LIGO/VIRGO/GEO or LISA might detectGW signals emitted by giant superstrings at cosmological distances.Let us now give an introduction to the physics behind the occurrence of thosecusps, and the associated emission of GW bursts. We consider the string position X µ as a function of the worldsheet coordinates τ and σ . We treat the string dynamics in a locally flat spacetime. Introducing thelightcone coordinates in conformal gauge, σ ± = τ ± σ, (5.5) X µ ( τ, σ ) satisfies ∂∂σ + ∂∂σ − X µ ( τ, σ ) = 0 (5.6)such that the generic string solution is the sum of left and right movers X µ ( τ, σ ) = 12 (cid:2) X µ + ( σ + ) + X µ − ( σ − ) (cid:3) (5.7)in which the factor / is introduced for convenience. The Virasoro constraintsread (cid:0) ∂ + X µ + (cid:1) = 0 (cid:0) ∂ − X µ − (cid:1) = 0 (5.8) T. Damour and M. Lilley
In the time gauge, the worldsheet is sliced by constant coordinate time hyper-planes X = x = τ , so that X ( τ, σ ) = τ = 12 ( σ + + σ − ) . (5.9)We thus have X = σ + and X − = σ − . Then ∂ ± X = 1 contributes a − in theVirasoro constraints, so that (cid:0) ∂ + X µ + (cid:1) = − (cid:0) ∂ + X i (cid:1) , (5.10)and similarly for the − equation. This means that the derivatives (w.r.t. theirargument) of the spatial components X i ± ( σ ± ) of the left and right modes areconstrained to be unit euclidean vectors : (cid:16) ˙ X i ± (cid:17) = 1 . (5.11)The X i ± are periodic and the time derivative of the spatial component of X areunit vectors. We may now use a representation first introduced in Ref. [79]. Thederivatives ˙ X i ± can be seen as drawing two curves on the unit sphere (the Turok-Kibble sphere). In addition, as X i ± is periodic in three-dimensional space (thereis no winding), we have (cid:82) d σ ± ˙ X i ± = 0 . As a result, the “center of mass” of bothleft and right moving curves must be at the center of the sphere. This impliesthat the two curves generically intersect twice [80]. Now, the main point isthat an intersection between the two curves represents a cusp . More technically,such an intersection corresponds to particular points on the string worldsheetat which the two null (see (5.8)) tangent vectors ˙ X µ + and ˙ X µ − are parallel inspacetime. In general, the string worldsheet intersects locally the light cone alongthe two distinct directions ˙ X µ + and ˙ X µ − . The cusps are special points where theworldsheet is tangent to the lightcone (see FIG. 8). This is a singularity of theclassical worldsheet at which a strong gravitational wave signal is emitted alongthe common null vector. Let us now indicate how one computes the emission ofgravitational wave bursts from cuspy strings. We consider Einstein’s theory inthe linearized approximation, g µν ( x ) = η µν + h µν ( x ) . (5.12)We use the harmonic gauge, ∂ ν ¯ h µν = 0 , so that Einstein’s equations simplify to (cid:3) ¯ h µν = − πGT µν ( x ) , (5.13) There exist, however, specially contrived curves that can avoid intersecting. tring theory, gravity and experiment X . µ− X . µ+ θ Figure 8. Left: The lightcone generically intersects the worldsheet into two separate null directionscorresponding to the velocity of the left and right movers. Right: When a cusp occurs, the two nullvectors are parallel and the worldsheet is tangent to the lightcone. There results a large burst ofoutgoing gravitational radiation. where ¯ h µν = h µν − hη µν . (5.14)The stress-energy tensor is obtained by differentiating the Nambu action w.r.t. g µν . Taking its Fourier transform, one finds T µν (cid:0) k λ (cid:1) = µT (cid:96) (cid:90) Σ (cid:96) d τ d σ ˙ X ( µ + ˙ X ν ) − e − i k. ( X + + X − ) (5.15)where ( µν ) indicates symmetrization over the indices µ, ν , and where, in theexponential, we have replaced X µ by the half-sum of the left and right movers.The fundamental period of a loop of length (cid:96) is T (cid:96) = (cid:96) . Note that (cid:96) is the invarianttotal length E µ , where µ is the string tension. In string theory, one usually uses aworldsheet gauge where (cid:96) is either or π but here one finds it more convenientto use a gauge where σ and τ are connected to an external definition of time(namely X = x = τ ).We wish to compute the integral giving T µν (cid:0) k λ (cid:1) over a periodic domain Σ (cid:96) in the τ , σ plane. We can rewrite the integral as an integral over d σ + d σ − . Thisyields the famous left-right factorization of closed string amplitudes and theFourier transform of the string stress-energy tensor reads T µν ( k ) = µ(cid:96) I ( µ + I ν ) − , (5.16) Though we are doing here a classical calculation, one recognizes that the result is given by thegraviton vertex operator. T. Damour and M. Lilley where I µ ± = (cid:90) (cid:96) dσ ± ˙ X µ ± ( σ ± ) e − i k.X ± . (5.17)By solving Einstein’s equation (5.13), one finds that the spacetime-Fourier trans-form of the source on the r.h.s. actually gives the time-Fourier transform of theasymptotic GW amplitude (emitted in the direction n i = k i /k ), i.e., the time-Fourier transform of the quantity κ µν ( t − r, (cid:126)n ) appearing in the asymptotic ex-pansion ¯ h µν ( t, (cid:126)x ) = κ µν ( t − r, (cid:126)n ) r + O (cid:18) r (cid:19) . (5.18)Here the /r decrease in amplitude as a function of distance away from the stringis caused by the retarded Green’s function in 3+1 dimensions. κ µν is a function ofboth the time variable and the angle of emission. As we just said, the time-Fouriertransform of κ µν is proportional to the spacetime-Fourier transform T µν ( k ) ofthe source, and is explicitly given by κ µν ( f, (cid:126)n ) = | f | (cid:90) d te πif ( t − r ) κ µν ( t − r, (cid:126)n )= 2 Gµ | f | I ( µ + ( ω, ω(cid:126)n ) I ν ) − ( ω, ω(cid:126)n ) . (5.19)This formula shows that we can compute what is observed in a GW detector as afunction of string tension, frequency, and the product of two integrals involvingleft and right moving modes.We can then estimate the generic features of the GW burst emitted by a cuspby noticing that, in the Fourier domain, each integral I µ ± is dominated (whenconsidering large frequencies: f (cid:29) T − (cid:96) ) by the singular behaviour of the twointegrands ˙ X µ ± ( σ ± ) e − i k.X ± near a cusp. The calculation proceeds by (Taylor)expanding the vectors X µ ± and ˙ X µ ± in powers of σ ± . One finds that the first fewleading terms in this expansion can be gauged away, so that the signal amplitudeis much smaller than what could have (and had) been initially thought. AfterFourier transforming back to the time domain, it is finally found that [74] κ ( t ) ∝ | t − t c | / , ¨ κ ( t ) ∝ | t − t c | − / . (5.20)As this result seems to crucially depend on the presence of a mathematically sin-gular behaviour of the classical string worldsheet at a cusp, one might worry thatquantum effects could blur away the sharp cusp, and make the above classicalburst signal disappear. It was checked that this is not the case [81] (the basic rea-son being that, finally, the strong GW burst signal is emitted by a large segmentof the string around the cusp). tring theory, gravity and experiment In order to understand the observational signature of a cosmic string network andnot just a single string, one must combine the analysis of the previous sectionwith the cosmological expansion of a Friedman-Lemaître universe and with anintegration over redshift. A crucial point is then to estimate the number density ofstring loops. This density can be analytically estimated as a function of the stringparameters, such as the reconnection probability p , and the string tension Gµ .Note that the reconnection probability is expected to be quite different for cosmicsupertrings compared to the traditionally considered field-theory strings. Fieldtheory strings are expected to reconnect, when they cross, with essentially unitprobability ( p (cid:39) ), while fundamental or D -strings are expected to reconnectwith a smallish probability, − < p < [82] (because of the presence of thestring coupling, and other factors).The loop number density can be approximately estimated as [83] n (cid:96) ∼ p Gµ t + . . . (5.21)where the first term on the r.h.s. comes from loops that were created at redshifts ≤ , while the ‘ . . . ’ denotes a possible additional contribution from high-redshiftstrings. When the loop-size parameter α is smaller or equal to the “tradition-ally expected” value Gµ , the contribution from high-redshift strings is neg-ligible (because strings decay in less than a Hubble time). By contrast, when α (cid:29) Gµ , the strings survive over many Hubble times, and the contributionof high-redshift strings starts to dominate the loop density. Note the somewhatunexpected feature displayed by the first term in n (cid:96) , namely that it increases bothas G µ and/or p are decreased. This feature is one of the features which allowGW signals from strings to be detected down to very small values of the stringtension (contrary to CMB effects). Indeed, as G µ is decreased, though eachindividual string signal will decrease proportionally to
G µ , there will be moreemitting loops. After integrating over redhifts, one finds that the observable sig-nal is a complicated, non monotonic function of
G µ . The numerical estimatesof Ref. [83] considered the case in which the loop size parameter α < Gµ ,in which case the first term in (5.21) is dominant. If, on the other hand, one as-sumes α ∼ . (as is suggested by some numerical simulations [75]) , stringssurvive longer so that higher redshift contributions are non-negligible. It hasbeen found that in such cases these contributions increase the number of loops(which increases the GW signal) but tend to drown the cusp signal within thequasi-Gaussiam random-mean-square GW background [84].Based on current detector capabilities and on the sensitivity estimates for fu-ture detectors, one finds that if α ≤ Gµ , LIGO could detect G µ ≥ − while LISA could detect G µ ≥ − . On the other hand, if α (cid:29) Gµ LISA T. Damour and M. Lilley could reach
G µ ≥ − . One has looked in the current LIGO data for the pos-sible presence of a background of GW’s, but without success so far [85]. Thebest current bound on G µ comes from pulsar timing [86] and is roughly at the
G µ ≤ − level (which is about three orders of magnitude more stringent thanthe limits than can be obtained from CMB data).Gravitational wave detectors are thus excellent probes of cosmic (super)strings.There is therefore the possibility that they could confirm or refute KKLMMT-type scenarios in a large domain of parameter space. However, there are largeuncertainties in string network dynamics which prevent one from being able tomake reliable analytical estimates. If one is in a region of parameter space wherethe rather specific cusp-related signals are well above the r.m.s. background onemight find rather direct experimental evidence for the existence of cosmic strings.There would however remain the task of discriminating between string theoreticstrings and field theoretic ones. One way would be (assuming one could stronglyreduce the string network uncertainties) to determine the reconnection probability p from its influence on the loop number density, and, thereby, on the recurrencerate of observed signals. Another more ambitious possibility would be to exploitthe presence of two populations of strings, namely D and F strings, in D-braneanti D-brane annihilation, and attempt to measure two different values of G µ ,the ratio of which satisfies µ D = µ F /g s .
6. Conclusion
We hope that these lectures have shown that gravity phenomenology is a poten-tially interesting arena for eventually confronting string theory to reality.
Acknowledgement
TD wishes to thank the organizers of this Les Houches session for putting to-gether a timely and interesting programme. He is especially grateful to NimaArkani-Hamed, Michael Douglas, Igor Klebanov and Eliezer Rabinovici for in-formative discussions. ML wishes to thank the organizers of the school and thelecturers for a very stimulating time in les Houches.
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