Grin of the Cheshire cat: Entropy density of spacetime as a relic from quantum gravity
aa r X i v : . [ g r- q c ] D ec Grin of the Cheshire cat: Entropy density ofspacetime as a relic from quantum gravity
Dawood Kothawala ∗ Department of Physics, IIT Madras, Chennai - 600 036, India.T. Padmanabhan † ,IUCAA, Post Bag 4, Ganeshkhind, Pune - 411 007, IndiaMarch 6, 2018 Abstract
There is considerable amount of evidence to suggest that the field equa-tions of gravity have the same status as, say, the equations describingan emergent phenomenon like elasticity. In fact, it is possible to de-rive the field equations from a thermodynamic variational principle inwhich a set of normalized vector fields are varied rather than the met-ric. We show that this variational principle can arise as a low energy( L P = ( G ~ /c ) / →
0) relic of a plausible non-perturbative effect ofquantum gravity, viz. the existence of a zero-point-length in the space-time. Our result is non-perturbative in the following sense: If we modifythe geodesic distance in a spacetime by introducing a zero-point-length,to incorporate some effects of quantum gravity, and take the limit L P → L P - independent) term. This term is identical to the ex-pression for entropy density of spacetime used previously in the emergentgravity approach.
This reconfirms the idea that the microscopic degreesof freedom of the spacetime, when properly described in the full theory,could lead to an effective description of geometry in terms of a thermody-namic variational principle. This is conceptually similar to the emergenceof thermodynamics from the mechanics of, say, molecules. The approachalso has important implications for the cosmological constant which arebriefly discussed. ∗ [email protected] † email: [email protected] Introduction, Motivation and Summary
Recent work has gone a long way in demonstrating that gravitational field equa-tions can be thought of as having the same conceptual status as equations inemergent phenomena like fluid dynamics or elasticity ([1, 2]). Two strongestpieces of evidence, amongst many, which support this point of view are thefollowing: • It is possible to obtain [3, 4] the field equations of a large class of gravi-tational theories including, but not limited to, Einstein’s theory from analternative, thermodynamic variational principle. In this approach, onestarts with a well defined thermodynamic potential S [ ∇ n, n ] which de-pends on a vector field n i of constant norm and has [2] the interpretationas the gravitational heat density of spacetime: S ∝ [( ∇ i n i ) − ∇ i n j ∇ j n i ] = R ab n a n b + (total divergence) (1)Extremising S with respect to all vector fields n i simultaneously, leadsto a constraint on the background metric which turns out to be identicalto the field equations. (Adding the appropriate matter heat density andextremising the total heat density will lead to field equations with source;in this paper, we will be mainly concerned about pure gravity.) • The time evolution of the spacetime geometry can be described through anequation which is mathematically equivalent to, say, Einstein’s equationbut can be written and interpreted entirely in terms of surface ( N sur ) andbulk ( N bulk ) degrees of freedom [2, 5]. When the metric is independentof time [5] we get, N sur = N bulk (“holographic equipartition”) and — inthe most general context — the time evolution of the metric is driven by[2] the difference ( N sur − N bulk ). Thus, not only the variational principle but even the resulting field equation can be expressed in a thermodynamiclanguage that brings to the fore the importance of microscopic degrees offreedom in the bulk and in the boundary.These results suggest that one should interpret the standard gravitational dy-namics as the thermodynamic limit of some underlying statistical physics whichdeals with the (as yet unknown) atoms of spacetime. Eventually, when we dis-cover the fundamental laws governing the dynamics of these microscopic degreesof freedom, we will also discover and understand the limiting process by whichthe thermodynamic variational principle, based on S , can be obtained. Whilethis possibility sounds natural, there are couple of puzzling features related tothis issue.First, it seems reasonable to assume that classical equations of gravity shouldarise in the L P → L P dependent) corrections. But, this does not seem to be the direction whichis suggested by the emergent gravity paradigm.
There are strong conceptualreasons to believe (including the problem of the cosmological constant) that themetric should not be the dynamical variable which is varied in the low energyeffective action of the theory and ipso facto
Einstein-Hilbert action will not bethe correct, L P → S ( ∇ n, n ) and varying n i . It is notclear how this transmutation in the variational principle (from R to R ab n a n b )arises.Let us elaborate on this point which suggests a radical departure from theconventional wisdom. The usual approaches to quantum gravity presuppose thatthe metric will continue to be a dynamical degree of freedom in the quantumgravitational domain (in some form or the other) because it is the dynamicalvariable in the classical limit. But the emergent gravity paradigm suggestssomething different. In the classical limit, it does not treat the usual metric asthe degree of freedom to be varied in an extremum principle and hence thereis no reason to expect this metric to play a direct role in the quantum theoryeither . If Einstein’s equations are like equations of elasticity, treating the metricas a quantum variable is like quantizing elasticity; we will then get gravitons asanalogues of phonons but the real microscopic degrees of freedom will be quitedifferent — in a solid or in a spacetime.Second, the variational principle based on S uses a vector field of constant norm.It is not clear, a priori, how a quantum gravitational variational principle willproduce such a vector field which survives in the low energy limit of L P → S in the L P → n i and the field equations of the low energy theory canbe obtained by varying this vector field. This demonstrates the possibility thatthe thermodynamic variational principle can indeed have a microscopic origin.We will see that the limiting process is subtle and mathematically non-trivial.It is this non-triviality which leads to a leading order ( L P →
0) variationalprinciple that is quite different from the Einstein-Hilbert action.The key new idea is to work with the biscalar σ ( p, P ) which is the geodesicdistance between two events p and P in any spacetime (with Ω ≡ (1 / σ being3he so called Synge world function). Locally, this function is related to themetric by the usual Hamilton-Jacobi equation, (1 / g ab ∇ a Ω ∇ b Ω = Ω, whichtakes the form: g ab ( x ) ∇ a σ ( x, x ′ ) ∇ b σ ( x, x ′ ) = 4 σ ( x, x ′ ) (2)Indeed, all the information about the spacetime geometry can be shown tobe encoded in σ . This is because the metric can be obtained [6] from thecoincidence limit of covariant derivatives of σ by: g ab = lim x ′ → x [ ∇ a ∇ b Ω( x, x ′ )] (3)and, of course, all other geometrical properties can be obtained from the metric.In other words, all of classical gravity and all of spacetime dynamics can bedescribed entirely in terms of the biscalar function σ ( p, P ). Therefore, onecan trade off the (local) degrees of freedom associated with the metric for thedegrees of freedom represented by the (nonlocal) object σ ( p, P ).The major advantage of using the geodesic distance instead of the metric isthe following. We have no clue how quantum gravitational effects modify thenotion of the metric at short distances and what kind of effective description iscalled for in the quasi-classical domain. However, there is significant amount ofevidence [7, 8] to suggest that quantum gravity introduces a zero-point-lengthto the spacetime in the sense of:lim p → P h σ ( p, P ) i = L P (4)where the h· · · i denotes averaging over metric fluctuations and L P is a funda-mental length scale, of the order of Planck length ( ≈ − cm). This suggeststhat one can capture the lowest order quantum gravitational effects by intro-ducing a zero-point-length to the spacetime by modifying σ → σ + L P . (5)For example, this changes the coincidence limit of Green’s functions and servesas a Lorentz-invariant UV regulator. The Euclidean propagator for the masslessscalar field, for example, gets replaced by: G ( σ ) ∝ ( σ ) − ( D − / → ( σ + L P ) − ( D − / (6)near the coincidence limit [7] (and the same holds for the massive propagator aswell). The logic and justification for such a result have been presented in severalprevious papers [7, 8, 9] and will not be repeated here. But the key point tonote is that the correction to σ in Eq. (5) is universal and captures some basicfeatures of quantum gravity. Equations (4) and (5) should be considered aspurely non-perturbative results in quantum gravity arising from the quantumgravitational averaging of σ ( p, P | g ab ) over the fluctuations of the metric g ab .4e have, of course, no clear idea as to how to describe the spacetime at Planckscales where the notions of differential manifold, metric etc. may break downcompletely. On the other hand, these features emerge in the long-wavelengthlimit, at scales much larger than L P . It seems reasonable to assume that therecould exist an in-between, quasi-classical domain, that interfaces classical andquantum gravity, in which we can still talk about a metric, spacetime intervaletc but with some modifications arising from the quantum gravity. The resultsin Eq. (4) and Eq. (5) motivate one to proceed along the following lines todescribe physics in the quasi-classical domain:1. Suppose we are interested in a classical spacetime with a given metric g ab or, equivalently, a given biscalar σ ( p, P ) corresponding to that metric.Assume for a moment that we can find another second rank, symmetric,bitensor q ab ( p, P ; L P ) (which we will call ‘qmetric’ for reasons which willbe clear soon) such that its geodesic distance is σ ( p, P ) + L P .2. Since it can be argued that σ → σ + L P captures the quantum gravita-tional effects in the quasi-classical domain, it seems reasonable to think ofa variational principle based on q ab ( p, P ; L P ) to achieve the same. This,in turn, motivates us to consider the Einstein-Hilbert action functional for q ab ( p, P ; L P ) which is constructed from the Ricci biscalar for the qmetric q ab ( p, P ; L P ) using the standard formula which connects Ricci scalar tothe metric.3. The qmetric is, of course, a non-local bitensor depending on two events p and P (and on L P ). Hence the Ricci biscalar R ( p, P ; L P ) obtained fromit will also depend on two events p and P and on L P . Obviously, we needto take a suitable limit of p → P as well as L P → L eff ≡ lim L P → lim p → P R ( p, P ; L P ) (7)can then be interpreted as the correct, low-energy, functional to be usedin the variational principle.Incredibly enough, strange — but nice — things happen along the way whenwe attempt to carry out the logical steps of the above program.To begin with, the qmetric is not a metric in the standard sense of differentialgeometry. This should be obvious because any geodesic interval obtained froma genuine metric, by integrating along a geodesic from event p to event P , isguaranteed to vanish when we take the coincidence limit of p → P . We arenever going to get the zero-point-length modification of the geodesic intervalin Eq. (5) if the qmetric is a genuine, local, metric and the limiting process isnon-singular. As we shall show, the qmetric depends explicitly on both p and P (making it a nonlocal bitensor) and has a singular structure in the coincidence5imit — which allows the geodesic interval constructed from the qmetric to benon-vanishing in the coincidence limit. The fact that this result arises in a verynatural fashion is noteworthy.Second — and the most surprising feature — is the form of the coincidencelimit of the Ricci biscalar. It turns out that when the coincidence limit p → P is taken in the Ricci biscalar for the qmetric, the leading term is precisely S given in Eq. (1) and not the Ricci scalar for the original metric! That is, weget: L eff ≡ lim L P → lim p → P R ( p, P ; L P ) ∝ S ( P ) (8)In other words, starting from the Ricci biscalar for the qmetric, taking the p → P limit and then taking L P → limit does not lead to the Ricci scalar for theoriginal metric but to S used in emergent gravity paradigm! The final resultdepends on an extra vector field of constant norm arising from the derivativesof the geodesic interval.The entire process involves taking the limits in a particular order so that thelocal quantities are defined from the non-local entities in a specific manner.Such operations are well justified by physical considerations at this stage andwill probably acquire a firmer mathematical basis when we understand quantumgravity better. What is obvious from our result — and is quite significant — isthe following conceptual fact: Quantum gravitational structures which dependon L P can lead to unexpected semi-classical relics when L P → L P . There is considerable amount ofevidence that this could indeed be the case.An analogy may be useful to clarify this point. We know that the classicaltheory of elasticity should be obtainable from the quantum dynamics of a solidby taking an appropriate continuum limit. One would have thought that such aclassical limit of a continuum solid can be obtained by taking the ~ → ~ → ~ → ~ inside the atoms” to be non-zero (ensuring the existenceof atoms) and let “all other ~ ’s” to vanish. The real classical world cannotbe described as a sum of a leading order, ~ -independent, phenomenon with ~ -dependent corrections. Figuratively speaking, classical world is non-analytic in ~ because matter made of atoms cannot exist in the strict ~ → ~ → L P ∝A P = ( G ~ /c ). If we write the Newton’s law of gravitation in two equivalentforms as F = G m m r = A P c ~ m m r (9)6t is clear that the limit ~ → G as independent of ~ andthe first form of the equation is oblivious to the process of taking ~ →
0. But if G is an emergent constant, like Young’s modulus or conductivity of a solid, thenit can have a non-trivial, implicit, dependence on ~ . If we further assume, basedon Eq. (5) that A P is the quantity which is independent of ~ , and characterizesquantum spacetime, then a completely different picture emerges [10]. Now weneed to use the second form of the equation to study, say, the planetary motion.The ~ → ~ ; there is no such thing as classical gravity just as there is no such thing asclassical solid since neither could exist in the strictly ~ → L P start-ing from quantum gravity. This, in turn, assumes that all quantum gravitationaleffects are analytic in L P and will lead to some sensible classical limits when L P → this is a highly questionableassumption and we should take seriously the possibility that quantum gravitycould have features which are non-analytic in L P . In that case, the process oftaking limits will involve manipulating singular quantities leading to unexpected(but interesting) results. The emergent nature of gravitational field equationscan arise from such a non-trivial limiting process which we illustrate here.There is a different conceptual bonus from this analysis. There are, arguably,three areas of contact and conflict between the principles of quantum theoryand gravity: (i) Thermodynamics of spacetime horizons. (ii) The singularityproblem in classical gravity, especially in cosmology and black hole physics.(iii) The problem of the cosmological constant. There have been repeated sug-gestions in the literature that quantum gravity will have something non-trivialto say about all these three issues. It is already known that emergent gravityparadigm provides deep insights into items (i) and (iii) but has been silent aboutthe singularity problem. On the other hand, the prescription in Eq. (5) has thepotentiality of tackling the singularity problem by making the coincidence limitfinite due to the fluctuations of the metric. We show here that the prescrip-tion in Eq. (5) — which has been discussed in the past in connection with thesingularity resolution and as a UV-regulator — can also lead to the variationalprinciple in emergent gravity. So the present work provides a unifying threadlinking the three issues listed above which is conceptually rather pleasing.We shall now describe some of the mathematical details of this procedure. Restof the sections of the paper are organized as follows. In Sec. 2, we present theexact form of the Ricci biscalar for the qmetric and obtain from it a local scalarwhich is the natural candidate for the quantum corrected Ricci scalar. We thencompare this object with the Ricci scalar of the original spacetime, which revealsthe non-triviality of the L P → Notation : We work in D dimensions, and use the sign convention ( − , + , + , . . . )for Lorentzian spaces. Most of our analysis is best viewed as done in a Euclideanspace, with analytic continuation done right in the end.
This is implicit in thecalculations even when we do not state in explicitly. In this sense, the parameter ǫ which is the norm of the normalized tangent vector n i to geodesics is simply+1. We have nevertheless kept it in the equations to help keep track of signissues arising for timelike geodesics after analytic continuation. We will now carry out the steps described in items (1) to (3) in page 5. Ourstarting point is the result derived in [11]. This result allows us to associate asecond rank, symmetric, bitensor q ab ( p, P ) with a spacetime having the metric g ab and the geodesic distance σ ( p, P ) such that the following two propertiesare satisfied: (i) The modified geodesic distance σ ( q )2 = σ + L P (10)(where σ ( q )2 ≡ σ ( p, P | q ab ) and σ ≡ σ ( p, P | g ab ) for simplifying the notation)satisfies the analogue of Eq. (2) with the metric replaced by the qmetric. Thatis: q ab ∇ a σ ( q )2 ∇ b σ ( q )2 = 4 σ ( q )2 (11)(ii) The Euclidean propagator for the massless scalar field, with the modificationin Eq. (6), satisfies the usual Green function equation √− g (cid:3) G = δ ( x, x ′ ) withthe qmetric replacing the metric in the operator on the left-hand-side. Thesetwo conditions allow us to determine the form of the qmetric to be: q ab ( p, P ; L P ) = A [ σ ; L P ] g ab ( p ) − ǫ (cid:18) A [ σ ; L P ] − A [ σ ; L P ] (cid:19) n a ( p ; P ) n b ( p ; P )(12)with A [ σ ; L P ] = 1 + L P σ ; n a = ∇ a σ √ ǫσ (13)In fact, it is possible to generalize the notion of qmetric for an arbitrary mod-ification of the geodesic interval σ → S ( σ ); an outline of derivation for this8eneral case is given in Appendix A which covers the above result as a specialcase. Derivation of the above result, for the modification of the geodesic intervalin Eq. (10), and its implications (such as the effect on short-distance behaviorof Green’s functions and on spacetime singularities) are discussed in ref.[11].One can think of Eq. (12) as associating a qmetric with every metric such thatthe geodesic distance computed from the qmetric is ( σ + L P ) if the geodesicdistance computed from the original metric is σ . Throughout the paper wethink of P as a fixed ‘base event’ and p as the variable ‘field event’. All deriva-tives, covariant or partial, are taken at the spacetime event p . We see thatwhen the geodesic distance between the two events is much larger than thePlanck length ( σ ( p, P ) /L P ≫ q ab ( p, P ; L P ) → g ab ( p ). In this sense we can think of qmetric as describingthe nonlocal, quantum gravitational effects near the Planck scale, arising fromthe existence of a zero-point-length.We will now attempt to relate the curvature bi-invariants (obtained by the usualformulas treating it like a metric) of the qmetric q with those of the originalmetric g . Needless to say, this is a formidable task, since the qmetric is a sumof two terms: (i) a piece which is conformal to g , and (ii) a piece depending onthe tangent vector n connecting P to p . This complicates the evaluation of thefull curvature tensor of q in terms of that of g . Fortunately, for the purpose ofthis paper, we only need the Ricci biscalar corresponding to the qmetric whichcan be derived by a few tricks involving the Gauss-Codazzi relation. To begin with, note that the deformation is characterized by the function A [ σ ; L P ] which is only a function(al) of σ . Further, as was shown in [11],the induced geometry on σ =const surface undergoes a conformal deformation h ab → A [ σ ; L P ] h ab which leads to a rather simple relation for the induced Ricciscalars which appear in the Gauss-Codazzi equations. The two facts above sug-gest that it might be mathematically convenient to consider the foliation definedby σ = constant surfaces . The extrinsic curvature for this foliation can be de-rived using the identities satisfied by the geodesic distance function which havebeen summarized, for example, in [13]. Using this procedure we can computethe explicit form of the Ricci scalar for the qmetric. The outline of the steps forthe general case is given in Appendix B and the explicit derivation of the finalresult for the special case are given in Appendix C.Thus, after some lengthy but straightforward algebra, the final expression isfound to be R ( p ; P, L P ) = R ( p ) − (cid:18) − A (cid:19) Y − ǫ (1 − A ) Z (14) This analysis also sheds some light on more general aspects of intrinsic and extrinsicgeometries of metrics related in a manner similar to Eq. (12), which might be relevant intheir own right from a purely differential geometric point of view; these will be presented ina separate paper [12]. Y = R Σ − D D σ (15)and Z = 2 R ab n a n b + K ab K ab − K + D D ǫσ + 2 D − √ ǫσ (cid:18) K − D √ ǫσ (cid:19) (16)where √ ǫσ K ab = √ ǫσ ∇ a n b = ∇ a ∇ b (cid:0) σ / (cid:1) − ǫn a n b (17)is the extrinsic curvature of the σ =constant foliation, R Σ is its induced scalarcurvature: R Σ = R − ǫ (cid:0) R ab n a n b + K ab K ab − K (cid:1) (18)and we have introduced the convenient notation D k = D − k . Also, ǫ = g ab n a n b = ± n a = g ab n b . (See Appendix D on a more geometricalform of the above expression.)The Eq. (14) is a non-local expression for the biscalar R ( p ; P, L P ). The final stepof the analysis is to extract a local scalar object from this biscalar by taking thecoincidence limit. In general, to study any particular curvature invariant K ( P )obtained from the qmetric (using the same algebraic expression that connectsthe corresponding scalar with the original metric), we start from the modified biscalar curvature invariant K ( q ) ( p ; P, L P ) evaluated for the qmetric, and takethe coincidence limit σ → regular curvature . Then we can use, for ∇ a ∇ b σ the well known expansion in a covariant Taylor series around the base point P given (see, e.g., [13]) by:12 ∇ a ∇ b σ = g ab − λ S ab + λ ∇ n S ab − λ (cid:18) ∇ n S ab + 43 S ia S ib (cid:19) + O ( λ ) (19)where λ is the arc-length along the geodesic which, of course, is numericallysame as λ = √ ǫσ , ∇ n ≡ n i ∇ i and S ab = R acbd n c n d . It is convenient to definea few related quantities as well: S = g ab S ab = R ab n a n b ; ˙ S = ∇ n S ; ¨ S = ∇ n ( ∇ n S ) (20)Note that S is the functional used in the variational principle in the emergentgravity paradigm.Carrying out this operation for the Ricci biscalar in D dimensions (see AppendixC for the details in a special case), we find that we get an object which dependsnot only on local tensorial objects such as g ab , R abcd etc at P , but also on a vector n a which, apart from being normalized, becomes arbitrary in the coincidence10imit: R ( P, L P ) = lim p → P R ( p ; P, L P )= α (cid:2) R ab n a n b (cid:3) P | {z } O (1) term − L P (cid:20) S ab S ab + 32 ¨ S + 53 S (cid:21) P | {z } O ( L P ) term (21)where α = 2 ǫ ( D +1) /
3. (In general, there is also a divergent O ( λ − ) term whichis well-known in the context of point-splitting regularization. This term can beregularized and dropped; one way to do so formally is described in AppendixE.) This is our final result.Note that, after the coincidence limit has been taken, the vector n a on the RHSabove must be treated as an arbitrary (normalized) vector. In other words, theobject R ( P, L P ) at any event P must be treated as a quantity which dependson local tensorial objects at P , such as g ab , R abcd etc., as well as on arbitrarynormalized vectors n a . In this sense, we end with an object which depends,in addition to standard geometric objects such as g ab , R abcd etc., on arbitraryvector degrees of freedom at each spacetime event. These vectors have to bethought of as the vestige of the small scale structure of spacetime, and will allowus to connect up with the emergent gravity paradigm; we will come back to thispoint, in a broader context, shortly.The most important aspect of the result, as far as this paper is concerned, isthe following: When we carry out the last step (3) of the procedure outlinedin page 5, by taking the L P → L P → lim p → P R ( p, P ; L P ) = αR ab ( P ) n a n b ∝ S ( P ) (22)In other words, R ( P, L P = 0) = R ( P ) (23)That is, we start with Ricci biscalar for the qmetric (which bears the samealgebraic relation to qmetric as the usual Ricci scalar does to the usual metric),take the coincidence limit p → P and then take the ‘classical limit’ of L P →
0; only to find that the resulting expression is not the Ricci scalar for thebackground metric!On the other hand, it is easy to see from Eq. (14) that if we take the ‘classicallimit’ first , we get: R ( p ; P, L P = 0) = R ( P ) (24) As a curious aside we mention the following: The condition αR ab n a n b = R numerically,requires the rather peculiar relation R ab = 3 / [2( D +1)] g ab R to be satisfied by the backgroundmetric. Even the maximally symmetric space(time)s satisfy this relation only for the specialcase D = 2; this includes, e.g., the 2-sphere. σ → L P → L P → lim σ → R ( p ; P, L P ) = lim σ → lim L P → R ( p ; P, L P ) (25)The origin of this non-commutativity of the limits can be traced back to thefactor (cid:0) − A − (cid:1) in the second term of Eq. (14); in fact, the function f ( σ, L P ) = 1 − A − = 1 −
11 + L P /σ (26)has no limit at ( σ, L P ) = (0 , L P → lim σ → f ( σ, L P ) | {z } =1 = lim σ → lim L P → f ( σ, L P ) | {z } =0 (27)In the spirit of conventional regularization techniques (in our case, the mostnatural one is, of course, point-splitting regularization), we must take the co-incidence limit p → P first, which results in the structure of R ( P ) given byEq. (21). The resulting expression in Eq. (22), arising as the leading order termis our key result.
Given the rather surprising and counter-intuitive nature of the result, we givean explicit demonstration of the same, in a simpler context, Appendix C. Thisis done by using the synchronous coordinate system for the background metricand taking p and P along radial direction in the Euclidean space. This is, ofcourse, not the most general case but this captures much of the subtleties in thecalculation. The interested reader is referred to Appendix C for the details asto how the non-commuting nature of the limits lead to this result.We shall now discuss the implications of our result for the emergent gravityparadigm. The standard action for general relativity is based on the Einstein-Hilbert La-grangian: L EH = R . The usual belief is that one has to somehow quantise thetheory based on this Lagrangian. But if gravity is an emergent phenomenon likeelasticity, this effort is like quantizing a theory based on a Lagrangian describingelastic vibrations. Obviously, this will not lead to lasting progress.The problem with the emergent gravity program, on the other hand, is thatit a ‘top-down’ approach (in length scales) and the next logical step of the12rogramme is something like trying to discover statistical mechanics from ther-modynamics. One simply cannot do this without additional postulates.
Further,what we are really interested is in the quasi-classical limit in which ideas likedifferential manifold, metric etc remain valid and only the dynamical descrip-tion changes. In this limit, the modification in Eq. (5) captures some of the keyaspects of quantum gravity. This modification, in turn, is equivalent to usingthe qmetric in place of the original metric.Given such a replacement, the most natural extremum principle will be the onebased on the Ricci biscalar R ( p, P ; L P ) for the qmetric, just as we think of theextremum of the usual Ricci scalar R (for the usual metric) as the natural choicein general relativity. To obtain a local variational principle from this biscalarwe take the limit of p → P in R ( p, P ; L P ), obtaining the result in Eq. (21).That is, in the limit of p → P , the extremum principle based on R is equivalentto the extremum principle based on the limiting expression (viz. the first termin Eq. (21) when L P → S ∝ R ab ( P ) n a n b . The variation ofthe metric, which is equivalent to the variation of geodesic distance, translatesinto varying n a in this expression since we treat the base point P as fixed. Inother words, there is a natural transmutation of the variational principle whenwe carry out the steps 1-3 outlined in page 5. We see that: L EH = R −→ L eff = αR ab n a n b (28)where n is a vector of constant norm. The L eff has precisely the form of the thermodynamic functional S first suggested by Padmanabhan et al. [3] as abasis for an alternate variational principle for describing emergent gravitationaldynamics, motivated by thermodynamics of causal horizons [14]. One can treatthis as a functional of the normalised vector field n a and the extremum mustnow hold for variations of all n a . As demonstrated in several previous papers[2, 3], this leads to standard field equations of gravity with the cosmologicalconstant arising as an undetermined integration constant. The results in Eq. (21) and Eq. (22) are highly non-trivial and we could nothave guessed them from any simple consideration. These results highlight the robustness of the framework for emergent gravity paradigm based on an alternatevariational principle by demonstrating that such a principle could be enforced upon us by the existence of a fundamental length scale in spacetime. Theconceptual loop closes nicely since the emergent gravity paradigm was itselflargely motivated by existence of thermal attributes of causal horizons. One ofthese attributes, the horizon entropy, however, turns out to be divergent whenviewed as entanglement entropy due to tracing over the vacuum fluctuationsof fields beyond a causal horizon. It was argued in [16] that this divergence The idea of gravity being described fundamentally by a non-local object with the geodesicdistance playing the key role (which was recently emphasized in [11]) seems to be conceptuallyin tune with some earlier work by Alvarez et al. [15]. In our framework, the geodesic distanceappears naturally if spacetime has a built in zero-point length which does not violate Lorentzinvariance.
One major conceptual bonus we obtain from the emergent gravity paradigmis the possibility of understanding the cosmological constant. It has been ar-gued [2, 17] that a clean solution to cosmological constant problem can beobtained only if the metric is not treated as a dynamical variable in a localextremum principle. The thermodynamic variational principle motivated byemergent gravity paradigm satisfies this condition and obtains the equationsby varying a vector field of constant norm. In this approach, the cosmologicalconstant arises as an undetermined integration constant in the solution [2, 17].Here, we have shown that such a variational principle can indeed arise as anon-trivial limit of a calculation that incorporates the zero-point-length. Theemergent gravity paradigm also suggests a particular approach towards deter-mining the numerical value of the cosmological constant [17]. This, in turn,depends on the existence of a minimal area L P in quantum gravity so that thenumber of surface degrees of freedom of a sphere with radius L P is given by N sur = (4 πL P /L P ) = 4 π . The existence of a zero-point length, on which entireanalysis of this paper is based, ties in nicely with the existence of a minimal areaand a minimal count for the surface degrees of freedom. This gives the hopethat a more sophisticated model will be able to put these results on a firmerfooting. The Table 1 summarizes the results of the paper in a thematic manner. Theanalysis involves four key steps, which are given in the middle column as steps1-4 and described in detail in the paper.
Step 1 : We choose to describe a classical spacetime, not in terms of a metric,but in terms of its geodesic interval, which is a biscalar and contains exactly thesame information as the metric. Upgrading the role of σ ( P, p ) as the descriptorof geometry is a key new aspect of this paper.
Step 2 : We recall that certain aspects of quantum gravity can be incorporatedby the ansatz σ q ) ( P, p, L P ) = σ ( P, p ) + L P . This description is valid in the14et L P → p = P (no surprises here!) The Strategy of this pa-per Let p → P with L P = 0(leads to entropy density ofemergent gravity paradigm) σ → σ ( P, p ) (1) Start with thegeodesic interval σ ( P, p ) for a metric g ab σ → σ q ) → σ ( P, p ) (2) Incorporate someQG effects by theansatz σ q ) ( P, p, L P ) = σ ( P, p ) + L P σ q ) → L P q ab ( P, p, L P ) → g ab ( P ) (3) Find the qmetric q ab ( P, p, L P ) related to σ q ) ( P, p, L P ) Diverges as ( L P /σ ) | σ → R ( P, p, L P ) → R ( P ) (4) Compute the Riccibiscalar R ( P, p, L P ) forthe q ab ( P, p, L P ) R ( P, p, L P ) → S + O ( L P )Table 1: Summary of the paper. The strategy adopted in the paper is describedin the middle column with the logical flow being from the top to the bottom. Wetake the coincidence limit to obtain local quantities from bitensors, biscalars etc.These limits are shown in the right column. As a crosscheck, we have includeda left column showing what happens when we set L P = 0. (The results in thiscolumn are trivial and as expected.) See text for more discussion.15uasi-classical region between the full quantum gravity domain (in which wedo not know what replaces a differential manifold, metric etc) and the classicaldomain (in which conventional GR holds). We expect the quasi-classical regionto admit a description in terms of effective geometrical variables built from theqmetric. Step 3 : We next determine the form of a symmetric, second rank, bitensor q ab ( P, p, L P ) which corresponds to the modified geodesic interval σ q ) ( P, p, L P ).(See Sec. 2). It bears the same relation to σ q ) as the background metric g ab bears to σ and serves as the analogue of the metric in the quasi-classicaldomain. Step 4 : We compute the Ricci biscalar R ( P, p, L P ) for the qmetric q ab ( P, p, L P ).The local object R ( P, L P ) obtained by p → P limit on this biscalar is a goodcandidate for the variational principle in quasi-classical domain. We find thatthe leading order, L P -independent term in R ( P, L P ) is given by the functional S used previously in the thermodynamic variational principle in the emergentgravity paradigm! This is our key result.In the right column, we have given the coincidence limit p → P of variousquantities keeping L P nonzero. To begin with, we see that, in the coincidencelimit σ → σ q ) → L P , by design of our ansatz. It is this difference whichis at the foundation of our result. The coincidence limit of q ab is divergent, whichis easy to understand because we do not expect notions like metric to survivewhen σ . L P ; this is encoded in the factor ( L P /σ ) in q ab . Finally, whenwe take the coincidence limit of the Ricci biscalar R ( P, p, L P ) for the qmetric,we get the final result shown in the boxed equation at bottom right. Just as acrosscheck, we have given the L P →
1. Can one trust the grins of the Cheshire cats?
The key step which leads to this result is the possibility that effects of a minimallength scale are not necessarily “small” and vanish when L P → ( x , x ). We will, however, find that among all such functions which satisfythe Schr¨odinger equation, only half of them — viz., those which satisfy theantisymmetric condition ψ ( x , x ) = − ψ ( x , x ) — occur in nature. No amountof study of the Schr¨odinger equation for helium atom will explain the peculiarphenomena related to the Fermi statistics. The actual explanation lies deeplyburied in the relativistic quantum field theory from which we can obtain theSchr¨odinger equation by a suitable limiting process involving the c → ∞ limit.In this limit, c disappears from the relevant equations but a peculiar featureof the relativistic quantum field theory remains as a leading order (i.e., c = 0limit) relic in the non-relativistic helium atom. What is more, this effect is quitedifferent from, say, the usual relativistic “corrections” to Schr¨odinger equationwhich will admit a Taylor series expansion in (1 /c ); we cannot say that Fermistatistics is obeyed with increasing accuracy in a similar Taylor series expansionin (1 /c )! It is a completely different kind of low energy relic from the highenergy theory.A moment of thought will show that this is very similar to what happens in ourcase. When L P → L P because theleading order term itself is different. But, this should not be conceptually anymore surprising than the behaviour of wave functions of helium atoms. Inneither case can one guess, staring at the low energy theory, the origin of therelic.A slightly more technical result of this kind was discussed in Ref. [18]. It wasshown that, when one proceeds from the action functional for a relativisticparticle to the action functional for the non-relativistic particle, in the pathintegral approach to quantum theory, a relic term arises in the c → ∞ limitleading to a well-defined phase factor in the non-relativistic wave function. Onceagain, the origin of this phase factor (which is independent of c though arises asa relativistic relic) in the non-relativistic limit is completely mysterious if onedid not know the relativistic version of the relevant expressions.Mathematically, our result arises from the non-commutativity of the limits inEq. (25). Such effects are also known in the literature and we give two exam-ples. First arises in computing curvature tensors of some rather simple metrics.Consider a metric g ab ( x ; ε ) which depends on some parameter ε , from which wecompute the curvature tensor R abcd ( x ; ε ) and then take the limit ε →
0. Wenow compare this result with the one obtained by first taking the limit ε → not , with the difference being a singular term (usuallya Dirac delta function). 17 more technical result of similar nature arises due to interplay between loopdivergences in conventional quantum field theory, and attempts to regularizethem using regulators which break Lorentz symmetry, as was first highlightedin Refs. [20]. These authors pointed out the inevitability of O (1) effects arisingdue to Lorentz violations (LV) at higher energies, whose effects can genericallyget dragged to lower energies unsuppressed, due to interplay between radiativecorrections , which involve large loop momenta k , and the fact that the LV termsare also expected to regulate the UV divergences. In a sense, this result is alsoa consequence of the non-commutativity of the two limits, corresponding towhether one sets the regulating function to zero before or after evaluating theloop integral. Our result points to something similar in the context of smallscale “geometry” of spacetime, where we have retained Lorentz invariance buthave abandoned locality . (The key difference, of course, is that in our case, the O (1) term we obtain turns out to have important physical implications.)Finally, even the familiar conformal anomaly, arising in theories which are classi-cally conformal invariant — and, in fact, many other symmetry breaking anoma-lies — can be thought of as quantum residues of similar nature. If we regularizea theory by dimensional regularization, we use expressions in D dimensions andfinally take, say, the D → D = 4 will not, in general, be conformally invariant in D = 4 and when weeventually take the limit D →
4, we get an anomalous result. This result, again,is difficult to understand working entirely in the D = 4 situation but is clearwhen we think of it as a limiting process leaving a residue.These examples show that when certain limits are taken in a theoretical modelthe resulting theory could contain relics of the more exact description. In allsuch cases no amount of study of the approximate theory will give us a clue asto where the relic came from (e.g., the study of Schr¨odinger equation for thehelium atom can never lead us to the Fermi statistics for the electrons). Webelieve the transmutation of the variational principle in Eq. (28) is of similarnature which is nearly impossible to understand within the context of classicalgravity itself. Our analysis throws light on this and shows that it could be avalid relic of quantum gravity.
2. How valid is the assumption that quantum gravity effects would modify lengthscales as: σ → σ + L P ? This was the key input for deriving the qmetric in [11] and forms the startingpoint of the entire analysis. There has been a good deal of evidence in favorof this modification, from several independent lines of analyses dating back to1960’s. The two aspects which need scrutiny in this regard are the following: (a)Can a c-number description with the replacement σ → σ + L P capture someof the quantum gravitational effects? (b) How unique is such a replacement tobring in Eq. (4) which probably is a more precisely stated result?18he first issue is similar to what is encountered in other contexts as well. As asimple example, consider a harmonic oscillator with the Hamiltonian H ( p, q ) =(1 / p + q ] (in convenient units). Classically H has minimum at p = 0 = q corresponding to the ground state in which the oscillator sits at the minimum ofthe potential with zero velocity. Quantum mechanically, uncertainty principleprevents us from giving precise values to q and p simultaneously and hencethis cannot be the description of the ground state. The exact analysis of thisproblem, of course, will involve treating H as a operator and finding its lowesteigenvalue etc. However we can take the point of view that when q is closeto zero, the uncertainty in the momentum is ~ /q and the relevant c-number tominimize is a ‘quantum-corrected’ Hamiltonian H qc ≡ (1 / ~ /q ) + q ] with theminimum value being O ( ~ ). We can indeed capture the essential feature by justmaking the replacement H → H qc . We consider the replacement σ → σ + L P to be similar in spirit, getting us some quantum results at c-number price! Notethat the existence of the zero-point-length can be demonstrated by consideringthe effect of the uncertainty principle on spacetime measurements [8], so theanalogy above is conceptually quite close.As regards question (b), one could say the following: The ansatz σ → σ + L P ,of course, is not the only modification possible to incorporate Eq. (4), and in factwe show in Appendix A how to get the corresponding results for a more generalmodification σ → S ( σ ). But in general, the resulting form of the qmetricwill not reduce to flat spacetime if the original spacetime is flat. The specificchoice σ → σ + L P has the additional advantage that it reduces the qmetricto flat spacetime when the original metric is flat. In this sense, the modificationwe have worked with is indeed special (and might even be unique). Consider,e.g., another modification [21] : σ → σ F ( σ /L P ) where F ( x ) = e /x (a naiveexpansion in L P gives σ ∼ σ + L P ). This is another example of a deformationwhich is non-analytic at x = 0. It would be interesting to see if there existsa specific sub-class of such functions which leaves a O (1) effect on curvatureinvariants; work along these lines is in progress.
3. Does analytic continuation from the Euclidean to Lorentzian spacetime leadto special difficulties?
Not really, in the mathematical sense. In the Euclidean sector a constantnorm vector is always ‘spacelike’ (we use the signature ( − , + , + , + , . . . ) in theLorentzian spacetime) while in the Lorentzian spacetime it can be timelike,spacelike or null. The previous results in emergent gravity paradigm are usu-ally presented for null vectors but the results continue to hold for any constantnorm vector. The introduction of the zero-point-length and its manipulationsare best done, however, in the Euclidean space which is what we have done. Itis easy to see that the results hold without any ambiguity for both spacelikeand timelike vectors in the Lorentzian space. Therefore one can take the nulllimit by a continuity argument after straddling the null surface by spacelikeand timelike vectors on the two sides with some care in working with the affine19istance along a null ray.
4. To what extent can we think of the qmetric q ab ( p, P ) as a metric? This issue is somewhat irrelevant to our results and — more generally — whenone treats a distance function d ( x, y ) = √ ǫσ defined on a manifold as morefundamental than the metric, and the metric g ab ( x ) as a derived quantity whichenables us to construct geometric invariants for a manifold. As explained indetail in [11], the non-local character of the qmetric is simply a physical charac-terization of quantum fluctuations which leave their imprint (in this approach)in the form of a lower bound on the intervals: d ( x, y ) ≥ L P . At the smallest ofthe scales, we do not expect any local tensorial object to describe the quantumspacetime geometry accurately anyway, and hence it is not unexpected thatmore general mathematical objects must replace the conventional ones. Theideas here can be thought of as a first step in this direction.
In fact, non-localeffective actions have been considered for quite some time in the context ofquantum gravity (e.g., DeWitt proposed such an action in [22]), and q ab ( p, P )might serve as an important mathematical object to build such actions.
5. What are the implications for spacetime singularities?
Some preliminary results along these lines were given in [11]. But to answer thequestion in full generality, one needs to obtain the full curvature tensor for theqmetric and then study typical curvature invariants. (We expect to do this in afuture work.) Even restricting to the Ricci scalar (which we do have), one needsto revert to the general equation, Eq. (14), instead of Eq. (21) which is arrivedat by assuming that one is working in regions of finite curvature. The transitionfrom Eq. (14) to Eq. (21) makes use of the covariant Taylor series expansionsof various bitensors involved, and the coefficients of such series depend on thecurvature and its derivatives. If the latter blow up, as they are expected tonear spacetime singularities, then one must be careful about using such series.Efforts are ongoing to evaluate Eq. (14) for some physically relevant singularspacetimes using valid expressions for the world function.
6. What about the O ( L P ) term ? Since the leading order term in Eq. (21) gives the entropy functional of emergentgravity, (leading to classical Einstein’s equations) one might be tempted toconsider the next term as some kind of quantum gravitational correction andexplore its consequences. This temptation must be resisted for the followingreason: The entire philosophy behind the analysis was that classical spacetimeis a non-perturbative limit of quantum spacetime and it is incorrect to do aperturbative model of quantum gravity. So, when the leading order ( L P =0) term itself is not the classical term, it is inconsistent to study quantumcorrections perturbatively, order by order in L P . The second, rather technicalreason, for not pursuing this line of attack has to do with the fact that weare so far dealing with pure gravity. We do not know how to get the matter20tress tensor from a corresponding quantum calculation by taking a suitablelimit. (In emergent gravity one can simply add the thermodynamic potentialof matter, T ab n a n b and get the correct result but we do not have, yet, thecorresponding quantum version.) So it is rather useless to compute quantumgravitational corrections in source-free spacetimes. (If one does this, in spite ofthese reservations, one finds that R ab = 0 continue to be a solution even withthe lowest order quantum correction terms.)The key insight coming from our analysis, going beyond the specifics, is thefollowing: It seems likely that non-local but (hopefully) Lorentz in(co)variantdeformation of the spacetime geometry at small scales, will be a consequence ofquantum fluctuations of the (as yet unknown) microscopic degrees of freedomof quantum gravity. This leads to an inevitable O (1) modification due to aminimal spacetime length. Regardless of the precise form of the deformation,the results will now essentially depend on the two limits, ξ → ξ → ∞ (with ξ = σ /L P ) being inequivalent. Any dimensionless deformation function A [ σ ; L P ] can only depend on ξ ; A [ σ ; L P ] = A [ ξ ]. Hence, unless A [ ξ ] has samelimit at ξ → ξ → ∞ , (e.g., it is symmetric under ξ → ξ − ) one willgenerically obtain the kind of non-trivial result that we have obtained. Theprecise form of the O (1) term would depend on the form of A [ ξ ]; our choice, A [ ξ ] = 1 + 1 /ξ , yields a residual term which happens to connect up with certainkey ideas concerning emergent gravity and cosmological constant that have beendeveloped over the past decade. Acknowledgments
Various expression(s) given/used in this work have been verified for special casesby symbolic computations in
Maple using GRTensorII [23]. The research workof TP is partially supported by the J.C.Bose Fellowship of DST, India. DKthanks IUCAA, Pune, where part of this work was done, for hospitality. Wethank the referee for several constructive comments.
A Determination of qmetric associated with ar-bitrary modifications of geodesic intervals
It is possible to generalize the qmetric approach to the cases in which thegeodesic intervals are modified in an arbitrary manner. We briefly commenton this possibility in this appendix, leaving a detailed analysis of such a gener-alization for future work.Let us consider the case in which the geodesic intervals are modified as σ → ( σ ) and the scalar propagator is correspondingly modified as ( σ ) − ( D − / → ( S ( σ )) − ( D − / where S ( σ ) is a given function. (The choice S ( x ) = x + L P will reproduce the results of this paper). We want to determine the form ofa qmetric q ab ( p, P ), (which is a second rank, symmetric, bitensor) built out of σ ( p, P ) , g ab and n a = g ab n b , such that the following two conditions are satisfied:(i) The modified geodesic interval S satisfies Eq. (2) with the metric replacedby the qmetric and (ii) The modified propagator ( S ( σ )) − ( D − / will satisfythe Green function equation with the metric replaced by the qmetric. This isindeed possible and we outline the steps here:Let us assume that such a qmetric has the form q ab = A − g ab + ǫQn a n b = A − h ab + ǫ ( A − + Q ) n a n b (29)where A and Q are (as yet arbitrary) functions of σ and h ab = g ab − ǫn a n b is theinduced metric on the hypersurface with normals n a . This ansatz is motivatedby the fact that qmetric q ab is a second rank, symmetric, bitensor built from g ab and n a . Note that the corresponding covariant components are given by: q ab = Ag ab − ǫBn a n b ; B = QA (1 + QA ) − (30)We want S to be the geodesic interval for q ab , and hence we substitute this ansatz in the Hamilton-Jacobi equation: q ab ∂ a S ∂ b S = 4 S (31)This gives, on using g ab ∂ a σ ∂ b σ = 4 σ , the following relation between coeffi-cients A, Q in the metric and the function S : A − + Q = 1 σ SS ′ (32)where S ′ = d S/ d σ . Obviously, the condition in Eq. (31) can only fix theprojection of the qmetric in the subspace spanned by n a n b ; so we have: q ab = A − h ab + ǫ (cid:18) σ SS ′ (cid:19) n a n b (33)To fix the form of A we use the condition on the Green function. For an arbitraryfunction S ( σ ), it is readily shown that the flat space propagator G ( σ ) will bemodified to G ( S ( σ )) provided that A = S ( σ ) /σ . To establish this, one usesthe above form of the qmetric and the relation p | det q | = A D − p A − + Q p | g | (34)which follows from the matrix determinant lemma :det (cid:0) M + uv T (cid:1) = (det M ) × (cid:0) v T M − u (cid:1) (35)22here M is an invertible square matrix, and u , v are column vectors (of samedimension as M ). Subsequent computation of (cid:3) ( q ) G ( S ( σ )) is straightforward(although lengthy). Enforcing the condition that (cid:3) ( q ) G ( S ( σ )) = 0 for p = P gives the differential equationd ln A d ln σ = σ S ′ S − A = S ( σ ) /σ . (The multiplicative constant is fixed bythe condition that A = 1 when S ( σ ) = σ ).This fixes the final form of the qmetric to be: q ab = σ S ( σ ) h ab + ǫ (cid:18) σ SS ′ (cid:19) n a n b (37)For the choice made in this paper, S = σ + L P , one recovers Eq. (12). Foran arbitrary S ( σ ), the resultant qmetric can (generically) yield a non-zerocurvature even when g ab represents a flat spacetime, which does not happen forthe choice S = σ + L P . The above formulation is quite promising in yielding aconsiderable generalization of the results in [11], and is currently being pursued. B Evaluation of Ricci scalar for the qmetric
One can employ a nice trick based on the Gauss-Codazzi decomposition toobtain the Ricci scalar for the modified metric. In fact, the same trick worksalso for the conformally related metrics, and we outline this case first since thelogic remains the same.Suppose two metrics are related by a conformal transformation: ˜ g ab = F ( x ) g ab .Then, one can imagine foliating the spacetime by vector fields normal to F =constant surfaces, given by n i = ∇ i F ( x ) / √ ǫ ∇ a F ∇ a F . The conformal trans-form of this vector field will be ˜ n i = √ F n i , whereas ˜ n i = n i / √ F where n i = g ij n j . (Note that one needs to be careful about indices; all indices on ˜ n are raised or lowered using ˜ g ab .)The advantage of doing this is that the induced geometries of this foliation in thetwo metrics are related in a simple manner. The relation between the extrinsiccurvatures ˜ K ab and K ab is easy to establish, and turns out to be˜ K ab = √ F K ab + (cid:16) n k ∇ k √ F (cid:17) h ab (38)where h ab is the induced metric. It is easy to show that ˜ h ab = F h ab . We nowuse the Gauss-Codazzi relation R ( q ) = R Σ ,q − ǫ (cid:16) ˜ K + ˜ K ab + 2˜ n i ˜ ∇ i ˜ K (cid:17) + 2 ǫ ˜ ∇ i ˜ a i (39)23here ˜ a i = ˜ n k ˜ ∇ k ˜ n i . Again, it can shown by straightforward computation that˜ a i = a i and ˜ a i = a i /F . Most importantly, since we are considering F =constant foliation, the induced Ricci scalars are related by a simple scaling: R Σ ,q = R Σ /F . Putting all these together, it can be shown that one recovers thewell known relation between the Ricci scalars of conformally related metrics.The above procedure can be generalized for the qmetric. This case is morecomplicated due to the presence of the n a n b term in the metric, but otherwisethe steps remain the same. One first shows that, even for this general case, theinduced metrics are related by a conformal transformation as above, and theextrinsic curvature ˜ K ab can be written easily in terms of K ab and h ab . (Theserelations are quoted in [11]). Therefore, one again has R Σ ,q = R Σ /F . However,the extrinsic curvature terms in the Gauss-Codazzi relation produce a morecomplicated relation, although there are no new conceptual points involved.The algebraic details, however, are a bit longer, and will be presented in [12]. C Explicit demonstration of our result in a spe-cial case
In this Appendix, we will provide an explicit demonstration of how our key resultarises in a special case which captures all the key features of the general case.We consider a Euclidean spacetime described in the synchronous coordinateswith the line element: ds = dt + h µν ( t, x α ) dx µ dx ν (40)This spacetime possesses a geodesic interval function σ ( x, x ′ ) from which all the metric coefficients can be obtained using Eq. (3). It is, however, well-known(see, e.g. [25], page 288) that x µ = constant are geodesics in this spacetime andthe geodesic interval between two points in such a geodesic is just ( t − t ′ ). So, ifwe confine our attention to two points p and P along the ‘radial’ direction [forwhich x µ = constant], then we can take the t coordinate as numerically equalto the geodesic interval and write the metric as: ds = dσ + h µν ( σ, x α ) dx µ dx ν (41)Note that we have now downgraded the function σ ( x, x ′ ) to a coordinate labeland this will work only for the radial geodesics. Our aim is to confine ourselvesto points p and P along the σ direction and illustrate our results when σ → n a dx a = dσ and the resulting24ine element is: ds q ) = dσ A + Ah µν ( σ, x α ) dx µ dx ν ; A = (cid:18) L P σ (cid:19) (42)Our task is now straightforward: Compute R for this metric, expand everythingrelated to the background in a Taylor series in σ assuming the backgroundspacetime is nonsingular, take the limit of σ → R ab n a n b ), identify the divergentterm (which we expect to be proportional to 1 /σ ) and identify the O ( L P ) termthereby demonstrating our result. We will now outline the key algebraic stepsin this calculation.Computing R ( q ) for the line element in Eq. (42) is straightforward using theformulas in, say, ref.[25] (after making the sign switch h µν → − h µν to go fromLorentzian to Euclidean signature). This gives: R ( q ) = A − R +(1 − A − )( R − R Σ )+( A − S + K ab − K ) − (5 KA ′ − A ′ A − A ′′ )(43)where R is the Ricci scalar of the background metric (viz. the one with A = 1), R Σ is the 3-dimensional Ricci scalar of σ = constant surface, S ≡ R ab n a n b , K ab ≡ ∇ a n b is the extrinsic curvature of σ = constant surface and primesdenote derivative with respect to σ . Obviously R ( q ) = R when A = 1.We now need to take σ → A → L P /σ diverges and the limit needsto be taken with care. In particular, since R (and all other curvature invariants)for the back ground ( A = 1) metric is assumed to be well-defined in the σ → A − in first term in Eq. (43) kills the background Ricciscalar. Since this is the only term that survives when L P = 0 , A = 1 , we cansee the first signs of why the two limits do not commute. To take the limit, weneed an Taylor series expansion of various quantities. We start with the generalresult, valid in any spacetime:12 ∇ a ∇ b ( σ ) = g ab − λ S ab + λ ∇ n S ab − λ {∇ n ∇ n S ab + 43 S ia S ib } + O ( λ ) (44)where λ is the arc length of the geodesic connecting p and P which is numeri-cally equal to σ ( p, P ) and we use the notation ∇ n = n j ∇ j . Further using thedefinition of n a in terms of σ ( x, x ′ ) (see second equation in Eq. (13)), it is easyto show that: λK ab = λ ∇ a n b = 12 ∇ a ∇ b σ − ǫn a n b (45)Using Eq. (44) in this we can get the corresponding expansion for K ab as: λK ab = h ab − λ S ab + 112 λ ∇ n S ab − λ F ab + O ( λ ) (46)where F ab = ∇ n ∇ n S ab + (4 / S ia S ib (47)25aking the trace we get: λK = D − λ S + 112 λ ∇ n S − λ F (48)where we use the notation D k = D − k and F = F ab g ab = ∇ q S + (4 / S ab S ab (49)Using these two results we can calculate the useful combination: λ ( K ab − K ) = − D D + 23 λ D S − λ D ˙ S + λ F + O ( λ ) (50)where we have defined, for ease of notation, the quantity: F ≡ (cid:20)(cid:18) D + 13 S ab + D ¨ S − S (cid:19)(cid:21) (51)and use overdot to denote the operation of n i ∇ i . Finally, we compute the Taylorseries expansion of R Σ : R Σ = R − ǫ S− ǫ ( K ab − k ) = R − ǫ S− ǫλ (cid:18) − D D + 23 λ D S + O ( λ ) (cid:19) (52)We are now in a position to evaluate all the terms which appear in the righthand side of Eq. (43). Note that, for our special case, we can set λ = σ in theexpansions. We will also set D = 4. Starting with the results1 − A − = 1 − σ + O ( σ /L p ) (53)and R − R Σ = 103 S − σ + O ( σ ) (54)we get the term (1 − A − )( R − R Σ ) in Eq. (43) to be:(1 − A − )( R − R Σ ) = 103 S − σ + 6 L p + O ( σ ) (55)Next consider the term ( A − S + K ab − K ) in Eq. (43). Using K ab − K = − σ + 43 S − σ ˙ S + σ F + O ( σ ) (56)we get:( A − S + K ab − K ) = − L p σ + 103 L p σ S − L p σ ˙ S + F L p + O ( σ ) (57)Similarly, using K = 3 σ − σ S + 112 σ ˙ S −
F σ + O ( σ ) (58)26he last bunch of terms [ − KA ′ − (3 / A ′ /A ) − A ′′ ] in Eq. (43) evaluates to: − KA ′ − (3 / A ′ /A ) − A ′′ = 6 L p σ − L p σ S + 56 L p σ ˙ S + 6 σ − L p + F L p + O ( σ )(59)where F = − (1 / F . We now substitute these expressions in Eq. (55), Eq. (57)and Eq. (59) into Eq. (43) to obtain: R ( q ) = A − R + (cid:26) S − σ + 6 L p + O ( σ ) (cid:27) + ( − L p σ + 103 L p σ S − L p σ ˙ S + F L p + O ( σ ) ) + ( L p σ − L p σ S + 6 σ − L p + 56 L p σ ˙ S + F L p + O ( σ ) ) (60)Miraculously, the terms diverging as σ − and σ − nicely cancel leaving only a σ − divergence which we know how to regularize in the point-splitting approach. Further, the non-analytic terms in Planck length /L P also cancel out. Finally,the A − factor kills the background Ricci scalar term, leading to the final result: R ( q ) → S + ( F + F ) L p + " ˙ S L p σ σ → (61)The prefactor is indeed (2 / D + 1) = 10 / D = 4 we are working in.Once the divergent expression is regularized and eliminated, the leading term isproportional to S = R ab n a n b as advertised.Exactly the same kind of analysis works in the most general background whenwe work in an arbitrary coordinate system. The above approach shows thattaking σ → D Aside on the geometrical structure of the mod-ified curvature scalar
Eq. (14) gives the full expression for the modified Ricci scalar, and as is ev-ident, its evaluation requires the knowledge of the geodesic interval biscalar.Nevertheless, it is instructive to rewrite it in a slightly different manner whichhighlights the elegant geometrical structure of the whole set-up, and might behelpful in further developments on purely geometric grounds. After a few alge-27raic manipulations, Eq. (14) can be written as R ( p ; P, L P ) = AR − (cid:0) A − A − (cid:1) (cid:0) R Σ − R (cid:1) + 2 ǫ ( A − (cid:18) D + 1 D − (cid:19) (cid:0) K − K (cid:1) K (62)where R = D D /σ and K = D / √ ǫσ are the induced and extrinsic cur-vatures of the σ =const. surfaces in the flat spacetime (which are maximallysymmetric spaces with positive or negative curvature). Note that the first twoterms mimic the relationship between g ab and q ab . It is worth investigating thegeometrical structure of the above expression in detail, since it might hold thekey to a generic study of the behavior of the modified curvature scalar nearthe spacetime singularities in terms of the focusing of geodesics. This work iscurrently in progress. E The divergent term in Eq. (21)
As mentioned in the paper, there is a divergent term in (21) which is given bylim λ → + + L P λ ˙ S ( p ) (63)This term has an odd number of factors of n i , since ˙ S = n i ∇ i (cid:0) R jk n j n k (cid:1) = n i n j n k ∇ i R jk . (Recall that n i ∇ i n j = 0). Therefore, if one defines the coin-cidence limit by considering the limits p, p ′ → P where p, p ′ are diametricallyopposite, i.e., R reg ( P, L P ) = 12 lim p → P R ( p ; P, L P ) + lim p ′ → P R ( p ′ ; P, L P ) ! the divergent term will cancel out. (Other terms have even number of factorsof n i , so their contribution remains unchanged). This way of taking the limitwas discussed by Christensen and Davies et al. (amongst others) in some olderworks related to the point-splitting regularization in curved spacetime [13, 24].We, therefore, obtain R reg ( P, L P ) = α (cid:2) R ab n a n b (cid:3) P − L P (cid:20) S ab S ab − ¨ S + 53 S (cid:21) P Note that the coefficient of the ¨ S gets modified (see Eq. (21)) since λ − ˙ S ( p ) = λ − h ˙ S ( P ) + λ ¨ S ( P ) + O ( λ ) i has a O ( λ ) contribution.28 eferences [1] T. Padmanabhan: AIP Conf.Proc. , 212 (2012) [arXiv:1208.1375];Rept. Prog. Phys., , 046901 (2010) [arXiv:0911.5004]; Phys.Rept. ,49 (2005) [gr-qc/0311036].[2] T. Padmanabhan, Gen.Rel.Grav, , 1673 (2014) [arXiv:1312.3253].[3] T. Padmanabhan, Gen. Rel. Grav. , 2031 (2008); T. Padmanabhan, A.Paranjape, Phys. Rev. D , 064004 (2007) [gr-qc/0701003].[4] T. Padmanabhan, Dawood Kothawala: Phys.Repts. ,115(2013)[arXiv:1302.2151].[5] T. Padmanabhan: Class.Quan.Grav., , 4485 (2004) [gr-qc/0308070];Mod.Phys.Letts., A 25 , 1129-1136 (2010) [arXiv:0912.3165]; Phys.Rev.,
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