aa r X i v : . [ g r- q c ] D ec Gravity and is Thermodynamics
T. PadmanabhanIUCAA, Pune University Campus,Ganeshkhind, Pune- 411 007. email: [email protected]
December 22, 2015
Abstract
The equations of motion describing all physical systems, excludinggravity, remain invariant if a constant is added to the Lagrangian. Inthe conventional approach, gravitational theories break this symmetryexhibited by all other physical systems. Restoring this symmetry togravity and demanding that gravitational field equations should alsoremain invariant under the addition of a constant to a Lagrangian, leads to the interpretation of gravity as the thermodynamic limit ofthe kinetic theory of atoms of space. This approach selects, in a verynatural fashion, Einstein’s general relativity in d = 4. Developing thisparadigm at a deeper level, one can obtain the distribution functionfor the atoms of space and connect it up with the thermodynamicdescription of spacetime. This extension relies on a curious fact thatthe quantum spacetime endows each event with a finite area but zerovolume. This approach allows us determine the numerical value of thecosmological constant and suggests a new perspective on cosmology. A crucial fact about normal matter, say, a glass of water, which is almostnever stressed in textbooks is the following:
You could have figured out thatwater must be made of discrete atoms without ever probing it at scales com-parable to atomic dimensions!
All you need to realize is that water can1e heated and hence must have an internal mechanism to store the energywhich you supply to it. This is the breakthrough in the understanding ofthe nature of heat and temperature [1] which came with the work of Boltz-mann, who essentially said: “If you can heat it, it has microscopic degreesof freedom”. This profound insight underscores the following fact: The exis-tence of microscopic degrees of freedom leaves a clear signature even at thelargest macroscopic scales in the form of temperature and heat. One can evencount the number of atoms , using purely macroscopic variables through therelation N k B = R = E (1 / T (1)Again, standard textbooks do not stress the beauty of this result. The vari-ables in the right hand side, E and T , have valid interpretations in thecontinuum (thermodynamic) limit, but N in the left hand side has no mean-ing in the same limit. The N actually counts the number of atoms in thesystem, the very existence of which is not recognized by continuum thermo-dynamics! So you don’t need the technology capable of probing matter atangstrom scales in order to figure out that matter is actually made of atoms.The mere fact that matter can be hot, is enough — if you are as clever asBoltzmann.The key new variable which distinguishes thermodynamics from pointmechanics is the heat content T S of matter which is the difference F − E between the free energy and internal energy of the system. Expressed interms of densities, the Gibbs-Duhem relation (for systems with zero chemicalpotential) tells us that T s = P + ρ where s is the entropy density, ρ is theenergy density and P is the pressure. The heat density is something uniquelythermodynamic and has no direct analog in point mechanics. More is indeed different.Let us now proceed from normal matter to the fabric of spacetime. Workdone in the last several decades [2–6] shows that even spacetime, like matter,can possess a heat density. As I will soon describe, it is possible to associatea temperature and entropy density with every event in spacetime just as youcould have done it to a glass of water. On the other hand, one traditionallydescribed the dynamics of spacetime through some field equation for gravity Throughout this article I will use the word ‘atoms’ when I mean ‘microscopic degreesof freedom’ or ‘number of relevant microstates’; they usually differ by an unimportantnumerical factor. • The Boltzmann principle tells us that if spacetime can be hot, it musthave microstructure. What is more, we should be able to count theatoms of spacetime without having the technology to do Planck scaleexperiments just as Boltzmann guessed the existence of atoms of matterwithout doing angstrom scale experiments. We would expect a relationlike Eq. (1) to exist for the spacetime. • If the spacetime is like a fluid made of atoms, the gravitational fieldequations must have the same status as the equations describing, say,the flow of water. Therefore, we should be able to derive them from apurely thermodynamic variational principle. Further, the equation it-self should allow a reinterpretation in a purely thermodynamic languagerather than in the conventional geometrical language. Consequently,we would expect several variables, which are usually considered geo-metrical, to have an underlying thermodynamic interpretation. • The discreteness of matter can be taken into account in the kinetictheory by introducing a distribution function f ( x i , p i ) such that dN = f ( x i , p i ) d xd p counts the number of atoms in a phase volume. Sucha description recognizes discreteness but yet works at scales such thatthe volume d x is large enough to, say, contain sufficient number ofatoms. We should be able to develop a similar concept for spacetimewhich recognizes the discreteness at the Planck scale and yet allows theuse of continuum mathematics to describe the phenomena.It turns out that the above goals can indeed be achieved thereby providinga thermodynamic description of (what we call) gravity. Further, such anapproach allows us to understand several aspects of conventional gravity ata deeper level and — most importantly — provides a novel perspective oncosmology capable of predicting the numerical value of cosmological constant.I will now describe how all these come about.3 Why are spacetimes hot?
The temperature of a material object is purely kinematic in the sense thata metal rod and a glass of water — having completely different structuralproperties — can possess the same temperature. Similarly, one can associatea temperature with an event in a spacetime which is completely indepen-dent of the field equations of gravity which determined the structure of thatspacetime. Let me briefly describe how one arrives at this concept just fromthe kinematics of spacetime.Principle of equivalence tells us that (i) the gravitational field is describedby the metric of a curved spacetime and (ii) one can determine the influenceof gravity on all other systems by a judicious application of the laws ofspecial relativity in the freely falling frame (FFF), around any event. Thisinfluence can be completely encoded in the equation ∇ a T ab = 0 where T ab isthe symmetric energy momentum tensor of the matter and the ∇ a dependson the background geometry describing a gravitational field. Applying thisto the electromagnetic field, one finds that gravity affects the propagation oflight rays and thus the causal structure. In particular, it is easy to constructobservers (i.e., timelike congruences) in any spacetime such that part of thespacetime will be inaccessible to them. A generic example of such an observeris provided by the local Rindler observers [8] constructed as follows: Startwith the FFF around any event P , with coordinates ( T, X ) and boost toa local Rindler frame (LRF) with coordinates ( t, x ) constructed using someacceleration a , through the transformations: X = x cosh( at ) , T = x sinh( at ).There will be a null surface passing though P which will be the X = T surface in the FFF; this null surface will now act as a patch of horizon to the x = constant Rindler observers.This LRF leads to the most beautiful result [4] we know of that ariseson combining the principles of general relativity and quantum field theory:The local vacuum state, of the freely falling observers around an event, willappear as a thermal state to the local Rindler observer with the temperature: k B T = (cid:18) ~ c (cid:19) (cid:16) a π (cid:17) (2)where a is the acceleration of the local Rindler observer which can be relatedto other geometrical variables of the spacetime if required. This temperaturetells us that around any event, in any spacetime, there exist observers whowill perceive the spacetime as hot. These local Rindler observers will also4otice that matter takes a very long time to cross the local Rindler horizonthereby allowing for thermalization to take place. Since the local Rindlerobserver attributes a temperature T to the horizon, she will interpret theenergy associated with the matter that crosses the null surface (asymptoti-cally), as some amount of energy ∆ E being dumped on a hot surface, therebycontributing a heat content ∆ Q m = ∆ E . One can show that the resultingheat density (energy per unit area per unit time) of the null surface, con-tributed by matter dumped a local Rindler horizon, as interpreted by thelocal Rindler observer, is given by H m [ ℓ a ] ≡ dQ m √ γd xdλ = T ab ℓ a ℓ b (3)The heat transfered by matter is obtained by integrating H m with the inte-gration measure d Σ ≡ √ γd xdλ over the null surface generated by the nullcongruence ℓ a , parametrized by λ . (The factor √ γd x is the transverse areaelement of the λ = constant cross-section of the null surface.) There are twofeatures which are noteworthy regarding H m . • If we add a constant to the matter Lagrangian (i.e., L m → L m + con-stant, the T ab changes by T ab → T ab + (constant) δ ab . The H m , definedby Eq. (3) remains invariant under this transformation. • The heat density vanishes if T ab ∝ δ ab . So the cosmological constanthas zero heat density though it has non-zero energy density. In fact,for an ideal, comoving fluid, T ab ℓ a ℓ b = ( ρ + P ) and hence the heatdensity vanishes only for the cosmological constant with equation ofstate ρ = − P .Thus the kinematics of spacetime allows us to associate an (observerdependent) temperature with every event in spacetime and a heat densitycontributed by matter with every null surface. Our next job is to develop athermodynamic variational principle to obtain the dynamics of the spacetime. Recall that the equations of motion for matter remain invariant when weadd a constant to the Lagrangian. It seems reasonable to postulate that5he gravity should not break this symmetry which is present in the mattersector. Since T ab is the most natural source for gravity (as can be arguedfrom the principle of equivalence and considerations of the Newtonian limit),this leads to the demand: ◮ The extremum principle which determines the dynamics of spacetimemust remain invariant under the shift T ab → T ab + (constant) δ ab .It can be easily proved [7] that this principle rules out the possibility ofvarying the metric tensor g ab (in an unrestricted manner) in a covariant,local, action principle to obtain the field equations! Therefore, our variationalprinciple cannot use g ab as the dynamical variables and we need to introducesome other auxiliary variables. Further, in the traditional approach, T ab arises as the source when we vary the metric in the matter Lagrangian. Butsince we are not varying g ab , but still want T ab to be the source, we need toexplicitly include T ab in the variational principle. So the variational principlehas to depend on T ab and yet be invariant under T ab → T ab + (constant) δ ab . The simplest choice (involving the least number of auxiliary degrees offreedom) will be to demand that the variational principle has the form: Q tot ≡ Z d Σ( H m + H g ); H m [ n a ] ≡ T ab n a n b (4)where the null vector n a acts as the auxiliary variable. Since Q tot depends(linearly) on T ab only through the heat density H m [ n a ] in Eq. (3), it is ob-viously invariant under the shift T ab → T ab + (constant) δ ab . The H g is thecorresponding contribution from gravity which is yet to be determined. Thisapproach introduces an arbitrary null vector n a into the variational principlewhich, at this stage, is just an auxiliary field. But since no null vector is spe-cial, the extremum condition should hold for all n a , leading to a constraint onthe background metric g ab thereby determining the dynamics of spacetime.Obviously, the form of the gravitational heat density H g determines thespacetime structure, just as the form of entropy functional determines thestructure of a material body. A natural choice [9, 10] for H g , which isquadratic in ∇ n will have the form: H g = − (cid:18) πL P (cid:19) (4 P abcd ∇ a n c ∇ b n d ) (5)where P abcd is a dimensionless tensor to be determined and L P is an arbitraryconstant, with the dimensions of area. (This expression, by itself, may not6ook thermodynamical but it is indeed the heat density of gravity — whichshould be obvious from the fact that we are adding it to the heat density ofmatter. This will become clearer later on, see Eq. (8).) We require that thecondition, δQ tot /δn a = 0 for all null vectors n a at a given event, should con-strain the background geometry. This requirement leads to the expression: P abcd ∝ δ aba b ...a m b m cdc d ...c m d m R c d a b . . . R c m d m a m b m (6)where δ aba b ...a m b m cdc d ...c m d m is the totally antisymmetric m -dimensional determinanttensor. The resulting field equation is identical to that of (what is known as)the Lanczos-Lovelock model [9–11] with the cosmological constant appearingas an integration constant. (These models have the interesting — and unique— feature that, the field equations are second degree in g ab !) The ‘entropytensor’ P abcd determines the entropy of horizons in the resulting theory throughthe expression [6, 11]: s = − √ γP abcd ǫ ab ǫ cd (7)(where ǫ ab is the binormal to the horizon surface) One can show that the on-shell value of Q tot is indeed (the difference in) the corresponding heat densityof the theory: Q tot = Z d x ( T loc s ) (cid:12)(cid:12) λ λ (8)(This result also confirms that the H g — which is added to the heat densityof matter — can indeed be interpreted as the heat density of gravity.) Thus,the specification of horizon entropy specifies the P abcd and selects the corre-sponding Lanczos-Lovelock model. The temperature of the spacetime, as wesaw before, is purely kinematic, but specifying the form of horizon entropyin Eq. (7), specifies the dynamics of the theory. This is precisely what weexpect in the thermodynamic description of a system.In d = 4 dimensions, P abcd reduces to P abcd = (1 / δ ac δ bd − δ bc δ ad ). Theresulting equation for the background spacetime is identical to Einstein’sequation: G ab = (8 πL P ) T ab + Λ δ ab (9)with an undetermined cosmological constant. By the very construction, thecosmological constant (for which T (Λ) ab n a n b = 0) cannot appear in the ex-tremum principle; but since the theory is invariant under the shift T ab → T ab + (constant) δ ab , it arises as an integration constant. (So we need a further static spacetime foliated by a seriesof spacelike hypersurfaces. Let V be a 3-dimensional region in a spacelikehypersurface with a 2-dimensional boundary ∂ V , which we could choose tobe an equipotential surface (corresponding to constant lapse function). Wecan then show that [12, 13] the gravitating (Komar) energy E Komar containedin V is equal to the equipartition heat energy of the surface ∂ V if we associate dN = dA/L P degrees of freedom with each area element dA . That is, we canshow: E Komar = Z √ γ d xL P (cid:18) k B T loc (cid:19) ≡ N sur ( k B T avg ) (10)where T avg is the average temperature of ∂ V and N sur = A sur /L P . So we canactually count the microscopic degrees of freedom through an equipartitionlaw which — since it relates bulk and boundary energies — could be calledholographic equipartition. (One can rescale (1 / k B T → ( ν/ k B T, N sur → A sur /νL P without changing the result; we have chosen ν = 1.)We can do better. Consider the most general spacetime rather than staticspacetimes. In this case, we can associate with the bulk energy E Komar thenumber N bulk , defined as the number of degrees of freedom in V if E Komar isat equipartition at the temperature T avg . That is: N bulk ≡ | E Komar | (1 / T avg (11)It then turns out that [14] the time evolution of the spacetime geometryin V is driven by the difference between the bulk and boundary degrees offreedom. Specifically:18 π Z d x √ hu a g ij £ ξ N aij = 12 T avg ( N sur − N bulk ) (12)8here N cab ≡ − Γ cab + (1 / δ ca Γ ddb + δ cb Γ dad ), ξ a ≡ N u a is the time evolution vec-tor, where u a is the velocity of the observers moving normal to the foliation.A simple corollary is that all static [12, 13] spacetimes maintain holographicequipartition in terms of the number of degrees of freedom in the bulk andboundary: N sur = N bulk (13)which, of course, is a nicer restatement of Eq. (10).The role of Planck constant ~ in this approach is worth emphasizing.Relativity brings in the speed of light c , the Davies-Unruh temperature bringsin ~ and the expression for heat density H g introduces the quantum of area L P . When we take the Newtonian limit of the gravitational field equations,we will end up getting the gravitational force to be: F = (cid:18) c L P ~ (cid:19) (cid:16) m m r (cid:17) (14)We should resist the temptation to call the combination ( c L P / ~ ) as G whichis independent of ~ . Equation (14) tells us that the ~ → Gravity is quantum mechanical at all scales!
Just as matter is quantummechanical at all scales — the individual atoms will collapse ~ → The above discussion highlights the clear analogy between, say, a fluid andthe spacetime from a thermodynamic perspective. The equipartition laws inEq. (1) and Eq. (10), in particular, allow us to count the number densityof atoms in either of these systems. The next logical step will be to takethese ideas one level deeper and obtain the gravitational heat density H g from microscopic considerations.To do this, we need to take into account the discreteness of spacetime,arising from the quantum of area L P , without losing the privilege of usingcontinuum mathematics in our description. In the case of normal fluid, theuse of a distribution function allows us to reconcile these two mutually con-tradictory requirements. When you say dN = f ( x i , p i ) d xd p counts the9umber of atoms around an event x i (with momentum p i ) in a small phasevolume, you are assuming that d x is small enough to be considered infinitesi-mal and yet big enough to contain sufficiently large number of atoms; by thevery process of counting, f ( x i , p i ) incorporates discreteness while allowingthe use of continuum mathematics. Proceeding by analogy, we are lookingfor a distribution function f ( x i , n j ) which could count the number of atomsof space at an event x i with an additional dynamical variable n j . One mightguess that n j could possibly be related to the null vector which occurs inthe gravitational heat density H g but this remains to be obtained from themicroscopic analysis.Since the distribution function has to be a primitive construct in thespacetime, it seems natural to assume that the number of atoms of space-time will be proportional either to the volume measure or the area measureassociated with a given event . In the continuum description of spacetime,an event has zero area or volume associated with it and hence we cannothope to obtain a f ( x i , n j ) from such a construction. This is, however, under-standable. Unless we incorporate the quantum of area into the descriptionof spacetime, one cannot hope to get a sensible distribution function for theatoms of spacetime. So, our strategy will be to incorporate the zero-pointarea L P into the fundamental description of spacetime in a suitable manner,define appropriate area and volume measures in such a quantum correctedspacetime and extract the distribution function from these primitive con-structs. Of course, there is no guarantee that the quantum spacetime willendow an event with a non-zero area (or volume) but, incredibly enough, itdoes. I will now describe this procedure in some detail.Before we start, it is convenient to re-write H g in an equivalent dimension-less form. Using the fact that H g ∝ P abcd ∇ a n c ∇ b n d and K g ∝ R ab n a n b differby an ignorable total divergence [14] , we could as well use the K g insteadof H g in our variational principle. Introducing the appropriate numericalfactors, it is convenient to work with the dimensionless combination d ( ¯ Q g /E P ) d ( √ γd xdλ/L P ) ≡ K g ≡ − L P π R ab n a n b (15)It is this quantity which we hope to obtain from some primitive construct inthe quantum spacetime.Our first task is to incorporate the zero-point area into the spacetime,which could be done in a model independent manner along the following10ines. There is considerable amount of evidence [15] to suggest that a primaryeffect of quantum gravity is to modify the geodesic interval σ ( x, x ′ ) in aspacetime to another form S ( σ ) such that S (0) ≡ L is a finite constantof the order of L P . For illustrative purposes, we will take S ( σ ) = σ + L though none of our results depend on this explicit form. One can show thatsuch a modification is equivalent to working with a renormalized spacetimemetric (called qmetric) q ab ( x, x ′ ; L ) instead of the original classical metric g ab ( x ). The explicit form of the qmetric is given by q ab = Ah ab + Bn a n b ; q ab = 1 A h ab + 1 B n a n b (16)where B = σ σ + L ; A = (cid:18) ∆∆ S (cid:19) /D σ + L σ ; n a = ∇ a σ (17)and ∆ is the Van Vleck determinant related to the geodesic interval σ by∆( x, x ′ ) = 12 1 p g ( x ) g ( x ′ ) det n ∇ xa ∇ x ′ b σ ( x, x ′ ) o (18)The ∆ S is the corresponding quantity computed with σ replaced by S ( σ )in Eq. (18).The qmetric is a bi-tensor depending on x and x ′ through σ ( x, x ′ ) andis singular everywhere in the spacetime in the limit of x ′ → x with finite L .On the other hand, q ab → g ab when L → g ab ( x )] constructed from the background metric and its derivatives, we cancompute the corresponding (bi)scalar Φ[ q ab ( x, x ′ ); L ] for the renormalizedspacetime by replacing g ab by q ab in Φ[ g ab ( x )] and evaluating all the deriva-tives at x , keeping x ′ fixed. The renormalized value of Φ[ q ab ( x, x ′ ); L ] is thenobtained by taking the limit x → x ′ in this expression keeping L non-zero.It turns out that many useful scalars like R , K etc. remain finite [16–18]and local in this limit even though the qmetric itself is singular when x → x ′ with non-zero L . The algebraic reason for this curious fact [16] is that thefollowing two limits do not commute:lim L → lim x → x ′ Φ[ q ab ( x, x ′ ); L ] = lim x → x ′ lim L → Φ[ q ab ( x, x ′ ); L ] (19)It is now easy to see how null surfaces and null vectors are singled out inthis approach. In all calculations we will eventually take the limit σ → σ →
0, will translate into a nullsurface in the Minkowski spacetime and the normal vector n i = ∇ i σ (whichoccurs in the qmetric and all the resulting constructs) will pick out the nullvector which is the normal to the null surface. More generally, σ ( x, x ′ ) → n a willcorrespond to a null vector in the Minkowski spacetime. This is how a nullvector field n i is introduced in the description from a microscopic point ofview.With this mathematical structure in place, we can define the volumeand area measure of the renormalized spacetime as follows. It is convenientto describe the Euclidean background spacetime in synchronous coordinates( σ, θ , θ , θ ) where σ (the geodesic distance from the origin) is the ‘radial’coordinates and θ i are the angular coordinates on the equi-geodesic surfacescorresponding to σ = constant. We next introduce the zero-point-area byconstructing the corresponding qmetric. (The equigeodesic surfaces remainequi-geodesic surfaces in the renormalized spacetime.) Using the qmetric onecan then compute the volume measure, √ qd x , as well as the area measureof the equi-geodesic surfaces, √ hd x . Both √ q and √ h will be now bi scalarsand we define their value at a given event by taking the limit of x ′ → x corresponding to σ →
0. As mentioned earlier, this will lead to a dependenceon a null vector n i which could be in any direction at the given event (andis reminiscent of the momentum variable which occurs in the distributionfunction of normal matter.) Such a computation shows that the volume andarea measures behave as follows: √ q = σ (cid:0) σ + L (cid:1) (cid:20) − E (cid:0) σ + L (cid:1)(cid:21) p h Ω (20) √ h = (cid:0) σ + L (cid:1) / (cid:20) − E (cid:0) σ + L (cid:1)(cid:21) p h Ω (21) The local Rindler observers who live on the hyperboloid r − t = σ see the null cone r − t = 0 as the horizon. In the Euclidean sector the hyperboloid becomes the sphere r + t E = σ E and approaching the Euclidean origin, σ E →
0, translates to approachingthe light cone in the Minkowski space. These results are somewhat subtle algebraically. The leading order behaviour of √ qdσ ≈ σdσ , which makes the volumes scale as σ (while the area measure is finite)produces the following result [22]: The effective dimension of the renormalized spacetimereduces to D = 2 close to Planck scales. I will not elaborate on this result here.
E ≡ R ab n a n b . When L → σ → L . Something remarkable happens when we do this. Thevolume measure √ q vanishes but the area measure √ h has a non-zero limitgiven by: √ h = L (cid:20) − E L (cid:21) p h Ω (22)The the renormalized spacetime attributes to every point in the spacetime afinite area measure but a zero volume measure! Since L √ h Ω is the volumemeasure of the σ = L surface, the dimensionless density of the atoms ofspacetime, contributing to the gravitational heat is given by: f ( x i , n a ) ≡ √ hL √ h Ω = 1 − E L = 1 − L R ab n a n b (23)This matches with what we need if we take L = (3 / π ) L P . Briefly stated,quantum gravity endows each event in spacetime with a finite area but zerovolume. It is this area measure which we compute to obtain a natural esti-mate for f ( x i , n a ). In the macroscopic limit, the contribution to the gravita-tional heat in any volume is obtained by integrating f ( x i , n j ) over the volume.So the expression for the heating rate, in dimensionless form is given by: L P dQ g dλ = Z √ γd xL P f ( x i , n j ) = Z √ γd xL P (cid:20) − π L P ( R ab n a n b ) (cid:21) (24)which gives the the correct expression — with the crucial minus sign — plusa constant. So one can indeed interpret the gravitational heat density as thearea measure of the renormalized spacetime.
While the second term in Eq. (23) gives what we want for the variationalprinciple, the first term tells us that there is a zero-point contribution tothe degrees of freedom in spacetime, which, in dimensionless form, is justunity. Therefore, it makes sense to ascribe
A/L P degrees of freedom to anarea A , which is consistent with what we saw in the macroscopic description.We also see that a two sphere of radius L P has 4 πL P /L P = 4 π degrees offreedom. This was the crucial input which was used in a previous work todetermine the numerical value of the cosmological constant for our universe.Using this result, one can show express the energy density corresponding to13he cosmological constant in the form [19, 20]: ρ Λ ≈ ρ / ρ / exp( − π ) (25)where ρ inf is the energy density during inflation and ρ eq is the energy densityat the epoch of matter radiation equality. From cosmological observations,we find that ρ / eq = (0 . ± . ρ / = (1 . − . × GeV, we get ρ Λ L P = (1 . − . × − , which is consistent with observations!This novel approach for solving the cosmological constant problem pro-vides a unified view of cosmic evolution, connecting all the three phasesthrough Eq. (25); this is to be contrasted with standard cosmology in whichthe three phases are put together in an unrelated, ad hoc manner. Further,this approach to the cosmological constant problem makes a falsifiable pre-diction , unlike any other approach I know of. From the observed values of ρ Λ and ρ eq we can constrain the energy scale of inflation to a very narrowband — to within a factor of about five, if we consider the ambiguities inre-heating. If future observations show that inflation took place at energyscales outside the band of (1 − × GeV, this model for explaining thevalue of cosmological constant is ruled out.
We have completed the program outlined in the introduction using essen-tially two ingredients: (a) We postulated that the extremum principle de-termining spacetime dynamics should be invariant under the shift T ab → T ab + (constant) δ ab . this allowed us to obtain an expression for gravitationalheat density which depended on a null vector that acted as an auxiliaryvariable. (b) We introduced the zero-point area into the spacetime by thereplacement σ → σ + L . The modified spacetime led to an area measurewhich, in dimensionless form, matched precisely with the gravitational heatdensity we needed. We interpreted the microscopic origin in terms of thedistribution function for the atoms of spacetime.This approach raises several important issues for further investigationsand let me mention a couple of them. First, we need to understand precisely what is counted by f ( x i , n j ). We called it atoms of space which stands for14he microscopic degrees of freedom of quantum space(time) parametrized bya null vector n i . One could equally well have thought of it as related tonumber of microscopic states available to quantum geometry. This suggeststhat, in the suitable limit, one can introduce a probability P ( x i , n a ) for n a at each event x i and define the partition function: e S ( x i ) ∝ Z D n i P ( x i , n a ) exp[ µL P T ab n a n b ] (26)where µ is a numerical factor of order unity. If we take P ( x i , n a ) ∝ exp[ µf ( x i , n a )] ∝ exp (cid:18) − µL P π R ab n a n b (cid:19) (27)then the saddle point evaluation will peak at the geometry determined byEinstein’s equation with an arbitrary cosmological constant. (The choice µ =1 / P to be interpreted as number of microstates.) Alternatively,one can think of P ( x i , n a ) to be such as to give the correlator h n a n b i ≈ (4 π/µL P ) R − ab which allows us to write the field equations in the form:2 µL P h ¯ T ab n a n b i ≈ µL P h ¯ T ab ih n a n b i = 1 (28)The averaging h· · · i now indicates both expectation values for the quantumoperator T ab as well as a probabilistic averaging of n a n b . Equation (28) has aMachian flavour. One cannot set h T ab i = 0 and study the resulting spacetimesince it will lead to 0 = 1!. Matter and geometry must emerge and co-existtogether in a manner we have not yet understood.
There is no such thing asflat spacetime existing in the absence of matter!Second, a thermodynamic approach to gravity strongly suggests that cos-mology should not be treated as a part of general relativity and we shouldlook at cosmic questions afresh. The study of thermodynamics, distribu-tion functions for atoms of space etc. pre-supposes some unstated notionof equilibrium at the microscopic scales, which, in turn, will involve certaintimescales over which such an equilibrium can be established. For normalsystems characterized by timescales much less than the age of the universe,one could possibly assume that Planck scale physics has established the nec-essary equilibrium conditions. But such an assumption is likely to breakdown when we consider the entire universe as a physical system. Instead,one is led to a picture in which larger and larger spatial scales achieve mi-croscopic equilibrium as the cosmic time evolves. In such a scenario, one15ould even argue that the space as we know itself emerges [21] as a conden-sate of the atoms of space as the cosmic time evolves. The deviations frommicroscopic equilibrium can then have important implications for the largescale dynamics of the universe, a glimpse of which was seen in the suggestedsolution to the cosmological constant problem.
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