Surfaces away from horizons are not thermodynamic

Since the 1970s, it has been known that black-hole (and other) horizons are truly thermodynamic. More generally, surfaces which are not horizons have also been conjectured to behave thermodynamically. Initially, for surfaces microscopically expanded from a horizon to so-called stretched horizons, and more recently, for more general ordinary surfaces in the emergent gravity program. To test these conjectures we ask whether such surfaces satisfy an analogue to the first law of thermodynamics (as do horizons). For static asymptotically flat spacetimes we find that such a first law holds on horizons. We prove that this law remains an excellent approximation for stretched horizons, but counter-intuitively this result illustrates the insufficiency of the laws of black-hole mechanics alone from implying truly thermodynamic behavior. For surfaces away from horizons in the emergent gravity program the first law fails (except for spherically symmetric scenarios), thus undermining the key thermodynamic assumption of this program.


Introduction
In order to attempt to derive a first law for ordinary surfaces within a spacetime we shall follow closely in the footsteps of Bardeen, Carter and Hawking's 1973 classic paper [1]. The first step is to obtain an integral equation for the net energy in a static system, where instead of an inner boundary located at a black hole horizon, this boundary will be an ordinary surface. In order to obtain a first law, we must convert this into a description of how small "changes" in the net energy may be accounted for via a differential version of energy balance. The changes here actually refer to the differences between nearby solutions of the field equations related by a small diffeomorphism of the initial metric. Of course such changes could be implemented physically via an adiabatic process involving weightless strings used to add, subtract, or otherwise rearrange matter within the screen. Alternatively, one might consider physical changes due to matter dynamically falling past the screen [2], but we shall not study this latter approach here. We set G = c = = k B = 1 throughout.
As already noted, we shall be considering diffeomorphisms between pairs of nearby static asymptotically-flat solutions with zero shift vector (β µ = 0). For simplicity, we shall suppose (from Eq. (10) onwards) that the spacetime is non-rotating. Finally, from Eq. (20) onwards, we shall assume that there is no matter exterior to the inner boundary (T µν = 0). For ease of navigation, we have placed in boxes all the key results used in the main manuscript.
The original 1973 paper [1] gives key signposts for deriving the first law. However, we have been unable to find a detailed derivation in any subsequent paper or textbook. Therefore, other than the above simplifications (which suffice for our purposes in studying ordinary surfaces) we give those details here. This is done partly to allow our claims to be checked in detail and partly as a resource for the community to better appreciate the original 1973 result.
We focus specifically on generalizing the first law of black hole mechanics to ordinary surfaces. Indeed, it is straightforward to derive an integral version of the first law associated with arbitrary surfaces. This allows one to identify how surface gravity (and hence presumably temperature) should be generalized away from a horizon in a manner consistent with the natural generalization of the area law. Next, following Bardeen et al., we study diffeomorphisms of this integral formulation in order to attempt to construct a differential first law. Everything follows through as in the 1973 analysis except that diffeomorphisms of the surface gravity of an ordinary surface does not reduce to the form found for horizons. We conclude that the first law of thermodynamics does not hold for ordinary 2-surfaces ∂Σ in and hence such surfaces do not behave thermodynamically.
We finish this introduction with a list of key symbols:  HereN µ is the spacelike 4-vector normal to the boundaries of Σ (note the direction convention on the inner boundary). In the absence of matter outside the inner boundary, the gravitating mass as measured at infinity, M , will be entirely inside the inner boundary.

Integral expression for net energy
Consider a static spacetime with a Killing vector K µ = ∂ t = (1, 0, 0, 0), with K µ K µ = −1 at spatial infinity. The Killing equation implies that Now recall that permuting the order of a pair of covariant derivatives acting on a 4-vector A µ may be expressed in terms of the Riemann curvature tensor as [3] Contracting the indices µ and α reduces this to an expression in terms of the Ricci tensor Since the Killing vector is anti-symmetric we must have K µ ;µ = 0 and we immediately find that Integrating this over a spacelike hypersurface Σ, yields (4) hereT µ is the timelike unit 4-vector normal to Σ, sô T µT µ = −1. The hypersurface is assumed to have an outer boundary at spatial infinity ∂Σ ∞ , and an inner boundary Σ in (see Fig. 1). In the original work of Bardeen et al. [1], this inner boundary corresponded to the black hole's horizon ∂Σ BH . Here we generalize this by taking it to be an arbitrary closed 2-surface ∂Σ in . The boundary of the hypersurface is assumed to be oriented, with unit normal N µ (see Fig. 1), soN µN µ = 1 andN µT µ = 0. Recalling Stokes's theorem for an anti-symmetric ten- and applying it to the left-hand-side of Eq. (4) we find At this stage, we wish to generalize the concept of surface gravity as a quantity defined anywhere. Assuming that the surface ∂Σ is non-rotating (corresponding to zero angular velocity of the spacetime itself) , we may interpret the integrand of the integral on the boundary in Eq. (6) to be the surface gravity, so It is worth noting that κ/(2π) is precisely the formula Verlinde gives (his Eq. (5.3) of [4]) for what he calls the local temperature of the holographic screen (ordinary surfaces of constant Newtonian potential φ) as measured with respect to a reference point at spatial infinity. This definition of surface gravity allows us to naturally extend the original 1973 analysis away from black hole horizons. In particular, the left-hand-side of Eq. (6) reduces to The integral over ∂Σ ∞ reduces to the Komar expression for the total gravitating mass within the system, M , [3] leading to Were we to consider spherically symmetric case, Eq. (9) would reduce to (This is exactly the first law given by Bardeen et al. [1] taking angular velocity of ∂Σ to vanish).
Just to emphasize what this represents, here the hypersurface, Σ, extends from an arbitrary inner boundary, ∂Σ in , out to spatial infinity. Thus, the generalized surface gravity, κ, and the area, A, are those associated with the inner boundary itself (rather than any horizon).
Eq. (9) has exactly the same form as the conventional formula for the total mass of the system [1] but extended to an arbitrary 2-dimensional surface (instead of a horizon). Finally, note that the matter inside the inner boundary need not be associated with a black hole, it may be ordinary matter, with no horizon present at all. Thus, were inner boundaries found to have thermodynamic properties (i.e., a well-defined entropy and temperature), it would not be because such properties were inherited from a real horizon behind the screen.
Differential "first law" The above straightforward generalization, especially in the spherically symmetric case, for net energy on a hypersurface might appear to suggest that a temperature and entropy can actually be defined for any surface by However, such quantities need to behave thermodynamically. In particular, for our static system, the net energy E, should admit changes which behave as (ignoring work terms) so that the temperature would be acting as an integrating factor relating changes in the (state function) entropy to changes in the energy. In other words, we must show that such changes lead to the expected form of the first-law of thermodynamics. Again here we follow in the footsteps of the original analysis and consider changes corresponding to parametric differences between diffeomorphicly nearby solutions. In particular, we will consider two nearby configurations corresponding to the metrics where h µν ≡ δg µν = −g µσ g ντ δg στ , i.e., δg στ = −h στ .
As with the original analysis and without loss of generality, we may assume that for the two diffeomorphicly related configurations, the hypersurfaces Σ and Σ are described by identical sets of coordinates; this is always possible due to "gauge" freedom in the choice of coordinate systems [1]. Henceforth we label both by Σ. Similarly, for their boundaries ∂Σ. Further, as in [1] we likewise assume that both configurations have the same Killing vector, so Finally, it will be sufficient for our purposes to consider only the case where there is no matter on Σ itself, so T µν = 0 there. Geometrically, this corresponds to all the matter lying behind or within the inner boundary ∂Σ in (see Fig. 1).
In order to consider diffeomorphisms which need not respect spherical symmetry, we return to Eq. (9). Using the Einstein field equations we start by rewriting this integral formula as Recall K µ = ∂ t andT µ is normal to Σ, so K µ = NT µ + β µ where β µ is the shift vector and N = 1/T t is the lapse function [5]. Assuming a zero shift vector β µ = 0, thenT µ =T t K µ andT µ =T t K µ . Since N |γ (Σ) | = √ −g on the hypersurface [5], the variation of the Ricci scalar term may be computed as where in the last step we have used the well-known result that [6] δ ;ν , a result quoted in Ref. [1], there without proof. Proof: we have This completes the proof of Lemma 1.
Using Lemma 1, the variation of the Ricci scalar term becomes since the first term is zero if we assume T µν = 0 on Σ outside the holographic screen.
Proof: Expanding out the right-hand-side (rhs) of the claim in Lemma 2, we get The Lie derivative along K µ vanishes since the pair of diffeomorphicly related metrics are assumed static [8]. This completes the proof of Lemma 2.
Applying Lemma 2 to Eq. (20), the variation of the term involving the Ricci scalar reduces to Thus, the variation in the total mass may be written Since the term inside the bracket is an anti-symmetric tensor, we may use Stokes's theorem, Eq. (5), to obtain where the boundary has be split into the inner boundary ∂Σ in and the boundary at infinity ∂Σ ∞ . The contribution for the term at infinity may be evaluated using the notation of tensorial volume elements [9] as where the orientation of ε βναµ is chosen so that ε βναµ = −6 ε [βν ε αµ] and ε αµ is the volume element of the boundary at infinity, and we have applied the result 1 8π ∂Σ∞ (h µ µ;ν − h µ ν;µ )K β ε βναµ = −δM in the final step [9].
Eq. (26) allows us to transform Eq. (25) into Or equivalently, where we have used K νN ν = NT νN ν = 0, which follows sinceT µ is normal to Σ andN µ lies in Σ. For the first law to be true, we would require that the first two boundary integrals of Eq. (28) exactly cancel. In order to further simplify these terms we start by considering the diffeomorphic changes in more detail.

Diffeomorphic conditions
As already discussed, we assume Recall that by "gauge" freedom the sets of coordinates of Σ and ∂Σ are unchanged by the diffeomorphism, so without loss of generality we may take [1] δ(dx µ ) = 0 , ∀dx µ in Σ.

Reduction to the first law
Since K µN µ = 0, then (K µN µ ) ;ν = 0, and so We then consider the expansion of null normal congruences on the inner boundary which may be written as where l µ =T µ +N µ is the outgoing null normal vector of the inner boundary, and we have used Eq. (41) in the fifth line and Eq. (50) in the sixth line. Using this relation we may express the variation of κT t as where δ √ −g = 1 2 √ −gg µν δg µν . Hence, the first term in Eq. (28) may be written as The second term h µ ν;µN ν in Eq.(28) can then be expressed as where we have used Eq. (50) in the fifth line, and Eqs. (7) and (48) in the last step. Next, consider the final term 1 2 δP µνN µ;ν in Eq. (54), using Eq. (46) we have where σ (l) × are the shears of l µ defined by [10] σ (l) Therefore, Finally, substituting Eq.(53) and Eq.(57) into Eq.(28), we find Or in summary, It is worth noting that which separately depends only on k 2 , k 3 , k 6 , and we will prove it next. Since P µνT µ;ν = P µν (K µT t ) ;ν = P µν K µ;νT t + P µν K µ (T t ) ;ν = 0, θ (l) can be simplified as θ (l) = P µν l µ;ν = P µν (T µ;ν +N µ;ν ) = P µνN µ;ν .

Vanishing extrinsic curvature tensor
Our analysis has been predicated on static screens. However, there is another way to define screens, so their normal direction remains parallel to the proper acceleration of a family of locally coincident timelike observers [12]. These observers are constrained to have constant 4-acceleration along with a number of other technical assumptions [12]. A first law is then obtained for these surfaces provided they additionally have a vanishing extrinsic curvature tensor K µν = 0 [12]. The first law obtained is of a form with energy and temperature measured locally instead of at spatial infinity, which for asymptotically-flat spacetimes are unambiguous. Finally, we note that there is no easy way in this other formalism [12] to investigate stretched horizons.
In our setting with zero shift vector β µ = 0, soT µ = T t K µ , and our hypersurfaces Σ are orthogonal toT µ , we find that K µν = 0 implies a vanishing expansion θ (l) = 0. Thus, for our setting, the formalism of Ref. 12 only yields a first law on horizons.
To see that this is the case, recall that the extrinsic curvature tensor of our inner boundary equals [3] K µν ≡N (λ;ρ) P λ µ P ρ ν .
Taking the trace of this yields the extrinsic curvature scalar as K = P µνN µ;ν = θ (l) , where in the final step we use Eq. (61). Thus, for our setting, the first law of Ref. 12 appears to occur at the horizon; a result which is naively consistent with the classic 1973 result.
Let us now consider a construction for a screen surrounding a gravitating body as proposed by Ref. 12: Construct a screen using a family of stationary timelike observers at fixed radius around a Schwarzschild black hole. It is easy to calculate the extrinsic curvature tensor for the screen and see, as noted above, that this curvature vanishes only on the horizon. Hence the screen is on the horizon and the observers are null instead of timelike observers. Next drop in a spherical shell of matter. As the shell passes the screen of observers, the horizon (where θ (l) = 0) discontinuously jumps, the surface gravity of the new horizon changes and the original screen of observers fall into the black hole. We must then conclude either that the construction using the methods of Ref. 12 of a screen surrounding the black hole is simply impossible (because the observers are not timelike), or it fails to continue to hold under perturbation.
Thus, although Ref. 12 purports to describe a dynamical first law for ordinary surfaces its conditions are either in general impossible to satisfy or are generally not preserved under perturbation.

Local temperature in emergent gravity
We focus here on the temperature defined in the original paper on emergent gravity [4] which is used there in a heuristic derivation of the Einstein field equations. In Fig. 2 we show a schematic of the hypersurface considered there. ∂Σ HS denotes the holographic screen (ordinary surfaces of constant Newtonian potential φ) which now is the outer boundary of the spacelike hypersurface Σ EG under study, andN µ is the unit normal vector of the holographic screen. The 'local' temperature of the holographic screen (as measured at spatial infinity) used in Ref. 4 is defined as where φ is the generalized Newtonian potential, given by φ = 1 2 ln(−K µ K µ ) = ln N , recalling that K µ K µ = −N 2 . It is now an easy matter to check that In summary, recall the definition of κ in Eq. (7), yielding For reference, the Unruh temperature associated with a stationary observer is just the magnitude of the observer's proper acceleration a µ over 2π. As their 4velocity is given byT µ we easily find a µ ≡T µ ;νT sinceT µ =T t K µ = K µ /N = e −φ K µ . Thus a µ is perpendicular to surfaces of constant φ. When Verlinde's temperature is measured locally (instead of referenced to spatial infinity) it is T local = 1 2π φ ;µN µ . For this to equal the Unruh temperature at the same point, the local unit normalN µ to the screen must be aligned with the proper acceleration a µ of our stationary observer there. Therefore, it trivially follows that only for surfaces of constant Newtonian potential φ would the holographic screens be in thermal equilibrium with stationary physical surfaces of the same shape, size and location. Hence, Thermodynamic equilibrium ⇒N µ φ ;µ Finally, we show that for surfaces of constant φ, we have δφ = k 1 /2. Indeed, sinceT t = 1/N = e −φ , we have where in the last step we have used Eqs. (36) and (37).