Local corner Energy

We saw, for the variation principle to be well defined the Holst action must acquire additional terms, one being the Gibbons–Hawking–York boundary term [66, 67] for first-order tetrad-connection variables [68], the other belonging to the two-dimensional corner, i.e. the boundary of the boundary. Having already explored the mathemat-ical properties of this additional corner term, it is now time to ask for its physmathemat-ical role. Studying an accelerated observer close to the corner S (see figure 2.1 for an illustration), we will see, the two-dimensional boundary integral measures the local gravitational energy. The energy, thus uncovered, will match what has recently been studied in a series of pioneering articles by E. Frodden, A. Gosh and A. Perez, who boosted the understanding of thermodynamical properties of accelerated observers in both classical and quantum gravity [73, 107]. In this section we will present another look at these results, and rederive them directly from the Hamilton–Jacobi equation of general relativity. This section is based upon the results partially published together with E. Bianchi [108].

First of all we must agree on some simplifying assumptions. We expand the metric g_ab =⁽⁰⁾g_ab+ε⁽¹⁾g_ab+. . . close to the cornerS(that have the topology of a two-sphere) in powers of the ratio

ε= L

√A ≪1 (2.77)

of the two typical length scales of the problem;Lis the proper distance from the corner S, and A is its area. Employing the principle of general covariance, we introduce a family of accelerated, static (non-rotating) observers that stay at fixed distance from the surfaceS, such that the line element assumes the asymptotic form of a two-dimensional Rindler metric plus the line element on the two-surfaceS. We can thus write:

ds²=−c²L²dΞ²+ dL²+ A

4πdσ²+O(ε). (2.78)

Here we have introduced the observers’ rapidity Ξ, defined just like in (2.5) given above, together with the induced two-dimensional metricA/(4π)dσ² on the corner. A typical example of such a geometry is given by the near-horizon approximation of the Schwarzschild spacetime. In this case (using the standard Schwarzschild coordinates in the exterior region of the black-hole solution)A = 16πM² is the area of the horizon, dσ² equals the induced metric dϑ²+ sin²ϑdϕ² thereon, the observers rapidity isΞ = t/(4M), while the asymptotic expansion of the Newton potential in powers ofε yields (1−2M/r) =L²/(4M)²(1 +O(ε)). There is, however, no need to restrict ourselves to this particular geometry, as shown for a wide class of black-hole solutions in reference [107].

Staying at a fixed distance L_o above the surface, the rapidity Ξ measures the ob-server’s proper timeτ according to

dτ = L_o(1 +O(ε))

c dΞ≈ L_o

c dΞ. (2.79)

Here, and in the following “≈” means equality up to terms of higher order in ε. Next, we match the time function^* t:M →R previously introduced with the proper timeτ of the observer at the distanceLo. If γ(τ) is the observer’s trajectory parametrised in proper timeτ we thus ask fort(γ(τ)) =τ.

With a notion of time, that agrees with physical duration as measured by an accel-erated observer, there should also come a notion of gravitational energy. The relation between time and energy becomes particularly clear when looking at the Hamilton–

Jacobi equation and realising one as the conjugate of the other. Let us thus briefly recall those aspects of the Hamilton–Jacobi formalism that we will need in the follow-ing. Consider a one-dimensional mechanical system, that shall share with the general theory of relativity the absence of a preferred notion of time. The configuration vari-ables be q ∈ R, that measure location, and proper time τ. The canonical momenta bepand E respectively, whereE stand for the energy. Call S(q_f, qi;τ_f, τi) Hamilton’s principal function, i.e. the action

S= Z 1

0

dt pq˙−τ H(p, q)˙

(2.80) evaluated on a solution of the equations of motion to the boundary value problem q(ti) = qi, q(t_f) = q_f, and τ(ti) = τi, τ(t_f) = τ_f. Hamilton’s principal function is a solution of the Hamilton–Jacobi equation:

p= ∂S(q, q_i;τ, τ_i)

∂q , E=H

q,∂S(q, q_i;τ, τ_i)

∂q

=−∂S(q, q_i;τ, τ_i)

∂τ . (2.81)

Since the energy is conserved, the solution of the Hamilton–Jacobi equation is only a function of the time interval τf −τi, and there is no dependence of τi +τf therein.

Performing a Legendre transformation, that amounts to keep the energy fixed while allowing for arbitrary variations ofτ_f−τ_i, we can remove theτ-dependence in favour of an energy dependence, eventually revealing what is sometimes called the characteristic Hamilton function S(q_f, q_i;E) = E·(τ_f −τ_i) +S(q_f, q_i;τ_f, τ_i). Taking the derivative with respect to the energy we get the conjugate variable, which is the observer’s proper time elapsed when passing fromq_i to q_f:

∂S(q_f, qi;E)

∂E =τ_f −τ_i. (2.82)

Looking at the analogous equation for the gravitational action (2.1) we will now read off the observer’s energy. Hamilton’s principal function is the action evaluated on a solution of the equations of motion, its functional differentials define energy and time through equations (2.81) and (2.82). We thus need to study variations of the action around a solution of the equations of motion. Working in a first-order formalism these are the Einstein equations together with the torsion-free condition, i.e. equations (2.22)

*In the beginning of this chapter we have fixedtto the valuest= 0, andt= 1on the initial and final slices respectively, this restriction must now be relaxed forΞto assume arbitrary values inR.

and (2.12) respectively. We look back at equations (2.26), (2.27) and (2.36) and see that we have already computed those variations explicitly, and thus readily find:

δ I_M +I_∂M+I_S Let us rewrite this expression in a more compact form. Using time gauge (2.58) and employing our definitions for the densitised triad Eia and for both the extrinsic cur-vature Kⁱ_a and the intrinsic so(3)-connection Γⁱ_a (collected in equations (2.63) and (2.62) respectively) we get: where we have also introduced the internal outwardly pointing normal zⁱ = eⁱaz^a of the two-dimensional corner S. The first term is a total divergence. This becomes immediate when first looking at the functional differential of the pullback of the torsion-free equation (2.12) onto the spatial slice. In fact:

Dηⁱ = 0⇒Deⁱ = deⁱ+ǫⁱ_lmΓ^l∧e^m = 0⇒Dδeⁱ+ǫⁱ_lmδΓ^l∧e^m= 0, (2.85) where have implicitly introduced the exterior covariant derivativeD= d + [Γ,·]on the spatial slice. Inserting (2.85) into the first term of (2.84), and once again using Stoke’s theorem, we arrive at an integral over the two-dimensional corner:

Z Let us also mention, that this additional corner term, often identified with a symplectic structure of a Chern–Simons connection, plays an important role in the semiclassical description of black-hole horizons in loop quantum gravity [109–112]. In our case we can drop this term, because the variations of the triad should be everywhere continuous.

This in turn implies: Z

We have thus achieved to compute the functional differential of the Holst action evaluated on a solution of the equations of motion, and immediately see Hamilton’s principal function is only a functional of the densitised triads, simply since no functional differential of the connection components ever appears:

δS_Holst

The last term is the desired expression, that we want to compare with (2.81) in order to read off the energy. This term consists of two elements, one being the rapidityΞ,

while the other measures the areaA of the corner in terms of the flux ofE throughS wherezⁱ denotes again the outwardly pointing normal of∂Σ =S in three-dimensional internal space. Next, we also need to better understand how the observer’s rapidity Ξ can actually measure the elapsing proper time. Taking our simplifying assumption on the asymptotic behaviour of the metric at a distance L_o close to the corner S, i.e.

employing equation (2.78), we find:

Looking back at the defining equation for energy and time, i.e. equations (2.82, 2.81), we can identify the energy E_bulk of the gravitational field as measured by the local observer to be:

E_bulk ≈ −~c ℓ²_P

A

L_o. (2.92)

Consider now a process where the observer can exchange gravitational energy with the region beyond the surfaceS, callE_S the energy stored therein, and letδE_bulk andδE_S be the change of energy in the two respective regions. If we assume energy conservation this process must obey

δE_bulk+δE_S = 0. (2.93)

If the observer moves without ever changing the distance from the corner we thus get δE_S ≈ c⁴

8πG δA

L_o. (2.94)

This equation coincides with the local form of the first law of black-hole thermody-namics as introduced by Frodden, Gosh and Perez. It states that for an uniformly accelerated observer flying at fixed distance L_o above the surface S, any process re-sulting in an increase δAof the area of the surface is accompanied by a change δE_S of the energy stored behind the surfaceS. We can see in (4.37) below and reference [113]

how this formula reappears also in the quantisation of the theory.