gr-qc/9504004

UMDGR-95-114

UMDGR-95-114

The Einstein Equation of State

College Park, MD 20742-4111, USA

The Einstein equation is derived from the
proportionality of entropy and horizon area
together with the fundamental relation
connecting heat, entropy, and temperature.
The key idea is to demand that this relation hold for all the
local Rindler causal horizons through each spacetime point, with
and *T*
interpreted as the energy flux and Unruh temperature
seen by an accelerated observer just inside the horizon.
This requires that gravitational lensing by matter energy
distorts the causal structure of spacetime in just such a
way that the Einstein equation holds.
Viewed in this way, the Einstein equation is an equation of state.
This perspective suggests that it may be
no more appropriate to canonically quantize the Einstein equation than
it would be to quantize the wave equation for sound in air.

The four laws of black hole mechanics, which are analogous to those
of thermodynamics, were originally
derived from the classical Einstein equation[1].
With the discovery of the quantum Hawking radiation[2],
it became clear that the analogy is in fact an identity.
How did classical General Relativity know that horizon area
would turn out to be a form of entropy, and that surface gravity
is a temperature? In this letter I
will answer that question by turning the logic around
and deriving the Einstein equation
from the proportionality of entropy and horizon
area together with the
fundamental relation
connecting heat *Q*
, entropy *S*
, and temperature *T*
.
Viewed in this way,
the Einstein equation is an equation of state. It is
born in the thermodynamic limit as a relation between thermodynamic
variables, and its validity is seen to depend on the existence of
local equilibrium conditions.
This perspective suggests that it may be no more
appropriate to quantize the Einstein equation than
it would be to quantize the wave equation for sound
in air.

The basic idea can be illustrated by thermodynamics
of a simple homogeneous system. If one knows the entropy
*S*(*E*,*V*)
as a function of energy and volume,
one can deduce the equation of state from
. The first law of thermodynamics
yields
, while differentiation yields
the identity
.
One thus infers that
and
that
. The latter equation is the
equation of state, and yields useful information if the
function *S*
is known. For example, for weakly interacting
molecules at low density, a simple counting argument yields
for some
function *f*(*E*)
. In this case,
,
so one infers that
, which is the equation of state
of an ideal gas.

In thermodynamics, heat is energy that flows between degrees of freedom that are not macroscopically observable. In spacetime dynamics, we shall define heat as energy that flows across a causal horizon. It can be felt via the gravitational field it generates, but its particular form or nature is unobservable from outside the horizon. For the purposes of this definition it is not necessary that the horizon be a black hole event horizon. It can be simply the boundary of the past of any set (for ``observer"). This sort of horizon is a null hypersurface (not necessarily smooth) and, assuming cosmic censorship, it is composed of generators which are null geodesic segments emanating backwards in time from the set . We can consider a kind of local gravitational thermodynamics associated with such causal horizons, where the ``system" is the degrees of freedom beyond the horizon. The outside world is separated from the system not by a diathermic wall, but by a causality barrier.

That causal horizons should be associated with entropy is suggested
by the observation that they hide information[3].
In fact, the overwhelming majority of the
information that is hidden resides
in correlations between vacuum fluctuations just inside and outside
of the horizon[4].
Because of the infinite number of short wavelength field degrees of
freedom near the horizon, the associated ``entanglement entropy" is divergent
in continuum quantum field theory. If, on the other hand, there is a
fundamental cutoff length *l*_{c}
, then the entanglement entropy
is finite and proportional to the horizon area in units of *l*_{c}^{2}
,
as long as the radius
of curvature of spacetime is much longer than *l*_{c}
.
(Subleading dependence on curvature
and other fields induces subleading terms in the gravitational
field equation.)
We shall thus assume for most of this letter
that the entropy is proportional to horizon area.
Note that the area is an extensive quantity for a horizon,
as one expects for entropy[5].

As we will see, consistency with thermodynamics requires that
*l*_{c}
must be of order the Planck length (10^{-33}
cm). Even
at the horizon of a stellar mass black hole, the radius of curvature
is 10^{38}
times this cutoff scale. Only near the big bang or
a black hole singularity, or in the final stages of evaporation
of a primordial black hole, would such a vast separation of scales
fail to exist. Our analysis relies heavily on this circumstance.

So far we have argued that energy flux across a causal horizon is a kind of heat flow, and that entropy of the system beyond is proportional to the area of that horizon. It remains to identify the temperature of the system into which the heat is flowing. Recall that the origin of the large entropy is the vacuum fluctuations of quantum fields. According to the Unruh effect[8], those same vacuum fluctuations have a thermal character when seen from the perspective of a uniformly accelerated observer. We shall thus take the temperature of the system to be the Unruh temperature associated with such an observer hovering just inside the horizon. For consistency, the same observer should be used to measure the energy flux that defines the heat flow. Different accelerated observers will obtain different results. In the limit that the accelerated worldline approaches the horizon the acceleration diverges, so the Unruh temperature and energy flux diverge, however their ratio approaches a finite limit. It is in this limit that we analyse the thermodynamics, in order to make the arguments as local as possible.

Up to this point we have been thinking of the system as
defined by any causal horizon. However, in general, such
a system is not in ``equilibrium" because the horizon
is expanding, contracting, or shearing. Since we wish to
apply equilibrium thermodynamics, the system is further
specified as follows. The equivalence principle is invoked to
view a small neighborhood of each spacetime point *p*
as a piece of flat spacetime. Through *p*
we consider a small
spacelike 2-surface element
whose past directed
null normal congruence to one side (which we call the ``inside")
has vanishing expansion and shear at *p*
. It is always
possible to choose
through *p*
so that the expansion and
shear vanish in a first order neighborhood of *p*
. We call
the past horizon of such a *P*
the ``local Rindler horizon of
", and we think of it as defining a system--the part of
spacetime beyond the Rindler horizon--that is instantaneously
stationary (in ``local equilibrium") at *p*
.
Through any spacetime point there are local Rindler horizons in
all null directions.

The fundamental principle at play in our analysis is this: The equilibrium thermodynamic relation , as interpreted here in terms of energy flux and area of local Rindler horizons, can only be satisfied if gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. We turn now to a demonstration of this claim.

First, to sharpen the the above definitions of temperature and heat,
note that in a small neighborhood of any spacelike 2-surface
element
one has an approximately flat
region of spacetime with the usual Poincaré symmetries.
In particular, there is an approximate Killing field
generating boosts orthogonal to
and vanishing at
.
According to the Unruh effect[8], the Minkowski
vacuum state of quantum fields--or any state at very short distances--
is a thermal state with respect to the boost hamiltonian
at temperature
, where
is the acceleration of the Killing orbit on which
the norm of
is unity (and we employ units with the
speed of light equal to unity.) The heat flow is to be defined by the
boost-energy current of the matter,
, where
*T*_{ab}
is the matter energy-momentum tensor.
Since the temperature and heat flow scale the same way under
a constant rescaling of
, this scale ambiguity will
not prevent us from inferring the equation of state.

Consider now any local Rindler horizon through a spacetime point *p*
.
(See Fig. 1.)

Let be an approximate local boost Killing field generating this horizon, with the direction of chosen to be future pointing to the ``inside" past of . We assume that all the heat flow across the horizon is (boost) energy carried by matter. This heat flux to the past of is given by

1 |

(In keeping with the thermodynamic limit, we assume the quantum fluctuations in

Assume now that the entropy is proportional to the horizon
area, so the entropy variation associated with a piece of the
horizon satisfies
, where
is
the area variation of a cross section of a pencil of generators
of
. The dimensional constant
is
undetermined by anything we have said so far
(although given a microscopic theory of spacetime structure
one may someday be able to compute
in terms of a
fundamental length scale.)
The area variation is given by

3 |

where is the expansion of the horizon generators.

The content of is essentially to require that the presence of the energy flux is associated with a focussing of the horizon generators. At the local Rindler horizon has vanishing expansion, so the focussing to the past of must bring an expansion to zero at just the right rate so that the area increase of a portion of the horizon will be proportional to the energy flux across it. This requirement imposes a condition on the curvature of spacetime as follows.

The equation of geodesic deviation applied to the null geodesic
congruence generating the horizon yields the Raychaudhuri equation

4 |

where is the square of the shear and

With the help of (2) and (5) we can now see that
can only be valid if
for all null *k*^{a}
, which implies that
for some function *f*
.
Local conservation of energy and momentum implies that
*T*_{ab}
is divergence free and therefore, using the contracted
Bianchi identity, that
for some
constant
. We thus deduce that the Einstein equation
holds:

6 |

The constant of proportionality between the entropy and the area determines Newton's constant as , which identifies the length as twice the Planck length . The undetermined cosmological constant remains as enigmatic as ever.

Changing the assumed entropy functional would change the implied gravitational field equations. For instance, if the entropy density is given by a polynomial in the Ricci scalar , then will imply field equations arising from a Lagrangian polynomial in the Ricci scalar[9]. It is an interesting question what ``integrability" conditions must an entropy density satisfy in order for to hold for all local Rindler horizons. It seems likely that the requirement is that the entropy density arises from the variation of a generally covariant action just as it does for black hole entropy. Then the implied field equations will be those arising from that same action.

Our thermodynamic derivation of the Einstein equation of state presumed the existence of local equilibrium conditions in that the relation only applies to variations between nearby states of local thermodynamic equilibrium. For instance, in free expansion of a gas, entropy increase is not associated with any heat flow, and this relation is not valid. Moreover, local temperature and entropy are not even well defined away from equilibrium. In the case of gravity, we chose our systems to be defined by local Rindler horizons, which are instantaneously stationary, in order to have systems in local equilibrium. At a deeper level, we also assumed the usual form of short distance vacuum fluctuations in quantum fields when we motivated the proportionality of entropy and horizon area and the use of the Unruh acceleration temperature. Viewing the usual vacuum as a zero temperature thermal state[11], this also amounts to a sort of local equilibrium assumption. This deeper assumption is probably valid only in some extremely good approximation. We speculate that out of equilibrium vacuum fluctuations would entail an ill-defined spacetime metric.

Given local equilibrium conditions, we have in the Einstein equation a system of local partial differential equations that is time reversal invariant and whose solutions include propagating waves. One might think of these as analogous to sound in a gas propagating as an adiabatic compression wave. Such a wave is a travelling disturbance of local density, which propagates via a myriad of incoherent collisions. Since the sound field is only a statistically defined observable on the fundamental phase space of the multiparticle system, it should not be canonically quantized as if it were a fundamental field, even though there is no question that the individual molecules are quantum mechanical. By analogy, the viewpoint developed here suggests that it may not be correct to canonically quantize the Einstein equations, even if they describe a phenomenon that is ultimately quantum mechanical.

For sufficiently high sound frequency or intensity one knows that the local equilibrium condition breaks down, entropy increases, and sound no longer propagates in a time reversal invariant manner. Similarly, one might expect that sufficiently high frequency or large amplitude disturbances of the gravitational field would no longer be described by the Einstein equation, not because some quantum operator nature of the metric would become relevant, but because the local equilibrium condition would fail. It is my hope that, by following this line of inquiry, we shall eventually reach an understanding of the nature of ``non-equilibrium spacetime".

I am grateful to S. Corley, J.C. Dell, R.C. Myers, J.Z. Simon, and L. Smolin for helpful comments on the presentation in earlier drafts of this letter. This work was supported in part by NSF grant PHY94-13253.