Thermodynamics of Spacetime: The Einstein Equation of State

Thermodynamics of Spacetime: The Einstein Equation of State

6 Jun 1995 | Ted Jacobson
The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation δQ = T dS. The key idea is to demand that this relation hold for all local Rindler causal horizons through each spacetime point, with δQ 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, analogous to those of thermodynamics, were originally derived from the classical Einstein equation. With the discovery of the quantum Hawking radiation, 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, the question is answered by turning the logic around and deriving the Einstein equation from the proportionality of entropy and horizon area together with the fundamental relation δQ = T dS. 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. 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 δQ = T dS. The first law of thermodynamics yields δQ = dE + pdV, while differentiation yields the identity dS = (∂S/∂E)dE + (∂S/∂V)dV. One thus infers that T⁻¹ = ∂S/∂E and that p = T(∂S/∂V). 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 S = ln(# accessible states) ∝ ln V + f(E) for some function f(E). In this case, ∂S/∂V ∝ V⁻¹, so one infers that pV ∝ T, 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,The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation δQ = T dS. The key idea is to demand that this relation hold for all local Rindler causal horizons through each spacetime point, with δQ 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, analogous to those of thermodynamics, were originally derived from the classical Einstein equation. With the discovery of the quantum Hawking radiation, 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, the question is answered by turning the logic around and deriving the Einstein equation from the proportionality of entropy and horizon area together with the fundamental relation δQ = T dS. 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. 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 δQ = T dS. The first law of thermodynamics yields δQ = dE + pdV, while differentiation yields the identity dS = (∂S/∂E)dE + (∂S/∂V)dV. One thus infers that T⁻¹ = ∂S/∂E and that p = T(∂S/∂V). 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 S = ln(# accessible states) ∝ ln V + f(E) for some function f(E). In this case, ∂S/∂V ∝ V⁻¹, so one infers that pV ∝ T, 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,
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[slides and audio] Thermodynamics of spacetime%3A The Einstein equation of state.