Nonequilibrium thermodynamics of spacetime: The role of gravitational dissipation

In [T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).] it was shown that the Einstein equation can be derived as a local constitutive equation for an equilibrium spacetime thermodynamics. More recently, in the attempt to extend the same approach to the case of $f(R)$ theories of gravity, it was found that a nonequilibrium setting is indeed required in order to fully describe both this theory as well as classical general relativity (GR) [C. Eling, R. Guedens, and T. Jacobson, Phys. Rev. Lett. 96, 121301 (2006).]. Here, elaborating on this point, we show that the dissipative character leading to nonequilibrium spacetime thermodynamics is actually related—both in GR as well as in $f(R)$ gravity—to nonlocal heat fluxes associated with the purely gravitational/internal degrees of freedom of the theory. In particular, in the case of GR we show that the internal entropy production term is identical to the so-called tidal heating term of Hartle-Hawking. Similarly, for the case of $f(R)$ gravity, we show that dissipative effects can be associated with the generalization of this term plus a scalar contribution whose presence is clearly justified within the scalar-tensor representation of the theory. Finally, we show that the allowed gravitational degrees of freedom can be fixed by the kinematics of the local spacetime causal structure, through the specific equivalence principle formulation. In this sense, the thermodynamical description seems to go beyond Einstein’s theory as an intrinsic property of gravitation.

Figure 1

A schematic view of the thermal Rindler wedge. The local causal horizon $H$ plays the role of a diathermic barrier for the thermal Rindler wedge. The thermal system is constituted by the set of Rindler observers ${\xi }^a$ moving along their isometry trajectories. For each of them the ensemble is described by $\rho =Z^{-1}\exp (-H_b/T)$ and perturbed by the energy $\delta Q$ that flows across $H$.