The Einstein–Langevin equation

8.5 The Einstein–Langevin equation

In this section we show how a semiclassical Einstein–Langevin equation can be derived from the previous thermal CTP effective action. This equation depicts the stochastic evolution of the perturbations of the black hole under the influence of the fluctuations of the thermal scalar field.

The influence functional $ℱIF ≡ exp(iSIF)$ previously introduced in Equation (16 ) can be written in terms of the the CTP effective action $β Seff[h ±μν]$ derived in Equation (174 ) using Equation (17 ). The Einstein–Langevin equation follows from taking the functional derivative of the stochastic effective action (27 ) with respect to $[hμν](x)$ and imposing $[hμν](x ) = 0$ . This leads to

1 ∫ 4 ′ μν,αβ ′ ′ 1 μν ------- d x L (o) (x − x )hαβ(x ) + -T (β) 16π∫GN 2 + d4x′ (H μν,αβ(x − x ′) − Dμν,αβ(x − x′)) h (x′) + ξμν(x ) = 0, (183 ) αβ

where

In the far field limit this equation should reduce to that obtained by Campos and Hu [54

, 55

]: For gravitational perturbations $h μν$ defined in Equation (165

) under the harmonic gauge $¯hμν,ν = 0$ , their Einstein–Langevin equation is given by

1 { μν ∫ ( ) } □ ¯hμν(x ) +------2 T(β) + 2P ρσ,αβ d4x′ H μν,αβ(x− x ′) − Dμν,αβ(x− x′) ¯hρσ(x′) + 2ξμν(x) = 0, 16 πG N (185 )

where the tensor $P ρσ,αβ$ is given by

The expression for $Pρσ,αβ$ in the near horizon limit of course cannot be expressed in such a simple form. Note that this differential stochastic equation includes a non-local term responsible for the dissipation of the gravitational field and a noise source term which accounts for the fluctuations of the quantum field. Note also that this equation in combination with the correlation for the stochastic variable (184

) determines the two-point correlation for the stochastic metric fluctuations $¯ ¯ ′ ⟨hμν(x)hαβ(x )⟩ξ$ self-consistently.

As we have seen before and here, the Einstein–Langevin equation is a dynamical equation governing the dissipative evolution of the gravitational field under the influence of the fluctuations of the quantum field, which, in the case of black holes, takes the form of thermal radiance. From its form we can see that even for the quasi-static case under study the backreaction of Hawking radiation on the black hole spacetime has an innate dynamical nature.

For the far field case, making use of the explicit forms available for the noise and dissipation kernels, Campos and Hu [54 , 55 ] formally proved the existence of a fluctuation-dissipation relation at all temperatures between the quantum fluctuations of the thermal radiance and the dissipation of the gravitational field. They also showed the formal equivalence of this method with linear response theory for lowest order perturbations of a near-equilibrium system, and how the response functions such as the contribution of the quantum scalar field to the thermal graviton polarization tensor can be derived. An important quantity not usually obtained in linear response theory, but of equal importance, manifest in the CTP stochastic approach is the noise term arising from the quantum and statistical fluctuations in the thermal field. The example given in this section shows that the backreaction is intrinsically a dynamic process described (at this level of sophistication) by the Einstein–Langevin equation.