Near horizon case

8.4 Near horizon case

In this case, since the perturbation is taken around the Schwarzschild spacetime, exact expressions for the corresponding unperturbed propagators $β ± G ab[hμν]$ are not known. Therefore apart from the approximation of computing the CTP effective action to certain order in perturbation theory, an appropriate approximation scheme for the unperturbed Green’s functions is also required. This feature manifested itself in York’s calculation of backreaction as well, where, in writing the $⟨Tμν⟩$ on the right-hand side of the semiclassical Einstein equation in the unperturbed Schwarzschild metric, he had to use an approximate expression for $⟨Tμν⟩$ in the Schwarzschild metric given by Page [230

]. The additional complication here is that while to obtain $⟨Tμν⟩$ as in York’s calculation the knowledge of only the thermal Feynman Green’s function is required; however, to calculate the CTP effective action one needs the knowledge of the full matrix propagator, which involves the Feynman, Schwinger, and Wightman functions.

It is indeed possible to construct the full thermal matrix propagator $β ± G ab[h μν]$ based on Page’s approximate Feynman Green’s function by using identities relating the Feynman Green’s function with the other Green’s functions with different boundary conditions. One can then proceed to explicitly compute a CTP effective action and hence the influence functional based on this approximation. However, we desist from delving into such a calculation for the following reason. Our main interest in performing such a calculation is to identify and analyze the noise term which is the new ingredient in the backreaction. We have mentioned that the noise term gives a stochastic contribution $ξμν$ to the Einstein–Langevin equation (14 ). We had also stated that this term is related to the variance of fluctuations in $T μν$ , i.e, schematically, to $⟨T2 ⟩ μν$ . However, a calculation of $2 ⟨Tμν⟩$ in the Hartle-Hawking state in a Schwarzschild background using the Page approximation was performed by Phillips and Hu [243 , 245 , 244 ], and it was shown that though the approximation is excellent as far as $⟨T μν⟩$ is concerned, it gives unacceptably large errors for $⟨Tμ2ν⟩$ at the horizon. In fact, similar errors will be propagated in the non-local dissipation term as well, because both terms originate from the same source, that is, they come from the last trace term in Equation (169 ) which contains terms quadratic in the Green’s function. However, the Influence Functional or CTP formalism itself does not depend on the nature of the approximation, so we will attempt to exhibit the general structure of the calculation without resorting to a specific form for the Green’s function and conjecture on what is to be expected. A more accurate computation can be performed using this formal structure once a better approximation becomes available.

The general structure of the CTP effective action arising from the calculation of the traces in equation (169 ) remains the same. But to write down explicit expressions for the non-local kernels one requires the input of the explicit form of $G β [h ± ] ab μν$ in the Schwarzschild metric, which is not available in closed form. We can make some general observations about the terms in there. The first line containing $L$ does not have an explicit Fourier representation as given in the far field case, neither will $T μν (β)$ in the second line representing the zeroth order contribution to $⟨Tμν⟩$ have a perfect fluid form. The third and fourth terms containing the remaining quadratic component of the real part of the effective action will not have any simple or even complicated analytic form. The symmetry properties of the kernels $H μν,αβ(x,x′)$ and $Dμν,αβ(x,x′)$ remain intact, i.e., they are even and odd in $x, x′$ , respectively. The last term in the CTP effective action gives the imaginary part of the effective action and the kernel $N (x,x′)$ is symmetric.

Continuing our general observations from this CTP effective action, using the connection between this thermal CTP effective action to the influence functional [272 , 44 ] via an equation in the schematic form (17 ), we see that the nonlocal imaginary term containing the kernel $N μν,αβ(x,x′)$ is responsible for the generation of the stochastic noise term in the Einstein–Langevin equation, and the real non-local term containing kernel $μν,αβ ′ D (x,x )$ is responsible for the non-local dissipation term. To derive the Einstein–Langevin equation we first construct the stochastic effective action (27 ). We then derive the equation of motion, as shown earlier in Equation (29 ), by taking its functional derivative with respect to $[h μν]$ and equating it to zero. With the identification of noise and dissipation kernels, one can write down a linear, non-local relation of the form

where $D (t,t′) = − ∂t′γ(t,t′)$ . This is the general functional form of a fluctuation-dissipation relation, and $K (t,s)$ is called the fluctuation-dissipation kernel [32

, 111

, 161

, 162

]. In the present context this relation depicts the backreaction of thermal Hawking radiance for a black hole in quasi-equilibrium.