Using the property , it is easy to see that the kernel is symmetric and is antisymmetric in its arguments; that is, and .

The physical meanings of these kernels can be extracted if we write the renormalized CTP effective action at finite temperature (169) in an influence functional form [32, 111, 161, 162]. , the imaginary part of the CTP effective action can be identified with the noise kernel and , the antisymmetric piece of the real part with the dissipation kernel. Campos and Hu [54, 55] have shown that these kernels identified as such indeed satisfy a thermal fluctuation-dissipation relation.

If we denote the difference and the sum of the perturbations defined along each branch of the complex time path of integration by and , respectively, the influence functional form of the thermal CTP effective action may be written to second order in as

The first line is the Einstein–Hilbert action to second order in the perturbation . is a symmetric kernel (i.e., = ). In the near flat case its Fourier transform is given by The 14 elements of the tensor basis , , are defined in [249, 250]. The second is a local term linear in . Only far away from the hole it takes the form of the stress tensor of massless scalar particles at temperature , which has the form of a perfect fluid stress-energy tensor, where is the four-velocity of the plasma and the factor is the familiar thermal energy density for massless scalar particles at temperature . In the far field limit, taking into account the four-velocity of the fluid, a manifestly Lorentz-covariant approach to thermal field theory may be used [292]. However, in order to simplify the involved tensorial structure, we work in the co-moving coordinate system of the fluid where . In the third line, the Fourier transform of the symmetric kernel can be expressed as where is a simple redefinition of the renormalization parameter given by , and the tensors and are defined by respectively.In the above and subsequent equations, we denote the coupling parameter in four dimensions by , and consequently means evaluated at . is the complete contribution of a free massless quantum scalar field to the thermal graviton polarization tensor [249, 250, 72, 27], and it is responsible for the instabilities found in flat spacetime at finite temperature [116, 249, 250, 72, 27]. Note that the addition of the contribution of other kinds of matter fields to the effective action, even graviton contributions, does not change the tensor structure of these kernels, and only the overall factors are different to leading order [249, 250]. Equation (177) reflects the fact that the kernel has thermal as well as non-thermal contributions. Note that it reduces to the first term in the zero temperature limit (),

and at high temperatures the leading term () may be written as where we have introduced the dimensionless external momentum . The coefficients were first given in [249, 250] and generalized to the next-to-leading order in [72, 27]. (They are given with the MTW sign convention in [54, 55].)Finally, as defined above, is the noise kernel representing the random fluctuations of the thermal radiance and is the dissipation kernel, describing the dissipation of energy of the gravitational field.

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