Near flat case

8.3 Near flat case

At this point we divide our considerations into two cases. In the far field limit $hμν$ represent perturbations about flat space, i.e., $(0) gμν = ημν$ . The exact “unperturbed” thermal propagators for scalar fields are known, i.e., the Euclidean propagator with periodicity $β$ . Using the Fourier transformed form (those quantities are denoted with a tilde) of the thermal propagators $β G&tidle;ab(k )$ , the trace terms of the form $tr[V(a1)Gβ V (1)G β] mn b rs$ can be written as [54

, 55

]

[ ] tr V (a1)Gβmn V (1)Gβrs = ∫ b ∫ n n ′ a b ′ -dnk---dnq-- ik(x−x′) &tidle; β &tidle;β μν,αβ d x d x hμν(x)h αβ(x ) (2π )n(2π )n e G mn(k + q)G rs(q)T (q,k), (170 )

where the tensor $μν,αβ T (q,k )$ is defined in [54

, 55

] after an expansion in terms of a basis of 14 tensors [249

, 250

]. In particular, the last trace of Equation (169

) may be split in two different kernels $N μν,αβ(x − x′)$ and $D μν,αβ(x − x′)$ ,

One can express the Fourier transforms of these kernels as

∫ 4 &tidle;N μν,αβ(k) = π2 -d-q-{ θ(ko + qo) θ(− qo) + θ(− ko − qo)θ(qo) + n β(|qo|) + nβ(|ko + qo|) (2π)4 +2 n (|qo|)n (|ko + qo|)} δ(q2) δ[(k + q)2] Tμν,αβ (q,k ), (172 ) ∫ 4 β β &tidle; μν,αβ 2 -d-q-- o o o o o o o o o D (k) = − iπ (2π )4 {θ(k + q )θ(− q ) − θ(− k − q )θ (q ) + sig (k + q )nβ(|q |) o o o 2 [ 2] μν,αβ − sig(q )n β(|k + q |)}δ(q )δ (k + q) T (q,k ), (173 )

respectively.

Using the property $μν,αβ μν,αβ T (q,k) = T (− q, − k )$ , it is easy to see that the kernel $μν,αβ ′ N (x − x )$ is symmetric and $D μν,αβ(x − x ′)$ is antisymmetric in its arguments; that is, $Nμν,αβ(x) = N μν,αβ(− x)$ and $D μν,αβ(x) = − D μν,αβ(− x)$ .

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 ]. $N$ , the imaginary part of the CTP effective action can be identified with the noise kernel and $D$ , 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 $h ±μν$ defined along each branch $C ±$ of the complex time path of integration $C$ by $[h μν] ≡ h+μν − h −μν$ and ${hμν} ≡ h+μν + h −μν$ , respectively, the influence functional form of the thermal CTP effective action may be written to second order in $h μν$ as

∫ β [ ± ] ----1----- 4 4 ′ μν,αβ ′ ′ Seff hμν ≃ 2(16πG ) d x d x [h μν](x) L(o) (x − x ){hα β}(x) ∫ N + 1- d4x [h ](x )Tμν 2 μν (β) 1 ∫ 4 4 ′ μν,αβ ′ ′ + -- d x d x [hμν](x)H (x − x ){hαβ}(x ) 2 ∫ − 1- d4x d4x′[h ](x)D μν,αβ(x − x′){h }(x ′) 2 μν αβ i ∫ + -- d4x d4x′[hμν](x)N μν,αβ(x − x′)[h αβ](x′). (174 ) 2

The first line is the Einstein–Hilbert action to second order in the perturbation $h ±μν(x )$ . $L μ(oν,)αβ(x)$ is a symmetric kernel (i.e., $μν,αβ L(o) (x )$ = $μν,αβ L(o) (− x )$ ). In the near flat case its Fourier transform is given by

The 14 elements of the tensor basis $μν,αβ T i (q,k)$ , $i = 1,...,14$ , are defined in [249

, 250

]. The second is a local term linear in $h±μν(x)$ . Only far away from the hole it takes the form of the stress tensor of massless scalar particles at temperature $β −1$ , which has the form of a perfect fluid stress-energy tensor,

where $uμ$ is the four-velocity of the plasma and the factor $2 π30β4-$ is the familiar thermal energy density for massless scalar particles at temperature $β−1$ . In the far field limit, taking into account the four-velocity $μ u$ 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 $uμ = (1, 0,0,0)$ . In the third line, the Fourier transform of the symmetric kernel $Hμν,αβ(x)$ can be expressed as

4{ 2 } H&tidle;μν,αβ(k) = − αk-- 1-ln |k-|Q μν,αβ(k) + 1-¯Qμν,αβ(k) 4 2 μ2 3 2 { } + --π--- − Tμ1ν,αβ(u,k) − 2T μ2ν,αβ(u,k) + T μ4ν,αβ(u,k) + 2T μ5ν,αβ(u,k) 180 β4 ξ { 2 μν,αβ 2 μν,αβ μν,αβ μν,αβ } + ----2 k T 1 (u,k) − 2k T 4 (u,k ) − T 8 (u,k ) + 2T 13 (u, k) 96∫β 4 { [ ] [ ]} -d-q-- 2 o ---1--- 2 o o 1- μν,αβ + π (2 π)4 δ(q )nβ(|q |)𝒫 (k +q )2 + δ[(k+q ) ]n β(|k + q |)𝒫 q2 T (q,k), (177 )

where $μ$ is a simple redefinition of the renormalization parameter $μ¯$ given by $μ ≡ ¯μ exp (2135 + 12 ln4 π − 12γ)$ , and the tensors $Q μν,αβ(k)$ and $Q¯μν,αβ (k )$ are defined by

{ } μν,αβ 3 μν,αβ 1 μν,αβ 2 μν,αβ Q (k) = 2- T 1 (q,k ) − k2T 8 (q,k) + k4T 12 (q,k) [ ( ) ]{ } 1 2 μν,αβ 1 μν,αβ 1 μν,αβ − 1 − 360 ξ − -- T 4 (q,k) + -4-T12 (q,k ) − -2T 13 (q,k) , (178 ) [ 6 k ] k ( 1)2 1 ¯Q μν,αβ(k) = 1 + 576 ξ − -- − 60 (ξ − -)(1 − 36ξ′) 6 6 { } × Tμ4ν,αβ(q,k) + 1-T μ1ν,2αβ(q,k) − -1-Tμ1ν3,αβ(q,k ) , (179 ) k4 k2

respectively.

In the above and subsequent equations, we denote the coupling parameter in four dimensions $ξ(4)$ by $ξ$ , and consequently $ξ′$ means $dξ(n )∕dn$ evaluated at $n = 4$ . $&tidle;H μν,αβ(k)$ 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 $&tidle;Hμν,α β(k )$ has thermal as well as non-thermal contributions. Note that it reduces to the first term in the zero temperature limit ( $β → ∞$ ),

and at high temperatures the leading term ( $β−4$ ) may be written as

where we have introduced the dimensionless external momentum $K μ ≡ kμ∕|⃗k| ≡ (r,kˆ)$ . The $Hi(r)$ coefficients were first given in [249 , 250 ] and generalized to the next-to-leading order $β −2$ in [72 , 27 ]. (They are given with the MTW sign convention in [54

, 55

].)

Finally, as defined above, $μν,αβ N (x)$ is the noise kernel representing the random fluctuations of the thermal radiance and $D μν,αβ(x)$ is the dissipation kernel, describing the dissipation of energy of the gravitational field.