CTP effective action for the black hole

8.2 CTP effective action for the black hole

We first derive the CTP effective action for the model described in Sec. (7.1). Using the metric (156

) (and neglecting the surface terms that appear in an integration by parts) we have the action for the scalar field written perturbatively as

where the first and second order perturbative operators $V (1)$ and $V(2)$ are given by

(1) 1 { (∘ ------μν) μν (1)} V ≡ − ∘----(0)- ∂μ − g (0)¯h ∂ν + ¯h ∂μ∂ν + ξ(n)R , − g { ( ) ( ) } (164 ) (2) ---1---- ∘ ---(0)ˆμν ˆμν (2) 1- (1) V ≡ − ∘ −-g(0)- ∂ μ − g h ∂ν + h ∂μ∂ν − ξ(n ) R + 2 hR .

In the above expressions, $R (k)$ is the $k$ -order term in the perturbation $hμν(x )$ of the scalar curvature $R$ , and $¯hμν$ and $ˆh μν$ denote a linear and a quadratic combination of the perturbation, respectively:

¯h ≡ h − 1hg (0), μν μν 2 μν 1 1 1 (165 ) ˆhμν ≡ hμαh αν − -hh μν + -h2g(μ0ν)− --hαβhαβg (0μ)ν. 2 8 4

From quantum field theory in curved spacetime considerations discussed above we take the following action for the gravitational field:

1 ∫ ∘ ------ α ¯μn−4 ∫ ∘ ------ Sg[gμν] = -------n−2- dnx − g(x )R (x) +--------- dnx − g(x) (16πG ){ 2 [ 4(n − 4) ] } ( 1 )2 × 3R μναβ(x)R μναβ (x ) − 1 − 360 ξ(n) − -- R2(x ) . (166 ) 6

The first term is the classical Einstein–Hilbert action, and the second term is the counterterm in four dimensions used to renormalize the divergent effective action. In this action $ℓ2P = 16πGN$ , $α = (2880π2)−1$ , and $¯μ$ is an arbitrary mass scale.

We are interested in computing the CTP effective action (163 ) for the matter action and when the field $φ$ is initially in the Hartle–Hawking vacuum. This is equivalent to saying that the initial state of the field is described by a thermal density matrix at a finite temperature $T = TH$ . The CTP effective action at finite temperature $T ≡ 1∕β$ for this model is given by (for details see [54 , 55 ])

where $±$ denote the forward and backward time path of the CTP formalism, and $¯Gβab[h±μν]$ is the complete $2 × 2$ matrix propagator ( $a$ and $b$ take $±$ values: $G++$ , $G+ −$ , and $G− −$ correspond to the Feynman, Wightman, and Schwinger Green’s functions respectively) with thermal boundary conditions for the differential operator $∘ ------ (1) (2) − g(0)(□ + V + V + ...)$ . The actual form of $β ¯G ab$ cannot be explicitly given. However, it is easy to obtain a perturbative expansion in terms of $Va(bk)$ , the $k$ -order matrix version of the complete differential operator defined by $(k) (k) V±± ≡ ±V ±$ and $(k) V±∓ ≡ 0$ , and $β G ab$ , the thermal matrix propagator for a massless scalar field in Schwarzschild spacetime. To second order $β G¯ab$ reads

Expanding the logarithm and dropping one term independent of the perturbation $± hμν(x)$ , the CTP effective action may be perturbatively written as

S β [h± ] = S [h+ ] − S [h − ] eff μν g μν[ g μν ] i- (1) β (1) β (2) β (2) β + 2tr V+ G ++ − V− G− − + V+ G ++ − V− G− − i [ (1) β (1) β (1) β (1) β (1) β (1) β ] − -tr V+ G ++ V+ G ++ + V − G− − V − G −− − 2V+ G +− V− G − + . (169 ) 4

In computing the traces, some terms containing divergences are canceled using counterterms introduced in the classical gravitational action after dimensional regularization.