Explicit form of the Einstein–Langevin equation

4.3 Explicit form of the Einstein–Langevin equation

We can write the Einstein–Langevin equation in a more explicit form by working out the expansion of $ˆab ⟨T [g + h]⟩$ up to linear order in the perturbation $hab$ . From Equation (19

), we see that this expansion can be easily obtained from Equation (23

). The result is

Here we use a subscript $n$ on a given tensor to indicate that we are explicitly working in $n$ dimensions, as we use dimensional regularization, and we also use the superindex $(1)$ to generally indicate that the tensor is the first order correction, linear in $hab$ , in a perturbative expansion around the background $gab$ .

Using the Klein–Gordon equation (2 ), and expressions (3 ) for the stress-energy tensor and the corresponding operator, we can write

where $ab ℱ [g;h]$ is the differential operator

( ) ( ) ℱ ab ≡ ξ − 1- hab − 1gabhc □ 4 2 c ξ [ + -- ∇c∇ahbc + ∇c ∇bhac − □hab − ∇a ∇bhcc − gab∇c ∇dhcd + gab□hcc 2 ( a b b a ab ab d ab d) c ab c d] + ∇ hc + ∇ hc − ∇ch − 2g ∇ hcd + g ∇ch d ∇ − g hcd∇ ∇ . (33 )

It is understood that indices are raised with the background inverse metric $gab$ , and that all the covariant derivatives are associated to the metric $gab$ .

Substituting Equation (31 ) into the $n$ -dimensional version of the Einstein–Langevin Equation (14 ), taking into account that $gab$ satisfies the semiclassical Einstein equation (7 ), and substituting expression (32 ), we can write the Einstein–Langevin equation in dimensional regularization as

[ ( ) ] --1--- (1)ab 1- ab cd ac b bc a ab 1-ab c 8πGB G − 2 g G hcd + G hc + G hc + ΛB h − 2g hc ( ) − 4αB- D (1)ab − 1gabDcdhcd + Dachb + Dbcha 3 2 c c ( 1 ) − 2βB B(1)ab − -gabBcdhcd + Bachbc + Bbchac 2 ∫ −(n−4) ab ˆ2 1- n ∘ ------ −(n−4) abcd − μ ℱx ⟨φ n[g;x )⟩ + 2 d y − g(y )μ H n [g;x,y)hcd(y) − (n− 4) ab = μ ξn , (34 )

where the tensors $Gab$ , $Dab$ , and $Bab$ are computed from the semiclassical metric $gab$ , and where we have omitted the functional dependence on $gab$ and $hab$ in $(1)ab G$ , $(1)ab D$ , $(1)ab B$ , and $ℱ ab$ to simplify the notation. The parameter $μ$ is a mass scale which relates the dimensions of the physical field $φ$ with the dimensions of the corresponding field in $n$ dimensions, $(n−4)∕2 φn = μ φ$ . Notice that, in Equation (34

), all the ultraviolet divergences in the limit $n → 4$ , which must be removed by renormalization of the coupling constants, are in $⟨ˆφ2n(x)⟩$ and the symmetric part $HabScnd(x,y )$ of the kernel $Hanbcd(x,y)$ , whereas the kernels $N nabcd(x,y)$ and $Habcd(x, y) An$ are free of ultraviolet divergences. If we introduce the bi-tensor $F abcd[g;x,y) n$ defined by

where $ˆab t$ is defined by Equation (12

), then the kernels $N$ and $HA$ can be written as

where we have used that

ab cd ab cd ab cd 2⟨ˆt (x )ˆt (y)⟩ = ⟨{ˆt (x),ˆt (y)}⟩ + ⟨[ˆt (x),ˆt (y)]⟩,

and the fact that the first term on the right-hand side of this identity is real, whereas the second one is pure imaginary. Once we perform the renormalization procedure in Equation (34 ), setting $n = 4$ will yield the physical Einstein–Langevin equation. Due to the presence of the kernel $abcd H n (x, y)$ , this equation will be usually non-local in the metric perturbation. In Section 6 we will carry out an explicit evaluation of the physical Einstein–Langevin equation which will illustrate the procedure.

4.3.1 The kernels for the vacuum state

When the expectation values in the Einstein–Langevin equation are taken in a vacuum state $|0⟩$ , such as, for instance, an “in” vacuum, we can be more explicit, since we can write the expectation values in terms of the Wightman and Feynman functions, defined as

These expressions for the kernels in the Einstein–Langevin equation will be very useful for explicit calculations. To simplify the notation, we omit the functional dependence on the semiclassical metric $gab$ , which will be understood in all the expressions below.

From Equations (36 ), we see that the kernels $abcd Nn (x,y)$ and $abcd H An (x, y)$ are the real and imaginary parts, respectively, of the bi-tensor $Fnabcd(x,y)$ . From the expression (4 ) we see that the stress-energy operator $Tˆab n$ can be written as a sum of terms of the form ${ } 𝒜 ˆφ (x ), ℬ φˆ (x) x n x n$ , where $𝒜 x$ and $ℬ x$ are some differential operators. It then follows that we can express the bi-tensor $F anbcd(x,y)$ in terms of the Wightman function as

F abcd(x,y) = ∇a ∇c G+ (x,y )∇b ∇d G+ (x,y ) + ∇a ∇d G+ (x,y)∇b ∇c G+ (x,y ) n x y (n x y n ) x y n( x y n ) +2 𝒟abx ∇cyG+n(x,y)∇dyG+n(x,y ) + 2𝒟cdy ∇axG+n(x,y )∇bxG+n(x, y) ab cd( +2 ) +2 𝒟 x 𝒟y Gn (x, y) , (38 )

where $𝒟ab x$ is the differential operator (5

). From this expression and the relations (36

), we get expressions for the kernels $Nn$ and $HAn$ in terms of the Wightman function $+ G n (x, y)$ .

Similarly the kernel $abcd HSn (x,y)$ can be written in terms of the Feynman function as

[ HaSbncd(x,y) = − Im ∇ax∇cyGFn (x,y)∇bx∇dyGFn (x,y) + ∇ax∇dyGFn(x, y)∇bx∇cyGFn (x,y ) ab e c x d − g (x)∇ x∇ yGFn(x, y)∇e∇ yGFn (x,y ) − gcd(y)∇a ∇eG (x,y )∇b ∇y G (x,y) x y Fn x e Fn + 1gab(x)gcd(y)∇e ∇f G (x,y)∇x ∇y G (x,y) 2 x y Fn e f Fn + 𝒦ab (2∇c G (x, y)∇d G (x, y) − gcd(y)∇e G (x,y)∇y G (x,y)) x ( y Fn y Fn y Fn e Fn ) + 𝒦cdy 2∇axGFn (x,y)∇bxGFn (x, y) − gab(x )∇exGFn (x,y)∇xeGFn (x,y) ab cd( 2 )] +2 𝒦 x 𝒦 y G Fn(x,y) , (39 )

where $ab 𝒦x$ is the differential operator

Note that, in the vacuum state $|0⟩$ , the term $ˆ2 ⟨φn(x)⟩$ in Equation (34

) can also be written as $⟨ˆφ2n (x )⟩ = iGFn (x,x) = G+n(x, x)$ .

Finally, the causality of the Einstein–Langevin equation (34 ) can be explicitly seen as follows. The non-local terms in that equation are due to the kernel $H (x, y)$ which is defined in Equation (22 ) as the sum of $H (x,y ) S$ and $H (x,y) A$ . Now, when the points $x$ and $y$ are spacelike separated, $ˆφ (x ) n$ and $ˆ φn (y)$ commute and, thus, $+ 1 ˆ ˆ Gn (x,y) = iGFn(x, y) = 2⟨0|{φn(x),φn (y )} |0⟩$ , which is real. Hence, from the above expressions, we have that $HabAcnd(x,y ) = HabScnd (x,y) = 0$ , and thus $Habncd(x,y) = 0$ . This fact is expected since, from the causality of the expectation value of the stress-energy operator [282 ], we know that the non-local dependence on the metric perturbation in the Einstein–Langevin equation, see Equation (14 ), must be causal. See [169 ] for an alternative proof of the causal nature of the Einstein–Langevin equation.