Perturbations around Minkowski spacetime

6.1 Perturbations around Minkowski spacetime

The Minkowski metric $ηab$ , in a manifold $ℳ$ which is topologically $4 IR$ , and the usual Minkowski vacuum, denoted as $|0⟩$ , are the class of simplest solutions to the semiclassical Einstein equation (7

), the so-called trivial solutions of semiclassical gravity [91

]. They constitute the ground state of semiclassical gravity. In fact, we can always choose a renormalization scheme in which the renormalized expectation value $⟨0 | ˆTab[η]|0⟩ = 0 R$ . Thus, Minkowski spacetime $4 (IR ,ηab)$ and the vacuum state $|0⟩$ are a solution to the semiclassical Einstein equation with renormalized cosmological constant $Λ = 0$ . The fact that the vacuum expectation value of the renormalized stress-energy operator in Minkowski spacetime should vanish was originally proposed by Wald [282 ], and it may be understood as a renormalization convention [100 , 113 ]. Note that other possible solutions of semiclassical gravity with zero vacuum expectation value of the stress-energy tensor are the exact gravitational plane waves, since they are known to be vacuum solutions of Einstein equations which induce neither particle creation nor vacuum polarization [107 , 73 , 104 ].

As we have already mentioned the vacuum $|0⟩$ is an eigenstate of the total four-momentum operator in Minkowski spacetime, but not an eigenstate of $ˆR Tab[η]$ . Hence, even in the Minkowski background, there are quantum fluctuations in the stress-energy tensor and, as a result, the noise kernel does not vanish. This fact leads to consider the stochastic corrections to this class of trivial solutions of semiclassical gravity. Since, in this case, the Wightman and Feynman functions (37 ), their values in the two-point coincidence limit, and the products of derivatives of two of such functions appearing in expressions (38 ) and (39 ) are known in dimensional regularization, we can compute the Einstein–Langevin equation using the methods outlined in Sections 3 and 4.

To perform explicit calculations it is convenient to work in a global inertial coordinate system ${xμ}$ and in the associated basis, in which the components of the flat metric are simply $ημν = diag (− 1,1, ...,1)$ . In Minkowski spacetime, the components of the classical stress-energy tensor (3 ) reduce to

where $□ ≡ ∂μ∂μ$ , and the formal expression for the components of the corresponding “operator” in dimensional regularization, see Equation (4

), is

where $μν 𝒟$ is the differential operator (5

), with $gμν = ημν$ , $R μν = 0$ , and $∇ μ = ∂μ$ . The field $φˆn (x)$ is the field operator in the Heisenberg representation in an $n$ -dimensional Minkowski spacetime, which satisfies the Klein–Gordon equation (2

). We use here a stress-energy tensor which differs from the canonical one, which corresponds to $ξ = 0$ ; both tensors, however, define the same total momentum.

The Wightman and Feynman functions (37 ) for $gμν = ημν$ are well known:

with

∫ dnk Δ+n(x ) = − 2πi ----n-eikx δ(k2 + m2 )θ (k0 ), ∫ (2π) (81 ) dnk eikx + ΔFn (x ) = − (2-π)n-k2 +-m2-−-iε- for ε → 0 ,

where $k2 ≡ ημνkμk ν$ and $kx ≡ ημνkμxν$ . Note that the derivatives of these functions satisfy $∂xμ Δ+n (x − y) = ∂μΔ+n(x − y)$ and $∂yμΔ+n(x − y) = − ∂μΔ+n(x − y)$ , and similarly for the Feynman propagator $Δ (x − y) Fn$ .

To write down the semiclassical Einstein equation (7 ) in $n$ dimensions for this case, we need to compute the vacuum expectation value of the stress-energy operator components (79 ). Since, from (80 ), we have that $⟨0|ˆφ2n(x)|0⟩ = iΔFn (0) = iΔ+n (0 )$ , which is a constant (independent of $x$ ), we have simply

where the integrals in dimensional regularization have been computed in the standard way [209

], and where $Γ (z)$ is Euler’s gamma function. The semiclassical Einstein equation (7

) in $n$ dimensions before renormalization reduces now to

This equation, thus, simply sets the value of the bare coupling constant $ΛB∕GB$ . Note, from Equation (82

), that in order to have $⟨0| ˆTμRν |0 ⟩[η] = 0$ , the renormalized and regularized stress-energy tensor “operator” for a scalar field in Minkowski spacetime, see Equation (6

), has to be defined as

which corresponds to a renormalization of the cosmological constant

where

with $γ$ being Euler’s constant. In the case of a massless scalar field, $2 m = 0$ , one simply has $ΛB ∕GB = Λ ∕G$ . Introducing this renormalized coupling constant into Equation (83

), we can take the limit $n → 4$ . We find that, for $(IR4, ηab,|0⟩)$ to satisfy the semiclassical Einstein equation, we must take $Λ = 0$ .

We can now write down the Einstein–Langevin equations for the components $hμν$ of the stochastic metric perturbation in dimensional regularization. In our case, using $⟨0|ˆφ2n(x)|0⟩ = iΔFn (0)$ and the explicit expression of Equation (34 ), we obtain

[ ] 1 ( 1 ) 4 ------ G (1)μν + ΛB h μν − --ημνh (x) − -αBD (1)μν(x) − 2βBB (1)μν(x) 8πGB 2 3 (1)μν −(n−4) 1 ∫ n −(n− 4) μναβ μν − ξG (x )μ iΔFn (0) + -- d yμ H n (x,y )h αβ(y) = ξ (x). (87 ) 2

The indices in $hμν$ are raised with the Minkowski metric, and $h ≡ h ρρ$ ; here a superindex $(1)$ denotes the components of a tensor linearized around the flat metric. Note that in $n$ dimensions the two-point correlation functions for the field $ξμν$ is written as

Explicit expressions for $(1)μν D$ and $(1)μν B$ are given by

with the differential operators $ℱ μxν ≡ ημν□x − ∂μx∂xν$ and $μ(α β)ν ℱ μxναβ≡ 3ℱ x ℱx − ℱxμνℱxαβ$ .