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6.1 Perturbations around Minkowski spacetime

The Minkowski metric jab, 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 (7View Equation), the so-called trivial solutions of semiclassical gravity [91Jump To The Next Citation Point]. They constitute the ground state of semiclassical gravity. In fact, we can always choose a renormalization scheme in which the renormalized expectation value ^ab <0 |TR [j]| 0> = 0. Thus, Minkowski spacetime 4 (IR ,jab) and the vacuum state |0> are a solution to the semiclassical Einstein equation with renormalized cosmological constant /\ = 0. The fact that the vacuum expectation value of the renormalized stress-energy operator in Minkowski spacetime should vanish was originally proposed by Wald [283], and it may be understood as a renormalization convention [100113]. 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 [10773104].

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 ^TRab[j]. 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 (37View Equation), their values in the two-point coincidence limit, and the products of derivatives of two of such functions appearing in expressions (38View Equation) and (39View Equation) 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 m {x } and in the associated basis, in which the components of the flat metric are simply jmn = diag (-1, 1,...,1). In Minkowski spacetime, the components of the classical stress-energy tensor (3View Equation) reduce to

1 1 T mn[j, f] = @mf@nf - -jmn@rf@rf - --jmnm2f2 + q(jmn[] - @m@n) f2, (78) 2 2
where [] =_ @m@m, and the formal expression for the components of the corresponding “operator” in dimensional regularization, see Equation (4View Equation), is
^mn 1-{ m^ n ^ } mn^2 Tn [j] = 2 @ fn,@ fn + D fn, (79)
where mn D is the differential operator (5View Equation), with gmn = jmn, Rmn = 0, and \~/ m = @m. The field ^ fn(x) is the field operator in the Heisenberg representation in an n-dimensional Minkowski spacetime, which satisfies the Klein-Gordon equation (2View Equation). We use here a stress-energy tensor which differs from the canonical one, which corresponds to q = 0; both tensors, however, define the same total momentum.

The Wightman and Feynman functions (37View Equation) for gmn = jmn are well known:

G+n(x,y) = iD+n(x - y), GFn(x, y) = DFn(x - y), (80)
with
integral dnk D+n(x) = - 2pi ----n-eikx d(k2 + m2) h(k0), integral (2p) (81) dnk eikx + DFn(x) = - (2p)n--k2 +-m2---ie- for e --&gt; 0 ,
where k2 =_ jmnkmkn and kx =_ jmnkmxn. Note that the derivatives of these functions satisfy @xmD+n (x- y) = @mD+n(x - y) and @ymD+n(x - y) = - @mD+n (x - y), and similarly for the Feynman propagator D (x- y) Fn.

To write down the semiclassical Einstein equation (7View Equation) in n dimensions for this case, we need to compute the vacuum expectation value of the stress-energy operator components (79View Equation). Since, from (80View Equation), we have that <0|^f2n(x)|0> = iDFn(0) = iD+n (0), which is a constant (independent of x), we have simply

&lt; | | &gt; integral dnk kmkn jmn(m2 )n/2 ( n ) 0||T^mnn[j]||0 = - i ----n--2-----2-----= ---- --- G - -- , (82) (2p) k + m - ie 2 4p 2
where the integrals in dimensional regularization have been computed in the standard way [209Jump To The Next Citation Point], and where G(z) is Euler’s gamma function. The semiclassical Einstein equation (7View Equation) in n dimensions before renormalization reduces now to
-/\B--- mn -(n-4)&lt; ||^mn || &gt; 8pG j = m 0 |T n [j]| 0 . (83) B
This equation, thus, simply sets the value of the bare coupling constant /\B/GB. Note, from Equation (82View Equation), that in order to have <0|^TmRn |0>[j] = 0, the renormalized and regularized stress-energy tensor “operator” for a scalar field in Minkowski spacetime, see Equation (6View Equation), has to be defined as
( )n-4- ^mn -(n-4) ^mn jmn--m4--- -m2-- 2 ( n) TR [j] = m T n [j] - 2 (4p)2 4pm2 G - 2 , (84)
which corresponds to a renormalization of the cosmological constant
/\B-- /\- 2----m4---- GB = G - p n(n - 2) kn + O(n - 4), (85)
where
( g 2)n-4- ( g 2 ) k =_ --1--- e-m-- 2 = --1---+ 1-ln e-m-- + O(n - 4), (86) n n - 4 4pm2 n - 4 2 4pm2
with g being Euler’s constant. In the case of a massless scalar field, 2 m = 0, one simply has /\B/GB = /\/G. Introducing this renormalized coupling constant into Equation (83View Equation), we can take the limit n --> 4. We find that, for (IR4,jab,|0>) to satisfy the semiclassical Einstein equation, we must take /\ = 0.

We can now write down the Einstein-Langevin equations for the components hmn of the stochastic metric perturbation in dimensional regularization. In our case, using <0|^f2n(x)|0> = iDFn(0) and the explicit expression of Equation (34View Equation), we obtain

[ ( )] --1--- (1)mn mn 1- mn 4- (1)mn (1)mn 8pGB G + /\B h - 2 j h (x) - 3 aBD (x)- 2bBB (x) integral - qG(1)mn(x)m -(n-4)iDF (0) + 1- dny m- (n- 4)Hmnab (x,y)hab(y) = qmn(x). (87) n 2 n
The indices in h mn are raised with the Minkowski metric, and h =_ hr r; 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 mn q is written as
&lt; &gt; qmn(x)qab(y) s = m -2(n- 4)N mnnab(x,y). (88)

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

(1)mn 1- mnab (1)mn mn ab D (x) = 2 Fx hab(x), B (x) = 2F x F x hab(x), (89)
with the differential operators mn mn m n F x =_ j []x - @x@x and mnab m(a b)n mn ab F x =_ 3F x Fx - F x F x.

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