11.1 Field equations

We already showed that under the conformal transformation &tidle;gμν = e−2Qκϕgμν the action (10.10View Equation) is transformed to the Einstein frame action:
∫ ∘ ---[ 1 1 ] ∫ SE = d4x − &tidle;g --2R&tidle;− -(∇&tidle;ϕ)2 − V (ϕ) + d4x ℒM (e2Qκϕ&tidle;gμν,ΨM ). (11.1 ) 2κ 2
Recall that in the Einstein frame this gives rise to a constant coupling Q between non-relativistic matter and the field ϕ. We use the unit κ2 = 8 πG = 1, but we restore the gravitational constant G when it is required.

Let us consider a spherically symmetric static metric in the Einstein frame:

d&tidle;s2 = − e2Ψ(r&tidle;)dt2 + e2Φ(&tidle;r)d&tidle;r2 + &tidle;r2(d𝜃2 + sin2𝜃d ϕ2), (11.2 )
where Ψ(&tidle;r) and Φ(&tidle;r) are functions of the distance &tidle;r from the center of symmetry. For the action (11.1View Equation) the energy-momentum tensors for the scalar field ϕ and the matter are given, respectively, by
[ ] &tidle;(ϕ) 1-αβ Tμν = ∂μϕ ∂νϕ − &tidle;gμν 2&tidle;g ∂ αϕ∂βϕ + V (ϕ) , (11.3 ) T&tidle;(M) = − √-2--δℒM--. (11.4 ) μν − &tidle;g δ&tidle;gμν

For the metric (11.2View Equation) the (00) and (11) components for the energy-momentum tensor of the field are

&tidle;0(ϕ) 1-−2Φ ′2 &tidle;1(ϕ) 1- −2Φ ′2 T0 = − 2e ϕ − V (ϕ ), T1 = 2e ϕ − V (ϕ), (11.5 )
where a prime represents a derivative with respect to &tidle;r. The energy-momentum tensor of matter in the Einstein frame is given by μ &tidle;Tν = diag(− &tidle;ρM ,P&tidle;M ,P&tidle;M , &tidle;PM ), which is related to T μν(M ) in the Jordan frame via &tidle;Tνμ(M )= e4Q ϕTμν(M ). Hence it follows that &tidle;ρM = e4QϕρM and P&tidle;M = e4Q ϕPM.

Variation of the action (11.1View Equation) with respect to ϕ gives

( √ --- ) √ --- ∂( −g&tidle;ℒϕ) ∂( − &tidle;gℒ ϕ) ∂ℒM − ∂μ ----------- + -----------+ -----= 0, (11.6 ) ∂(∂μϕ) ∂ϕ ∂ϕ
where ℒϕ = − (&tidle;∇ ϕ)2∕2 − V (ϕ) is the field Lagrangian density. Since the derivative of ℒM in terms of ϕ is given by Eq. (2.41View Equation), i.e., ∂ℒM ∕∂ ϕ = √ −-&tidle;gQ (− &tidle;ρM + 3P&tidle;M ), we obtain the equation of the field ϕ [594Jump To The Next Citation Point42Jump To The Next Citation Point]:
( ) [ ] ϕ ′′ + 2-+ Ψ ′ − Φ ′ ϕ′ = e2Φ V + Q(&tidle;ρ − 3P&tidle; ) , (11.7 ) &tidle;r ,ϕ M M
where a tilde represents a derivative with respect to &tidle;r. From the Einstein equations it follows that
[ ] ′ 1 − e2Φ 1 ′2 2Φ 2Φ Φ = --2r&tidle;---+ 4πG &tidle;r 2ϕ + e V (ϕ) + e &tidle;ρM , (11.8 ) [ ] ′ e2Φ-−-1- 1- ′2 2Φ 2Φ &tidle; Ψ = 2r&tidle; + 4πG &tidle;r 2ϕ − e V (ϕ) + e PM , (11.9 ) ′ ′ [ ] Ψ ′′ + Ψ′2 − Ψ′Φ′ + Ψ--−-Φ--= − 8πG 1-ϕ′2 + e2ΦV (ϕ) − e2Φ &tidle;PM . (11.10 ) &tidle;r 2
Using the continuity equation μ ∇ μT1 = 0 in the Jordan frame, we obtain
P&tidle;M′ + (&tidle;ρM + &tidle;PM )Ψ′ + Q ϕ′(&tidle;ρM − 3P&tidle;M ) = 0. (11.11 )
In the absence of the coupling Q this reduces to the Tolman–Oppenheimer–Volkoff equation, ′ ′ P&tidle;M + (&tidle;ρM + &tidle;PM )Ψ = 0.

If the field potential V (ϕ) is responsible for dark energy, we can neglect both V (ϕ) and ϕ′2 relative to &tidle;ρM in the local region whose density is much larger than the cosmological density (ρ ∼ 10− 29 g∕cm3 0). In this case Eq. (11.8View Equation) is integrated to give

[ ] ∫ 2Φ(&tidle;r) 2Gm (&tidle;r) −1 r&tidle; 2 e = 1 − -------- , m (&tidle;r) = 4π¯r &tidle;ρM d¯r. (11.12 ) &tidle;r 0
Substituting Eqs. (11.8View Equation) and (11.9View Equation) into Eq. (11.7View Equation), it follows that
[ 2Φ ] [ ] ϕ′′ + 1 +-e--− 4πG &tidle;re2Φ(&tidle;ρM − &tidle;PM ) ϕ ′ = e2Φ V,ϕ + Q (&tidle;ρM − 3 &tidle;PM ) . (11.13 ) r&tidle;

The gravitational potential Φ around the surface of a compact object can be estimated as Φ ≈ G &tidle;ρ &tidle;r2 M c, where &tidle;ρ M is the mean density of the star and &tidle;r c is its radius. Provided that Φ ≪ 1, Eq. (11.13View Equation) reduces to Eq. (5.15View Equation) in the Minkowski background (note that the pressure &tidle; PM is also much smaller than the density &tidle;ρM for non-relativistic matter).


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