5.3 Mach’s Principle
5.3.3 Universe with an outer edge
giving our universe an edge, we will extend these radial spacelike geodesicsGinto the vacuum exterior and onward to spacelike infinity, use them to carry our angular coordinates out to infinity, and thereby use them to define the angular velocity of our universe’s fluid with respect to infinity.
in whichAR =A(R). Notice that the term in square brackets in Eq. (5.25) is equal to the inner product of the 4-velocity of the fluid element, i.e.,−1. To match this 3-metric with the one computed from the interior, Eq. (5.23), we must letR(τ)≡ar¯}. This amounts to selecting, from a family of radial geodesics, a solution that has the sameτdependence asa¯r}∼τ2/3. This solution can be found, e.g., in Ref. [7] Eq. (25.38) with time reversed:
R(τ)=2M 3τ 4M
!2/3
. (5.26)
EquatingR(τ) (5.26) anda¯r}(5.15), we obtain
M =4π
3 ρo(aor¯})3, (5.27)
in which we have used the relationτ2o =1/(6πρo). [This can be obtained by combing the second equation in (5.14) with Eq. (5.15).] The corresponding expression forT(τ) is
T(τ)=τ+4M R
2M 1/2
+2Mlog
(R/2M)1/2−1 (R/2M)1/2+1
. (5.28)
Note that in matching Eq. (5.25) with Eq. (5.23), we must also identify the angular coordinates asθ=θ,¯ ϕ = ϕ. These angular coordinates match up smoothly at the boundary because of the spherical symmetry¯ throughout the interior universe and the exterior vacuum region.
It remains to check that the extrinsic curvature ofΣis the same as computed using the interior and exterior 4-metrics. In Appendix 5.B, we verify that this is indeed so.
5.3.3.2 Exterior of the rotating universe
When we set our model universe into slow rotation, the exterior spacetime is dragged by its motion, giving rise to a frame dragging angular velocityσ(r) in the metric
ds2+ = −(1−A)dt2+ dr2
1−A+r2dθ2+r2sin2θ(dϕ−σdt)2 (5.29) [cf. Eq. (5.4)]. The vacuum field equation forσ, to first order inσ, is (e.g. Eq. (9) in Ref. [24])
r−4 d dr r4dσ
dr
!
=0, (5.30)
which impliesσ =const./r3. Comparing this with the standard asymptotic form for the metric far from a source (e.g. Eq. (19.5) of Ref. [7]), we see that the constant is 2J, whereJis the spin angular momentum of the slowly rotating universe (as measured via frame dragging by distant observers):
σ=2J
r3 . (5.31)
The frame dragging prevents the azimuthal coordinateϕin the exterior from matching up smoothly atΣ with the interior azimuthal coordinate ¯ϕ, and it also aggravates the coordinate singularity atr=2M(A=0) in the exterior. To achieve a smooth match atΣand remove the singularity atA=0, we use the radial spacelike geodesicsGdescribed at the end of Sec. 5.3.2 to define new exterior azimuthal and time coordinates ˜ϕand t. Specifically, we extend the radial geodesics˜ Gfrom the universe’s interior into the exterior and on outward to radial infinity. Then we carry our angular azimuthal coordinate ¯φfrom the interior into and throughout the exterior, along the geodesicsG(giving it the name ˜φin the exterior); and we carry our new time coordinate ˜t inward from infinity (where we set it equal to the Schwarzschildtcoordinate) toΣ. In Appendix 5.A we carry out this geometric construction, thereby arriving at the following relationship between the Schwarzschild (ϕ,t) and the new coordinates ( ˜ϕ,t):˜
dϕ = dϕ˜ +2J r3
√AR
1−A
√ dr
1−A+AR
, (5.32a)
dt = dt˜+
√AR
1−A
√ dr
1−A+AR
, (5.32b)
whereAR≡A(R). In terms of the new, geometrically defined coordinates, the exterior metric takes the form
ds2+=−(1−A)dt˜2−2 AR
1−A+AR
!1/2
dtdr˜ + dr2
1−A+AR +r2dθ2+r2sin2θ dϕ˜−2J r3dt˜
!2
, (5.33)
In this coordinate system, each spacelike geodesicG, starting orthogonal to the surface of the universe, travels along fixed (˜t, θ,ϕ); just like it travels along fixed (τ,˜ θ,¯ ϕ) inside the universe.¯
5.3.3.3 Matching interior and exterior of rotating universe
For the slowly rotating universe, the hypersurfaceΣswept out by its moving boundary is still at the inte- rior coordinate location ¯r = r¯}(to first order in the rotation) and is described by the same geodesic func- tions, Eq. (5.26) and (5.28). However, since we have adopted (˜t,ϕ) coordinates in the exterior region,˜ T(τ) [Eq. (5.28)] must be transformed into ˜T(τ) using Eq. (5.32b) withr =R(τ). By imposing continuity of the intrinsic 3-metric and extrinsic curvature acrossΣ, we will obtain the unknown functionζ1(τ) and the constant
$oin the frame dragging equation (5.22).
Calculated from the exterior metric [Eq. (5.33)], the 3-metric of the hypersurfaceΣis
(3)ds2+ = −dτ2+R2
dθ2+sin2θdϕ˜2
−4J
R sin2θdϕdτ .˜ (5.34) Calculated from the interior metric [Eq. (5.17)], the 3-metric is
(3)ds2− = −dτ2+(a¯r})2
dθ¯2+sin2θ¯dϕ¯2
−2ω(a¯r})2sin2θ¯dϕdτ ,¯ (5.35)
whereωis to be evaluated at ¯r}. Setting ˜ϕ=ϕ¯ everywhere onΣ(for allτ), and correspondingly matching thedϕdτ˜ term in Eq. (5.34) to thedϕdτ¯ term in Eq. (5.35), we obtain
ω(¯r}, τ)= 2J
R3(τ). (5.36)
To compute the extrinsic curvature onΣ, we follow the prescription given in Appendix 5.B. The result is:
Ki j−dxidxj = −ar¯}
dθ¯2+sin2θ¯2dϕ¯2
+a¯r} 2ω+r¯}ω,¯rsin2θ¯dϕdτ ,¯ (5.37) Ki j+dxidxj = −R
dθ2+sin2θ2dϕ˜2
−2J
R2sin2θdϕdτ .˜ (5.38)
Note thatωandω,¯rin Eq. (5.37) should be evaluated at ¯r}. The matching of thedϕdτ¯ anddϕdτ˜ terms leads to
2ω+rω¯ ,¯r r¯
}=−2J
R3. (5.39)
Inserting Eq. (5.22) into Eq. (5.36) and (5.39), we can determine the two unknownsζ1and$oinω[Eq. (5.22)]:
ω(¯r, τ) = −3J R3
¯ r
¯ r}
!2
+5J
R3 , (5.40a)
Ω(¯r, τ) = ω+$0/a2=ω+ 5J
2MR2 . (5.40b)
Note that their dependence onτis entirely contained inR(τ) andAR=2M/R(τ).
The frame dragging can be manifested by calculating the ratio of (Ω−ω) [the angular velocity of the fluid as measured by local inertial-guidance gyroscopes] toΩ[the angular velocity of the fluid relative to inertial frames at infinity]. This ratioχcis to be evaluated at the center of the universe ¯r=0:
χc≡ Ωc−ωc Ωc
!
= 1
1+2M/R. (5.41)
At very early times, whenR(τ)/M → 0, the ratioχcvanishes and the frame dragging is perfect: the spin direction of a gyroscope is locked to the rotational motion of the universe’s fluid. At very late times,χc becomes unity, so the universe’s rotation has no significant influence on gyroscopes. Figure 5.1 depicts the evolution ofχc as the universe expands. Two special times are shown on the graph: (i) The crossing of the cosmological horizon happens at the timeτ, at which an observer at the center can first see out to the universe’s edge; that is, when a null ray emitted from the edge atτ = 0 reaches the center. This horizon crossing timeτHCis determined mathematically by
Z τHC 0
dτ a(τ) =Z r¯s
0
dr¯, (5.42)
witha(τ) given by Eq. (5.15). The radius of the universe at that time isRHC =a(τHC)×r¯s =M/2. (ii) The second special timeτis the one at which the edge of the universe passes through the white-hole horizon:
R=2M. For a discussion of Fig. 5.1, see Sec. 5.3.1 above.