5.3 Mach’s Principle
5.3.2 Spatially homogeneous model universe that rotates slowly and rigidly
function of time: no lock-step rotation [(Ωc−ωc)/Ωc=0] before causal contact with the universe’s edge is achieved; no sudden change when causal contact is reached. This seems rather non-Machian.
This non-Machian behavior is intimately tied to a truly non-causal aspect of frame dragging. In any spherically symmetric situation, such as this one, the frame-dragging angular velocityω(viewed as a vector when its direction is taken into account) has a dipolar angular dependence; and in general relativity, dipolar fields are not radiative, i.e. they do not obey wave equations. This remarkable fact is tied to the spin-two nature of the graviton: only fields with quadrupolar and higher-order angular forms (i.e. with angular order greater than or equal to the graviton spins=2) are governed by wave equations.
The frame-dragging angular velocity ω, like the non-propagating, spherical coulomb field of a charge distribution in electromagnetic theory, is laid down at some initial time and evolves forward thereafter in a nonradiative manner. For a detailed general relativistic analysis and discussion of how the matter’s angular momentum distribution governs the evolution ofω, see Refs. [16, 18] and references therein.
The most serious way in which our model problem differs from the physical universe is in its neglect of
“dark energy”, which constitutes today about 65% of the universe’s energy density. In Sec. 5.3.4, we rectify this neglect by inserting dark energy into our universe-with-edge model. We assume (as is somewhat likely) that the dark energy takes the mathematical form of a cosmological constant in Einstein’s equations.
By contrast with the matter, we cannot cut offthis dark energy at an outer edge for the universe. It extends out of the universe into the vacuum exterior, undiminished, i.e. with a continuing-constant cosmological term in Einstein’s equations. This radically alters the geometry of spacetime outside the universe: the geometry is no longer asymptotically flat and so no longer has the asymptotic inertial properties and influence familiar from special relativity. Instead, the geometry is asymptotically that of the DeSitter solution to Einstein’s equations, and it has an asymptotic “DeSitter horizon”. Despite this change of spacetime geometry, we find that the fractional slippage of the inertial axes, (Ωc−ωc)/Ωc, remains qualitatively the same as in the absence of dark energy [Eq. (5.57) and Fig. 5.2 of Sec. 5.3.4 below].
natesxα−={τ,r,¯ θ,¯ ϕ}¯ , the universe’s spacetime metric can be written as ds2−=−dτ2+a2(τ)
dr¯2+r¯2dθ¯2+r¯2sin2θ¯dϕ¯2
. (5.13)
Herea(τ) is the scale factor, whose form is determined by the field equations. The density of the universe satisfies ρa3 = ρoa3o = constant, where the subscript “o” indicates quantities evaluated at some reference timeτo. We presume that the universe has negligible pressure so its energy-momentum tensor takes the form Tαβ =ρuα−uβ−, whereuα−is the 4-velocity of its fluid. Since the fluid elements are at rest with respect to the spatial coordinates (¯r, θ, φ), the components ofuα−are (1,0,0,0). Given thisTαβand the metric (5.13), the Einstein field equations yield
˙ a2= 8π
3 ρa2= 8π 3
ρoa3o
a , (5.14)
where dots indicate derivative with respect toτ. The solution to Eq. (5.14) is
a(τ)=ao(τ/τo)2/3. (5.15)
Now set the universe into slow rotation so that the fluid elements, instead of being at rest in the coordinates (¯r,θ,¯ ϕ), move with angular velocity¯
dϕ¯
dτ = Ω(¯r, τ). (5.16)
We seek a solution to the Einstein field equations, accurate to first order inΩ, that has the following metric ds2−=−dτ2+a2h
dr¯2+r¯2dθ¯2+r¯2sin2θ(d¯ ϕ¯−ωdτ)2i
. (5.17)
Hereω = ω(¯r, τ) is the angular velocity of local inertial frames with respect to the coordinates anda(τ) takes the unperturbed form (5.15). The term quadratic inωin Eq. (5.17) should be ignored. The energy- momentum tensor that goes with this metric is still that of a pressureless fluid : Tαβ=ρuα−uβ−, in whichρis therestmass density anduα−now takes the values (1,0,0,Ω). The energy-momentum conservation law leads to the following first-order nontrivial equation:
Tφβ;β = 1 a3
d dτ
ha5ρ(Ω−ω)i
=0. (5.18)
Sinceρa3=constant [Eq. (5.14)], equation (5.18) implies Ω−ω=ao
a 2
$o(¯r), (5.19)
where$o is an arbitrary function of ¯r, representing the difference betweenΩandω at timeτo. It can be verified that Eq. (5.19) is consistent with the geodesic equation for the fluid elements and with the conser- vation of angular momentum per unit mass for each fluid element. It can also be verified that, if (Ω−ω) is
independent of ¯r, then the vorticity is 2(Ω−ω)=2(ao/a)2$o.
In general relativity, as in Newtonian theory, the local angular velocity of rotation of a fluid element relative to local inertial frames is equal to half its vorticity.5 We shall impose the homogeneity condition that the universe fluid rotates uniformly relative to local inertial frames, i.e., its vorticity is the same everywhere.
This implies that the vorticity is
vorticity=2(Ω−ω)=2$o
ao
a 2
, (5.20)
and so it decays in magnitude at the same rate as it would in Newtonian physics: vorticity∝1/a2. At time τo, the vorticity of the fluid is simply 2$o.
Next we compute and solve the Einstein field equations to determine the relationship betweenωandΩ.
The non-trivial components of the field equations are found to be
τφ: 16πρa2r(ω¯ −Ω)=4ω,¯r+rω¯ ,¯r¯r, (5.21a) rφ¯ : p
24πρ ω,¯r+ω,¯rτ=0. (5.21b)
Equation (5.19) can be inserted into Eq. (5.21a), and the resulting differential equation has the following solution forω
ω(¯r, τ)=
"
−8π
5 ρ¯r2+ζ1(τ)−ζ2(τ)
¯ r3
#
$o.
Hereζ1 andζ2 are two dimensionless functions ofτto be determined. Since the 1/¯r3 term diverges at the origin, we shall setζ2(τ)≡0, which makesωsatisfy Eq. (5.21b) automatically and gives
ω(¯r, τ)=
"
−8π
5 ρ¯r2+ζ1(τ)
#
$o . (5.22)
Since$o=(a/ao)2(Ω−ω), this is the desired relation betweenωandΩ.
To help in the next subsection, we elucidate the nature of the coordinates (τ,r,¯ θ,¯ ϕ) used in defining the¯ angular velocityΩ = dϕ/dτ¯ =(∂ϕ/∂τ)¯ ¯r,θ¯ of the universe’s fluid. The interior metric (5.13) shows that the 3-surfaces of constantτare homogeneous and have a flat 3-metric, and that radial lines of constant (τ,r,¯ θ)¯ are geodesics in the homogeneous 3-surfaces, and also spacelike geodesics of the 4-dimensional spacetime.
These radial, spacelike geodesics Gcan be thought of as carrying the angular coordinates (¯θ,ϕ) from the¯ origin outward, throughout the universe; i.e., they can be thought of as mapping the angles (¯θ,ϕ) from the¯ origin to other points in the universe. Schmid [18] has shown that this mapping of of angles by means of spacelike geodesicsGgives rise to a remarkably simple and elegant description of frame dragging not only when the vorticity is uniform (as in our model universe) but also when it is nonuniform. In Sec. 5.3.3, when
5In the local rest frame of a fluid element, the velocities of neighboring fluid elements are small compared to the speed of light, so the general relativistic viewpoint on fluid motions and inertia reduces to the Newtonian viewpoint. For a Newtonian proof that the local fluid angular velocity relative to local inertial frames is half its vorticity, see Sec. 6.4 of [20].
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.