Cosmological perturbation theory, instantaneous gauges, and local inertial frames

Linear perturbations of Friedmann-Robertson-Walker universes with any curvature and cosmological constant are studied in a general gauge without decomposition into harmonics. Desirable gauges are selected as those which embody best Mach’s principle: in these gauges local inertial frames can be determined instantaneously via the perturbed Einstein field equations from the distributions of energy and momentum in the universe. The inertial frames are identified by their “accelerations and rotations” with respect to the cosmological frames associated with the “Machian gauges.” In closed spherical universes, integral gauge conditions are imposed to eliminate motions generated by the conformal Killing vectors. The meaning of Traschen’s integral-constraint vectors is thus elucidated. For all three types of Friedmann-Robertson-Walker universes the Machian gauges admit much less residual freedom than the synchronous or generalized harmonic gauge. Mach’s principle is best exhibited in the Machian gauges in closed spherical universes. Independent of any Machian motivation, the general perturbation equations and discussion of gauges are useful for cosmological perturbation theory.

Figure 1

The Fermi-Walker time derivative $D_F{S}^{\mu }$ (based in part on Fig. 2 in 44). The cosmological observer $o$ with four-velocity $e_{(0)}^{\mu }$ carries with himself a gyroscope, represented by the spatial vector $W^{\mu }$ (dashed arrow), and a spatial vector $S^{\mu }$ which are both perpendicular to $e_{(0)}^{\mu }$ and identical at the observer’s proper time $\tau$. After $d\tau$, both $S^{\mu }$ and $W^{\mu }$ remain perpendicular to $e_{(0)}^{\mu }$ but the generally propagating vector $S^{\mu }$ will differ from the nonrotating, Fermi-Walker transported gyroscope by the Fermi-Walker time derivative $(D_F{S}^{\mu })d\tau$.

Figure 2

Because of the shear with the principal axes $a$, $b$, almost orthogonal unit vectors ${\mathbf{m}}_{(1)}$, ${\mathbf{m}}_{(2)}$ change their directions but they do not, on average, rotate with respect to the fixed axes $x^1$, $x^2$.