Rigid covariance as a natural extension of Painlevé–Gullstrand space-times: gravitational waves

Abstract

The group of rigid motions is considered to guide the search for a natural system of space-time coordinates in General Relativity. This search leads us to a natural extension of the space-times that support Painlevé-Gullstrand synchronization. As an interesting example, here we describe a system of rigid coordinates for the cross mode of gravitational linear plane waves.

Rigid covariance, equivalence principle and Fermi rigid coordinates: gravitational waves

“…We will consider a general space-time which admits rigid covariant coordinates in the sense explained in [4]. The metric of such a space-time in a rigid reference system S can be expressed as [4]:…”

“…Given a metric of the form (40), with (41), we will say that a transformation of the space-time coordinates, {λ, x} → {ζ, y}, is a general rigid transformation if the transformed metric has the same rigid form as the original, but it may have a different rigid time ζ. In Section §4 of the reference [4], we saw how to find rigid coordinates from a metric expressed in arbitrary space-time coordinates. If the starting metric in Section §4 of reference [4] is already in rigid coordinates, the procedure described there will define a general rigid transformation.…”

“…In Section §4 of the reference [4], we saw how to find rigid coordinates from a metric expressed in arbitrary space-time coordinates. If the starting metric in Section §4 of reference [4] is already in rigid coordinates, the procedure described there will define a general rigid transformation. Given all the general rigid transformations, we distinguish the set that transform time, which we will call λ-rigid transformations, from the rest that do not transform time, ζ = λ.…”

“…We have to find rigid coordinates for (84) such that the geodesic congruence is G-isochronous. Part of this work was already performed in reference [4] for the same kind of waves. Equations (17) of [4] were solved with τ (…”

“…In a series of recent papers, [1,2,3,4], we have presented a formulation of the general theory of relativity with a covariance group that is smaller than usual. General covariance implies having ten potentials to describe gravitation; but four of them can be eliminated by means of coordinate transformations.…”