Cylindrical Gravitational Waves in Einstein-Aether Theory

TL;DR

This work derives the cylindrical gravitational wave equation within Einstein-Aether theory, generalizing the classical Einstein-Rosen framework from GR. It shows that in EA both metric functions Ψ(r,t) and H(r,t) are strictly periodic in time, in contrast to GR where H can exhibit a secular drift, and it reveals aether-induced modifications to wave propagation, including a velocity c_{ea} = 1/√(1 - c_{13}). The authors also compute the EA energy-momentum pseudotensor, demonstrating nontrivial aether contributions that yield finite, well-defined energies at infinity. A combination of monochromatic and pulsed solutions, plus a careful GR limit comparison, exposes observable signatures that could distinguish EA from GR in cylindrical gravitational-wave scenarios.

Abstract

Along the lines of the Einstein-Rosen wave equation of General Relativity (GR), we derive a gravitational wave equation with cylindrical symmetry in the Einstein-aether (EA) theory. We show that the gravitational wave in the EA is periodic in time for both the metric functions $Ψ(r,t)$ and $H(r,t)$. However, in GR, $Ψ(r,t)$ is periodic in time, but $H(r,t)$ is semi-periodic in time, having a secular drifting in the wave frequency. The evolution of wave pulses of a given width is entirely different in both theories in the $H(r,t)$ metric function due to this frequency drifting. Another fundamental difference between the two theories is the gravitational wave velocity. While in GR, the waves propagate with the speed of light, in EA, there is no upper limit to the wave velocity, reaching infinity if $c_{13} \rightarrow 1$ and zero if $c_{13} \rightarrow -\infty$. We also show that energy-momentum pseudotensor and superpotential get contributions from aether in addition to the usual gravitational field part. All these characteristics are observational signatures that differentiate GR and EA.

Figures (10)

• Figure 1: Wave components $\Psi({\mathrm w})$ and $H(\omega)$ using $r=1$, $t=1$ and the parameters of Table \ref{['table1']} in equations (\ref{['Psirt']}) and (\ref{['Hrt']}).
• Figure 2: Wave component pulse $\Psi$ of width $\beta$, using $C_1=1$ in equation (\ref{['Psirt']}) and $\alpha=t$, $\beta=1$, $\gamma=\sqrt{1-c_{13}}\,r$ in equation (\ref{['intpulse']}), for four values of $t=0$, $t=2$, $t=6$ and $t=10$ and the parameters of Table \ref{['table1']}.
• Figure 3: Wave component pulse $\Psi$ of width $\beta$, using $C_1=1$ in equation (\ref{['Psirt']}) and $\alpha=t$, $\beta=1$, $\gamma=\sqrt{1-c_{13}}\,r$ in equation (\ref{['intpulse']}), for four values of $t=0$, $t=2$, $t=6$ and $t=10$, and the parameters of Table \ref{['table1']}.
• Figure 4: Wave component pulse $\Psi$ of width $\beta$, using $C_1=1$ in equation (\ref{['Psirt']}) and $\alpha=t$, $\beta=1$, $\gamma=\sqrt{1-c_{13}}\,r$ in equation (\ref{['intpulse']}), for four values of $t=0$, $t=2$, $t=6$ and $t=10$ and the parameters of Table \ref{['table1']}.
• Figure 5: Wave component pulse $H$ of width $\beta$, using equation (\ref{['Hrt']}) and $\alpha={t\,\sqrt {c_{14}}}/{\sqrt {{c_{13}}+{c_2}}}$, $\beta=1$, $\gamma=r$ in equation (\ref{['intpulse']}), for four values of $t=0$, $t=2$, $t=6$ and $t=10$ and the parameters of Table \ref{['table1']}.