Gravitational Lorentz Violations and Adjustment of the Cosmological Constant in Asymmetrically Warped Spacetimes

TL;DR

This work explores asymmetrically warped five-dimensional spacetimes in which time and space warp factors differ along the extra dimension, leading to bulk-induced Lorentz violations on a flat 4D brane and potentially faster propagation of gravitational waves compared to light. The authors derive the most general bulk solutions consistent with a bulk cosmological constant and a bulk U(1) field, identifying AdS–Schwarzschild and AdS–RN metrics, and solve the Israel junction conditions for a Z2-symmetric brane, showing that a self-tuning mechanism for the 4D cosmological constant can arise when a bulk gauge field is present. They analyze horizon structures, the possibility of shielding singularities, and the resulting phase diagrams, noting that horizon-containing solutions require exotic brane matter. They further study the observable consequences, including geodesic analyses indicating potential superluminal graviton propagation, perturbative graviton zero modes, and cosmological expansion on the brane, where μ and Q^2 terms modify Friedmann dynamics and demand a fast relaxation of bulk parameters to achieve a viable late-time cosmology. Overall, the paper proposes a novel route to address the cosmological constant problem and cosmic acceleration via gravitational Lorentz violations in extra dimensions, while outlining observational constraints from gravitational-wave speeds and brane cosmology.

Abstract

We investigate spacetimes in which the speed of light along flat 4D sections varies over the extra dimensions due to different warp factors for the space and the time coordinates (``asymmetrically warped'' spacetimes). The main property of such spaces is that while the induced metric is flat, implying Lorentz invariant particle physics on a brane, bulk gravitational effects will cause apparent violations of Lorentz invariance and of causality from the brane observer's point of view. An important experimentally verifiable consequence of this is that gravitational waves may travel with a speed different from the speed of light on the brane, and possibly even faster. We find the most general spacetimes of this sort, which are given by AdS-Schwarzschild or AdS-Reissner-Nordstrom black holes, assuming the simplest possible sources in the bulk. Due to the gravitational Lorentz violations these models do not have an ordinary Lorentz invariant effective description, and thus provide a possible way around Weinberg's no-go theorem for the adjustment of the cosmological constant. Indeed we show that the cosmological constant may relax in such theories by the adjustment of the mass and the charge of the black hole. The black hole singularity in these solutions can be protected by a horizon, but the existence of a horizon requires some exotic energy densities on the brane. We investigate the cosmological expansion of these models and speculate that it may provide an explanation for the accelerating Universe, provided that the timescale for the adjustment is shorter than the Hubble time. In this case the accelerating Universe would be a manifestation of gravitational Lorentz violations in extra dimensions.

Figures (4)

• Figure 1: Phase diagram of a brane sitting in a bulk black hole background. In the upper left zig-zagged region, the brane will expand. In the other parts of the diagram, a brane can remain static and its induced metric is 4D Lorentz invariant. In the other filled regions, the black hole singularity is hidden by two horizons; however for $\omega>0$ these horizons are developed in the part of the BH space-time cut by the $\mathbb{Z}_2$ orbifold. Fig. \ref{['fig:Horizons']} shows the displacement of the horizons when moving along the dashed lines varying the energy density or the equation of state on the brane. The region $\omega<-1$ has been magnified.
• Figure 2: Displacement of the horizons when varying the energy density or the equation of state on the brane. Both the inner and outer horizons are shown in the plots. (a) $\omega$ is varying in the region $\omega \leq -1$. When $\omega$ reaches the critical point of a vacuum energy equation of state, the two horizons hit the black hole singularity; (b) the energy density is varying with $\omega<-1$ fixed. The two horizons are between the black hole singularity and the brane. When $\rho$ goes to $\rho_-$, the two horizons degenerate and when $\rho$ goes to $\rho_0$, the inner horizon reaches the singularity; (c) $\rho$ fixed, $\omega$ varying in the region $\omega>0$; (d) $\omega>0$ fixed, $\rho$ varying. The two horizons belong to the cut region of the BH space-time. When $\rho$ goes to $\rho_0$, the two horizons degenerate.
• Figure 3: Shapes of the Newtonian potential. The functions $\rho_-, \rho_0$ and $\rho_\mu$ defined in (\ref{['eq:rho_pm']}), (\ref{['eq:rho0']}) and (\ref{['eq:rhoMu']}) border the different regions in the plane $(\omega,\rho)$. The asymptotic value of the potential at infinity corresponds to the momentum $|\vec{p}|^2$ along the brane. The space is however cut at the brane sitting at $r=r_0$. We identify five different shapes for the potential. (a) $\omega<-1$ and $\rho_0>\rho>\rho_-$: the potential vanishes at the two horizons. The geodesics will come back to the brane but, from a brane observer's point of view, it already takes an infinite time to cross the horizon and so the geodesics will never be seen as coming back. (b) $\omega<-1$ and $\rho_->\rho$. The 4D local speed of graviton starts decreasing when the geodesic leaves the brane. (c) $-1<\omega<0$ and $\rho>\rho_\mu$. (d) $-1<\omega<0$ and $\rho_\mu>\rho$ or $0<\omega$ and $\rho_->\rho$. (e) $0<\omega$ and $\rho>\rho_-$. In the three last configurations, the 4D speed of the graviton is increasing when going into the bulk, and on its return the geodesic may reach a point on the brane where light emitted with the graviton has not yet arrived.
• Figure 4: The Average speed of gravitons propagating along a geodesic off the brane as function of the distance on the brane. We clearly see that in the region of the plane $(\omega,\rho)$ where the Newtonian potential behaves like the shapes (c),(d) and (e) of Fig. \ref{['fig:pot']}, the graviton can propagate faster than the light and its speed increases with the distance to the source on the brane. We also note in the case (d) that there are various ways for gravity to propagate between the same points on the brane, i.e. different geodesics in the bulk with different values of $E/|\vec{p}|$ can return to the same point on the brane. When $E/|\vec{p}|$ is too large, then the average speed becomes lower than the speed of light. In the case (b), the graviton will always propagate along the brane with a speed faster than that in the bulk.