Global structure of bigravity solutions

TL;DR

This work analyzes the global causal structure of static, spherically symmetric bigravity solutions with interacting metrics $f$ and $g$. It shows that Type I solutions reduce to Schwarzschild–(A)dS forms for both metrics and can be maximally extended into stair-case diagrams linking multiple copies of the individual geometries, often at the expense of global hyperbolicity, while preserving causality (no closed timelike curves). A Type II subclass with one metric satisfying Einstein’s equations yields $f$ proportional to $g$ and standard cosmological-constant-driven dynamics. Overall, the study reveals rich, nontrivial causal structures in bigravity and motivates further dynamical and phenomenological investigations.

Abstract

We discuss the causal diagrams of static and spherically symmetric bigravity vacuum solutions, with interacting metrics $f$ and $g$. Such solutions can be classified into type I (or "non-diagonal") and type II (or "diagonal"). The general solution of type I is known, and leads to metrics $f$ and $g$ in the Schwarzschild-(Anti)de Sitter family. The two metrics are not always diagonalizable in the same coordinate system, and the light-cone structure of both metrics can be quite different. In spite of this, we find that causality is preserved, in the sense that closed time-like curves cannot be pieced together from geodesics of both metrics. We propose maximal extensions of Type I bigravity solutions, where geodesics of both metrics do not stop unless a curvature singularity is encountered. Such maximal extensions can contain several copies (or even an infinite number of them) of the maximally extended "individual" geometries associated to $f$ and $g$ separately. Generically, we find that the maximal extensions of bigravity solutions are not globally hyperbolic, even in cases when the individual geometries are. The general solution of type II has not been given in closed form. We discuss a subclass where $g$ is an arbitrary solution of Einstein's equations with a cosmological constant, and we find that in this case the only solutions are such that $f\propto g$ (with trivial causal structure).

Figures (10)

• Figure 1: Causal diagrams when the $f$ metric is de Sitter (left diagram) while the $g$ metric is Minkowski (right diagram) and $\beta=1$. The dashed curly vertical line of the left diagram represents a sphere of constant radial coordinate $r$. The solid curly vertical line of the right diagram represents the de Sitter horizon $r=r_H$ plotted in the Minkowski space-time. We also plotted three radial geodesics of Minkowksi space-time emanating from the origin $r=0$ at $t=0$: the thick dashed (blue) curve is a future-directed radial null ray from the origin (notice it is also a null geodesic ($V=$ constant) of the de Sitter space-time), the thin solid (green) curve with two arrows is a $t=0$ radial geodesic, the thin dashed (red) curve is a past-directed null ray from the origin. The last two curves are radial geodesics of Minkowski space-time but not of de Sitter space-time. The whole of the Minkowski space-time is mapped onto the half of the de Sitter diagram verifying $V>0$. Note that the past directed null geodesics of Minkowski turn around and start moving towards the future boundary of de Sitter space. This behaviour, however, does not lead to closed time-like curves, as discussed in Section \ref{['3.4']}
• Figure 2: Causal diagram for de Sitter with Minkowski, for $\beta=1$. The left diagram is for de Sitter with horizon radius $r_H$, while the right diagram is for Minkowski. The dashed thin lines (with no arrows) are $t =$ constant lines. The dashed thick line with one (resp. two) arrow is an $r =$ constant curve, with $r<r_H$ (resp. $r>r_H$). The thin solid line with three arrows represents the trajectory of an observer sitting at constant radius $r=r_H$ in Minkowski spacetime. The thick solid lines with arrows are past directed null geodesics of de Sitter space time $U= constant$ curves. The mapping of the infinities (null, spacelike, timelike) of Minkowski spacetimes ($i^{\pm,0}$, ${\cal I}^{\pm}$) has been indicated on the de Sitter diagram. One of the stricking feature of those diagrams, is that the past time-like infinity of Minkowski is split between the upper left corner (for $r>r_H$), the lower right corner (for $r<r_H$) and the diagonal ($r=r_H$) of the de Sitter space-time.
• Figure 3: Causal diagrams when the $f$ metric is de Sitter (left diagram) while the $g$ metric is Minkowski (right diagram) and $\beta=1/6$. Thick dashed (blue) curve, thin dashed (red) curve, and thin solid (green) curve with two arrows, are respectively null (for the two first) and spacelike (for the last) radial geodesics of Minkowski space-time. The dashed curly vertical line in both diagram is an $r=$ constant curve which is the boundary of the region where one of the metrics becomes complex.
• Figure 4: Diagram showing the extension proposed in the text for the de Sitter/Minkowski solution. Notations are the same as in Fig. \ref{['figure1']}. By using a second Minkowski space, we can extend the de Sitter diagram of Fig. \ref{['figure1']}, represented by region I and II above, to the lower half, represented by region III and IV above. The de Sitter space-time is now geodesically complete, however the whole space-time it is not globally hyperbolic, when both metric are considered on the same footing. If we draw a Cauchy surface for all the de Sitter geodesics [such as a horizontal line cutting accross the diagram $(b)$], this surface will intersect some of the Minkowski geodesics twice, while it will fail to intersect some others.
• Figure 5: Same as Fig. \ref{['figure3']}, with radial geodesics of de Sitter plotted instead of those of Minkowski. The thick dashed (blue) curve is a future-directed radial null ray from the origin $(r=0, \tilde{t}=0$). The thin solid (green) curve is a $\tilde{t}=0$ radial geodesic of de Sitter. The thin dashed (red) with one arrow curve a the past-directed null geodesic from the origin. We also plotted, as thin solid (green) curves with two and three arrows, the continuation of the $\tilde{t}=0$ curve beyond the horizon $r=r_H$. When mapped into the Minkowski diagram, the past directed null geodesics of de Sitter, of region I, reach the timelike past infinity of the Minkowski space-time at a finite value of their affine parameter in de Sitter, namely when they cross the de Sitter horizon $r=r_H$. Nevertheless, we can "smoothly" continue them in the newly added Minkowski solution onto which regions III and IV of de Sitter space-time are mapped.