Causal propagation of constraints in bimetric relativity in standard 3+1 form

Abstract

The goal of this work was to investigate the propagation of the constraints in the ghost-free bimetric theory where the evolution equations are in standard 3+1 form. It is established that the constraints evolve according to a first-order symmetric hyperbolic system whose characteristic cone consists of the null cones of the two metrics. Consequently, the constraint evolution equations are well-posed, and the constraints stably propagate.

Bimetric cosmology is compatible with local tests of gravity

“…[14] seems to differ by a numerical factor (which is not manifestly positive) from (12), the scale is the same as long as the parameters satisfy β n ∼ O(1). 5 Well inside the Vainshtein radius, deviations from the inverse-square law of the gravitational force scale like (r/r V ) 3 [20,22] and hence, in order to satisfy (10), we require (for α 1),…”

“…However, we still have the free parameter α, which we can use to suppress the extra term in the Newtonian potential. The most stringent 5 We emphasize that the value for the Vainshtein radius is derived assuming r < m −1 FP . For the Solar System, this is not an issue since the intersection of the lines corresponding to r V and m −1 FP lies close to the observable threshold of ∼ 10 µm.…”

“…Furthermore, bimetric theory can give rise to cosmological solutions with accelerated expansion even in the absence of vacuum energy [11][12][13]. These selfaccelerating solutions require the mass scale which is associated to the breaking of independent diffeomorphisms * mlueben@mpp.mpg.de † edvard@fysik.su.se ‡ angnissm@mpp.mpg.de 1 An important additional point to consider is the Cauchy problem, which is known to be well-posed in GR, but still the subject of ongoing work within bimetric theory [4][5][6][7].…”

On constraint preservation and strong hyperbolicity

“…This field h A ∆ ( hA ∆ ) appear in the equations of the SS (31), showing that the SS always exists and how the information of the reductions h α B ( hα B ) is propagated to this equation. The reason for this simple answer is that equation (30) follows directly from the integrability conditions ( 25) and ( 26), which do not depend on any reduction h α B . As it is shown in the proof of the theorem, these reductions appear only as a trick…”

“…Most well-known physical systems have quasi-linear first-order partial differential subsidiary systems. Some examples are: Maxwell [15], and Non-linear [2] Electrodynamics, Einstein [20], [42], Einstein-Christoffel [17] ,BSSN [48], [7], ADM [29], [47], n+1 ADM [43], f (R)-Gravity [38], [35], Bimetric Relativity [30] theories, etc. The process for deriving the SS and verifying its strong hyperbolicity is conducted separately for each physical theory and usually involves very cumbersome calculations.To assess this problem, we present a theory that simplifies and automatizes the process.…”