Constraint propagation equations of the 3+1 decomposition of f(R) gravity

Theories of gravity other than general relativity (GR) can explain the observed cosmic acceleration without a cosmological constant. One such class of theories of gravity is f(R). Metric f(R) theories have been proven to be equivalent to Brans-Dicke (BD) scalar-tensor gravity without a kinetic term. Using this equivalence and a 3+1 decomposition of the theory it has been shown that metric f(R) gravity admits a well-posed initial value problem. However, it has not been proven that the 3+1 evolution equations of metric f(R) gravity preserve the (hamiltonian and momentum) constraints. In this paper we show that this is indeed the case. In addition, we show that the mathematical form of the constraint propagation equations in BD-equilavent f(R) gravity and in f(R) gravity in both the Jordan and Einstein frames, is exactly the same as in the standard ADM 3+1 decomposition of GR. Finally, we point out that current numerical relativity codes can incorporate the 3+1 evolution equations of metric f(R) gravity by modifying the stress-energy tensor and adding an additional scalar field evolution equation. We hope that this work will serve as a starting point for relativists to develop fully dynamical codes for valid f(R) models.Comment: 25 pages, matches published version in CQG, references update

Black hole-neutron star mergers and short gamma-ray bursts: A relativistic toy model to estimate the mass of the torus

The merger of a binary system composed of a black hole (BH) and a neutron star (NS) may leave behind a torus of hot, dense matter orbiting around the BH. While numerical-relativity simulations are necessary to simulate this process accurately, they are also computationally expensive and unable at present to cover the large space of possible parameters, which include the relative mass ratio, the stellar compactness, and the BH spin. To mitigate this and provide a first reasonable coverage of the space of parameters, we have developed a method for estimating the mass of the remnant torus from BH-NS mergers. The toy model makes use of an improved relativistic affine model to describe the tidal deformations of an extended tri-axial ellipsoid orbiting around a Kerr BH and measures the mass of the remnant torus by considering which of the fluid particles composing the star are on bound orbits at the time of the tidal disruption. We tune the toy model by using the results of fully general-relativistic simulations obtaining relative precisions of a few percent and use it to investigate the space of parameters extensively. In this way, we find that the torus mass is largest for systems with highly spinning BHs, small stellar compactnesses, and large mass ratios. As an example, tori as massive as M-b,M-tor similar or equal to 1.33 M-circle dot can be produced for a very extended star with compactness C similar or equal to 0.1 inspiralling around a BH with dimensionless spin parameter a = 0.85 and mass ratio q similar or equal to 0.3. However, for a more astrophysically reasonable mass ratio q similar or equal to 0.14 and a canonical value of the stellar compactness C similar or equal to 0.145, the toy model sets a considerably smaller upper limit of M-b,M-tor less than or similar to 0.34 M-circle dot