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.
