Symmetries from Locality. III. Massless Spin 2 Gravitons and Time
  Translations
  We relax the assumption of time translation and Lorentz boost symmetry in
theories involving massless spin 2 gravitons, while maintaining a basic notion
of locality that there is no instantaneous signaling at a distance. We project
out longitudinal modes, leaving only two degrees of freedom of the graviton.
Our previous work, which assumed time translation symmetry, found that the
Lorentz boost symmetry is required to ensure locality at leading order. In this
work, without assuming time translations or Lorentz boosts, we show that
locality of the exchange action between matter sources demands that massless
spin 2, at leading order, organizes into Einstein-Hilbert plus a Gauss-Bonnet
term with a prefactor that is constrained to be a particular function of time;
while in the matter sector we recover time translation and Lorentz boost
symmetry. Finally, we comment on whether the time dependence of the
Gauss-Bonnet prefactor may be forbidden by going to higher order in the
analysis and we mention that other possibilities are anticipated if graviton
mass terms are included.
