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We study the degrees of freedom in new general relativity---a flat and metric compatible family of theories---around the Minkowski background in a gauge-invariant manner. First, we confirm the decoupling case, in which the theory reduces to linearized gravity plus a massless Kalm-Ramond field. We then show that, unless they vanish, the vector modes of this theory will be unstable. In addition, we find two new branches of the theories, which are instability-free and propagate linearly two tensor modes and in one of the cases also a massless scalar field. This shows that while the generic theory is ill behaved, there are three possible realizations of instability-free cases, in contradiction to the previous literature, which states that there is only one healthy theory in addition to general relativity.
Bahamonde et al. (Mon,) studied this question.