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The de Rham-Gabadadze-Tolley massive gravity admits pp-wave backgrounds on which linear fluctuations are shown to undergo time advances for all values of the parameters.
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The situation is analogous to having a macroscopic (super-Planckian) black hole in General Relativity
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The absence of time advances has been used before in the literature, as a legitimate and necessary requirement for a theory to be healthy [ 24 ] . These are global time advances that should be understood as a violation of the asymptotic causal structure, i.e., Lorentz symmetry. Strictly speaking, the pp-wave background does break asymptotic Lorentz invariance. Such a background, however, ought to be considered as an idealization of a coherent bunch of finite energy wave packets. In a more realistic setting, the wave packets would have to get focused from infinity to form a mildly deformed pp-wave, and then disperse after some time. Thus, instead of an energy source that remains concentrated in a small region from null past to null future, we would have “radiation” coming from null past to form a transitory pp-wave before dispersing. In this way the energy would be dispersed both in the past and the future, and the solution would respect asymptotic Lorentz invariance. For the sake of simplicity one may just analyze the transient pp-wave, and this is enough for our pourpose. Given this, the light cone at infinity is fixed, and so the time advances we found correspond to causally connecting spacelike-separated points. Since the pp-wave vanishes before and after u = ± λ u=\pm\lambda , a mode propagating from asymptotic infinity to asymptotic infinity would undergo the same time advances as found above. These time advances are problematic if one uses the notion of causality introduced by Gao and Wald [ 25 ] : one cannot send signals faster than what is allowed by the asymptotic causal structure of the spacetime (see also [ 26 ] ). This notion has already been used in Ref. [ 8 ]
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S. F. Hassan and R. A. Rosen, Phys. Rev. Lett. 108
2070
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