RevModPhys.84.671

# RevModPhys.84.671

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Unformatted text preview: n. The linear theory violates the principle of continuity of the physics in the parameters of the theory. This is known as the vDVZ discontinuity. The discontinuity was soon traced to the fact that not all of the degrees of freedom introduced by the graviton mass decouple as the mass goes to zero. The massive graviton has ﬁve spin states, which in the massless limit become the two helicity states of a massless graviton, two helicity states of a massless vector, and a single massless scalar. The scalar is essentially the longitudinal graviton, and it maintains a ﬁnite coupling to the trace of the source stress tensor even in the massless limit. In other words, the massless limit of a massive graviton is not a massless graviton, but rather a massless graviton plus a coupled scalar, and the scalar is responsible for the vDVZ discontinuity. Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 If the linear theory is accurate, then the vDVZ discontinuity represents a true physical discontinuity in predictions, violating our intuition that physics should be continuous in the parameters. Measuring the light bending in this theory would be a way to show that the graviton mass is mathematically zero rather than just very small. However, the linear theory is only the start of a complete nonlinear theory, coupled to all the particles of the standard model. The possible nonlinearities of a real theory were studied several years later by Vainshtein (1972), who found that the nonlinearities of the theory become stronger and stronger as the mass of the graviton shrinks. What he found was that around any massive source of mass M, such as the Sun, there is a new length scale known as the Vainshtein radius rV \$ 2 ðM=m4 MP Þ1=5 . At distances r & rV , nonlinearities begin to dominate and the predictions of the linear theory cannot be trusted. The Vainshtein radius goes to inﬁnity as m ! 0, so there is no radius at which the linear approximation tells us something trustworthy about the massless limit. This opens the possibility that the nonlinear effects cure the discontinuity. To have some values in mind, if we take M the mass of the Sun and m a very small value, say the Hubble constant m \$ 10À33 eV, the scale at which we might want to modify gravity to explain the cosmological constant, we have rV \$ 1018 km, about the size of the Milky Way. Later the same year, Boulware and Deser (1972) studied some speciﬁc fully nonlinear massive gravity theories and showed that they possess a ghostlike instability. Whereas the linear theory has 5 degrees of freedom, the nonlinear theories they studied turned out to have 6, and the extra degree of freedom manifests itself around nontrivial backgrounds as a scalar ﬁeld with a wrong sign kinetic term, known as the Boulware-Deser ghost. Meanwhile, the ideas of effective ﬁeld theory were being developed, and it was realized that a nonrenormalizable theory, even one with apparent instabilities such as massive gravity, can be made s...
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