Gravity from Breaking of Local Lorentz Symmetry

We present a model of gravity based on spontaneous Lorentz symmetry breaking. We start from a model with spontaneously broken symmetries for a massless 2-tensor with a linear kinetic term and a nonderivative potential, which is shown to be equivalent to linearized general relativity, with the Nambu-Goldstone (NG) bosons playing the role of the gravitons. We apply a bootstrap procedure to the model based on the principle of consistent coupling to the total energy energy-momentum tensor. Demanding consistent application of the bootstrap to the potential term severely restricts the form of the latter. Nevertheless, suitable potentials exists that permit stable vacua. It is shown that the resulting model is equivalent, at low energy, to General Relativity in a fixed gauge. 1. Symmetry vs. Broken Symmetry Masslessness often arises as a consequence of the existence of a symmetry. In quantum electrodynamics the masslessness of the photon is normally attributed to gauge invariance, or symmetry under local changes of phase. In quantum chromodynamics, the theory of the strong interaction, masslessness of the gluons is likewise attributed to a gauge invariance, albeit a nonlinear one. In general relativity, the masslessness of gravitons can be traced to symmetry under active diffeomorphisms: no diffeomorphism-invariant mass term exists. In some circumstances, however, there exists an alternative reason why a field might be massless. Surprisingly, this alternative explanation involves the breaking of a symmetry rather than its existence. A general result, the Nambu-Goldstone theorem [1], states under mild assumptions that there must be a massless particle whenever a continuous global symmetry of an action isn’t a symmetry of the vacuum. In this talk, based on ongoing work with Alan Kostelecký [2, 3], we show that an alternative description of gravity can be constructed from a symmetric two-tensor without the assumption of masslessness. In this picture, masslessness is a consequence of symmetry breaking rather than of exact symmetry: diffeomorphism symmetry and local Lorentz symmetry are spontaneously broken, but the graviton remains massless because it is a Nambu-Goldstone mode. The cardinal object in the theory is a symmetric two-tensor-density, denoted by C . Starting point is the Lagrange density L = 12C KμναβC αβ + V (C , ημν). (1) Here, Kμναβ is the usual quadratic kinetic operator for a massless spin-2 field, Kμναβ = 1 2 [(−ημνηαβ + 1 2ημαηνβ + 1 2ημβηνα)∂ 2 + ημν∂α∂β + ηαβ∂μ∂ν − 1 2ημα∂ν∂β − 1 2ηνα∂μ∂β − 1 2ημβ∂ν∂α − 1 2ηνβ∂μ∂α], (2) and V is a potential which is built out of the four scalars X1 = C ηνμ, (3) X2 = (C · η · C · η) μ μ, (4) X3 = (C · η · C · η · C · η) μ μ, (5) X4 = (C · η · C · η · C · η · C · η) μ μ. (6) We will suppose that V has a local minimum at C = c 6= 0, and that C acquires an expectation value 〈C〉 = c . Note that this implies spontaneous breaking of Lorentz invariance. We can now decompose C = c + C̃ where C̃ are the fluctuations of C . At low energy, the values of the scalars Xn will be approximately fixed to their values xn in the local minimum. Then, the linearized form of the potential can be taken equivalent to a sum of Lagrange multiplier terms that fix the values of the four scalars Xn to the values xn: