General Relativity as the effective theory of GL(4,R) spontaneous symmetry breaking

Abstract

We assume a GL(4,R) space-time symmetry which is spontaneously broken to SO(3,1). We carry out the coset construction of the effective theory for the non-linearly realized broken symmetry in terms of the Goldstone fields and matter fields transforming linearly under the unbroken Lorentz subgroup. We then identify functions of the Goldstone and matter fields that transform linearly also under the broken symmetry. Expressed in terms of these quantities the effective theory reproduces the vierbein formalism of General Relativity with general coordinate invariance being automatically realized non-linearly over GL(4,R). The coset construction makes no assumptions about any underlying theory that might be responsible for the assumed symmetry breaking. We give a brief discussion of the possibility of field theories with GL(4,R) rather than Lorentz space-time symmetry providing the underlying dynamics.