The Third Way to 3D Gravity

Consistency of Einstein’s gravitational field equation Gμν ∝ Tμν imposes a “conservation condition” on the T -tensor that is satisfied by (i) matter stress tensors, as a consequence of the matter equations of motion, and (ii) identically by certain other tensors, such as the metric tensor. However, there is a third way, overlooked until now because it implies a “nongeometrical” action: one not constructed from the metric and its derivatives alone. The new possibility is exemplified by the 3D “minimal massive gravity” model, which resolves the “bulk vs boundary” unitarity problem of topologically massive gravity with anti-de Sitter asymptotics. Although all known examples of the third way are in three spacetime dimensions, the idea is general and could, in principle, apply to higher-dimensional theories. Essay written for the Gravity Research Foundation 2015 Awards for Essays on Gravitation. It is now 100 years since Einstein wrote down, after a long struggle, his gravitational field equations, Gμν = (8πG) Tμν , (1) where Tμν is the matter stress tensor, G is Newton’s gravitational constant and Gμν is the Einstein tensor, defined in terms of the metric gμν and its Ricci tensor Rμν , and Ricci scalar R, by Gμν = Rμν − 1 2 gμνR . (2) The stress tensor satisfies, as a consequence of the matter field equations, the “conservation condition” DμTμν = 0 , (3) where D is the usual covariant derivative in a gravitational background. In flat space this condition implies conservation of energy and momentum in the matter fields. It was this condition that led Einstein to the Einstein tensor: he needed to construct from the metric a tensor Gμν satisfying the Bianchi identity DμGμν ≡ 0 . (4) However, one can turn the logic around: it is because the Einstein tensor satisfies the Bianchi identity that the tensor Tμν appearing on the right-hand side of the Einstein field equations (1) must satisfy the conservation condition (3). This perspective leads to the following question. How many ways are there to construct a tensor Tμν satisfying (3)? Certainly, one may take Tμν to be a matter stress tensor, but another obvious possibility is to choose Tμν ∝ gμν . The conservation condition is an identity for this tensor, which can be viewed as a dark-energy contribution to the stress tensor, but it can also be taken over to the left-hand side to give us the modified field equation Gμν + Λgμν = (8πG)Tμν , (5) where Λ is a constant, the “cosmological constant” introduced by Einstein himself in 1917. But this is just a special case: for any diffeomorphism invariant functional I[g], the tensor Iμν = 1 √ − det g δI[g] δg (6) satisfies the Bianchi-type identity DμIμν ≡ 0 . (7) Taking all such tensors over to the left hand side we arrive at a generalisation of the Einstein field equations of the form Eμν = (8πG)Tμν , (8) where Eμν is a symmetric tensor satisfying D μEμν ≡ 0. It will have the form Eμν = Λgμν + σGμν + . . . , (9)