Torsion-induced gravitational $$\theta $$ term and gravitoelectromagnetism

Abstract

Motivated by the analogy between a weak field expansion of general relativity and Maxwell’s laws of electrodynamics, we explore physical consequences of a parity violating $$\theta $$ θ term in gravitoelectromagnetism. This is distinct from the common gravitational $$\theta $$ θ term formed as a square of the Riemann tensor. Instead it appears as a product of the gravitoelectric and gravitomagnetic fields in the Lagrangian, similar to the Maxwellian $$\theta $$ θ term. We show that this sector can arise from a quadratic torsion term in nonlinear gravity. In analogy to the physics of topological insulators, the torsion-induced $$\theta $$ θ parameter can lead to excess mass density at the interface of regions where $$\theta $$ θ varies and consequently it generates a correction to Newton’s law of gravity. We discuss also an analogue of the Witten effect for gravitational dyons.

Duality and Higher Buscher Rules in p‐Form Gauge Theory and Linearized Gravity

“…Duality of topological insulators has been reported theoretically in [31, 32] and experimentally in [50, 51]. Some theoretical possibilities for analog topological magnetoelectric effects in gravity were considered in [14]. It would be interesting to study further physical consequences of this setting in the future, also in light of the results of the present paper.…”

“…[ 39 ] It becomes particularly interesting in situations where θ is not the same in all regions of spectime or when it is promoted to a background field, resulting in a gravitational analogue of axion electrodynamics. This viewpoint can lead to topological magnetoelectric effects, similar to topological insulators, as suggested in [14]. It is also worth noting that θ could be promoted to a dynamical axion‐like field in nonlinear gravity with torsion, in which case it can have important physical implications e.g.…”

“…It turned out that it is a topological term formed as a product of the Newtonian gravitoelectric field and the gravitomagnetic field of gravitoelectromagnetism, whose nonlinear origin can be traced in quadratic torsion invariants of Nieh‐Yan type in spacetimes equipped with the Weitzenböck connection. [ 14 ]…”

“…However, this would result in more derivatives than one requires for the on‐shell duality relation. Indeed, we propose that the relevant topological θ‐term is what was recently suggested in [14] in the context of modified gravitoelectromagnetism and its origin may be traced in the Nieh‐Yan invariants of gravity with torsion. [ 15–17 ] This is the second main result of the present paper.…”

“…However, it does not arise in the linearized limit of the Pontryagin density, which involves the Riemann tensor instead and therefore second derivatives of . As shown in [14], it is the linearized version of a topological invariant in a spacetime with torsion equipped with the Weitzenböck connection. Invariants formed through the torsion tensor are the so‐called Nieh‐Yan terms [ 15–17 ] and hence the term we encounter here is a special case of them.…”