Abstract
It is well-known that the presence of a spacetime boundary requires the conventional Einstein-Hilbert (EH) action to be supplemented by the Gibbons-Hawking (GH) boundary term in order to retain the standard variational procedure. When the EH action is amended by the diffeomorphism-invariant graviton mass and potential terms, it naively appears that no further boundary terms are needed since all the new fields of massive gravity enter the action with the first derivative. However, we show that such a formulation would be inconsistent, even when the bulk action is ghost-free. The theory is well-defined only after introducing novel boundary counterterms, which dominate over the GH term in the massless limit and cancel the problematic boundary terms induced by the bulk action. The number of boundary counterterms equals the number of total derivatives one could construct in the bulk using positive powers of two derivatives of the longitudinal mode of the massive graviton.
Highlights
AND SUMMARYThe conventional action,1 − R M ð∂φÞ2, for a scalar field φ on a D-dimensional flat spacetime M endowed with a boundary ∂M does not require a boundary term for the standard variational principle to be well defined
When the Einstein-Hilbert action is amended by the diffeomorphism-invariant graviton mass and potential terms, it naively appears that no further boundary terms are needed since all the new fields of massive gravity enter the action with the first derivative
The theory is well defined only after introducing novel boundary counterterms, which dominate over the Gibbons-Hawking term in the massless limit and cancel the problematic boundary terms induced by the bulk action
Summary
For a scalar field φ on a D-dimensional flat spacetime M endowed with a boundary ∂M does not require a boundary term for the standard variational principle to be well defined. III, to the case of the Fierz-Pauli theory of a free massive graviton and show that this theory requires a novel boundary term [in addition to the Gibbons-Hawking term of general relativity (GR)] for consistency, once the background spacetime is endowed with a boundary We derive this term in two different ways: first by studying consistency of the boundary effective action in the 5D theory and by considering the KaluzaKlein modes of a six-dimensional (6D) massless graviton in the presence of a boundary, with one spatial dimension along the boundary compactified on a circle.
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