Abstract
Main results of our recent investigations on five-dimensional scenarios of massive (bi-)gravity will be summarized in this article. In particular, we will show how to construct higher dimensional massive graviton terms from the characteristic equation of square matrix, which is a consequence of the Cayley-Hamilton theorem. Then, we will show whether massive graviton terms of five-dimensional massive (bi-)gravity behave as effective cosmological constants for a number of physical metrics compatible with fiducial ones such as the Friedmann-Lemaitre-Robertson-Walker, Bianchi type I, and Schwarzschild-Tangherlini metrics. Finally, we will show the corresponding cosmological solutions for the five-dimensional massive (bi-)gravity.
Highlights
Massive gravity, in which a graviton is assumed to have tiny but non-vanishing mass, has a rich and long story since a seminal paper by Fierz and Pauli (FP) [1]. It was shown by van Dam and Veltman [2] and Zakharov [3] that the FP theory cannot recover the general relativity by Einstein in the massless limit
We have mainly summarized our results on both five-dimensional massive gravity and bigravity, which have been published in two recent papers in [17, 18]
We have shown the useful method based on the Cayley-Hamilton theorem to construct arbitrary dimensional graviton terms Li and the five-dimensional graviton term L5
Summary
In which a graviton is assumed to have tiny but non-vanishing mass, has a rich and long story since a seminal paper by Fierz and Pauli (FP) [1]. It is to the noted dynamical property (or the existence of Ricci scalar R(f )), field equations of reference metric in bi-gravity will no longer be algebraic but differential as that of physical metric. We have been able to show that the four-dimensional graviton terms, L2, L3, and L4, can be reconstructed by applying the well-known Cayley-Hamilton theorem in linear algebra [19] for a 4 × 4 matrix Kμν. This theorem states that any n × n matrix K must obey its characteristic equation: P(K) ≡ Kn − Dn−1Kn−1 + Dn−2Kn−2 − .
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