Bigravity in Kuchar̆’s Hamiltonian formalism: The special case

Abstract

It is proved, that, in order to avoid the ghost mode in bigravity theory, it is sufficient to impose four conditions on the potential of interaction of the two metrics. First, the potential should allow its expression as a function of components of the two metrics' 3+1-decomposition. Second, the potential must satisfy the first order linear differential equations which are necessary for the presence of four first class constraints in bigravity. Third, the potential should be a solution of the Monge-Ampère equation, where the lapse and shift are considered as variables. Fourth, the potential must have a nondegenerate Hessian in the shift variables. The proof is based on the explicit derivation of the Hamiltonian constraints, the construction of Dirac brackets on the base of a part of these constraints, and calculation of other constraints' algebra in these Dirac brackets. As a byproduct, we prove that these conditions are also sufficient in the massive gravity case.

Analysis of constraints and their algebra in bimetric theory

“…When rewriting them this way, one makes use of 1 ∇ 2 0 = 0, (A. 25) which holds for the vanishing boundary conditions at infinity [44]. Let us now define,…”

Massive gravity: a general analysis

“…In section 6 we will discuss in detail this approach, which does not change the result of this section and of our work. 4 One could also give a dynamical character to the extra metric adding its Einstein-Hilbert action, see for instance [17,21,[23][24][25][26][27], leading to a bimetric theory. In the bimetric formulation there is no absolute object but clearly the dynamics is more complicated; for instance, at the linearized level there are two gravitons, one massive and one massless, and the interaction terms break the "relative" diffs acting on the two metrics.…”

Бигравитация В Гамильтоновом Формализме Кухаржа. Общий Случай

“…Позднее этот автор представил другую работу [17], где подтвердил свои результаты для более общего потенциала, выражающегося через полный набор инвариантов матрицы g αµ f µβ . Эта последняя работа касается также и обсуждения потенциала де Рама-Габададзе-Толи, наши выводы относительно этого потенциала будут опубликованы в следующей статье (ее предварительная версия имеется в архиве [18]), соответствующее сравнение под-ходов и результатов будет сделано там. Отметим здесь только то, что потенциал де Рама-Габададзе-Толи строится с помощью инвариантов матричного квадратного корня из g αµ f µβ .…”