We explore the multimetric theory of gravitation, also known as multigravity. We derive additional new exact solutions for the theory in proportional Kerr-Schild and double Kerr-Schild forms. We extend several solutions from the theory of General Relativity, characterized by a constant Ricci scalar in single and double Kerr-Schild forms, to derive solutions in the multi-gravity context. We also examine and extend the classical double copy relations that can be constructed out from these solutions in multigravity exploring the dynamics of the single copy and zero copy fields.
The path to quantum gravity has several subtleties and candidates (see [1] for a review). One of them modifies Einstein's theory of General Relativity (GR) at the classical level due to effective corrections coming from quantum effects. From the perspective of particle physics, we think of the gravitational field as a spin-2 field with massless excitations known as gravitons that mediate the gravitational interaction. In fact, it can be shown that that GR is the unique theory that can be constructed from a spin-2 massless field theory with local interactions, which is Lorentz invariant and unitary, as can be seen in [2][3][4][5][6].
It is also possible to examine this process in reverse, beginning with an altered classical gravitation theory and tackling the physical phenomena that GR fails to explain. The fundamental problems of dark matter and dark energy are some of the sources of the need for a modified theory of gravity. Modifications to GR include Lorentz violating and non-local theories (see [7,8]); however, the conservation of Lorentz symmetry has ever been found to be valid experimentally. Another possible modification consists of a theory with a massive graviton. Massive gravity has been explored since the work of Fierz and Pauli [9], where an additional mass term was introduced in the perturbation around the background metric, resulting in a linearized version of a theory of massive gravity. The non-linear extension of the Fierz-Pauli theory leads to two issues: the van Dam-Veltman-Zakharov (vDVZ) discontinuity [10,11] and the Boulware-Deser (BD) ghost problem [12,13].
For the Boulware-Deser problem, a theory of a spin-2 massive field with six degrees of freedom exists. One of these modes is a scalar mode known as the BD ghost. The energy of the BD ghost is negative, which leads to an unstable vacuum, making the theory inconsistent. A Boulware-Deser ghost free theory of massive gravity was formulated by de Rham-Gabadadze-Tolley and is commonly known as dRGT gravity (see [14]). In dRGT gravity there is a fixed spin-2 metric field that interacts with the GR metric field. The discontinuity of vDVZ is fixed in this theory by the Vainshtein mechanism [15]. A massive spin-2 metric field propagates 5 degrees of freedom; however, through the Vainshtein mechanism, this extra degree of freedom responsible for the vDVZ discontinuity gets screened by its own interactions.
The consideration of massive gravity has resulted in the development of bimetric gravity, a theory with two general metrics, one of them is dynamical, and the other is a reference metric. This extension for two gravitational fields (see [16]) involves the interaction of two dynamic spin-2 fields (for some important reviews on massive gravity and bigravity, see [17][18][19]). This has further evolved into multimetric gravity theory, also referred to as multigravity, which describes the interaction among N dynamic spin-2 fields. All these generalizations of massive gravities into interacting spin-2 fields can be obtained from the deconstruction paradigm (see [18]). Multigravity was an alternative proposal to GR first motivated by trying to explain the origin of the late time accelerated expansion of the universe [20,21], but later was found useful to explore dark matter [22][23][24][25] and the hierarchy problem [26,27] by the introduction of additional massive spin-2 gravitons above the massless graviton of GR. In the search for solutions in multigravity, some black-hole-like solutions have been considered [28,29]. These solutions are of the proportional type, in which all metrics are proportional to the one of GR differing only in a constant conformal factor. Non-proportional (non-diagonal) solutions also arise in this context with a more complicated structure and properties. It is known that multigravity solutions present certain instabilities for some values of the graviton mass [30,31]. Moreover, some of the non-proportional black holes will be stable [32,33] (for a more detailed account on this topic, see [34]). For simplicity in the present work we will focus only on solutions of the proportional type.
On the other hand, in Refs. [35][36][37] it was proved that the n-leg tree-level amplitude in Yang-Mills theory can be expressed as a summation over the amplitudes of cubic diagrams. Although Yang-Mills theory includes interaction diagrams with both cubic and quartic vertices, one can always restructure the color factors to describe quartic-vertex diagrams as a sum of cubic-vertex diagrams [35][36][37]. The color factors in the gauge amplitudes are configured such that they encompass the weights of the rearranged quartic-vertex diagrams. Thus, once the Yang-Mills amplitude is expressed solely as a sum of cubic diagrams, they fulfill the color-kinematics duality, or the BCJ duality [35][36][37]. Then the color and the kinematic factors have a very similar structure (Jacobi identity) and can be exchanged. Using BCJ duality it becomes f
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