Rethinking the link between matter and geometry

Abstract

In the present manuscript, I examine an intriguing relation at the classical level between general relativity and a theory where matter couples uniquely multiplicatively to geometry in the Lagrangian density. Interestingly, the gravitational constant G is replaced by a novel fundamental constant, whose value is not tied to any classical phenomenon; while the value of G itself becomes related to the dynamics of the universe. I concentrate on different aspects of the Equivalence Principle, as the theory is expected to violate all of its different formulations.

Charged black hole and radiating solutions in entangled relativity

“…Entangled relativity [1][2][3][4] is a theory of relativity that fulfills Einstein's original idea that "there can be no G-field [space-time] without matter" [5], 1 while at the same time it recovers many predictions of general relativity -without any novel field, see [2] and references therein. Einstein originally thought that general relativity augmented with a cosmological constant λ would possess the wanted property that "physical qualities of space are completely determined by matter alone" [6] 2 ; whereas it was obviously not the case with general relativity without a cosmological constant since a flat space-time is obviously solution of vacuum in that case [5].…”

“…R is the usual Ricci scalar constructed upon the space-time metric g αβ , with determinant g; while L m is a scalar Lagrangian representing the matter fields. ξ defines a novel fundamental scale that is relevant at the quantum level only, and therefore is notably not related to the size of black holes; whereas the Planck scaledefined from κ and which is related to the size of black holesno longer is fundamental, nor constant, in entangled relativity [2]. 3 The impossible existence of gravity without matter, and vice versa, is obvious from the action.…”

“…But entangled relativity is more easily understood in its dilaton equivalent 4 form that reads [1,2]…”

“…whereκ is a positive effective coupling constant between matter and geometry, with the dimension of κ.κ takes its value from the asymptotic behavior of the effective scalar degree of freedom in Eq. ( 1) [1,2], as well as the considered normalisation of φ.κ/ √ φ, which defines an effective Planck scale, notably fixes the size of black holes with a given mass. The equivalence between the two actions is similar to the equivalence between f (R) theories and the corresponding specific scalar-tensor theories [19].…”

“…Indeed, the existence of space-time without matter means a e-mail: ominazzoli@gmail.com (corresponding author) b e-mail: edison_cesar@hotmail.com 1 A translation in English of the original paper in German is available online at https://einsteinpapers.press.princeton.edu/vol7-trans/49. 2 A translation in English of the original paper in German is available online at https://einsteinpapers.press.princeton.edu/vol7-trans/52. that inertia can be defined relative to space only, whereas Einstein had in mind that inertia -and, of course, angular momentum -could only be defined relatively to surrounding matter [7][8][9].…”