User:Fluctuating metric/draft entangled relativity

Like general relativity, entangled relativity is a theory that describes how spacetime and matter interact with each other.

The theory is based on the same fundamental principles and mathematical tools, but posit a different coupling between spacetime curvature and matter. While spacetime curvature and matter can be described independently at the fundamental level in general relativity, they cannot in entangled relativity.

General relativity is one specific limiting case of entangled relativity, which however happens in fairly generic situations. At the level of classical field theory, entangled relativity do not posit the existence of a coupling constant between matter and spacetime curvature. It means that the constant of Newton $$G$$ is not a fundamental constant in entangled relativity, but a field that converges toward a constant during the expansion of the universe. As a consequence, at the classical level, entangled relativity has one parameter less than general relativity without a cosmological constant, and two parameters less than general relativity with a cosmological constant.

Entangled relativity has been published for the first time in 2015 in Physics Letters B, although it has been named in 2021 only. Although the theory has not been widely studied yet, it has been shown that it can be rewritten effectively as another type of theories that has been widely studied since the first superstring revolution. Indeed, the field equations of entangled relativity match those of a specific form of theories with a dilaton. As a consequence, several aspects of the phenomenology of entangled relativity can be inferred from previous studies.

The name "entangled relativity '' is meant to indicate that the definition of the spacetime curvature and matter cannot be separated from each other in this novel formulation of a general theory of relativity.

Entangled relativity is not related to the quantum entanglement phenomenon.

Mathematical formulation
Entangled relativity is defined by its action:

$$S = - {\xi \over 2 c} \int \sqrt{-g} ~d^4x {\mathcal{L}_m^2 \over R}, (1)$$

where


 * $$\mathcal{L}_m$$ is the Lagrangian density of matter fields,
 * $$R$$ is the scalar curvature,
 * $$g$$ is the metric tensor determinant: $$g=det(g_{\mu \nu})$$,
 * $$\xi$$ is a new fundamental constant of nature that plays a role at the level of quantum field theory only.
 * $$c$$ is the speed of light in vacuum.

Unlike general relativity, entangled relativity cannot even be defined without a definition of the matter Lagrangian density $$\mathcal{L}_m$$.

For the metric
The metric field equation that derives from the action (1) read, for $$\mathcal{L}_m \neq 0$$

$$R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R=-\frac{R}{\mathcal{L}_{m}} T_{\mu \nu}+\frac{R^{2}}{\mathcal{L}_{m}^{2}}\left(\nabla_{\mu} \nabla_{\nu}-g_{\mu \nu} \square\right) \frac{\mathcal{L}_{m}^{2}}{R^{2}}. (2)$$

where the stress-energy tensor is defined as usual as

$$T_{\mu \nu} = -\frac{2}{\sqrt{-g}} \frac{\delta\left(\sqrt{-g} \mathcal{L}_{m}\right)}{\delta g^{\mu \nu}}$$.

General relativity as a limit
$$\mathcal{L}_m/R$$ behaves as a new degree of freedom that freezes when the matter field solutions are such that $$\mathcal{L}_m = T$$. When it is frozen, $$\mathcal{L}_m/R$$ becomes effectively a constant and one recovers the metric field equation of general relativity without a cosmological constant

$$R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R=\kappa~ T_{\mu \nu}, (3)$$

where $$\kappa = -R/ \mathcal{L}_m$$. One can note that it is consistent with the trace of the equation (3) that reads $$\kappa = - R/T$$. In general relativity, one has $$\kappa = 8 \pi G / c^4$$ where $$G$$ is the constant of Newton.

An effective constant of Newton
Because $$\mathcal{L}_m/R$$ is a degree of freedom, the constant of Newton $$G$$ is actually not a constant in entangled relativity, but a field whose value depends on the spatiotemporal location. Nevertheless, it has been shown that its variability actually is almost nonexistent at our cosmic age, and in any situation where pressure of matter is weak with respect to its energy density. This is notably the case in the solar system.

The value of $$G$$ has not been predicted from the theory so far, as it would rely on a model of the primordial universe within this framework, which has not been studied at the moment.

Explicit example of equivalence between entangled relativity and general relativity
For a pressure-less perfect fluid (also know as dust), one has $$\mathcal{L}_m = -\rho = T$$, where $$\rho$$ is the energy density of the fluid. A fictional universe filled by dust only would be the same for both theories.