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Register a new user1+3 formulation of Newtons equations
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Quentin Vigneron
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We present in this paper a 4-dimensional formulation of the Newton equations for gravitation on a Lorentzian manifold, inspired from the 1+3 and 3+1 formalisms of general relativity. We first show that the freedom on the coordinate velocity of a general time-parametrised coordinate system with respect to a Galilean reference system is similar to the shift freedom in the 3+1-formalism of general relativity. This allows us to write Newtons theory as living in a 4-dimensional Lorentzian manifold $M^N$. This manifold can be chosen to be curved depending on the dynamics of the Newtonian fluid. In this paper, we focus on a specific choice for $M^N$ leading to what we call the textit{1+3-Newton equations}. We show that these equations can be recovered from general relativity with a Newtonian limit performed in the rest frames of the relativistic fluid. The 1+3 formulation of the Newton equations along with the Newtonian limit we introduce also allow us to define a dictionary between Newtons theory and general relativity. This dictionary is defined in the rest frames of the dust fluid, i.e. a non-accelerating observer. A consequence of this is that it is only defined for irrotational fluids. As an example supporting the 1+3-Newton equations and our dictionary, we show that the parabolic free-fall solution in 1+3-Newton exactly translates into the Schwarzschild spacetime, and this without any approximations. The dictionary might then be an additional tool to test the validity of Newtonian solutions with respect to general relativity. It however needs to be further tested for non-vacuum, non-stationary and non-isolated Newtonian solutions, as well as to be adapted for rotational fluids. One of the main applications we consider for the 1+3 formulation of Newtons equations is to define new models suited for the study of backreaction and global topology in cosmology.
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Theories of gravity other than general relativity (GR) can explain the observed cosmic acceleration without a cosmological constant. One such class of theories of gravity is f(R). Metric f(R) theories have been proven to be equivalent to Brans-Dicke (BD) scalar-tensor gravity without a kinetic term. Using this equivalence and a 3+1 decomposition of the theory it has been shown that metric f(R) gravity admits a well-posed initial value problem. However, it has not been proven that the 3+1 evolution equations of metric f(R) gravity preserve the (hamiltonian and momentum) constraints. In this paper we show that this is indeed the case. In addition, we show that the mathematical form of the constraint propagation equations in BD-equilavent f(R) gravity and in f(R) gravity in both the Jordan and Einstein frames, is exactly the same as in the standard ADM 3+1 decomposition of GR. Finally, we point out that current numerical relativity codes can incorporate the 3+1 evolution equations of metric f(R) gravity by modifying the stress-energy tensor and adding an additional scalar field evolution equation. We hope that this work will serve as a starting point for relativists to develop fully dynamical codes for valid f(R) models.
A C-metric type solution for general relativity with cosmological constant is presented in 2+1 dimensions. It is interpreted as a three-dimensional black hole accelerated by a strut. Positive values of the cosmological constant are admissible too. Some embeddings of this metric in the 3+1 space-time are considered: accelerating BTZ black string and a black ring where the gravitational force is sustained by the acceleration.
We analyze the IGETS based gravitational acceleration measurements to search for a variability in Newtons constant $G$, as a complement to the analysis carried out in arXiv:2103.11157. To understand the variation in gravitational acceleration, we fit the aforementioned data to a constant model, two sinusoidal models, and a linear model. We find that none of the four models provide a good fit to the data, showing that there is no evidence for periodicity or linear temporal variation in the acceleration measurements. Based on the sinusoidal models, we infer that the relative variance of $G$ is $leqslant 2 times 10^{-9}$, which is about four orders of magnitude more stringent than the amplitude of periodic variations inferred from previous $G$ measurements. From the linear model, we obtain $frac{dot{G}}{G} < 2.14 times 10^{-10} rm{yr^{-1}}$.
We perform a covariant 1+3 split of the Newton-Cartan equations. The resulting 3-dimensional system of equations, called textit{the 1+3-Newton-Cartan equations}, is structurally equivalent to the 1+3-Einstein equations. In particular it features the momentum constraint, and a choice of adapted coordinates corresponds to a choice of shift vector. We show that these equations reduce to the classical Newton equations without the need for special Galilean coordinates. The solutions to the 1+3-Newton-Cartan equations are equivalent to the solutions of the classical Newton equations if space is assumed to be compact or if fall-off conditions at infinity are assumed. We then show that space expansion arises as a fundamental field in Newton-Cartan theory, and not by construction as in the classical formulation of Newtonian cosmology. We recover the Buchert-Ehlers theorem for the general expansion law in Newtonian cosmology.
Bonazzola, Gourgoulhon, Grandclement, and Novak [Phys. Rev. D {bf 70}, 104007 (2004)] proposed a new formulation for 3+1 numerical relativity. Einstein equations result, according to that formalism, in a coupled elliptic-hyperbolic system. We have carried out a preliminary analysis of the mathematical structure of that system, in particular focusing on the equations governing the evolution for the deviation of a conformal metric from a flat fiducial one. The choice of a Diracs gauge for the spatial coordinates guarantees the mathematical characterization of that system as a (strongly) hyperbolic system of conservation laws. In the presence of boundaries, this characterization also depends on the boundary conditions for the shift vector in the elliptic subsystem. This interplay between the hyperbolic and elliptic parts of the complete evolution system is used to assess the prescription of inner boundary conditions for the hyperbolic part when using an excision approach to black hole spacetime evolutions.
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