Ether theory of gravitation: why and how?

Gravitation might make a preferred frame appear, and with it a clear space/time separation--the latter being, a priori, needed by quantum mechanics (QM) in curved space-time. Several models of gravitation with an ether are discussed: they assume metrical effects in an heterogeneous ether and/or a Lorentz-symmetry breaking. One scalar model, starting from a semi-heuristic view of gravity as a pressure force, is detailed. It has been developed to a complete theory including continuum dynamics, cosmology, and links with electromagnetism and QM. To test the theory, an asymptotic scheme of post-Newtonian approximation has been built. That version of the theory which is discussed here predicts an internal-structure effect, even at the point-particle limit. The same might happen also in general relativity (GR) in some gauges, if one would use a similar scheme. Adjusting the equations of planetary motion on an ephemeris leaves a residual difference with it; one should adjust the equations using primary observations. The same effects on light rays are predicted as with GR, and a similar energy loss applies to binary pulsars.Comment: Standard LaTeX, 60 pages. Invited contribution to the book ``Ether, Spacetime and Cosmology" (M. C. Duffy, ed.), to appear at Hadronic Press. v2: minor improvements, new refs., post-scriptum summarizing later wor

Towards an interpretation of MOND as a modification of inertia

We explore the possibility that Milgrom's Modified Newtonian Dynamics (MOND) is a manifestation of the modification of inertia at small accelerations. Consistent with the Tully-Fisher relation, dynamics in the small acceleration domain may originate from a quartic (cubic) velocity-dependence of energy (momentum) whereas gravitational potentials remain linear with respect to mass. The natural framework for this interpretation is Finsler geometry. The simplest static isotropic Finsler metric of a gravitating mass that incorporates the Tully-Fisher relation at small acceleration is associated with a spacetime interval that is either a homogeneous quartic root of polynomials of local displacements or a simple root of a rational fraction thereof. We determine the low energy gravitational equation and find that Finsler spacetimes that produce a Tully-Fisher relation require that the gravitational potential be modified. For an isolated mass, Newton's potential $Mr^{-1}$ is replaced by $Ma_0\log (r/r_0)$ where $a_0$ is MOND's acceleration scale and $r_0$ is a yet undetermined distance scale. Orbital energy is linear with respect to mass but angular momentum is proportional to $M^{3/4}$. Asymptotic light deflection resulting from time curvature is similar to that of a singular isothermal sphere implying that space curvature must be the main source of deflection in static Finsler spacetimes possibly through the presence of the distance scale $r_0$ that appears in the asymptotic form of the gravitational potential. The quartic nature of the Finsler metric hints at the existence of an underlying area-metric that describes the effective structure of spacetime.Comment: Revised version, 9 pages, 1 figure. Accepted for publication in Monthly Notices of the Royal Astronomical Societ

Computationally Tractable Riemannian Manifolds for Graph Embeddings

Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic geometry. However, going beyond embedding spaces of constant sectional curvature, while potentially more representationally powerful, proves to be challenging as one can easily lose the appeal of computationally tractable tools such as geodesic distances or Riemannian gradients. Here, we explore computationally efficient matrix manifolds, showcasing how to learn and optimize graph embeddings in these Riemannian spaces. Empirically, we demonstrate consistent improvements over Euclidean geometry while often outperforming hyperbolic and elliptical embeddings based on various metrics that capture different graph properties. Our results serve as new evidence for the benefits of non-Euclidean embeddings in machine learning pipelines.Comment: Submitted to the Thirty-fourth Conference on Neural Information Processing System