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Introduction to the mathematics of general relativity
About: Introduction to the mathematics of general relativity is a research topic. Over the lifetime, 2583 publications have been published within this topic receiving 73295 citations.
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TL;DR: Tangent space null rotations are used to give a straightforward classification of the Ricci tensor in general relativity theory as discussed by the authors, and they are used for the classification of Ricci Tensor.
Abstract: Tangent space null rotations are used to give a straightforward classification of the Ricci tensor in general relativity theory.
64 citations
TL;DR: In this paper, it was shown that it is possible to construct a Hamiltonian description for Lorentzian General Relativity in terms of two real $SO(3) connections.
Abstract: I show in this letter that it is possible to construct a Hamiltonian description for Lorentzian General Relativity in terms of two real $SO(3)$ connections. The constraints are simple polynomials in the basic variables. The present framework gives us a new formulation of General Relativity that keeps some of the interesting features of the Ashtekar formulation without the complications associated with the complex character of the latter.
64 citations
TL;DR: In this article, a new approach that leads to the end results of the general theory of relativity via just the law of conservation of energy (broadened to embody the mass and energy equivalence of the special theory of the relativity) and quantum mechanics is presented.
Abstract: Herein we present a whole new approach that leads to the end results of the general theory of relativity via just the law of conservation of energy (broadened to embody the mass and energy equivalence of the special theory of relativity) and quantum mechanics. We start with the following postulate. Postulate: The rest mass of an object bound to a celestial body amounts less than its rest mass measured in empty space, and this, as much as its binding energy vis-a-vis the gravitational field of concern. The decreased rest mass is further dilated by the Lorentz factor if the object in hand is in motion in the gravitational field of concern. The overall relativistic energy must be constant on a stationary trajectory. This yields the equation of motion driven by the celestial body of concern, via the relationship e
α / √ 1 − r
0
2
/ e
0
2
= constant, along with the definition α = GM / re
0
2
; here M is the mass of the celestial body creating the gravitational field of concern; G is the universal gravitational constant, measured in empty space it comes into play in Newton's law of gravitation, which is assumed though to be valid for static masses only; r points to the location picked on the trajectory of the motion, the center of M being the origin of coordinates, as assessed by the distant observer; v
0 is the tangential velocity of the object at r; c
0 is the ceiling of the speed of light in empty space; v
0 and c
0 remain the same for both the local observer and the distant observer, just the same way as that framed by the special theory of relativity. The differentiation of the above relationship leads to − GM / r
2 (1 − v
0
2
/ c
0
2
) = v
0
dr
dv0
or, via v
0 = dr / dt, − GM / r
2 (1 − v
0
2
/ c
0
2
) ṟ / r = d??
0 / dt: ?? is the outward looking unit vector along r; the latter differential equation is the classical Newton's Equation of Motion, were v
0, negligible as compared to c
0; this equation is valid for any object, including a light photon. Taking into account the quantum mechanical stretching of lengths due to the rest mass decrease in the gravitational field, the above equation can be transformed into an equation written in terms of the proper lengths, yielding well the end results of the general theory of relativity, though through a completely different set up.
64 citations
TL;DR: In this article, the authors present a general relativistic variable Eddington tensor radiation transport equation, which is intended for use in simulations involving numerical relativity, particularly in the absence of spherical symmetry.
Abstract: We present conservative $3+1$ general relativistic variable Eddington tensor radiation transport equations, including greater elaboration of the momentum space divergence (that is, the energy derivative term) than in previous work. These equations are intended for use in simulations involving numerical relativity, particularly in the absence of spherical symmetry. The independent variables are the lab frame coordinate basis spacetime position coordinates and the particle energy measured in the comoving frame. With an eye towards astrophysical applications---such as core-collapse supernovae and compact object mergers---in which the fluid includes nuclei and/or nuclear matter at finite temperature, and in which the transported particles are neutrinos, we pay special attention to the consistency of four-momentum and lepton number exchange between neutrinos and the fluid, showing the term-by-term cancellations that must occur for this consistency to be achieved.
63 citations
TL;DR: In this paper, the authors discuss the difficulties appearing in explaining a viable and realistic cosmology within the $f(R,T)$ class of theories and challenge the viability of the proposed model as an alternative modification of gravity.
Abstract: Among many alternative gravitational theories to General Relativity (GR), $f(R,T)$ gravity (where $R$ is the Ricci scalar and $T$ the trace of the energy-momentum tensor) has been widely studied recently. By adding a matter contribution to the gravitational Lagrangian, $f(R,T)$ theories have become an interesting extension to GR displaying a broad phenomenology in astrophysics and cosmology. In this paper, we discuss, however, the difficulties appearing in explaining a viable and realistic cosmology within the $f(R,T)$ class of theories. Our results challenge the viability of $f(R,T)$ as an alternative modification of gravity.
63 citations