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.

The geodesic equation and planetary motion from a quaternion representation of general relativity

Abstract: In this paper, the quaternion representation of general relativity derived in a previous study is applied to the problem of planetary motion It is found that while the outward appearance of the geodesic equation is the same in this formalism as it is in the conventional Einstein formulation, when the derivatives are expressed with respect to the differential increment in four-space, ds, the special frame of reference in which this equation is recast in terms of derivatives with respect to the time changes (ie equations of motion) are somewhat different in the two formulations The reason has to do with the fact that 1) in this theory scalar invariant ds obeys the algebra of a quaternion-number field while in the Einstein formulation it belongs to a real-number field and 2) the bound states of planets are described here by quaternion-field variables that depend on the time parameter in a phase factor while there is no time dependence at all in the metric-tensor formulation for this physical situation As a result of this alteration in the description of a planetary orbit, it is found that the angular momentum,Lq,as compared with the angular momentum in the Einstein and Newtonian theories is as follows:Lq:LE:LN=mK exp [−2γ/r]:mK(1−2γ/r)1/2:mK, whereK is a constant that characterizes a particular orbit,m is the planetary mass andγ is the Schwarzschild radius On the other hand, it is found that to the order of perturbation that is required to compare with the observations of the anomalous part of the perihelion precession of Mercury’s orbit, the theoretical prediction that comes from this analysis is in numerical agreement with its prediction from Einstein’s theory and with the data Based on this analysis, it is suggested that future experimentation involving controlled artificial-satellite orbits could possibly be used to differentiate between the predictions of this theory and those of Einstein’s tensor formulation and the classical theory

Vacuum stress tensor for a slightly squashed Einstein Universe

Abstract: The effect of the deviation from spherical symmetry on the effective Lagrangian of a scalar field is investigated.

Lovelock black holes with a nonconstant curvature horizon

Abstract: This paper studies a class of $D=n+2(\ge 6)$ dimensional solutions to Lovelock gravity that is described by the warped product of a two-dimensional Lorentzian metric and an $n$-dimensional Einstein space. Assuming that the angular part of the stress-energy tensor is proportional to the Einstein metric, it turns out that the Weyl curvature of an Einstein space must obey two kinds of algebraic conditions. We present some exact solutions satisfying these conditions. We further define the quasilocal mass corresponding to the Misner-Sharp mass in general relativity. It is found that the quasilocal mass is constructed out of the Kodama flux and satisfies the unified first law and the monotonicity property under the dominant energy condition. Making use of the quasilocal mass, we show Birkhoff's theorem and address various aspects of dynamical black holes characterized by trapping horizons.

Plane symmetric cosmological micro model in modified theory of Einstein’s general relativity

Abstract: In this paper, we have investigated an anisotropic homogeneous plane symmetric cosmological micro-model in the presence of massless scalar field in modified theory of Einstein's general relativity. Some interesting physical and geometrical aspects of the model together with singularity in the model are discussed. Further, it is shown that this theory is valid and leads to Ein­stein's theory as the coupling parameter λ →>• 0 in micro (i.e. quantum) level in general.