Abstract

The paper is concerned with the problem of reduction of the general relativity theory to the Newton gravitation theory for a gravitation field with relatively low intensity. This problem is traditionally solved on the basis of linearized equations of general relativity which, being matched to the Newton theory equations, allow us to link the classical gravitation constant with the constant entering the general relativity equations. Analysis of the linearized general relativity equations shows that it can be done only for empty space in which the energy tensor is zero. In solids, the set of linearized general relativity equations is not consistent and is not reduced to the Newton theory equations. Specific features of the problem are demonstrated with the spherically symmetric static problem of general relativity which has the closed-form solution.

Highlights

  • General Relativity EquationsThe basic equation of general relativity which specifies the Einstein tensor has the following form: E=ij Rij − 1 2 δi j R (1)in which Rij ( R = Rii ) are the components of the Ricci curvature tensor

  • The paper is concerned with the problem of reduction of the general relativity theory to the Newton gravitation theory for a gravitation field with relatively low intensity

  • This problem is traditionally solved on the basis of linearized equations of general relativity which, being matched to the Newton theory equations, allow us to link the classical gravitation constant with the constant entering the general relativity equations

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Summary

Introduction

The basic equation of general relativity which specifies the Einstein tensor has the following form: E=ij. In which Rij ( R = Rii ) are the components of the Ricci curvature tensor (we use mixed components because for the spherically symmetric problem considered further they coincide with the physical components). The Einstein tensor is associated with the energy tensor as. V. Fedorov where χ is the relativity gravitational constant. The energy tensor expressed with the aid of Equations (1) and (2) identically satisfies the conservation equation. Where σ j i is the stress tensor and μ is the density. For gravitation with relatively low intensity, the general relativity must reduce to the Newton theory in which the gravitation potential ψ satisfies the Poisson equation. In which G is the classical gravitation constant. The linearized version of Equation (1) is obtained and matched to Equation (5).

Linearized Equations and the Reduction Problem
General Relativity Equations
Linearized Solution
Second-Order Asymptotic Approximation
Light Ray Deviation in the Vicinity of Sun
Conclusion

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