2023-12-23   99922

Determining the Relative Sign in the Einstein Field Equation

The Einstein field equation describes the dynamics of the spacetime metric field by relating it to the matter tensor (stress-energy tensor). In this relationship, the relative sign of the matter tensor term should be such that test objects are attracted by positive masses. This article uses that criterion to determine the appropriate sign. The approach used here starts with a simple ansatz for the spacetime metric tensor that is static, spherically symmetric, and diagonal, generalized to an arbitrary number N of spacetime dimensions. The components of the Einstein tensor are calculated, and those results are used to do three things: to determine the appropriate sign of the matter tensor term, to derive an N-dimensional generalization of the Schwarzschild solution (article 24902), and to highlight an exceptional feature of the case N=3.

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