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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: In this article, it was shown that when metric lengths replace coordinate lengths in Dirac's wave equation, it has a covariant form under a metric transformation of the physically measured distances themselves, rather than a coordinate transformation.
Abstract: When Euclidean coordinate lengths are replaced by the metric lengths of a curved geometry within Newton’s second law of motion, the metric form of the second law can be shown to be identical to the geodesic equation of motion of general relativity. The metric coefficients are contained in the metric lengths and satisfy the field equations of general relativity. Because metric lengths are the physically measured lengths, their use makes it possible to understand general relativity directly in terms of physical quantities such as energy and momentum within a curved space–time. The metric form of the second law contains gravitational effects in exactly the same manner as occurs in relativity. Its mathematical derivation uses vectors rather than tensors, and nongravitational forces can occur in this modified second law without a tensor form. Because quantum mechanics is based on Newtonian concepts of energy and momentum, it is shown that when metric lengths replace coordinate lengths in Dirac’s wave equation, it has a covariant form under a metric transformation of the physically measured distances themselves, rather than a coordinate transformation. Metric transformations are also used to describe the Dirac equation for the gravitational central field in a Schwarzschild metric.
16 citations
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TL;DR: The use of potentials describing perfect fluids illuminates the role of time in general relativity as discussed by the authors, and using Hamilton-Jacobi methods, one can find solutions for inhomogeneous situations of interest to cosmology without making an explicit time choice until the very end of the calculation.
Abstract: The use of potentials describing perfect fluids illuminates the role of time in general relativity. Using Hamilton-Jacobi methods, one can find solutions for inhomogeneous situations of interest to cosmology without making an explicit time choice until the very end of the calculation. We compute exact general solutions of long-wavelength matter and radiation interacting with gravity. Hamilton-Jacobi methods can describe adiabatic and isothermal fluctuations as well as the scalar, vector, and tensor modes.
16 citations
01 Jan 2002
16 citations
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TL;DR: In this article, it was shown that the relationship between general relativity and topological field theories of the $BF$ type is also present in the first-order formalism for general relativity.
Abstract: Cartan's first and second structure equations together with first and second Bianchi identities can be interpreted as equations of motion for the tetrad, the connection and a set of two-form fields ${T}^{I}$ and ${R}_{J}^{I}$. From this viewpoint, these equations define by themselves a field theory. Restricting the analysis to four-dimensional spacetimes (keeping gravity in mind), it is possible to give an action principle of the $BF$ type from which these equations of motion are obtained. The action turns out to be equivalent to a linear combination of the Nieh-Yan, Pontrjagin, and Euler classes, and so the field theory defined by the action is topological. Once Einstein's equations are added, the resulting theory is general relativity. Therefore, the current results show that the relationship between general relativity and topological field theories of the $BF$ type is also present in the first-order formalism for general relativity.
16 citations
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16 citations