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# Metric tensor (general relativity)

About: Metric tensor (general relativity) is a research topic. Over the lifetime, 1570 publications have been published within this topic receiving 28420 citations.

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TL;DR: In this paper, the number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.

Abstract: The Einstein tensorGij is symmetric, divergence free, and a concomitant of the metric tensorgab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.

2,821 citations

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TL;DR: In this paper, a convenient way of splitting the metric tensor and the Einstein field equations, applicable in any space-time, is first introduced, and suitable boundary conditions are set.

Abstract: Gravitational fields containing bounded sources and gravitational radiation are examined by analyzing their properties at spatial infinity. A convenient way of splitting the metric tensor and the Einstein field equations, applicable in any space-time, is first introduced. Then suitable boundary conditions are set. The group of co-ordinate transformations that preserves the boundary conditions is analyzed. Different possible gravitational fields are characterized intrinsically by a combination of (i) characteristic initial data, and (ii) Dirichlet data at spatial infinity. To determine a particular solution one must specify four functions of three variables and three functions of two variables; these functions are not subject to constraints. A method for integrating the field equations is given; the asymptotic behaviour of the metric and Riemann tensors for large spatial distances is analyzed in detail; the dynamical variables of the radiation modes are exhibited; and a superposition principle for the radiation modes of the gravitational field is suggested. Among the results are: (i) the group of allowed co-ordinate transformations contains the inhomogeneous orthochronous Lorentz group as a subgroup; (ii) each of the five leading terms in an asymptotic expansion of the Riemann tensor has the algebraic structure previously predicted from analyzing the Petrov classification; (iii) gravitational waves appear to carry mass away from the interior; (iv) time-dependent periodic solutions of the field equations which obey the stated boundary conditions do not exist. It was found that the general fields studied in the present work are in many ways very similar to the axially symmetric fields recently studied by Bondi, van der Burg & Metzner.

1,716 citations

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01 Jan 1924

TL;DR: In this article, the tensor calculus and the law of gravitation have been studied in a tensor tensor geometry setting, and the authors present a review of the literature.

Abstract: Introduction 1. Elementary principles 2. The tensor calculus 3. The law of gravitation 4. Relativity mechanics 5. Curvature of space and time 6. Electricity 7. World geometry Supplementary notes Bibliography Index.

917 citations

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TL;DR: In this article, it is shown that a coordinate-dependent transformation of the units of measure can be used to throw the theory into a form for which the gravitational field appears in the conventional form, as a metric tensor, such that the Einstein field equation is satisfied.

Abstract: A gravitational theory compatible with Mach's principle was published recently by Brans and Dicke. It is characterized by a gravitational field of the Jordan type, tensor plus scalar field. It is shown here that a coordinate-dependent transformation of the units of measure can be used to throw the theory into a form for which the gravitational field appears in the conventional form, as a metric tensor, such that the Einstein field equation is satisfied. The scalar field appears then as a "matter field" in the theory. The invariance of physical laws under coordinate-dependent transformations of units is discussed.

745 citations

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TL;DR: In this paper, the Beltrami-Laplace operator on a Riemannian manifold was introduced and the boundary of the manifold was defined by S. If V is not closed, its boundary was defined in terms of S. In terms of local coordinates (x i ), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … x N ) are the components of the metric tensor on V.

Abstract: Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (x i ), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … x N ) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

688 citations