In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.
In the mathematical field of Riemannian geometry, the scalar curvature is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry.
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. This tensor has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor that is a linear expression in the Ricci tensor.
In mathematics, a symmetric space is a Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.
When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity.
In differential geometry and theoretical physics, the Petrov classification describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold.
The Gödel metric, also known as the Gödel solution or Gödel universe, is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a negative cosmological constant.
A frame field in general relativity is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by and the three spacelike unit vector fields by . All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field.
In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. This decomposition is of fundamental importance in Riemannian and pseudo-Riemannian geometry.
In theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually two distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913 respectively. The first was quickly dismissed, but the second became the first known example of a metric theory of gravitation, in which the effects of gravitation are treated entirely in terms of the geometry of a curved spacetime.
In relativistic classical field theories of gravitation, particularly general relativity, an energy condition is a generalization of the statement "the energy density of a region of space cannot be negative" in a relativistically-phrased mathematical formulation. There are multiple possible alternative ways to express such a condition such that can be applied to the matter content of the theory. The hope is then that any reasonable matter theory will satisfy this condition or at least will preserve the condition if it is satisfied by the starting conditions.
In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors formed from these by the operations of taking dual contractions and covariant differentiations.
In general relativity, a congruence is the set of integral curves of a vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation.
In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter.
In general relativity, curvature invariants are a set of scalars formed from the Riemann, Weyl and Ricci tensors - which represent curvature, hence the name, - and possibly operations on them such as contraction, covariant differentiation and dualisation.
In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann.
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is adapted to these nested round spheres. The defining characteristic of Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and Gaussian curvature of each sphere. However, radial distances and angles are not accurately represented.
