Related Research Articles
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978) who proved the Calabi conjecture.
In the general theory of relativity, the Einstein field equations relate the geometry of spacetime to the distribution of matter within it.
In the mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a (pseudo-)Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental interest, as they are the solutions of Einstein's field equations in vacuum with vanishing cosmological constant.
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations, although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to Lorentzian manifolds. Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons.
Teleparallelism, was an attempt by Albert Einstein to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, a spacetime is characterized by a curvature-free linear connection in conjunction with a metric tensor field, both defined in terms of a dynamical tetrad field.
In theoretical physics, the Brans–Dicke theory of gravitation is a theoretical framework to explain gravitation. It is a competitor to Einstein's theory of general relativity. It is an example of a scalar–tensor theory, a gravitational theory in which the gravitational interaction is mediated by a scalar field as well as the tensor field of general relativity. The gravitational constant G is not presumed to be constant but instead 1/G is replaced by a scalar field which can vary from place to place and with time.
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 differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for n = 3 is necessary and sufficient condition for the manifold to be conformally flat. By contrast, in dimensions n ≥ 4, the vanishing of the Cotton tensor is necessary but not sufficient for the metric to be conformally flat; instead, the corresponding necessary and sufficient condition in these higher dimensions is the vanishing of the Weyl tensor, while the Cotton tensor just becomes a constant times the divergence of the Weyl tensor. For n < 3 the Cotton tensor is identically zero. The concept is named after Émile Cotton.
In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non-gravitational fields are present. These are distinct from the electrovacuum solutions, which take into account the electromagnetic field in addition to the gravitational field. Vacuum solutions are also distinct from the lambdavacuum solutions, where the only term in the stress–energy tensor is the cosmological constant term.
In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einstein's field equation. The term pp stands for plane-fronted waves with parallel propagation, and was introduced in 1962 by Jürgen Ehlers and Wolfgang Kundt.
An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski spacetime.
In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field.
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.
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 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 differential geometry and general relativity, the Bach tensor is a trace-free tensor of rank 2 which is conformally invariant in dimension n = 4. Before 1968, it was the only known conformally invariant tensor that is algebraically independent of the Weyl tensor. In abstract indices the Bach tensor is given by
The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches the speed of light, meaning that more variables and more complicated mathematics are required to calculate the object's motion. As a result, relativity requires the use of concepts such as vectors, tensors, pseudotensors and curvilinear coordinates.
The Kasner metric is an exact solution to Albert Einstein's theory of general relativity. It describes an anisotropic universe without matter. It can be written in any spacetime dimension and has strong connections with the study of gravitational chaos.
Campbell's theorem, also known as Campbell’s embedding theorem and the Campbell-Magaarrd theorem, is a mathematical theorem that evaluates the asymptotic distribution of random impulses acting with a determined intensity on a damped system. The theorem guarantees that any n-dimensional Riemannian manifold can be locally embedded in an (n + 1)-dimensional Ricci-flat Riemannian manifold.
In mathematical physics, vanishing scalar invariant (VSI) spacetimes are Lorentzian manifolds with all polynomial curvature invariants of all orders vanishing. Although the only Riemannian manifold with VSI property is flat space, the Lorentzian case admits nontrivial spacetimes with this property. Distinguishing these VSI spacetimes from Minkowski spacetime requires comparing non-polynomial invariants or carrying out the full Cartan–Karlhede algorithm on non-scalar quantities.
References
- ↑ H. Stephani et al., Exact solutions of Einstein's field equations, 2nd Edition, Cambridge Univ. Press 2003 (Kundt spacetimes are studied in Chapter 31)
- ↑ Pravda, V.; Pravdova, A.; Coley, A.; Milson, R. (2002), "All spacetimes with vanishing curvature invariants", Classical and Quantum Gravity, 19 (23): 6213–6236, arXiv: gr-qc/0209024 , Bibcode:2002CQGra..19.6213P, doi:10.1088/0264-9381/19/23/318, S2CID 11958495 .
 | This relativity-related article is a stub. You can help Wikipedia by expanding it. |
 | This differential geometry related article is a stub. You can help Wikipedia by expanding it. |
 | This mathematical physics-related article is a stub. You can help Wikipedia by expanding it. |