Vacuum solution (general relativity)

vacuum solutionvacuum solutionsVacuum
A third equivalent condition follows from the Ricci decomposition of the Riemann curvature tensor as a sum of the Weyl curvature tensor plus terms built out of the Ricci tensor: the Weyl and Riemann tensors agree, in some region if and only if it is a vacuum region. Since T^{ab} = 0 in a vacuum region, it might seem that according to general relativity, vacuum regions must contain no energy. But the gravitational field can do work, so we must expect the gravitational field itself to possess energy, and it does.

Cartan–Karlhede algorithm

Cartan-Karlhede algorithmCartan–Karlhede classification
The main strategy of the algorithm is to take covariant derivatives of the Riemann tensor. Cartan showed that in n dimensions at most n(n+1)/2 differentiations suffice. If the Riemann tensor and its derivatives of the one manifold are algebraically compatible with the other, then the two manifolds are isometric. The Cartan–Karlhede algorithm therefore acts as a kind of generalization of the Petrov classification. The potentially large number of derivatives can be computationally prohibitive. The algorithm was implemented in an early symbolic computation engine, SHEEP, but the size of the computations proved too challenging for early computer systems to handle.

Symplectic geometry

symplectic topologysymplecticsymplectic structure
Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature. This is a consequence of Darboux's theorem which states that a neighborhood of any point of a 2n-dimensional symplectic manifold is isomorphic to the standard symplectic structure on an open set of ℝ 2n. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For example, every symplectic manifold is even-dimensional and orientable.

Lanczos tensor

Lanczos potential
Note further that the full Riemann tensor cannot in general be derived from derivatives of the Lanczos potential alone. The Einstein field equations must provide the Ricci tensor to complete the components of the Ricci decomposition. The Weyl–Lanczos equations express the Weyl tensor entirely as derivatives of the Lanczos tensor: where C_{abcd} is the Weyl tensor, the semicolon denotes the covariant derivative, and the subscripted parentheses indicate symmetrization. Although the above equations can be used to define the Lanczos tensor, they also show that it is not unique but rather has gauge freedom under an affine group.

BKL singularity

BKL conjectureBKLBKL analysis
This would also resolve another aspect of the problem: the characteristics of spacetime metric evolution in the general solution when it reaches the physical singularity, understood as a point where matter density and invariants of the Riemann curvature tensor become infinite. One of the principal problems studied by the Landau group (to which BKL belong) was whether relativistic cosmological models necessarily contain a time singularity or whether the time singularity is an artifact of the assumptions used to simplify these models.

Scalar field solution

Such a field may or may not be massless, and it may be taken to have minimal curvature coupling, or some other choice, such as conformal coupling. In general relativity, the geometric setting for physical phenomena is a Lorentzian manifold, which is physically interpreted as a curved spacetime, and which is mathematically specified by defining a metric tensor g_{ab} (or by defining a frame field). The curvature tensor R^{a}_{bcd} of this manifold and associated quantities such as the Einstein tensor G_{ab}, are well-defined even in the absence of any physical theory, but in general relativity they acquire a physical interpretation as geometric manifestations of the gravitational field.

Mathematics of general relativity

directly involvingmathematical formulationmathematical relativity
The electromagnetic field tensor F^{ab}, a rank-two antisymmetric tensor. 1) the Riemann curvature tensor: or. 2) the Ricci tensor:. 3) the scalar curvature: R.

Magnetogravitic tensor

In general relativity, the magnetogravitic tensor is one of the three pieces appearing in the Bel decomposition of the Riemann tensor. The magnetogravitic tensor can be interpreted physically as a specifying possible spin-spin forces on spinning bits of matter, such as spinning test particles. Papapetrou-Dixon equations. Curvature invariants.

Conformally flat manifold

conformally flatflat
Then (M, g) is conformally flat if for each point x in M, there exists a neighborhood U of x and a smooth function f defined on U such that (U, e 2f g) is flat (i.e. the curvature of e 2f g vanishes on U). The function f need not be defined on all of M. Some authors use locally conformally flat to describe the above notion and reserve conformally flat for the case in which the function f is defined on all of M. Every manifold with constant sectional curvature is conformally flat. Every 2-dimensional pseudo-Riemannian manifold is conformally flat. A 3-dimensional pseudo-Riemannian manifold is conformally flat if and only if the Cotton tensor vanishes.

Gödel metric

Gödel universeGödel dust solutionGödel dust
In any Lorentzian spacetime, the fourth-rank Riemann tensor is a multilinear operator on the four-dimensional space of tangent vectors (at some event), but a linear operator on the six-dimensional space of bivectors at that event. Accordingly, it has a characteristic polynomial, whose roots are the eigenvalues.

Special relativity (alternative formulations)

This residual curvature is caused by a cosmological constant to be determined by observation. Due to the small magnitude of the constant, then special relativity with the Poincaré group is more than accurate enough for all practical purposes, although near the Big Bang and inflation de Sitter relativity may be more useful due to the cosmological constant being larger back then. Note this is not the same thing as solving Einstein's field equations for general relativity to get a de Sitter Universe, rather the de Sitter relativity is about getting a de Sitter Group for special relativity which neglects gravity. ''Equivalent to the original ? Yes.''

Relationship between string theory and quantum field theory

Strings in this mode couple to the worldsheet curvature of other strings, so their abundance through space-time determines the measure by which an average string worldsheet will be curved. This determines its probability to split or connect to other strings: the more a worldsheet is curved, it has a higher chance of splitting and reconnecting. Spin: each particle in quantum field theory has a particular spin s, which is an internal angular momentum. Classically, the particle rotates in a fixed frequency, but this cannot be understood if particles are point-like.

Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
In all of the following theorems we assume some local behavior of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at "sufficiently large" distances. * * From Riemann to Differential Geometry and Relativity (Lizhen Ji, Athanase Papadopoulos, and Sumio Yamada, Eds.) Springer, 2017, XXXIV, 647 p. ISBN: 978-3-319-60039-0 * * Metric tensor. Riemannian manifold. Levi-Civita connection. Curvature. Curvature tensor. List of differential geometry topics.


By the definition p-curvature measures the failure of the map to be a homomorphism of restricted Lie algebras, just like the usual curvature in differential geometry measures how far this map is from being a homomorphism of Lie algebras. * Grothendieck–Katz p-curvature conjecture. Restricted Lie algebra. Katz, N., "Nilpotent connections and the monodromy theorem", IHES Publ. Math. 39 (1970) 175–232. Ogus, A., "Higgs cohomology, p-curvature, and the Cartier isomorphism", Compositio Mathematica, 140.1 (Jan 2004): 145–164.

Coordinate conditions

coordinate condition
However, due to the second Bianchi identity of the Riemann curvature tensor, the divergence of the Einstein tensor is zero which means that four of the ten equations are redundant, leaving four degrees of freedom which can be associated with the choice of the four coordinates. The same result can be derived from a Kramers-Moyal-van-Kampen expansion of the Master equation (using the Clebsch–Gordan coefficients for decomposing tensor products). A particularly useful coordinate condition is the harmonic condition (also known as the "de Donder gauge"): Here, gamma is a Christoffel symbol (also known as the "affine connection"), and the "g" with superscripts is the inverse of the metric tensor.

Jürgen Ehlers

The analysis presented there uses tools from differential geometry such as the Petrov classification of Weyl tensors (that is, those parts of the Riemann tensor describing the curvature of space-time that are not constrained by Einstein's equations), isometry groups and conformal transformations. This work also includes the first definition and classification of pp-waves, a class of simple gravitational waves. The following papers in the series were treatises on gravitational radiation (one with Sachs, one with Trümper). The work with Sachs studies, among other things, vacuum solutions with special algebraic properties, using the 2-component spinor formalism.

Gauss–Bonnet gravity

Gauss-Bonnet gravityGauss–Bonnet term
Despite being quadratic in the Riemann tensor (and Ricci tensor), terms containing more than 2 partial derivatives of the metric cancel out, making the Euler–Lagrange equations second order quasilinear partial differential equations in the metric. Consequently, there are no additional dynamical degrees of freedom, as in say f(R) gravity. Gauss–Bonnet gravity has also been shown to be connected to classical electrodynamics by means of complete gauge invariance with respect to Noether's theorem. More generally, we may consider :term for some function f. Nonlinearities in f render this coupling nontrivial even in 3+1D. Therefore, fourth order terms reappear with the nonlinearities.

Lagrangian (field theory)

LagrangianLagrangian densityLagrangian field theory
The Lagrange density for general relativity in the presence of matter fields is :R is the curvature scalar, which is the Ricci tensor contracted with the metric tensor, and the Ricci tensor is the Riemann tensor contracted with a Kronecker delta. The integral of is known as the Einstein-Hilbert action. The Riemann tensor is the tidal force tensor, and is constructed out of Christoffel symbols and derivatives of Christoffel symbols, which are the gravitational force field.

Einstein–Hilbert action

Einstein-Hilbert actionEinstein–Hilbert LagrangianEinstein-Hilbert Lagrangian
So, the Riemann curvature tensor is defined as Since the Riemann curvature depends only on the Levi-Civita connection, the variation of the Riemann tensor can be calculated as Now, since is the difference of two connections, it is a tensor and we can thus calculate its covariant derivative, We can now observe that the expression for the variation of Riemann curvature tensor above is equal to the difference of two such terms, We may now obtain the variation of the Ricci curvature tensor simply by contracting two indices of the variation of the Riemann tensor, and get the Palatini identity: The Ricci scalar is defined as Therefore, its variation with respect to the inverse metric is given by In

Classical unified field theories

Generalized Theory of Gravitationunified field theory
Einstein's proposed unified-field equations (fundamental laws of physics) were generally derived from a variational principle expressed in terms of the Riemann curvature tensor for the presumed space-time manifold. In field theories of this kind, particles appear as limited regions in space-time in which the field strength or the energy density are particularly high. Einstein and coworker Leopold Infeld managed to demonstrate that, in Einstein's final theory of the unified field, true singularities of the field did have trajectories resembling point particles.

Directional derivative

normal derivativedirectionalderivative
The translation operator for δ is thus :and for δ′, :The difference between the two paths is then :It can be argued that the noncommutativity of the covariant derivatives measures the curvature of the manifold: :where R is the Riemann curvature tensor and the sign depends on the sign convention of the author.

Holst action

As with the first order tetradic Palatini action where and are taken to be independent variables, variation of the action with respect to the connection (assuming it to be torsion-free) implies the curvature be replaced by the usual (mixed index) curvature tensor (see article tetradic Palatini action for definitions). Variation of the first term of the action with respect to the tetrad gives the (mixed index) Einstein tensor and variation of the second term with respect to the tetrad gives a quantity that vanishes by symmetries of the Riemann tensor (specifically the first Bianchi identity), together these imply Einstein's vacuum field equations hold.

Introduction to general relativity

general relativityembedding diagramEinstein's theory of gravity
The metric function and its rate of change from point to point can be used to define a geometrical quantity called the Riemann curvature tensor, which describes exactly how the space or spacetime is curved at each point. In general relativity, the metric and the Riemann curvature tensor are quantities defined at each point in spacetime. As has already been mentioned, the matter content of the spacetime defines another quantity, the energy–momentum tensor T, and the principle that "spacetime tells matter how to move, and matter tells spacetime how to curve" means that these quantities must be related to each other.

Regge calculus

The deficit angles can be computed directly from the various edge lengths in the triangulation, which is equivalent to saying that the Riemann curvature tensor can be computed from the metric tensor of a Lorentzian manifold. Regge showed that the vacuum field equations can be reformulated as a restriction on these deficit angles. He then showed how this can be applied to evolve an initial spacelike hyperslice according to the vacuum field equation. The result is that, starting with a triangulation of some spacelike hyperslice (which must itself satisfy a certain constraint equation), one can eventually obtain a simplicial approximation to a vacuum solution.

Spin connection

For example, its action on V_\nu^{\ a} is : In the Cartan formalism, the spin connection is used to define both torsion and curvature. These are easiest to read by working with differential forms, as this hides some of the profusion of indexes. The equations presented here are effectively a restatement of those that can be found in the article on the connection form and the curvature form. The primary difference is that these retain the indexes on the vierbein, instead of completely hiding them.