HOME

TheInfoList



OR:

In the
general theory of relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
to the distribution of
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...
within it. The equations were published by Einstein in 1915 in the form of a
tensor equation In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
which related the local ' (expressed by the
Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field eq ...
) with the local energy,
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
and stress within that spacetime (expressed by the
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
). Analogously to the way that
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
s are related to the distribution of
charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqua ...
s and
current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
s via
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
, the EFE relate the
spacetime geometry In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
to the distribution of mass–energy, momentum and stress, that is, they determine the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s when used in this way. The solutions of the EFE are the components of the metric tensor. The
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
l trajectories of particles and radiation (
geodesics In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
) in the resulting geometry are then calculated using the
geodesic equation In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection ...
. As well as implying local energy–momentum conservation, the EFE reduce to
Newton's law of gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
in the limit of a weak gravitational field and velocities that are much less than the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. Exact solutions for the EFE can only be found under simplifying assumptions such as
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
. Special classes of
exact solutions In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
are most often studied since they model many gravitational phenomena, such as
rotating black hole A rotating black hole is a black hole that possesses angular momentum. In particular, it rotates about one of its axes of symmetry. All celestial objects – planets, stars (Sun), galaxies, black holes – spin. Types of black holes There a ...
s and the
expanding universe The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not exp ...
. Further simplification is achieved in approximating the spacetime as having only small deviations from
flat spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
, leading to the linearized EFE. These equations are used to study phenomena such as
gravitational waves Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that Wave propagation, propagate as waves outward from their source at the speed of light. They were first proposed by Oliv ...
.


Mathematical form

The Einstein field equations (EFE) may be written in the form: :G_ + \Lambda g_ = \kappa T_ where is the
Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field eq ...
, g_ is the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, is the
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
, is the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
and is the Einstein gravitational constant. The
Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field eq ...
is defined as :G_ = R_ - \frac R g_, where is the
Ricci curvature tensor 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 ...
, and is the
scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) 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 ...
. This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives. The Einstein gravitational constant is defined as :\kappa = \frac \approx 2.076647442844\times10^ \, \textrm^ , where is the
Newtonian constant of gravitation The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in th ...
and is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in vacuum. The EFE can thus also be written as :R_ - \frac R g_ + \Lambda g_ = \kappa T_. In standard units, each term on the left has units of 1/length2. The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime. These equations, together with the
geodesic equation In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection ...
, which dictates how freely falling matter moves through spacetime, form the core of the mathematical formulation of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four
Bianchi identities In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algebra ...
reduce the number of independent equations from 10 to 6, leaving the metric with four
gauge-fixing In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, which correspond to the freedom to choose a coordinate system. Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when is everywhere zero) define
Einstein manifold 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 i ...
s. The equations are more complex than they appear. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor g_, since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s.


Sign convention

The above form of the EFE is the standard established by Misner, Thorne, and Wheeler (MTW). The authors analyzed conventions that exist and classified these according to three signs ( 1 2 3: \begin g_ & = 1\times \operatorname(-1,+1,+1,+1) \\ pt_ & = 2\times \left(\Gamma^\mu_ - \Gamma^\mu_ + \Gamma^\mu_\Gamma^\sigma_ - \Gamma^\mu_\Gamma^\sigma_\right) \\ ptG_ & = 3\times \kappa T_ \end The third sign above is related to the choice of convention for the Ricci tensor: R_ = 2\times 3\times _ With these definitions Misner, Thorne, and Wheeler classify themselves as , whereas Weinberg (1972) is , Peebles (1980) and Efstathiou et al. (1990) are , Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) and Peacock (1999) are . Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative: R_ - \frac R g_ - \Lambda g_ = -\kappa T_. The sign of the cosmological term would change in both these versions if the metric
sign convention In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...
is used rather than the MTW metric sign convention adopted here.


Equivalent formulations

Taking the trace with respect to the metric of both sides of the EFE one gets R - \frac R + D \Lambda = \kappa T , where is the spacetime dimension. Solving for and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form: R_ - \frac \Lambda g_ = \kappa \left(T_ - \fracTg_\right) . In dimensions this reduces to R_ - \Lambda g_ = \kappa \left(T_ - \fracT\,g_\right) . Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace g_ in the expression on the right with the
Minkowski metric In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
without significant loss of accuracy).


The cosmological constant

In the Einstein field equations G_ + \Lambda g_ = \kappa T_ \,, the term containing the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
was absent from the version in which he originally published them. Einstein then included the term with the cosmological constant to allow for a universe that is not expanding or contracting. This effort was unsuccessful because: * any desired steady state solution described by this equation is unstable, and * observations by
Edwin Hubble Edwin Powell Hubble (November 20, 1889 – September 28, 1953) was an Americans, American astronomer. He played a crucial role in establishing the fields of extragalactic astronomy and observational cosmology. Hubble proved that many objects ...
showed that our universe is expanding. Einstein then abandoned , remarking to
George Gamow George Gamow (March 4, 1904 – August 19, 1968), born Georgiy Antonovich Gamov ( uk, Георгій Антонович Гамов, russian: Георгий Антонович Гамов), was a Russian-born Soviet and American polymath, theoreti ...
"that the introduction of the cosmological term was the biggest blunder of his life". The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recent
astronomical Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxies ...
observations have shown an
accelerating expansion of the universe Observations show that the expansion of the universe is accelerating, such that the velocity at which a distant galaxy recedes from the observer is continuously increasing with time. The accelerated expansion of the universe was discovered durin ...
, and to explain this a positive value of is needed. The cosmological constant is negligible at the scale of a galaxy or smaller. Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor: T_^\mathrm = - \frac g_ \,. This tensor describes a
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
with an
energy density In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or . Often only the ''useful'' or extract ...
and isotropic pressure that are fixed constants and given by \rho_\mathrm = - p_\mathrm = \frac, where it is assumed that has SI unit m and is defined as above. The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.


Features


Conservation of energy and momentum

General relativity is consistent with the local conservation of energy and momentum expressed as \nabla_\beta T^ = _ = 0. which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition.


Nonlinearity

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example,
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
are linear in the
electric Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, which is linear in the
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
.


The correspondence principle

The EFE reduce to
Newton's law of gravity Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
by using both the
weak-field approximation In the theory of general relativity, linearized gravity is the application of perturbation theory to the Metric tensor (general relativity), metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effect ...
and the
slow-motion approximation The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
. In fact, the constant appearing in the EFE is determined by making these two approximations.


Vacuum field equations

If the energy–momentum tensor is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting in the trace-reversed field equations, the vacuum equations can be written as R_ = 0 \,. In the case of nonzero cosmological constant, the equations are R_ = \frac g_ \,. The solutions to the vacuum field equations are called vacuum solutions. Flat
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
is the simplest example of a vacuum solution. Nontrivial examples include the
Schwarzschild solution In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
and the
Kerr solution The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of gene ...
.
Manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s with a vanishing
Ricci tensor 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 ...
, , are referred to as
Ricci-flat manifold In the mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are ...
s and manifolds with a Ricci tensor proportional to the metric as
Einstein manifold 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 i ...
s.


Einstein–Maxwell equations

If the energy–momentum tensor is that of an
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
in
free space A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
, i.e. if the
electromagnetic stress–energy tensor In relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electrom ...
T^ = \, -\frac \left( ^\psi ^\beta + \tfrac g^ F_ F^\right) is used, then the Einstein field equations are called the ''Einstein–Maxwell equations'' (with
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
, taken to be zero in conventional relativity theory): G^ + \Lambda g^ = \frac \left( ^\psi ^\beta + \tfrac g^ F_ F^\right). Additionally, the covariant Maxwell equations are also applicable in free space: \begin _ &= 0 \\ F_&=\tfrac\left(F_ + F_+F_\right)=\tfrac\left(F_ + F_+F_\right)= 0. \end where the semicolon represents a
covariant derivative 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 different ...
, and the brackets denote anti-symmetrization. The first equation asserts that the 4-
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of the
2-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
is zero, and the second that its
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
is zero. From the latter, it follows by the
Poincaré lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another diff ...
that in a coordinate chart it is possible to introduce an electromagnetic field potential such that F_ = A_ - A_ = A_ - A_ in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived. However, there are global solutions of the equation that may lack a globally defined potential.


Solutions

The solutions of the Einstein field equations are
metrics Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as
post-Newtonian approximation In general relativity, the post-Newtonian expansions (PN expansions) are used for finding an approximate solution of the Einstein field equations for the metric tensor (general relativity), metric tensor. The approximations are expanded in small ...
s. Even so, there are several cases where the field equations have been solved completely, and those are called
exact solutions In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
. The study of exact solutions of Einstein's field equations is one of the activities of
cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...
. It leads to the prediction of
black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
s and to different models of evolution of the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
. One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum. In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright, self-similar solutions to the Einstein field equations are fixed points of the resulting
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
. New solutions have been discovered using these methods by LeBlanc and Kohli and Haslam.


The linearized EFE

The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
is very weak and the
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
approximates that of
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the
Minkowski metric In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of
gravitational radiation Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
.


Polynomial form

Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written \det(g) = \tfrac \varepsilon^ \varepsilon^ g_ g_ g_ g_ using the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for some ...
; and the inverse of the metric in 4 dimensions can be written as: g^ = \frac\,. Substituting this definition of the inverse of the metric into the equations then multiplying both sides by a suitable power of to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.


See also

*
Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the act ...
*
Equivalence principle In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (suc ...
*
Exact solutions in general relativity 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 sh ...
*
General relativity resources General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. G ...
*
History of general relativity General relativity is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915, with contributions by many others after 1915. According to general relativity, the observed gravitational attraction between masses results ...
*
Hamilton–Jacobi–Einstein equation In general relativity, the Hamilton–Jacobi–Einstein equation (HJEE) or Einstein–Hamilton–Jacobi equation (EHJE) is an equation in the Hamiltonian formulation of geometrodynamics in superspace, cast in the "geometrodynamics era" around t ...
*
Mathematics of general relativity 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 man ...
*
Numerical relativity Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars and ...
*
Ricci calculus In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be cal ...


Notes


References

See
General relativity resources General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. G ...
. * * *


External links

*
Caltech Tutorial on Relativity
— A simple introduction to Einstein's Field Equations.

— An explanation of Einstein's field equation, its derivation, and some of its consequences
Video Lecture on Einstein's Field Equations
by
MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
Physics Professor Edmund Bertschinger.
Arch and scaffold: How Einstein found his field equations
Physics Today November 2015, History of the Development of the Field Equations


External images



*
Suzanne Imber Suzanne Mary Imber (born May 1983) is a British planetary scientist specialising in space weather at the University of Leicester. She was the winner of the 2017 BBC Two television programme ''Astronauts, Do You Have What It Takes?''.
, ttps://imaggeo.egu.eu/view/886/ "The impact of general relativity on the Atacama Desert" Einstein field equation on the side of a train in Bolivia. {{DEFAULTSORT:Einstein Field Equations Albert Einstein Equations of physics General relativity Partial differential equations