In the
general theory of relativity, 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 diffe ...
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 par ...
within it.
The equations were published by Einstein in 1915 in the form of a
tensor equation which related the local ' (expressed by the
Einstein tensor) 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 ...
and stress within that spacetime (expressed by the
stress–energy tensor).
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 classica ...
s are related to the distribution of
charges and
currents 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.
Th ...
, the EFE relate the
spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the
metric tensor 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 function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
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 o ...
l trajectories of particles and radiation (
geodesics) 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 distan ...
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 fo ...
.
Exact solutions for the EFE can only be found under simplifying assumptions such as
symmetry. 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 holes and the
expanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations from
flat spacetime, leading to the
linearized EFE. These equations are used to study phenomena such as
gravitational waves.
Mathematical form
The Einstein field equations (EFE) may be written in the form:
:
where is the
Einstein tensor,
is the
metric tensor, is the
stress–energy tensor, is the
cosmological constant and is the Einstein gravitational constant.
The
Einstein tensor is defined as
:
where is the
Ricci curvature tensor, and is the
scalar curvature. 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
:
where is the
Newtonian constant of gravitation 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 fo ...
in vacuum.
The EFE can thus also be written as
:
In standard units, each term on the left has units of 1/length
2.
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 reduce the number of independent equations from 10 to 6, leaving the metric with four
gauge-fixing 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 is e ...
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
, 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 function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
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:
The third sign above is related to the choice of convention for the Ricci tensor:
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:
The sign of the cosmological term would change in both these versions if the metric
sign convention 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
where is the spacetime dimension. Solving for and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form:
In dimensions this reduces to
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
in the expression on the right with the
Minkowski metric without significant loss of accuracy).
The cosmological constant
In the Einstein field equations
the term containing the
cosmological constant 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 showed that our universe is
expanding.
Einstein then abandoned , remarking to
George Gamow "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 observations have shown an
accelerating expansion of the universe, 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:
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 a ...
with an
energy density and isotropic pressure that are fixed constants and given by
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
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.
Th ...
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 o ...
are linear in the
electric 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 t ...
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, q ...
, which is linear in the
wavefunction.
The correspondence principle
The EFE reduce to
Newton's law of gravity by using both the
weak-field approximation and the
slow-motion approximation. 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
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
. By setting in the
trace-reversed field equations, the vacuum equations can be written as
In the case of nonzero cosmological constant, the equations are
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 ...
is the simplest example of a vacuum solution. Nontrivial examples include the
Schwarzschild solution and the
Kerr solution.
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 ...
s with a vanishing
Ricci tensor, , are referred to as
Ricci-flat manifolds 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 is e ...
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 classica ...
in
free space, i.e. if the
electromagnetic stress–energy tensor
is used, then the Einstein field equations are called the ''Einstein–Maxwell equations'' (with
cosmological constant , taken to be zero in conventional relativity theory):
Additionally, the
covariant Maxwell equations are also applicable in free space:
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 differ ...
, and the brackets denote
anti-symmetrization. The first equation asserts that the 4-
divergence of the
2-form is zero, and the second that its
exterior derivative 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 ...
that in a coordinate chart it is possible to introduce an electromagnetic field potential such that
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 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 diffe ...
. 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 approximations. 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's ''Glossographia'', and in 1731 taken up in Latin by German philosophe ...
. It leads to the prediction of
black holes 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. A ...
.
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 describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
. 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 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 diffe ...
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 ...
. 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, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of
gravitational radiation.
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
using the
Levi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as:
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
*
Equivalence principle
*
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
*
Hamilton–Jacobi–Einstein equation
*
Mathematics of general relativity
*
Numerical relativity
*
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 b ...
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 Equationsby
MIT Physics Professor Edmund Bertschinger.
Arch and scaffold: How Einstein found his field equationsPhysics Today November 2015, History of the Development of the Field Equations
External images
*
Suzanne Imber,
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