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The Einstein-Hilbert action in general relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
 is the action
Action (physics)

In modern physics, action is an attribute of the development of a physical system over a period of time, namely amount by which the Phase of the wave function has changed....
 that yields the Einstein's field equations when varied to obtain equations of motion for the spacetime metric. The action was first proposed by David Hilbert
David Hilbert

David Hilbert was a Germany mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries....
 in 1915 (Hilbert, 1915). The derivation of the Einstein equations from an action has several advantages. First of all, it allows for easy unification of general relativity with other classical fields theories (such as Maxwell theory), which are also formulated in terms of an action.






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The Einstein-Hilbert action in general relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
 is the action
Action (physics)

In modern physics, action is an attribute of the development of a physical system over a period of time, namely amount by which the Phase of the wave function has changed....
 that yields the Einstein's field equations when varied to obtain equations of motion for the spacetime metric. The action was first proposed by David Hilbert
David Hilbert

David Hilbert was a Germany mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries....
 in 1915 (Hilbert, 1915). The derivation of the Einstein equations from an action has several advantages. First of all, it allows for easy unification of general relativity with other classical fields theories (such as Maxwell theory), which are also formulated in terms of an action. In the process the derivation from an action identifies a natural candidate for the source term coupling the metric to matter fields. Moreover, the action allows for the easy identification of conserved quantities through Noether's theorem
Noether's theorem

Noether's theorem states that any derivative Symmetry in physics of the action of a physical system has a corresponding conservation law. The action of a physical system is an integral of a so-called Lagrangian function, from which the system's behavior can be determined by the principle of least action....
 by studying symmetries of the action.

In general relativity, the action is usually assumed to be a functional
Functional (mathematics)

In mathematics, a functional is traditionally a map from a vector space to the Field underlying the vector space, which is usually the real numbers....
  of the metric (and matter fields), and the connection
Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner....
 is given by the Levi-Civita connection
Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is the Torsion -free Riemannian connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric....
. The Palatini formulation of general relativity assumes the metric and connection to be independent, and varies with respect to both independently, which makes it possible to include fermionic matter fields with non-integral spin.

The action which gives rise to the vacuum Einstein equations is given by the following integral of the Lagrangian
Lagrangian

The Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics....


where ' is the determinant of a spacetime Lorentz metric (with units corrected), R is the Ricci scalar, is a universal constant, the Lagrangian being , and the integral is taken over a region of spacetime
Spacetime

In physics, spacetime is any mathematical model that combines space and Time in physics into a single continuum . Spacetime is usually interpreted with space being Three-dimensional space and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions....
. The Einstein equations in the presence of matter are given by adding the Lagrangian for the matter into the integral. (In this article "Lagrangian" means "Lagrangian density", a scalar density with SI units of Joules per cubic meter; other authors occasionally use it to mean the integral of the Lagrangian density over space or over spacetime.)

Note that is an invariant 4-volume element, so the action can be also written in the following (somewhat more elegant) fashion:

Derivation of Einstein's field equations

Suppose that the full action of the theory is given by the Einstein-Hilbert term plus a term describing any matter fields appearing in the theory.

The action principle then tells us that the variation of this action with respect to the inverse metric is zero, yielding

Since this equation should hold for any variation , it implies that

is the equation of motion
Equation of motion

In physics, equations of motion are equations that describe the behavior of a system as a function of time. Sometimes the term refers to the differential equations that the system satisfies , and sometimes to the solutions to those equations....
 for the metric field. The right hand side of this equation is (by definition) proportional to the stress-energy tensor
Stress-energy tensor

The stress-energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress of Newtonian physics....
,

To calculate the left hand side of the equation we need the variations of the Ricci scalar R and the determinant of the metric. These can be obtained by standard text book calculations such as the one given below, which is strongly based on the one given in .

Variation of the Riemann tensor, the Ricci tensor, and the Ricci scalar

To calculate the variation of the Ricci scalar we calculate first the variation of the Riemann curvature tensor
Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann?Christoffel tensor is the most standard way to express curvature of Riemannian manifolds....
, and then the variation of the Ricci tensor. So, the Riemann curvature tensor is defined as,

Since the Riemann curvature depends only on the Levi-Civita connection
Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is the Torsion -free Riemannian connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric....
 , 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
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 differential operator, to be contrasted with the approach given by a connection on the frame bundle &mdas...
,

We can now cleverly 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,

The Ricci scalar is defined as Therefore, its variation with respect to the inverse metric is given by

In the second line we used the previously obtained result for the variation of the Ricci curvature and the metric compatibility of the covariant derivative, .

The last term is a total derivative
Total derivative

In the mathematics of differential calculus, the term total derivative has a number of closely related meanings.* The total derivative of a function, f, of several variables, e.g., t,x,y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative....
 and thus by Stokes' theorem
Stokes' theorem

In differential geometry, Stokes' theorem is a statement about the integral of differential forms which generalizes several theorems from vector calculus....
 only yields a boundary term when integrated. Hence when the variation of the metric vanishes at infinity, this term does not contribute to the variation of the action. And we thus obtain,

Variation of the determinant

Jacobi's formula
Jacobi's formula

In matrix calculus, Jacobi's formula expresses the differential of the determinant of a matrix A in terms of the adjugate of A and the derivative of A....
, the rule for differentiating a determinant
Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
, gives:

or one could transform to a coordinate system where is diagonal and then apply the product rule to differentiate the product of factors on the main diagonal.

Using this we get

and thus conclude that

Equation of motion

Now that we have all the necessary variations at our disposal, we can insert them into the equation of motion for the metric field to obtain,

which is Einstein's field equation and

has been chosen such that the non-relativistic limit yields the usual form of Newton's gravity law
Newton's law of universal gravitation

Isaac Newton's law of universal gravitation is an empirical physical law describing the gravitational attraction between bodies with mass. It is a part of classical mechanics and was first formulated in Newton's work Philosophiae Naturalis Principia Mathematica, first published on July 5 1687....
, where G is the gravitational constant
Gravitational constant

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitation between objects with mass....
.

Cosmological constant

Sometimes, a cosmological constant
Cosmological constant

In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a Einstein's universe....
 ? is included in the Lagrangian
Lagrangian

The Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics....
 so that the new action

yields the field equations:

See also


  • Belinfante-Rosenfeld tensor
  • Brans-Dicke theory
    Brans-Dicke theory

    In theoretical physics, the Brans-Dicke theory of gravitation is a theoretical framework to explain gravitation. It is a well-known competitor of Albert Einstein's more popular theory of general relativity....
     (in which the constant k is replaced by a scalar field).
  • Einstein-Cartan theory
  • f(R) gravity
    F(R) gravity

    f gravity is a type of modified gravity theory proposed as an alternative to Einstein's General Relativity. Although it is an active field of research, there are known problems with the theory....
  • Gibbons-Hawking-York boundary term
    Gibbons-Hawking-York boundary term

    In general relativity, the Gibbons-Hawking-York boundary term is a term that needs to be added to the Einstein-Hilbert action when the underlying spacetime manifold has a boundary....
  • Palatini action
  • Teleparallelism
    Teleparallelism

    Teleparallelism , was an attempt by Albert Einstein to unify electromagnetism and gravity. The idea is to use a geometry with a pseudo-Riemannian metric of metric signature , vanishing curvature, and non-vanishing torsion, and to use Cartan connection applicationss, rather than the metric, as basic variables....
  • Variational methods in general relativity
    Variational methods in general relativity

    Variational methods in general relativity refers to various mathematical techniques that employ the use of variational calculus in Albert Einstein's theory of general relativity....