# Einstein–Hilbert action

**Einstein-Hilbert actionEinstein–Hilbert LagrangianEinstein-Hilbert LagrangianEinstein–HilbertEinstein–Hilbert functionalEinstein–Hilbert termgravityHilbert's actionHilbert's action principleHilbert-Einstein Lagrangian**

The Einstein–Hilbert action (also referred to as [[Relativity priority dispute#Did Hilbert claim priority for parts of General Relativity?|Hilbert action]] ) in general relativity is the action that yields the Einstein field equations through the principle of least action.wikipedia

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### Principle of least action

**principle of stationary actionleast action principleleast action**

The Einstein–Hilbert action (also referred to as [[Relativity priority dispute#Did Hilbert claim priority for parts of General Relativity?|Hilbert action]] ) in general relativity is the action that yields the Einstein field equations through the principle of least action.

The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action).

### Scalar curvature

**Ricci scalarcurvaturecurvature scalar**

where is the determinant of the metric tensor matrix, R is the Ricci scalar, and is Einstein's constant (G is the gravitational constant and c is the speed of light in vacuum).

In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action.

### Relativity priority dispute

**Hilbert actionprioritythe article on this dispute**

The Einstein–Hilbert action (also referred to as [[Relativity priority dispute#Did Hilbert claim priority for parts of General Relativity?|Hilbert action]] ) in general relativity is the action that yields the Einstein field equations through the principle of least action.

### David Hilbert

**HilbertHilbert, DavidD. Hilbert**

The action was first proposed by David Hilbert in 1915.

Nearly simultaneously David Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (see Einstein–Hilbert action).

### Gibbons–Hawking–York boundary term

**GHY boundary termsGibbons-Hawking-York boundary term**

The boundary term is in general non-zero, because the integrand depends not only on but also on its partial derivatives ; see the article Gibbons–Hawking–York boundary term for details.

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.

### General relativity

**general theory of relativitygeneral relativity theoryrelativity**

The Einstein–Hilbert action (also referred to as [[Relativity priority dispute#Did Hilbert claim priority for parts of General Relativity?|Hilbert action]] ) in general relativity is the action that yields the Einstein field equations through the principle of least action.

### Palatini identity

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:

A proof can be found in the entry Einstein–Hilbert action.

### Action (physics)

**actionaction principleaction integral**

The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle.

### Einstein–Cartan theory

**Einstein-CartanEinstein-Cartan gravityEinstein–Cartan**

The theory of general relativity was originally formulated in the setting of Riemannian geometry by the Einstein-Hilbert action, out of which arise the Einstein field equations.

### Tetradic Palatini action

The Einstein–Hilbert action for general relativity was first formulated purely in terms of the space-time metric.

### Komar superpotential

In general relativity, the Komar superpotential, corresponding to the invariance of the Hilbert-Einstein Lagrangian, is the tensor density:

### Stress–energy tensor

**energy–momentum tensorenergy-momentum tensorstress-energy tensor**

The right hand side of this equation is (by definition) proportional to the stress-energy tensor,

See Einstein–Hilbert action for more information.

### F(R) gravity

**f(R) theoryf''(''R'') modified gravityf''(''R'') gravity**

In f(R) gravity, one seeks to generalize the Lagrangian of the Einstein–Hilbert action:

### Lagrangian (field theory)

**LagrangianLagrangian densityLagrangian field theory**

When a cosmological constant Λ is included in the Lagrangian, the action:

The integral of is known as the Einstein-Hilbert action.

### Kaluza–Klein theory

**Kaluza–KleinKaluza–Klein theoriesKaluza-Klein**

Next, one takes this scalar curvature as the Lagrangian density, and, from this, constructs the Einstein–Hilbert action for the bundle, as a whole.

### Einstein field equations

**Einstein field equationEinstein's field equationsEinstein's field equation**

### Variational methods in general relativity

**variational approach to general relativityvariational-principle formulation**

This used the functional now called the Einstein-Hilbert action.

### Metric tensor

**metricmetricsround metric**

where is the determinant of the metric tensor matrix, R is the Ricci scalar, and is Einstein's constant (G is the gravitational constant and c is the speed of light in vacuum).

### Einstein's constant

**Einstein constantEinstein's constant of gravitation**

where is the determinant of the metric tensor matrix, R is the Ricci scalar, and is Einstein's constant (G is the gravitational constant and c is the speed of light in vacuum).

### Gravitational constant

**Newton's constantGuniversal gravitational constant**

### Speed of light

**clight speedspeed of light in vacuum**

### Spacetime

**space-timespace-time continuumspace and time**

If it converges, the integral is taken over the whole spacetime.

### Euler–Lagrange equation

**Euler–Lagrange equationsEuler-Lagrange equationLagrange's equation**

If it does not converge, S is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the Euler–Lagrange equation of the Einstein–Hilbert action.

### Maxwell's equations

**Maxwell equationsMaxwell equationMaxwell’s equations**

First of all, it allows for easy unification of general relativity with other classical field theories (such as Maxwell theory), which are also formulated in terms of an action.