Einstein–Hilbert action

Einstein–Hilbert action

The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein's field equations through the principle of least action. With the (+ − − −) metric signature the action is given as[1]

S= - {1 \over 2\kappa}\int R \sqrt{-g} \, d^4x \;,

where g = det(gμν) is the determinant of the metric tensor, R is the Ricci scalar, and κ = 8πGc − 4, where G is the Newton's gravitational constant and c is the speed of light in vacuum. The integral is taken over the whole spacetime if it converges. 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.

The action was first proposed by David Hilbert in 1915.

Contents

Discussion

The derivation of 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 by studying symmetries of the action.

In general relativity, the action is usually assumed to be a functional of the metric (and matter fields), and the connection is given by the Levi-Civita connection. 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 Einstein equations in the presence of matter are given by adding the matter action to the Hilbert-Einstein action.

Derivation of Einstein's field equations

Suppose that the full action of the theory is given by the Einstein-Hilbert term plus a term \mathcal{L}_\mathrm{M} describing any matter fields appearing in the theory.

S = \int \left[ {1 \over 2\kappa} \, R + \mathcal{L}_\mathrm{M} \right] \sqrt{-g} \, \mathrm{d}^4 x

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


\begin{align}
0 & = \delta S \\
  & = \int 
         \left[ 
            {1 \over 2\kappa} \frac{\delta (\sqrt{-g}R)}{\delta g^{\mu\nu}} + 
            \frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}}
         \right] \delta g^{\mu\nu}\mathrm{d}^4x \\
  & = \int 
        \left[ 
           {1 \over 2\kappa} \left( \frac{\delta R}{\delta g^{\mu\nu}} +
             \frac{R}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu\nu} } 
            \right) +
           \frac{1}{\sqrt{-g}} \frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}} 
        \right] \delta g^{\mu\nu} \sqrt{-g}\, \mathrm{d}^4x.
\end{align}

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

  \frac{\delta R}{\delta g^{\mu\nu}} + \frac{R}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu\nu}} 
= - 2 \kappa \frac{1}{\sqrt{-g}}\frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}},

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

 T_{\mu\nu}:=  \frac{-2}{\sqrt{-g}}\frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}} 
= -2 \frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{\mu\nu}} + g_{\mu\nu} \mathcal{L}_\mathrm{M}.

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 Carroll 2004.

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, and then the variation of the Ricci tensor. So, the Riemann curvature tensor is defined as,

 {R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma}
    - \partial_\nu\Gamma^\rho_{\mu\sigma}
    + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}
    - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma},

Since the Riemann curvature depends only on the Levi-Civita connection  \Gamma^\lambda_{\mu\nu}, the variation of the Riemann tensor can be calculated as,

\delta{R^\rho}_{\sigma\mu\nu} = \partial_\mu\delta\Gamma^\rho_{\nu\sigma} - \partial_\nu\delta\Gamma^\rho_{\mu\sigma} + \delta\Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} + \Gamma^\rho_{\mu\lambda} \delta\Gamma^\lambda_{\nu\sigma}
- \delta\Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma} - \Gamma^\rho_{\nu\lambda} \delta\Gamma^\lambda_{\mu\sigma}.

Now, since \delta\Gamma^\rho_{\nu\mu} is the difference of two connections, it is a tensor and we can thus calculate its covariant derivative,

\nabla_\lambda (\delta \Gamma^\rho_{\nu\mu} ) = \partial_\lambda (\delta \Gamma^\rho_{\nu\mu} ) + \Gamma^\rho_{\sigma\lambda} \delta\Gamma^\sigma_{\nu\mu} - \Gamma^\sigma_{\nu\lambda} \delta \Gamma^\rho_{\sigma\mu} - \Gamma^\sigma_{\mu\lambda} \delta \Gamma^\rho_{\nu\sigma}

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,

\delta R^\rho{}_{\sigma\mu\nu} = \nabla_\mu (\delta \Gamma^\rho_{\nu\sigma}) - \nabla_\nu (\delta \Gamma^\rho_{\mu\sigma}).

We may now obtain the variation of the Ricci curvature tensor simply by contracting two indices of the variation of the Riemann tensor,

 \delta R_{\mu\nu} \equiv \delta R^\rho{}_{\mu\rho\nu} = \nabla_\rho (\delta \Gamma^\rho_{\nu\mu}) - \nabla_\nu (\delta \Gamma^\rho_{\rho\mu}).

The Ricci scalar is defined as

 R = g^{\mu\nu} R_{\mu\nu}.\!

Therefore, its variation with respect to the inverse metric gμν is given by


\begin{align}
\delta R &= R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu} \delta R_{\mu\nu}\\
         &= R_{\mu\nu} \delta g^{\mu\nu} + \nabla_\sigma \left( g^{\mu\nu} \delta\Gamma^\sigma_{\nu\mu} - g^{\mu\sigma}\delta\Gamma^\rho_{\rho\mu} \right) 
\end{align}

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, \nabla_\sigma g^{\mu\nu} = 0 .

The last term, \nabla_\sigma ( g^{\mu\nu} \delta\Gamma^\sigma_{\nu\mu} - g^{\mu\sigma}\delta\Gamma^\rho_{\rho\mu} ) , multiplied by \sqrt{-g} becomes a total derivative, since


\sqrt{-g}A^a_{;a} = (\sqrt{-g}A^a)_{,a} \;\mathrm{or}\;
\sqrt{-g}\nabla_\mu A^\mu = \partial_\mu\left(\sqrt{-g}A^\mu\right)

and thus by Stokes' theorem only yields a boundary term when integrated. Hence when the variation of the metric δgμν vanishes at infinity, this term does not contribute to the variation of the action. And we thus obtain,

\frac{\delta R}{\delta g^{\mu\nu}} = R_{\mu\nu}.

Variation of the determinant

Jacobi's formula, the rule for differentiating a determinant, gives:

\,\! \delta g = \delta det(g_{\mu\nu}) = g \, g^{\mu\nu} \delta g_{\mu\nu}

or one could transform to a coordinate system where g_{\mu\nu}\! is diagonal and then apply the product rule to differentiate the product of factors on the main diagonal.

Using this we get

\begin{align}
\delta \sqrt{-g} 
&= -\frac{1}{2\sqrt{-g}}\delta g 
&= \frac{1}{2} \sqrt{-g} (g^{\mu\nu} \delta g_{\mu\nu})
&= -\frac{1}{2} \sqrt{-g} (g_{\mu\nu} \delta g^{\mu\nu}),\end{align}

In the last equality we used the fact that: gμνδgμν = − gμνδgμν Which follows from the requirement that the inverse of the variationed metric matrix gμν + δgμν is gμν − δgμν.

Thus we conclude that

\frac{1}{\sqrt{-g}}\frac{\delta \sqrt{-g}}{\delta g^{\mu\nu} } = -\frac{1}{2} g_{\mu\nu} .

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,

R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \frac{8 \pi G}{c^4}  T_{\mu\nu},

which is Einstein's field equation and

\kappa = \frac{8 \pi G}{c^4}

has been chosen such that the non-relativistic limit yields the usual form of Newton's gravity law, where G is the gravitational constant (see here for details).

Cosmological constant

Sometimes, a cosmological constant Λ is included in the Lagrangian so that the new action

S = \int  \left[ {1 \over 2\kappa} \left( R - 2 \Lambda \right) + \mathcal{L}_\mathrm{M} \right] \sqrt{-g} \, \mathrm{d}^4 x

yields the field equations:

R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4}  T_{\mu \nu} \,.

See also

References

  1. ^ Richard P. Feynman, Feynman Lectures on Gravitation, Addison-Wesley, 1995, ISBN 0-201-62734-5, p. 136, eq. (10.1.2)

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Action d’Einstein-Hilbert — Action d Einstein Hilbert Pour les articles homonymes, voir Einstein (homonymie). L action d Einstein Hilbert est un objet mathématique homogène à une action. Elle est utilisée pour dériver les équations du champ de la relativité générale d… …   Wikipédia en Français

  • Action d'Einstein-Hilbert — Pour les articles homonymes, voir Einstein (homonymie). L action d Einstein Hilbert est un objet mathématique homogène à une action. Elle est utilisée pour dériver les équations du champ de la relativité générale d Einstein au moyen d un principe …   Wikipédia en Français

  • Einstein–Cartan theory — in theoretical physics extends general relativity to correctly handle spin angular momentum. As the master theory of classical physics general relativity has one known flaw: it cannot describe spin orbit coupling , i.e., exchange of intrinsic… …   Wikipedia

  • Einstein, Albert — born March 14, 1879, Ulm, Württemberg, Ger. died April 18, 1955, Princeton, N.J., U.S. German Swiss U.S. scientist. Born to a Jewish family in Germany, he grew up in Munich, and his family moved to Switzerland in 1894. He became a junior examiner …   Universalium

  • Einstein-aether theory — In physics the Einstein æther theory , also called æ theory is a controversial generally covariant generalization of general relativity which describes a spacetime endowed with both a metric and a unit timelike vector field named the æther. In… …   Wikipedia

  • Einstein field equations — General relativity Introduction Mathematical formulation Resources Fundamental concepts …   Wikipedia

  • Action (physics) — In physics, the action is a particular quantity in a physical system that can be used to describe its operation. Action is an alternative to differential equations. The action is not necessarily the same for different types of systems.The action… …   Wikipedia

  • Einstein (Homonymie) — Cette page d’homonymie répertorie les différents sujets et articles partageant un même nom. L einstein est une unité de mesure correspondant à l énergie lumineuse absorbée par une mole de réactif. Personnes Albert Einstein est le célèbre… …   Wikipédia en Français

  • Einstein (homonymie) — Cette page d’homonymie répertorie les différents sujets et articles partageant un même nom. L einstein est une unité de mesure correspondant à l énergie lumineuse absorbée par une mole de réactif. Personnes Albert Einstein est le célèbre… …   Wikipédia en Français

  • Action (Physique) — Pour les articles homonymes, voir Action. En physique théorique, l’action est une grandeur physique scalaire ayant pour dimension le produit d une énergie par un temps. Par exemple, le moment cinétique est une action. Sommaire 1 Principe de moind …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”