4.1. Gravitation and Electromagnetism as a Field Theory

The action for macroscopic gravity, electromagnetism and (possibly) charged relativistic dust is:

S = S_H + S_M + S_{EM} + S_q

where:

S_H[g^{\mu\nu}] = {c^4\over 16\pi G} \int R \sqrt{-g} \d^4 x S_M[g^{\mu\nu}, x^{\mu}] = -c\int \rho \sqrt{v_\mu v^\mu} \sqrt{-g} \d^4 x S_{EM}[g^{\mu\nu}, A^{\mu}] = -{1\over4\mu_0} \int F_{\alpha\beta} F^{\alpha\beta} \sqrt{-g} \d^4 x S_q[x^\mu, A^{\mu}] = -\int \rho_{EM} v^\mu A_\mu \sqrt{-g} \d^4 x

where x^\mu is the field of the matter, A^{\mu} is the electromagnetic field and g^{\mu\nu} is the gravitational field. We vary with respect to each of them to obtain (interacting) equations of motion. c is the speed of light, G is the gravitational constant, \mu_0 the permeability of vacuum. \rho is the mass density of the dust, \rho_{EM} is the charge density of the dust, v^\mu={\d x^\mu\over \d\tau} is 4-velocity of the dust, F_{\alpha\beta}=\nabla_\alpha A_\beta-\nabla_\beta A_\alpha is the electromagnetic field tensor, R is the Ricci scalar.

4.1.1. Gravitation

We vary with respect to g^{\mu\nu}. By changing the metric, we also change the invariant volume element (thus also \rho), so we need to be careful to vary properly. We start with S_H:

\delta S_H = \delta {c^4\over 16\pi G} \int R \sqrt{-g} \d^4 x = = {c^4\over 16\pi G} \int (\delta g^{\mu\nu}) R_{\mu\nu} \sqrt{-g} +g^{\mu\nu} (\delta R_{\mu\nu}) \sqrt{-g} +R (\delta \sqrt{-g}) \d^4 x= = {c^4\over 16\pi G} \int (\delta g^{\mu\nu}) R_{\mu\nu} \sqrt{-g} +g^{\mu\nu} \left( \nabla_\lambda(\delta \Gamma^\lambda_{\nu\mu}) -\nabla_\nu(\delta \Gamma^\lambda_{\lambda\mu}) \right)\sqrt{-g} +R ( -\half \sqrt{-g}\, g_{\mu\nu} (\delta g^{\mu\nu})) \d^4 x= = {c^4\over 16\pi G} \int (\delta g^{\mu\nu}) R_{\mu\nu} \sqrt{-g} + \left( \nabla_\lambda g^{\mu\nu}(\delta \Gamma^\lambda_{\nu\mu}) -\nabla_\nu g^{\mu\nu}(\delta \Gamma^\lambda_{\lambda\mu}) \right)\sqrt{-g} -\half R g_{\mu\nu} \sqrt{-g}\, (\delta g^{\mu\nu}) \d^4 x= = {c^4\over 16\pi G} \int (\delta g^{\mu\nu}) R_{\mu\nu} \sqrt{-g} -\half R g_{\mu\nu} \sqrt{-g}\, (\delta g^{\mu\nu}) \d^4 x= = {c^4\over 16\pi G} \int \left( R_{\mu\nu} -\half R g_{\mu\nu} \right) (\delta g^{\mu\nu}) \sqrt{-g} \d^4 x

Variation of S_M is:

\delta S_M = -c \delta \int \rho \sqrt{v_\mu v^\mu} \sqrt{-g} \d^4 x = = - \delta \int c \sqrt{\textgoth{p}_\mu \textgoth{p}^\mu} \d^4 x = = - \int c {\delta(g^{\mu\nu} \textgoth{p}_\mu \textgoth{p}_\nu) \over 2\sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} \d^4 x = = - \int c { \textgoth{p}_\mu \textgoth{p}_\nu \over 2\sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} \delta(g^{\mu\nu})\d^4 x = = - \int c { \rho v_\mu \rho v_\nu \sqrt{-g}^2 \over 2 \rho c \sqrt{-g} } \delta(g^{\mu\nu})\d^4 x = = - \int \half \rho v_\mu v_\nu \delta(g^{\mu\nu}) \sqrt{-g} \d^4 x

The variation of S_{EM} is:

\delta S_{EM} = -\delta \int {1\over 4\mu_0} F_{\alpha\beta} F^{\alpha\beta} \sqrt{-g}\d^4 x = = -\delta \int {1\over 4\mu_0} g^{\alpha\lambda} g^{\beta\rho} F_{\alpha\beta} F_{\lambda\rho} \sqrt{-g}\d^4 x = = -{1\over 4\mu_0} \int \left(\delta (g^{\alpha\lambda} g^{\beta\rho}) F_{\alpha\beta} F_{\lambda\rho} \sqrt{-g} + g^{\alpha\mu} g^{\beta\rho} F_{\alpha\beta} F_{\lambda\rho} \left(\delta \sqrt{-g} \right) \right)\d^4 x = = -{1\over 4\mu_0} \int \left(2(\delta g^{\alpha\lambda}) g^{\beta\rho} F_{\alpha\beta} F_{\lambda\rho} \sqrt{-g} + g^{\alpha\lambda} g^{\beta\rho} F_{\alpha\beta} F_{\lambda\rho} \left(-\half \sqrt{-g} g_{\mu\nu} (\delta g^{\mu\nu}) \right) \right)\d^4 x = = -{1\over 4\mu_0} \int \left(2(\delta g^{\alpha\lambda}) F_{\alpha\beta} F_\lambda{}^\beta -\half F_{\alpha\beta} F^{\alpha\beta} g_{\mu\nu} (\delta g^{\mu\nu}) \right) \sqrt{-g} \d^4 x = = -{1\over 2\mu_0} \int \left( F_{\mu\beta} F_\nu{}^\beta -{1\over 4} F_{\alpha\beta} F^{\alpha\beta} g_{\mu\nu} \right) (\delta g^{\mu\nu}) \sqrt{-g} \d^4 x

The variation of \delta S_q=0.

The equations of motion are:

{c^4\over 16\pi G} \left( R_{\mu\nu} -\half R g_{\mu\nu} \right) - \half \rho v_\mu v_\nu -{1\over 2\mu_0} \left( F_{\mu\beta} F_\nu{}^\beta -{1\over 4} F_{\alpha\beta} F^{\alpha\beta} g_{\mu\nu} \right) = 0

We rearrange:

R_{\mu\nu} -\half R g_{\mu\nu} = {8\pi G\over c^4} \rho v_\mu v_\nu +{8\pi G\over c^4}{1\over\mu_0} \left( F_{\mu\beta} F_\nu{}^\beta -{1\over 4} F_{\alpha\beta} F^{\alpha\beta} g_{\mu\nu} \right)

We define the stress energy tensor as:

(4.1.1.1)R_{\mu\nu} -\half R g_{\mu\nu} = {8\pi G\over c^4} T_{\mu\nu}

(4.1.1.2)T_{\mu\nu} = - {2\over\sqrt{-g}}{\delta (S_M + S_{EM} + S_q) \over \delta g^{\mu\nu}}

And we get:

(4.1.1.3)T_{\mu\nu} = T^M_{\mu\nu} + T^{EM}_{\mu\nu} T^M_{\mu\nu} = \rho v_\mu v_\nu T^{EM}_{\mu\nu} = {1\over \mu_0} \left( F_{\mu\beta} F_\nu{}^\beta -{1\over 4} F_{\alpha\beta} F^{\alpha\beta} g_{\mu\nu} \right)

The equations (4.1.1.1) are called Einstein’s equations and the equations (4.1.1.3) are stress energy tensors for the relativistic dust and electromagnetism. The equation (4.1.1.2) is the stress energy tensor corresponding to the given action. Sometimes it is not possible to write an action for more complex matter (perfect fluid, Navier-Stokes equations for fluid, …) in which case we cannot use (4.1.1.2), but we can still specify the stress energy tensor directly and (4.1.1.1) are the equations of motion.

4.1.2. Electromagnetism

We vary with respect to A^\mu. The variation of \delta S_H=0. The variation of \delta S_M = 0. The variation of S_{EM} is:

\delta S_{EM} = = -{1\over4\mu_0} \delta \int F_{\mu\nu} F^{\mu\nu} \sqrt{-g} \d^4 x = = -{1\over2\mu_0} \int F^{\mu\nu} (\delta F_{\mu\nu}) \sqrt{-g} \d^4 x = = -{1\over\mu_0} \int F^{\mu\nu} (\delta \partial_\nu A_\mu) \sqrt{-g} \d^4 x = = -{1\over\mu_0} \int F^{\mu\nu} \partial_\nu (\delta A_\mu) \sqrt{-g} \d^4 x = = {1\over\mu_0} \int \partial_\nu (F^{\mu\nu}\sqrt{-g}) (\delta A_\mu) \d^4 x = = {1\over\mu_0} \int \left( {1\over\sqrt{-g}}\partial_\nu (F^{\mu\nu}\sqrt{-g}) \right) (\delta A_\mu) \sqrt{-g}\d^4 x = = {1\over\mu_0} \int \nabla_\mu F^{\mu\nu} (\delta A_\nu) \sqrt{-g} \d^4 x

The variation of S_q is:

\delta S_q = =-\delta\int \rho_{EM} v^\nu A_\nu \sqrt{-g} \d^4 x = =-\int \rho_{EM} v^\nu (\delta A_\nu) \sqrt{-g} \d^4 x =

The equation of motion is:

{1\over\mu_0} \nabla_\mu F^{\mu\nu} - \rho_{EM} v^\nu = 0

Rearranging:

\nabla_\mu F^{\mu\nu} = \mu_0 \rho_{EM} v^\nu

4.1.3. Relativistic Dust

We vary the whole action with respect to x^\mu. The variation of \delta S_H=0. The variation of S_M is:

\delta S_M = -c\delta \int \rho \sqrt{v_\mu v^\mu} \sqrt{-g} \d^4 x = = - \delta \int c \sqrt{\textgoth{p}_\mu \textgoth{p}^\mu} \d^4 x = = - \int c {\delta(g^{\mu\nu} \textgoth{p}_\mu \textgoth{p}_\nu) \over 2\sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} \d^4 x = = - \int c { 2 g^{\mu\nu} \textgoth{p}_\mu (\delta \textgoth{p}_\nu) \over 2\sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} \d^4 x = = - \int c { \textgoth{p}_\mu \over \sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} (\delta \textgoth{p}^\mu) \d^4 x = = - \int c { \textgoth{p}_\mu \over \sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} \partial_\nu \left(\textgoth{p}^\nu(\delta x^\mu) - \textgoth{p}^\mu (\delta x^\nu)\right) \d^4 x = = \int c \partial_\nu \left({ \textgoth{p}_\mu \over \sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} \right) \left(\textgoth{p}^\nu(\delta x^\mu) - \textgoth{p}^\mu (\delta x^\nu)\right) \d^4 x = = \int c \left( \partial_\nu \left({ \textgoth{p}_\mu \over \sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} \right) -\partial_\mu \left({ \textgoth{p}_\nu \over \sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} \right) \right) \textgoth{p}^\nu(\delta x^\mu) \d^4 x = = \int c \left( \nabla_\nu \left({ \textgoth{p}_\mu \over \sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} \right) -\nabla_\mu \left({ \textgoth{p}_\nu \over \sqrt{\textgoth{p}_\alpha \textgoth{p}^\alpha}} \right) \right) \textgoth{p}^\nu(\delta x^\mu) \d^4 x = = \int \left( \nabla_\nu v_\mu -\nabla_\mu v_\nu \right) \rho v^\nu (\delta x^\mu) \sqrt{-g} \d^4 x = = \int \rho (\nabla_\nu v_\mu) v^\nu (\delta x^\mu) \sqrt{-g} \d^4 x

The variation of \delta S_{EM}=0. The variation of S_q is:

\delta S_q = - \delta \int \rho_{EM} v^\mu A_\mu \sqrt{-g} \d^4 x = = - \delta \int \textgoth{j}^\mu A_\mu \d^4 x = = - \int (\delta \textgoth{j}^\mu) A_\mu \d^4 x = = - \int \partial_\nu \left(\textgoth{j}^\nu (\delta x^\mu) - \textgoth{j}^\mu (\delta x^\nu)\right) A_\mu \d^4 x = = \int \left(\textgoth{j}^\nu (\delta x^\mu) - \textgoth{j}^\mu (\delta x^\nu)\right) \partial_\nu A_\mu \d^4 x = = \int \textgoth{j}^\nu (\delta x^\mu) (\partial_\nu A_\mu -\partial_\mu A_\nu) \d^4 x = = \int \rho_{EM} v^\nu (\nabla_\nu A_\mu -\nabla_\mu A_\nu) (\delta x^\mu) \sqrt{-g} \d^4 x = = -\int \rho_{EM} v^\nu F_{\mu\nu} (\delta x^\mu) \sqrt{-g} \d^4 x

The equation of motion is:

\rho (\nabla_\nu v_\mu) v^\nu -\rho_{EM} v^\nu F_{\mu\nu} = 0

Rearranging:

\rho (\nabla_\nu v_\mu) v^\nu = \rho_{EM} v^\nu F_{\mu\nu}

This is the geodesic equation with Lorentz force.

4.1.4. Equations of Motion

All together, the equations of motion are:

R_{\mu\nu} -\half R g_{\mu\nu} = {8\pi G\over c^4} \rho v_\mu v_\nu +{8\pi G\over c^4}{1\over\mu_0} \left( F_{\mu\beta} F_\nu{}^\beta -{1\over 4} F_{\alpha\beta} F^{\alpha\beta} g_{\mu\nu} \right) \nabla_\mu F^{\mu\nu} = \mu_0 \rho_{EM} v^\nu \rho (\nabla_\nu v_\mu) v^\nu = \rho_{EM} v^\nu F_{\mu\nu}

The first equation determines g_{\mu\nu} from the given sources (the stress energy tensors) on the right hand side, that depend on \rho, v^\mu, A^\mu and g_{\mu\nu}. The second equation determines A^\mu from the sources (\rho_{EM} and v^\mu) and from g_{\mu\nu} (through the covariant derivative). Finally, the last equation determines x^\mu and v^\mu from the given fields A^\mu (through the electromagnetic field tensor) and g_{\mu\nu} (through the covariant derivative).

Conservation

We apply covariant 4-divergence and use Bianci identities on the first equation:

0 = \nabla_\mu T^{\mu\nu} = \nabla_\mu (T^{\mu\nu}_M + T^{\mu\nu}_{EM})

So the total stress energy tensor is conserved. This fact makes the equations of motion (that follow from the action principle) not all independent. The third equation can be derived from the fist two as follows.

We calculate:

\nabla_\mu T^{\mu\nu}_{M} = \nabla_\mu (\rho v^\mu v^\nu) \nabla_\mu T^{\mu\nu}_{EM} = F^{\alpha\nu} \rho_{EM} v_\alpha

and we get:

\nabla_\mu (\rho v^\mu v^\nu) + F^{\alpha\nu} \rho_{EM} v_\alpha = 0 \nabla_\mu (\rho v^\mu) v^\nu + \rho v^\mu \nabla_\mu v^\nu + F^{\alpha\nu} \rho_{EM} v_\alpha = 0

The first term vanishes, because:

v_\nu \nabla_\mu (\rho v^\mu) v^\nu + v_\nu \rho v^\mu \nabla_\mu v^\nu + v_\nu F^{\alpha\nu} \rho_{EM} v_\alpha = 0 v_\nu \nabla_\mu (\rho v^\mu) v^\nu + v_\nu F^{\alpha\nu} \rho_{EM} v_\alpha = 0 c^2 \nabla_\mu (\rho v^\mu) + v_\nu F^{\alpha\nu} \rho_{EM} v_\alpha = 0 c^2 \nabla_\mu (\rho v^\mu) = 0

where we used v_\nu \nabla_\mu v^\nu=0 (follows from differentiating c^2 = v_\nu v^\nu) and v_\nu F^{\alpha\nu} v_\alpha=0 (contracting symmetric and antisymmetric tensors). We are left with:

\rho v^\mu \nabla_\mu v^\nu + F^{\alpha\nu} \rho_{EM} v_\alpha = 0 \rho v^\mu \nabla_\mu v^\nu = -F^{\alpha\nu} \rho_{EM} v_\alpha \rho v^\mu \nabla_\mu v^\nu = F^{\nu\alpha} \rho_{EM} v_\alpha

Which is the third equation.