Einstein-Maxwell Theory

EQUATIONS OF MOTION

Einstein gravity coupled to the Maxwell electromagnetic field with SI units and MTW sign conventions:

(1)   \begin{equation*} S[g_{\mu\nu}, A_\sigma]  = \frac{c^3}{16\pi G} \int d^4 x \sqrt{-g} R - \frac{1}{4}\sqrt{\frac{\epsilon_0}{\mu_0}} \int d^4 x\sqrt{-g} F^{\mu\nu} F_{\mu\nu} + \frac{1}{c} \int d^4x \sqrt{-g} A_\mu J^\mu \end{equation*}

where G is Newton’s constant. The speed of light is c, the permittivity of free space is \epsilon_0, and the permeability of free space is \mu_0; these are related by \epsilon_0 \mu_0 c^2 = 1. The field strength tensor is F_{\mu\nu} \equiv \partial_\mu A_\nu - \partial_\nu A_\mu and J^\sigma is an external charge current.

The variation of the action gives

(2)   \begin{eqnarray*} \frac{\delta S}{\delta g_{\mu\nu}} & = & \sqrt{-g} \biggl\{ -\frac{c^3}{16\pi G} G^{\mu\nu} + \frac{1}{2}\sqrt{\frac{\epsilon_0}{\mu_0}} \biggl(F^{\mu\alpha} F^{\nu\beta} g_{\alpha\beta} - \frac{1}{4} F^{\alpha\beta} F_{\alpha\beta}g^{\mu\nu} \biggr) + \frac{1}{2c} A_\sigma J^\sigma g^{\mu\nu} \biggr\} \\ \frac{\delta S}{\delta A_\sigma} & = & \sqrt{-g} \biggl\{ \sqrt{\frac{\epsilon_0}{\mu_0}} \nabla_\mu F^{\mu\sigma} + \frac{1}{c} J^\sigma \biggr\} \end{eqnarray*}

so the equations of motion are

(3)   \begin{eqnarray*} G^{\mu\nu} & = & \frac{8\pi G}{c^4} T^{\mu\nu} \\ \nabla_\mu F^{\mu\sigma} & = & -\mu_0 J^\sigma \end{eqnarray*}

with the stress–energy–momentum tensor

(4)   \begin{equation*} T^{\mu\nu} \equiv \frac{2c}{\sqrt{-g}} \frac{\delta S_{matter}}{\delta g_{\mu\nu}} = \frac{1}{\mu_0} \left( F^{\mu\alpha}F^{\nu\beta} g_{\alpha\beta} - \frac{1}{4} F^{\alpha\beta}F_{\alpha\beta} g^{\mu\nu} \right) + A_\sigma J^\sigma g^{\mu\nu} \end{equation*}

Dimensions

Let M, L, T, and I denote mass (in kilograms), length (in meters), time (in seconds) and current (in Amperes), respectively. The dimensions of the constants are

(5)   \begin{eqnarray*} c & \sim & L/T \\ G & \sim & L^3/(M\cdot T^2) \\ \epsilon_0 & \sim & I^2\cdot T^4/(M\cdot L^3) \\ \mu_0 & \sim & M\cdot L/(T^2\cdot I^2) \end{eqnarray*}

Tensor components are coordinate dependent, so their dimensions depend on the dimensions of the coordinates. Let the coordinates have dimensions of length:

(6)   \begin{equation*} x^\mu \sim L \end{equation*}

Then

(7)   \begin{equation*} g_{\mu\nu} \sim g^{\mu\nu} \sim 1 \end{equation*}

and we can raise and lower indices without changing dimensions. Dimensions of other tensors:

(8)   \begin{eqnarray*} R^\mu{}_{\alpha\nu\beta} & \sim & 1/L^2 \\ A_\mu & \sim & M\cdot L/(T^2\cdot I) \\ F_{\mu\nu} & \sim & M/(T^2\cdot I) \\ J^\mu & \sim & I/L^2\\ T^{\mu\nu} & \sim & M/(T^2\cdot L) \end{eqnarray*}

 

ELECTRIC AND MAGNETIC FIELDS

The velocity of an observer is

(9)   \begin{equation*} U^\mu \equiv \frac{\partial x^\mu}{\partial \tau} \sim L/T \end{equation*}

where \tau \sim T is proper time along the worldline. It satisfies the normalization condition U^\mu U_\mu = -c^2. The electric and magnetic fields as seen by this observer are

(10)   \begin{eqnarray*} E^\mu & \equiv & F^{\mu\nu}U_\nu \sim M\cdot L/(T^3 \cdot I)\\ B^\mu & \equiv & -\frac{1}{2c} \epsilon^{\mu\nu\sigma\rho} U_\nu F_{\sigma\rho} \sim M/(T^2\cdot I) \end{eqnarray*}

The field strength tensor is

(11)   \begin{equation*} F^{\mu\nu} = \frac{1}{c^2}\left( U^\mu E^\nu - U^\nu E^\mu\right) + \frac{1}{c} \epsilon^{\mu\nu\sigma\rho}U_\sigma B_\rho \end{equation*}

The scalar potential and electric charge density are:

(12)   \begin{eqnarray*} \Phi & \equiv & A_\sigma U^\sigma \sim M\cdot L^2/(T^3 \cdot I) \\ \rho_e & \equiv & \frac{1}{c^2} J_\sigma U^\sigma \sim I\cdot T/L^3  \end{eqnarray*}

The 3-vector potential and the 3-current defined by:

(13)   \begin{eqnarray*} A_\mu^{(3)} & \equiv & \perp_\mu^\nu A_\nu  \sim M\cdot L/(T^2\cdot I) \\ J_\mu^{(3)} & \equiv & \perp_\mu^\nu J_\nu \sim I/L^2 \end{eqnarray*}

where

(14)   \begin{equation*} \perp_\mu^\nu \equiv \delta_\mu^\nu + U_\mu U^\nu/c^2 \end{equation*}

is the spatial projection tensor for the observer.

Stress, Energy and Momentum

The stress-energy-momentum tensor is

(15)   \begin{eqnarray*} T^{\mu\nu} & = & \frac{1}{2} \left(\perp^{\mu\nu} + U^\mu U^\nu/c^2 \right) (\epsilon_0 E^2 + B^2/\mu_0)  - (\epsilon_0 E^\mu E^\nu + B^\mu B^\nu/\mu_0 ) \\ & & - \frac{2\epsilon_0}{c} U^{[\mu} \epsilon^{\nu] \alpha\beta\gamma} U_\alpha E_\beta B_\gamma - (\Phi \rho_e - A^{(3)}_\sigma J^\sigma_{(3)} ) g^{\mu\nu} \end{eqnarray*}

The energy density as seen by the observer is

(16)   \begin{equation*} (energy\ density) \equiv \frac{1}{c^2} T^{\mu\nu}U_\mu U_\nu = \frac{1}{2} ( \epsilon_0 E^2 + B^2/\mu_0) + \rho_e \Phi \sim M/(L\cdot T^2)  \end{equation*}

The momentum density and energy flux,

(17)   \begin{eqnarray*} (momentum\ density)^\mu & \equiv & -\frac{1}{c^2} \perp^\mu_\alpha T^{\alpha\beta} U_\beta =\frac{\epsilon_0}{c} \epsilon^{\mu\nu\alpha\beta} U_\nu E_\alpha B_\beta \sim M/(L^2\cdot T) \\ (energy\ flux)^\mu & \equiv & - \perp^\mu_\alpha T^{\alpha\beta} U_\beta =\epsilon_0 c \, \epsilon^{\mu\nu\alpha\beta} U_\nu E_\alpha B_\beta \sim M/T^3  \end{eqnarray*}

are related by (energy\ flux) = c^2(momentum\ density). The momentum flux and spatial stress are defined by

(18)   \begin{eqnarray*} & & (momentum\ flux)^{\mu\nu} = (spatial\ stress)^{\mu\nu} = \perp^\mu_\alpha T^{\alpha\beta} \perp^\nu_\beta \\ & = & \frac{1}{2} (\epsilon_0 E^2 + B^2/\mu_0) \perp^{\mu\nu} - (\epsilon_0 E^\mu E^\nu + B^\mu B^\nu/\mu_0 ) - (\Phi \rho_e - A^{(3)}_\sigma J^\sigma_{(3)} ) \perp^{\mu\nu} \sim M/(T^2\cdot L) \end{eqnarray*}

Fermi Normal Coordinates

Let \bar x^\mu denote Fermi normal coordinates (FNC) defined by the observer whose velocity is U^\mu. Thus, \bar x^0/c is the proper time along the observer’s worldline. The observer defines a triad of vectors, e^\mu_i that are Fermi-Walker transported along the worldline. These vectors are spatial, e^\mu_i g_{\mu\nu} U^\nu = 0, and orthogonal, e^\mu_i g_{\mu\nu} e^\nu_j = \delta_{ij}. The spatial coordinates of an event, \bar x^i, are the coefficients in the expansion of the vector \bar x^i e^\mu_i that is tangent to the spacelike geodesic that connects the worldline to the event, and has magnitude equal to the length of the geodesic.

In Fermi normal coordinates, the metric on the observer’s wordline is \bar g_{\mu\nu}= \eta_{\mu\nu} = {\rm diag}(-1,1,1,1) and

(19)   \begin{eqnarray*} \bar U^\mu & = & (c,0,0,0) \\ \bar e^\mu_1 & = & (0,1,0,0) \\ \bar e^\mu_2 & = & (0,0,1,0) \\ \bar e^\mu_3 & = & (0,0,0,1) \end{eqnarray*}

so that \bar U_\mu = (-c,0,0,0). We have

(20)   \begin{eqnarray*} \bar E^i & = & \bar F^{0i} \\ \bar B^i & = & \frac{1}{2} \bar\epsilon^{ijk}\bar F_{jk} \\ \bar\Phi & = & c \bar A_0 \\ \bar\rho_e & = & \bar J_0/c  \\ \bar A^{(3)}_i & = & \bar A_i \\ \bar J^{(3)}_i & = & \bar J_i  \end{eqnarray*}

where the 3D Levi-Civita tensor is defined by \epsilon_{\nu\alpha\beta} \equiv U^\mu \epsilon_{\mu\nu\alpha\beta}/c. Thus, in FNC, \epsilon^{ijk} = \pm 1 if i,j,k is an even/odd permutation of 1,2,3. The components of the stress-energy-momentum tensor are:

(21)   \begin{eqnarray*} (energy\ density) & = & \bar T^{00} \\ (momentum\ density)^i & = & \frac{1}{c} \bar T^{i0} \\ (energy\ flux)^i & = & c \bar T^{i0} \\ (momentum\ flux)^{ij} & = & \bar T^{ij} \end{eqnarray*}