Matter stress-energy (from SRHD worksheet, which is from Marti & Mueller SRHD finite volume solvers):

$u^a =$ four-velocity of fluid, has units $[1]$.
$u_a = g_{ab} u^b$,
$u_0 = g_{0a} u^a = g_{00} u^0 + g_{0j} u^j = (-1 + h_{00}) u^0 + h_{0j} u^j$
$u_i = g_{ia} u^a = g_{i0} u^0 + g_{ij} u^j = h_{i0} u^0 + (\delta_{ij} + h_{ij}) u^j$

For an object at rest this takes the value of $u^0 = 1, u^i = 0$,
and we find: $u_0 = h_{00} - 1$, $u_i = h_{i0}$

$\rho =$ density, which has units of $[\frac{kg}{m^3}]$
$P =$ pressure, which has units of $[\frac{kg}{m \cdot s^2}]$
$h = c^2 + \epsilon + P / \rho$ = specific enthalpy, which has units of $[\frac{m^2}{s^2}]$

ideal gas law:
$P = (\Gamma - 1) \rho e_{int}$
$e_{int} = \frac{P}{(\Gamma - 1) \rho}$ = internal specific energy, which has units of $[\frac{m^2}{s^2}]$
$\Gamma =$ heat capacity ratio, which has units of $[1]$

stress-energy for perfect fluids:
$T^{uv} = \rho h u^u u^v + P g^{uv}$ = stress-energy, has units $[\frac{kg}{m \cdot s^2}]$
$T_{00} = \rho h (u_0)^2 = \rho h \gamma^2$
$T_{0i} = \rho h u_0 u_i = \rho h \gamma^2 \frac{v_i}{c}$

Stress energy of matter in curved space:
$T_{ab} = c^2 \rho u_a u_b + P (g_{ab} + u_a u_b) $
$T_{ab} = (c^2 \rho + P) u_a u_b + P g_{ab}$

splitting into space+time components:
$T_{00} = (c^2 \rho + P) (u_0)^2 + P g_{00}$
$T_{0i} = (c^2 \rho + P) u_0 u_i$
$T_{ij} = c^2 \rho u_i u_j + P g_{ij}$

substituting low-velocity approximations:
$T_{00} \approx c^2 \rho$
$T_{0i} \approx -c \rho v_i$
$T_{ij} \approx \rho v_i v_j + \delta_{ij} P$

$T_{ab} = \left[\begin{matrix} c^2 \rho & -c \rho v_i \\ -c \rho v_i & \rho v_i v_j + \delta_{ij} P \end{matrix}\right]$

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What is a typical pseudocartesian Schwarzschild metric magnitude at the Earth's surface?
$G =$ $\frac{m^3}{kg \cdot s^2} =$ gravitational constant.
$c =$ $\frac{m}{s} =$ speed of light.
$r =$ $m =$ Earth's radius.
$M =$ $kg =$ Earth's mass.
$R = 2 \frac{G}{c^2} M =$ {{earth_Schwarzschild_radius_in_m = 2 * gravitational_constant_in_m3_per_kg_s2 / speed_of_light_in_m_per_s / speed_of_light_in_m_per_s * earth_mass_in_kg}} $m =$ Earth Schwarzschild radius
$h_{00} = R / r \approx$ {{earth_R_r_ratio = earth_Schwarzschild_radius_in_m / earth_radius_in_m}}
$h_{rr} = 1 / (1 / h_{00} - 1) \approx$ {{1 / (1 / earth_R_r_ratio - 1)}}

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Now comparing the ratios between matter and charge...
In fact, the Maxwell charge is the electric charge, however the Einstein charge is the $T_{00}$ term, which is both the matter and the electromagnetic energy.
Same deal with $T_{0i} = $ fluid momentum plus Poynting vector, versus $A_i = $ the vector potential of the magnetic field.
So I suspect that $E$ vs $E_g$ may not always line up with one another...