stress-energy tensors

c =

\frac{m}{s} =

the speed of light.

G =

\frac{m^{3}}{k g \cdot s^{2}} =

gravitational constant

k_{e} =

\cdot \frac{k g \cdot m^{3}}{C^{2} \cdot s^{2}} =

8.9875517873682e+9

\frac{k g}{C} \cdot \frac{m^{3}}{C \cdot s^{2}} =

Coulomb's constant.
(Coulomb's constant units are the same as the gravitational constant's units, but with

k g

swapped with

C

, and then with an extra ratio of

k g

C

.)

Derived values:

ϵ_{0} = \frac{1}{4 π k_{e}} =

8.854187817620367e-12

\frac{C^{2} \cdot s^{2}}{k g \cdot m^{3}} =

8.854187817620367e-12

\frac{C}{k g} \cdot \frac{C \cdot s^{2}}{m^{3}} =

vacuum permittivity.

μ_{0} = \frac{1}{ϵ_{0} c^{2}} =

1.2566370614359207e-6

\frac{k g \cdot m}{C^{2}} =

1.2566370614359207e-6

\frac{k g}{C} \cdot \frac{m}{C} =

vacuum permeability.

To boot, how about those gravitomagnetic permittivity and permeability:

ϵ_{G} = \frac{1}{4 π G} =

1.1923790733063374e+9

\frac{k g \cdot s^{2}}{m^{3}}

= gravitational vacuum permittivity.

μ_{G} = \frac{1}{ϵ_{G} c^{2}} =

9.331345047580892e-27

\frac{m}{k g} =

gravitational vacuum permeability.

In both cases,

\frac{1}{c^{2}} = μ_{0} ϵ_{0} = μ_{g} ϵ_{G} =

1.1126500560536185e-17

\frac{s^{2}}{m^{2}}

.

natural units, setting

c = G = k_{e} = 1

1 = \sqrt{k_{e} \cdot G} =

7.744771308477055e-1

\cdot \frac{m^{3}}{C \cdot s^{2}}

1 = \sqrt{\frac{k_{e} \cdot G}{c^{4}}} =

8.617220230499451e-18

\cdot \frac{m}{C}

, so

1 C =

8.617220230499451e-18

m

1 = \sqrt{\frac{k_{e}}{G}} =

1.1604670337432505e+10

\cdot \frac{k g}{C}

, so

1 C =

1.1604670337432505e+10

k g

, and

1 k g =

8.617220230499428e-11

C

.
Therefore, using natural units,

k_{e} =

7.744771308477054e-1

\frac{m^{3}}{C \cdot s^{2}} =

6.67384e-11

\frac{m^{3}}{k g \cdot s^{2}}

by unit conversion.
(Which makes sense if

k_{e} = G = 1

, then of course their conversion to identical units will have an identical value).

equilibrium fluid:
TODO the derivation

T_{a b} = (c^{2} ρ + P) u_{a} u_{b} + P g_{a b}

[T_{a b}] = \frac{k g}{m \cdot s^{2}}

for:

ρ = [\frac{k g}{m^{3}}] =

density

P = [\frac{k g}{m \cdot s^{2}}] =

pressure

u_{a} = [1] =

4-velocity.

electromagnetic field:

T_{a b} = \frac{1}{μ_{0}} (F_{a u} F_{b v} g^{u v} - \frac{1}{4} g_{a b} F_{u v} F^{u v})

[T_{a b}] = \frac{k g}{m \cdot s^{2}}

for:
Faraday tensor:

F_{a b} = \frac{1}{c} 2 E_{[a} t_{b]} + ϵ_{a b c d} t^{c} B^{d}

[F_{a b}] = \frac{k g}{C \cdot s}

t_{a} = [1] =

timelike unit vector, such that

t^{a} t_{a} = - 1

E_{a} = [\frac{k g \cdot m}{C \cdot s^{2}}] =

spatial electric field, such that

t^{a} E_{a} = 0

B_{a} = [\frac{k g}{C \cdot s}] =

spatial magnetic field, such that

t^{a} B_{a} = 0

Substituting:

T_{a b} = \frac{1}{μ_{0}} ((\frac{1}{c} 2 E_{[a} t_{u]} + ϵ_{a u c d} t^{c} B^{d}) (\frac{1}{c} 2 E_{[b} t_{v]} + ϵ_{b v e f} t^{e} B^{f}) g^{u v} - \frac{1}{4} g_{a b} (\frac{1}{c} 2 E_{[u} t_{v]} + ϵ_{u v c d} t^{c} B^{d}) (\frac{1}{c} 2 E^{[u} t^{v]} + ϵ^{u v e f} t_{e} B_{f}))

T_{a b} = \frac{1}{μ_{0}} (4 \frac{1}{c^{2}} E_{[a} t_{u]} E_{[b} t_{v]} g^{u v} + 2 \frac{1}{c} E_{[a} t_{u]} ϵ_{b v e f} t^{e} B^{f} g^{u v} + 2 \frac{1}{c} ϵ_{a u c d} t^{c} B^{d} E_{[b} t_{v]} g^{u v} + ϵ_{a u c d} t^{c} B^{d} ϵ_{b v e f} t^{e} B^{f} g^{u v} - \frac{1}{c^{2}} g_{a b} E^{[u} t^{v]} E_{[u} t_{v]} - \frac{1}{2} \frac{1}{c} g_{a b} E^{[u} t^{v]} ϵ_{u v c d} t^{c} B^{d} - \frac{1}{2} \frac{1}{c} g_{a b} ϵ^{u v e f} t_{e} B_{f} E_{[u} t_{v]} - \frac{1}{4} g_{a b} ϵ^{u v e f} t_{e} B_{f} ϵ_{u v c d} t^{c} B^{d})

T_{a b} = \frac{1}{μ_{0}} (\frac{1}{c^{2}} E_{a} t_{u} E_{b} t_{v} g^{u v} - \frac{1}{c^{2}} E_{u} t_{a} E_{b} t_{v} g^{u v} - \frac{1}{c^{2}} E_{a} t_{u} E_{v} t_{b} g^{u v} + \frac{1}{c^{2}} E_{u} t_{a} E_{v} t_{b} g^{u v} + \frac{1}{c} E_{a} t_{u} ϵ_{b v e f} t^{e} B^{f} g^{u v} - \frac{1}{c} E_{u} t_{a} ϵ_{b v e f} t^{e} B^{f} g^{u v} + \frac{1}{c} ϵ_{a u c d} t^{c} B^{d} E_{b} t_{v} g^{u v} - \frac{1}{c} ϵ_{a u c d} t^{c} B^{d} E_{v} t_{b} g^{u v} + ϵ_{a u c d} t^{c} B^{d} ϵ_{b v e f} t^{e} B^{f} g^{u v} - \frac{1}{4} \frac{1}{c^{2}} g_{a b} E^{u} t^{v} E_{u} t_{v} + \frac{1}{4} \frac{1}{c^{2}} g_{a b} E^{v} t^{u} E_{u} t_{v} + \frac{1}{4} \frac{1}{c^{2}} g_{a b} E^{u} t^{v} E_{v} t_{u} - \frac{1}{4} \frac{1}{c^{2}} g_{a b} E^{v} t^{u} E_{v} t_{u} - \frac{1}{4} \frac{1}{c} g_{a b} E^{u} t^{v} ϵ_{u v c d} t^{c} B^{d} + \frac{1}{4} \frac{1}{c} g_{a b} E^{v} t^{u} ϵ_{u v c d} t^{c} B^{d} - \frac{1}{4} \frac{1}{c} g_{a b} ϵ^{u v e f} t_{e} B_{f} E_{u} t_{v} + \frac{1}{4} \frac{1}{c} g_{a b} ϵ^{u v e f} t_{e} B_{f} E_{v} t_{u} - \frac{1}{4} g_{a b} ϵ^{u v e f} t_{e} B_{f} ϵ_{u v c d} t^{c} B^{d})

T_{a b} = \frac{1}{μ_{0}} (t_{a} t_{b} (\frac{1}{c^{2}} E_{u} E^{u} + B_{u} B^{u}) + \frac{1}{c} (t_{a} ϵ_{b c d e} + t_{b} ϵ_{a c d e}) t^{c} E^{d} B^{e} - \frac{1}{c^{2}} E_{a} E_{b} - B_{a} B_{b} + \frac{1}{2} g_{a b} (\frac{1}{c^{2}} E_{u} E^{u} + B_{u} B^{u}))

What about, alternatively, an equilibrium fluid but with electromagnetic terms instead?

T_{a b} = \sqrt{\frac{k_{e}}{G}} ((c^{2} ρ_{c h a r g e} + ϕ) u_{a} u_{b} + g_{a b} ϕ)

ρ_{c h a r g e} = [\frac{C}{m^{3}}] =

charge density.

c^{2} ρ_{c h a r g e} = [\frac{C}{m \cdot s^{2}}]

ϕ = [\frac{C}{m \cdot s^{2}}]

is the electrical equivalent of pressure.

J_{u} = (c ρ, j^{i})

in units of

[\frac{C}{m^{2} \cdot s}] =

current density.

u_{a} = \frac{1}{c \cdot ρ_{c h a r g e}} J_{a}

is in units

[1]