Maxwell's Eqns for Inertial Sys on Minkowski Space: Rindler (1977)

[Rasalhague]

Rindler (1977): Essential Relativity, 2nd ed., p. 249, deriving Maxwell's equations for orthonormal/inertial coordinate systems on Minkowski space:

Hence we are led to posit the following field equations

$$A^{\mu\nu}_{\enspace,\nu} = k \rho_0U^\mu = k J^\mu,$$

where $k$ is some constant, $U^\mu$ the 4-velocity of the source distribution, $\rho_0$ its proper charge density (i.e. the charge per unit comoving volume--a scalar), and where the 4-current density is defined by this equation.

And $A^{\mu\nu}_{\enspace,\nu}$ is the electromagnetic field tensor.

Just checking if I've understood this right. Is the "4-velocity of the source distribution" that of the worldlines of particles at rest in the centre-of-momentum coordinate system of the sources? And is comoving volume the spatial (3-dimensional) volume (rather than a 4d volume of spacetime), as measured in these coordinates?

 

[bcrowell]

Is the "4-velocity of the source distribution" that of the worldlines of particles at rest in the centre-of-momentum coordinate system of the sources?

I don't see why you want to constrain it to the c.m. frame. Since the expression involving U is stated to be a scalar multiplied by a 4-vector, the whole equation is tensorial, so it should have the same form regardless of what frame you choose, or even what coordinates.

And is comoving volume the spatial (3-dimensional) volume (rather than a 4d volume of spacetime), as measured in these coordinates?

What do you find when you check the units?

 

[Rasalhague]

I don't see why you want to constrain it to the c.m. frame. Since the expression involving U is stated to be a scalar multiplied by a 4-vector, the whole equation is tensorial, so it should have the same form regardless of what frame you choose, or even what coordinates.

I probably worded my question clumsily. I'm just trying to confirm whether I guessed correctly which 4-velocity is intended. I only mentioned this particular coordinate system as a way of defining what "source distribution" (and hence what its 4-velocity) is. There's a 4-velocity naturally associated with each inertial coordinate system, namely the 4-velocity of particles at rest in it, or equivalently the 4-velocity of lines of constant spatial position. I just meant: is the 4-velocity referred to here that of the c.m. frame of the sources?

What do you find when you check the units?

Chaos! But good idea: I'll keep working at that. Possibly I'm making one or more wrong assumptions about which unit-balancing conventions he's using.

 

[Rasalhague]

On the left, for Rindler's contravariant EM field tensor (as defined on p. 99), I get units of

Mass x Charge^-1 x Time^-1.

On the right, I get different units for time and space components of the 4-velocity U (as defined on p. 67, eq. 4.10). On p. 250, he gives the constant k the value 4pi/c. So the time component of

$$\rho_0 \frac{4 \pi}{c} \textbf{U}= \rho_0 \frac{4 \pi}{c} \gamma[\textbf{u}](1,\textbf{u}),$$

supposing he means 3-volume, I think would have units of

Charge x Time x Length^-4.

And the space components,

Charge x Length^-3.

Neither of these matches the LHS. Nor would either with an extra factor of Length^-1.

 

[DrGreg]

I probably worded my question clumsily. I'm just trying to confirm whether I guessed correctly which 4-velocity is intended. I only mentioned this particular coordinate system as a way of defining what "source distribution" (and hence what its 4-velocity) is. There's a 4-velocity naturally associated with each inertial coordinate system, namely the 4-velocity of particles at rest in it, or equivalently the 4-velocity of lines of constant spatial position. I just meant: is the 4-velocity referred to here that of the c.m. frame of the sources?

Unless I've misunderstood, the "source distribution" is (effectively) a collection of particles, and the 4-velocity is that of whichever particle is at each event in the manifold.

(There isn't a single 4-velocity, there's a vector field assigning a 4-velocity to each event.)

(I say "effectively" because really it's a continuous distribution rather than a set of discrete particles.)

 

[bcrowell]

I would suggest working the units in a system where c=1 and G=1. That leave you with only two base units, charge and distance(=mass=time). Then you should have the same units for all 4 components of the equation, which would be simpler to analyze.

A should have units of electromagnetic field. In SI, this would be N/C, and in c=G=1 units that becomes charge^-1.

So we then have charge^-1/distance=charge^1/distance^n, where we want to know whether n is 3 or 4.He's using units where the Coulomb constant k is unitless (if c=1), so charge^2/length^2 is unitless (because force is unitless), so charge and length have the same units. So this means n=3. His charge density is the density in three dimensions, not four.

 

[Rasalhague]

Thanks for your help. c=G=1 definitely sounds like the way to go! Very neat that length, time, mass and charge can have the same units this way.

 

[Rasalhague]

Unless I've misunderstood, the "source distribution" is (effectively) a collection of particles, and the 4-velocity is that of whichever particle is at each event in the manifold.

(There isn't a single 4-velocity, there's a vector field assigning a 4-velocity to each event.)

(I say "effectively" because really it's a continuous distribution rather than a set of discrete particles.)

Ah, so is this like an instantaneous version of the drift velocity? It seems that these equations just don't apply (become meaningless) in the case of a single, discrete charge following one world line, because, for events not on the world line of such a source, no value is defined for the field at events remote from any source. And similarly for current in a wire, with no continuous charge distribution outside of the wire. Is that anywhere near right?

 

[DrGreg]

Ah, so is this like an instantaneous version of the drift velocity? It seems that these equations just don't apply (become meaningless) in the case of a single, discrete charge following one world line, because, for events not on the world line of such a source, no value is defined for the field at events remote from any source. And similarly for current in a wire, with no continuous charge distribution outside of the wire. Is that anywhere near right?

This is where I get a bit out of my depth. I think there is a way of approaching this using distribution or measure theory. For example, an isolated single worldline could be represented by a Dirac delta distribution, I think. But I've never been happy with Dirac deltas because they're not actually functions. Maybe someone who understands this better could comment.

I guess the equation in post #1 still makes sense for the directional derivative along the worldline, or in all directions at events not on the worldline, but I'm not sure how you differentiate orthogonally across the worldline (for example). (I suppose that gives you the boundary conditions for the differential equation $A^{\mu\nu}_{\enspace,\nu} = 0$.)