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In the spacelike case, there is a frame in which the charge density vanishes but current density does not. Additionally, the drift velocity of a free current would propagate superluminally. For this reason I would expect such a current to be impossible.

In the lightlike case, the charge and current densities would come in equal proportions in all frames. The drift velocity would propagate at the speed of light. It's unclear to me whether this is forbidden outright.

Is the four-current necessarily a timelike vector?

Edit: Do Maxwell's equations admit solutions for lightlike or spacelike currents?

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    $\begingroup$ Four-current can be written in terms of the four-velocity as $J= \gamma \rho_0 (c,\vec{u})$. Now, how can there be an inertial frame where $\gamma\rho_0 c =0$? $\endgroup$ Dec 9 '16 at 9:53
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    $\begingroup$ @G.Bergeron By having more than one type of charge carrier: electrons going one way, protons going the other way, which gives you $\rho=0$ because it's a charge density, not a mass density. $\endgroup$ Jan 12 '17 at 13:06
  • $\begingroup$ @EmilioPisanty First, $\rho_0$ is the charge density in the rest frame of the current and multiplies the whole expression... Second, those two currents would interact and cancel out, no external field could be stronger than the mutual attraction of ''superposed'' point-like particles. Third, a zero charge density would imply a continuous charge density of both species and is unphysical. $\endgroup$ Jan 13 '17 at 7:13
  • $\begingroup$ Finally, electromagnetism doesn't describe charges, just their interactions. As such, you cannot decide on the possibility of massless charges. You need to add in considerations from QFT to conclude that. In any case, you would have these charges always ''moving'' at c and hence no co-moving frame and an ill-defined density. Spacelike 4-currents and massless charges can exist at the effective level in condensed matter, but then Lorentz invariance is violated anyway and you already swim deeply in QFT, making the application of classical EM questionable. $\endgroup$ Jan 13 '17 at 7:47
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    $\begingroup$ My main point in the first comment was that there needs to be a non-vanishing charge density in the co-moving frame, unless the charges are massless and ''travelling'' at $c$. This is related to the idealized cases being only useful in non-relativistic settings. Since the question was related to special relativity, I felt it was relevant to go beyond those ideal non-relativistic cases in which case, I would say that the 4-currents has to be timelike. $\endgroup$ Jan 13 '17 at 19:05
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Case 1: Current Densities Comprising Charge Carriers of the Same Sign

For currents comprising a lone charged particle, or charge carriers of the same sign, the current four vector must be timelike. But Maxwell's Equations don't impose the limit - relativity does. The limit arises because we postulate it (as I discuss here) to force causality - so that no boost can switch the sign of the time component of a four-vector. Maxwell's equations are not causal: for any given solution you can construct a time-reversed one, by mapping $t\mapsto-t$. When we solve antenna problems, we must explicitly exclude the advanced wave solution as unphysical - nothing in Maxwell's equations rules it out. If you write down and study the Liénard-Wiechert Potentials (well-known, "building block" solutions of Maxwell's Equations) for a supraluminal particle, you'll explicitly see causality violated in that the direction of the Poynting vector is switched for supraluminal observers relative to the direction for subluminal ones. This is because, for an accelerated particle, energy pulses, from a supraluminal standpoint, run inwards to the charged particle and bring about an acceleration of the particle, switching the causal field-particle relationship apparent to the subluminal observer.

Lightlike current densities are excluded because you can't boost such a particle to $c$ with finite energy, even for a charged particle of zero rest mass (we have solid theoretical reasons for believing that they don't exist. See the answers to this question here, especially Marek's and Lubos's answers). This is owing to the Abraham-Lorentz-Dirac force, i.e. the radiation resistance. This is readily seen from the Liénard relativistic generalization

$$P = \frac{\mu_o q^2 a^2 \gamma^6}{6 \pi c}$$

of the Larmor radiation formula (i.e. it diverges like $\gamma^6$) that follows from the Abraham-Lorentz-Dirac formula. Alternatively, use the Larmor formula from the frame momentarily co-moving with the charged particle to show that the same small change in rapidity always takes the same amount of energy (measured from the co-moving frame). Also note that the Liénard-Weichert potentials diverge for lightlike charge motion.

Case 2: Current Densities Comprising Charge Carriers of the Both Signs (Hat Tip: Emilio Pisanty for pointing out a vicious error on my part)

In this case, trivially, we can have a DC current distribution comprising charge carriers of opposite sign such that the charge density at any point is nought. Thus a four current density of the form $(0,\,\vec{J})$, i.e. a spacelike four current density. Another example is a sinusoidally varying with time current density comprising opposite signed charges which is orthogonal to the spatial part of the wave four vector and with a zero charge density so that the continuity equation $J^\alpha\,k_\alpha = 0$ is fulfilled.

Note that neither of these distributions entail faster than light particles, so causality is not threatened by them.

Lightlight distributions can arise as well. Consider again the sinusoidally varying with time case. Lightlike vectors by definition are null (self Minkowski orthogonal) so that put $J_\alpha$ parallel to the wave four co-vector $k_\alpha$ then $J^\alpha\,k_\alpha = 0$ and the current and charge density automatically fulfill the continuity equation.

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  • $\begingroup$ I am skeptical of that second part, see my comment on the OP. I wouldn't count displacement as a current. Also, in the lightlike case, you seem to propose longitudinally polarized EM waves, which do not exist in vacuum, precisely because photons are massless. If you are considering rather propagation in a medium, then some possibilities exists, but I don't see any not involving physics beyond Maxwell's equations. $\endgroup$ Jan 13 '17 at 7:37
  • $\begingroup$ Also, I agree Maxwell's equation in vacuum would not limit the types of 4-currents possible, but then that vacuum part implies no currents at all. If you introduce source particles, then the conformal invariance is lost. In the case of the antenna, I would argue that the rejected solutions are rejected because Maxwell's equations coupled to matter ARE causal, but we solve the idealized conformal case first. With the matter coupling, all the entropic considerations appear and break time inversion. $\endgroup$ Jan 13 '17 at 8:00
  • $\begingroup$ @G.Bergeron Let me think on these a bit. However, for now, with regards to the reversibility / otherwise of ME, yes, I agree with you in principle - once you couple with particulate matter then the reversibility is lost but I'm reading between the lines and guessing that what the OP has in mind is the idealized case, especially the antenna kind of problem with idealized, nonthermalizing currents as you'd find in the pages of Stratton, Purcell or Collin (I'm really showing my age here!). $\endgroup$ Jan 13 '17 at 11:47
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Suppose the current consists of only negative charges. If $j^μ$ is spacelike, $c^2 ρ^2-j^2<0$. In 2-dim there is a frame where $j^μ= \begin{bmatrix}0\\j\end{bmatrix}$. Under Lorentz boost $Λj=\begin{bmatrix}γ & γβ \\ γβ & γ\end{bmatrix}$ $\begin{bmatrix}0\\j\end{bmatrix}$ $=\begin{bmatrix}γβj\\γj\end{bmatrix}$ $=\begin{bmatrix}cρ^{'}\\j^{'}\end{bmatrix}$, so the sign of $ρ^{'}$ depends on the sign of $β$. We can reverse the sign of the charges by reversing the speed direction. This is a contradiction. On the other hand, if $j^μ$ is timelike, $c^2 ρ^2-j^2>0$. There is a frame where $j^μ= \begin{bmatrix}cρ\\0\end{bmatrix}$. Under boost $Λj=\begin{bmatrix}γ & γβ \\ γβ & γ\end{bmatrix}$ $\begin{bmatrix}cρ\\0\end{bmatrix}$ $=\begin{bmatrix}γcρ\\γβcρ\end{bmatrix}=$ $\begin{bmatrix}cρ^{'}\\j^{'}\end{bmatrix}$. The charge sign never changes.

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Timelike and spacelike are characteristics of a pair of points in the Minkowsky space. How can a four-vector possibly timelike or spacelike?

I guess you mean that: whether four-current $J^\alpha$ lies in the light cone. If so, then you call it timelike. If not, then you call it spacelike. We recall that $J^\alpha$ is the product of $\rho$ and four velocity $u^\alpha$. Then, because the moving charged particles cannot exceed speed of light (thus lying in the light cone), and that $\rho$ is real, we see that $J^\alpha$ is timelike indeed.

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  • $\begingroup$ the velocity $u$ is the difference between to two points, and therefore it can be timelike/spacelike. $\endgroup$ Dec 9 '16 at 11:03
  • $\begingroup$ @AccidentalFourierTransform : ...of course you mean the 4-velocity $\:\mathbf{U}=\gamma(u)\left(c,\mathbf{u}\right)\:$ of a massive particle is timelike. $\endgroup$
    – Frobenius
    Dec 9 '16 at 12:08
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    $\begingroup$ @Frobenius Im not sure what you mean by that. I was just addressing the phrase "How can a four-vector possibly timelike or spacelike?". Vectors can be timelike/spacelike. $\endgroup$ Dec 9 '16 at 12:12
  • $\begingroup$ @AccidentalFourierTransform : I think that the terms timelike/spacelike/lightlike characterize 4-vectors, not 3-vectors, but may be I am wrong. $\endgroup$
    – Frobenius
    Dec 9 '16 at 13:31
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    $\begingroup$ @Frobenius oh, now I see what you mean. To me (and most people discussing relativistc mechanics), a vector is just a 4-vector. I just drop the 4, but it's there. "Velocity" means "4-velocity", "momentum" means "4-momentum", etc. $\endgroup$ Dec 9 '16 at 13:36

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