Can we interpret timelike, rest frame motion as velocity in the time direction? If so, does it cause charge to generate a ''$B$ field'' perp. to time? For background: I've had courses on SR/GR from Hartle, E&M from Griffiths/Purcell, and know E/B fields are unified/transform with lorentz boost.
My question is less about 'does SR account for E/B fields' and moreso 'how far can we push the idea that four velocity in the time direction is genuine motion' and 'how far can we push the classical picture of B fields as being generated by motion'
I've been running the following thought experiment in my head; I'm aware I'm being loosey-goosey with things, and again, am mostly looking for where this analogy would break/ if it does.
From what I know, every object with mass moves with a four-velocity that has normalized magnitude 1, or unnormalized magnitude c. So if we're in the rest frame of an object, another way to interpret this is that the four velocity is entirely in the time direction.
Say we have two charges in one another's rest frame; if we can interpret the pure time-like motion of the charges as genuine motion (whatever that may be), they should both generate a B field.
The B field components generated by moving charges always only have components in the plane perpendicular to the direction of motion; In this case, the 'plane' perpendicular to the time direction is all of 3-dimensional space.
Moreover, we know that for two current sources parallel to one another (e.g. two wires), the force between them is radial and in the plane perpendicular to their current direction. Two charges in their rest frame, with pure time like motion, could be interpreted as current sources moving parallel in the time-like direction; then the 'B-fields' of these stationary charges should cause radial forces between the charges purely in 3-space, perpendicular to the time-like current direction.
the force on each charge would then be of the form v^2 * mu_0 * q1q2 /4pi r^2 , by combining the qV x B equation and approximating the B field with the Biot-savart law, where v is the speed in the time direction. Since the motions purely in the time direction, the full magnitude of the four-velocity is in that direction, so v = c. Then since mu_0= 1/c^2 * eps_0 , plugging this all in would give a radial force between charges with the same magnitude as coulombs law.
So if all this works, The time-like current would generate a force with the magnitude of coulombs law, which only had components in the 3 spatial directions, and where the resulting force would be purely radial; e.g. it gives coulombs law.
Then the E-field could be interpreted as a B-field resulting from pure time-like current motion.
Since E & B fields transform into one another with SR, this is what we'ed expect, but I'm really just wondering if it's valid to interpret things this way. For example, one thing that's tripping me up is that like currents attract, while like charges repel; the thought experiment I ran through with would predict protons attract , while electrons and protons repel, unless the opposite sign on time terms in the metric could somehow account for it.
 A: "Then the E-field could be interpreted as a B-field resulting from pure time-like current motion."
Yes. The electromagnetic field in 4D space can be thought of as 2D 'surfaces' radiating out from the worldline of a charged particle. The orientation of the surface can be defined as the combination of six 'basis' planes along the coordinate axes: $xt$, $yt$, $zt$, $yz$, $xz$, and $xy$. The components in the time direction ($xt$, $yt$, $zt$) are the electric field, the components in the purely spatial directions ($yz$, $xz$, $xy$) are the magnetic field. If you choose a different time direction as a basis for your coordinate system, the division into electric and magnetic field components changes.

If the charge is stationary, the current and therefore the field planes are purely timelike; a pure electric field. If the charge is moving (or equivalently, if we are moving past the stationary charge), the exact same field planes now have a spatial-only component in these coordinates as well, which is the magnetic field. Most people would probably put it the other way round: "the B-field could be interpreted as an E-field resulting from non-pure time-like current motion," but that surely is an equivalent statement.
[Technically, these 'surfaces' are entities called bivectors which are like vectors with more dimensions. Where a vector represents an oriented  1D length, a bivector represents an oriented 2D area. Because vector algebra can't handle bivectors, we turn them into vectors with a couple of mathematical tricks. First we multiply by the constant unit time vector, which because these things have the property that $t^2=1$, cancels the time part of $(xt,yt,zt)t=(x,y,z)$ for the E vector and turns the magnetic field into a trivector $(yz,xz,xy)t=(yzt,xzt,xyt)$, and then find the dual of the trivector by multiplying by the constant unit quadvector $xyzt$ to get $(yzt,xzt,xyt)xyzt=(x,y,z)$. This 'taking the dual' process is why magnetic fields are not vectors but pseudovectors. They give the opposite result when you apply a mirror reflection.]
The spatial and temporal parts of the electromagnetic bivector field are perpendicular to one another, but the trick we apply to turn them both into 3D vectors puts them both in the spatial 3D $xyz$ plane, both of them perpendicular to the time direction.
"For example, one thing that's tripping me up is that like currents attract, while like charges repel"
The usual treatment of electric currents can be a bit of a misleading comparison. We usually think of a 'current' in term of a positively charged wire of atomic nuclei surrounded by a stream of negatively charged electrons, so the charge cancel out, and we just get left with the correction terms due to the electron motion. In this situation, the wires attract. But this doesn't correspond to the moving-frame version of two static charges.
If we start with two like static charges repelling, and switch to a moving frame, we get currents, and hence magnetic forces. But the charges remain uncancelled and still repel electrically. This is the situation we need to look at if we want to compare with the static charges.
In both static and moving frames they repel - the electric repulsion between like currents is much bigger than the magnetic attraction. In a moving frame, the repulsive force reduces because of time dilation/length-contraction. We can treat the correction we have to apply as an additional force, and call it 'magnetism'.
