What is the point of 'retarded time'? I was just reading this explanation for the physical meaning of 'retarded time':

This is the usual argument for explaining retarded time -
Consider a charge moving with a constant velocity along a straight line. If the charge suddenly comes to a halt, there will be a change in the electric field due to the acceleration. But this change in the electric field isn't communicated instantaneously through the whole universe, that's prohibited by special relativity and the finite travel time of light.
Therefore one has to infer that observers that are closer to the charge "see" the change in the field earlier than observers farther away.
What that means for an observer too far away to see the change is this - If you are calculating a current/some other property dependent on the velocity of the particle, you have information that is old. The particle has already stopped moving, but you just haven't seen it as yet. So to get the physically accurate picture in the frame of the particle, you have to use retarded time, because otherwise you will infer that the particle is still moving at a constant velocity now. But you don't possess that information. You only know that it was moving at time $t - \frac{r}{c}$ earlier, so all calculations about the state of the system must be carried out with respect to that time.
So to answer your questions : $t_r$ is a time variable because it still depends on t $$t_r = t - \frac{r}{c}$$ which is clearly dependent on $t$. It isn't that you have two time axes, you are just evaluating your system at a finite point in the past (i.e., $t_r \leq t$ always).

I don't understand this explanation. If an observer too far away is calculating a current/some other property dependent on the velocity of the particle, then there are two possibilities: either (1) the particle is continuing unimpeded on its path, or (2) the particle has been impeded in some way (such as the case where it comes to a sudden stop). But, if we have knowledge of (1) and have no knowledge of (2), then it seems that there would be no point in using retarded time; and if we know that (2) occurred, then clearly the information has had time to reach us; so what is the point of retarded time?
 A: I think it would be helpful to see an example.  In the Lorenz gauge, the scalar potential can be written
$$\varphi(\mathbf r, t) = \frac{1}{4\pi\epsilon_0}\int d^3 r' \frac{\rho(\mathbf r',t_r)}{|\mathbf r-\mathbf r'|}, \quad t_r \equiv t-\frac{|\mathbf r-\mathbf r'|}{c}$$
The interpretation is straightforward - the potential at time $t$ and position $\mathbf r$ is obtained by summing the contributions from the charges at each point $\mathbf r'$ as they were a time $\frac{|\mathbf r-\mathbf r'|}{c}$ in the past.  That is, when we look 300 meters away, we see the charge distribution as it was $\frac{300 \text{ m}}{c} \approx 10^{-6} \ \text{s}$ ago.
A: Retarded time $t_r$ in EM theory is just time coordinate of observation $t$ decreased by appropriate time interval which it takes for the wavefront to get from the source to the point of interest. It is used in the context of standard EM theory because it is assumed EM field at any point is just a sum of retarded fields (which are functions of past positions, velocities and accelerations of charged particles), or very close to that (background field of unknown sources being small or zero).
Use of retarded time and retarded fields has nothing to do with knowledge or information of the past motion. It is a general theoretical concept which we use to express particular solutions of wave equations. It does not matter whether the observer knows the particle accelerated or not, the fields are always expressed using the retarded time.
