I don’t recall the name, but here is a neat java applet visualizing the situation. In essence, for uniform motion the field lines are always straight, pointing away from the charge, while the direction of light received from the charge points to the retarded position. There is a standard detailed calculation (like the one in Griffiths or Jackson) here, with the following conclusion:
which confirms that the electric field at R points along the direction from R to the present (not the retarded) position of the charge.
But it still doesn’t change the causal relationship: that alignment only applies if nothing happens to the other thing in the time since the center of that retarded cone. If no new information has been generated, then sure, you can use the new information instead of the old information. But if anything happens, you had better use the old information!
To get more formal about it:
Consider or charge (equivalently, a mass) whose worldline coincides with (t, 0,0,0) for all t ⇐ 0) in some reference frame
The field at the event (10, 10, 0, 0 ) occurs after (0,0,0,0) in every subluminal reference frame.
The field at (10, 10, 0, 0 ) is independent of whatever happens at (1, 0,0,0). If a laser comes in and knocks that charge aside, there’s zero difference. None at all.
Suppose the charge was deflected so that it passes through (10, 1, 0, 0). You can’t get the electrical field at event (10, 10, 0, 0) by looking at what the charge is doing at (10, 1, 0, 0) - you need to look at (0,0, 0,0).
This is what causality looks like. The causal influences propagate at lightspeed. Even electrostatic ones.
But it still doesn’t change the causal relationship: that alignment only applies if nothing happens to the other thing in the time since the center of that retarded cone.
Absolutely. As I said
This breaks down once you add acceleration, hence EM waves.
However I am not sure I agree with the part in bold:
This is what causality looks like. The causal influences propagate at lightspeed. Even electrostatic ones.
When you say “Suppose the charge was deflected”, you have broken the electrostatic assumptions, since the charge is now accelerating. Depending on the distance, you either get the near-field effects or the radiative effects, which do indeed propagate at lightspeed. Once the acceleration disappears and the light-speed transients died down, you are back in the electrostatic/magnetostatic mode with lag-free fields.
That was the point of the example—by time 10, the charge was no longer accelerating, but you know that you’re not clear to use electrostatics being ‘noncausal’ yet because the light cone hasn’t reached that far. Being lag free is a computational convenience that sometimes applies, and you need to know when by applying causality.
So, it is ALWAYS fair to say that fields are causal influences, whether they’re static or dynamic. That was why I objected to your complaint in the first place.
I don’t recall the name, but here is a neat java applet visualizing the situation. In essence, for uniform motion the field lines are always straight, pointing away from the charge, while the direction of light received from the charge points to the retarded position. There is a standard detailed calculation (like the one in Griffiths or Jackson) here, with the following conclusion:
Very interesting...
But it still doesn’t change the causal relationship: that alignment only applies if nothing happens to the other thing in the time since the center of that retarded cone. If no new information has been generated, then sure, you can use the new information instead of the old information. But if anything happens, you had better use the old information!
To get more formal about it: Consider or charge (equivalently, a mass) whose worldline coincides with (t, 0,0,0) for all t ⇐ 0) in some reference frame
The field at the event (10, 10, 0, 0 ) occurs after (0,0,0,0) in every subluminal reference frame.
The field at (10, 10, 0, 0 ) is independent of whatever happens at (1, 0,0,0). If a laser comes in and knocks that charge aside, there’s zero difference. None at all.
Suppose the charge was deflected so that it passes through (10, 1, 0, 0). You can’t get the electrical field at event (10, 10, 0, 0) by looking at what the charge is doing at (10, 1, 0, 0) - you need to look at (0,0, 0,0).
This is what causality looks like. The causal influences propagate at lightspeed. Even electrostatic ones.
Absolutely. As I said
However I am not sure I agree with the part in bold:
When you say “Suppose the charge was deflected”, you have broken the electrostatic assumptions, since the charge is now accelerating. Depending on the distance, you either get the near-field effects or the radiative effects, which do indeed propagate at lightspeed. Once the acceleration disappears and the light-speed transients died down, you are back in the electrostatic/magnetostatic mode with lag-free fields.
That was the point of the example—by time 10, the charge was no longer accelerating, but you know that you’re not clear to use electrostatics being ‘noncausal’ yet because the light cone hasn’t reached that far. Being lag free is a computational convenience that sometimes applies, and you need to know when by applying causality.
So, it is ALWAYS fair to say that fields are causal influences, whether they’re static or dynamic. That was why I objected to your complaint in the first place.