Those who think they understand General Relativity might warm up to the task of retconning Time-Turners by solving the following conundrum, which takes place in the real world, and so must be solved under the constraint that “magic” is not an allowed answer.
The Earth orbits at 18 miles per second, and causal influences from the Sun travel no faster than light, which takes 8 minutes to reach the Earth from the Sun. Therefore the Earth “sees” the Sun 18x8x60 miles behind the straight line joining the two (using a Sun-centred frame of reference). This is an angle to the radial line of 18x8x60/93000000 = 0.0001 radians, or about 1⁄3 of a minute of angle. The gravitational attraction, being a causal influence transmitted at lightspeed, must be along that line also, thus retarding the Earth in its orbit with an acceleration of 0.0001 times the radial acceleration. This is enough to cancel out the Earth’s orbital velocity in about 1000 years.
The Earth revolves around the point where the sun was 8 minutes ago. In a solar-centric frame of reference, this is where the sun is now. In any other frame of reference, “8 minutes” has changed.
I don’t know the relativistic math, but: Gur Fha vf abg npgvat ba gur Rnegu; vg jbhyq npg vqragvpnyyl ba nal znff gung unccraf gb or naljurer nybat gung beovg. Gb hfr lbhe jbeqf, “Gur tenivgngvbany nggenpgvba, orvat n pnhfny vasyhrapr genafzvggrq ng yvtugfcrrq, zhfg or nybat gung yvar”, ohg vg vf ba rirel bgure yvar vagrefrpgvat gur Fha nf jryy. Gurersber yvtugfcrrq vf abg n pbafgenvag ba jung unccraf.
I’m pretty sure there’s no general relativity in the error:
Qenj gur gevnatyr sebz gur rnegu gb gur fha gb gur cbvag jurer gur rnegu jvyy or jura vg vagreprcgf yvtug sebz gur fha gung vf orvat rzvggrq ‘abj’.
Gur shgher Rnegu gung vf frrvat gur yvtug sebz gur fha abj vf frrvat gur yvtug gung gur Rnegu rzvggrq nybat gur ulcbgrahfr, juvpu vf abg ng gur fnzr qverpgvba n gur Rnegu vf ‘abj’, ohg engure vf va gur qverpgvba sebz gur fha gb jurer gur Rnegu jvyy or jura vg vagreprcgf gur yvtug. Va bgure jbeqf, V qvfnterr gung gur Rnegu ‘frrf’ gur fha va gur jebat fcbg.
Because it’s an inertial frame, to the accuracy required, but the Earth’s frame isn’t.
Mind you, I can’t claim to know anything about GR, certainly not by this standard. My expectation is that when one does the mathematics, GR does not predict planets spiralling to destruction on thousand-year timescales, and that shminux has located the problematic step. But I do not know the mathematics. My expectation is also that verbal stories to justify why it does or does not happen are only helpful once one has worked out the real mathematical story, and cannot by themselves resolve the matter.
Umm, what? I studied GR way less than you, but so far as I know, there’s nothing even slightly problematic about that particular phrase. Perhaps you’re referring to its failure to imply the phrase after it? Because the following phrase is definitely wrong.
“The gravitational attraction, being a causal influence transmitted at lightspeed” is wrong here because static fields do not propagate, they are just there. That’s why black hole can retain electric charge and mass, even though “not even light can escape it”. EM and gravitational waves propagate at the speed of light, but that’s not related to gravitational attraction or the coulomb force between bodies. But you know all that. So it’s a wrong premise which motivates a wrong conclusion, which is the next phrase.
I won’t say anything about gravitational waves, but in EM, what you just said is unrelated, totally is related. If you just take the rules for electrostatics and the magnetic field from charges, and apply time-retardation, you recover the electromagnetic waves produced by that charge’s motion. No corrections or adjustments are needed.
The electrical and magnetic fields at a given point is precisely the sum over all charged particles of their momentary electrical and magnetic fields at that point, causally offset… (… plus any background fields that the universe was shipped with, that did not arise from charges. I’m not saying that that’s impossible)
But the point is, it’s totally completely fair to talk about causal influences of a charged particle or mass propagating at lightspeed. If a particle’s sitting there and then gets kicked and settles down, the electrostatic and gravitational fields update to their new values by the particle emitting waves.
I won’t say anything about gravitational waves, but in EM, what you just said is unrelated, totally is related. If you just take the rules for electrostatics and the magnetic field from charges, and apply time-retardation, you recover the electromagnetic waves produced by that charge’s motion. No corrections or adjustments are needed.
That’s true, but that’s waves, not static attraction. In essence, electric attraction is “magically” corrected for the straight-line motion, so the electric field from a moving but non-accelerating charge points exactly in the direction of the charge, not in the direction where the charge was after accounting for time-retardation. This breaks down once you add acceleration, hence EM waves. The original post made this (possibly deliberate) mistake: calculated retarded field for (nearly) uniform motion when calculating the direction of the attractive force. Unless I misunderstood it.
For gravity the corrections resulting in radiation appear even later, its “predictive power” is one order in time derivatives better than that of EM.
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.
Not necessarily. This is somewhat counter-intuitive, but the light lags the direction of the Coulomb’s law’s attraction if a charge moves past you with a constant velocity (and has been doing so for some time). The attraction force points to the “true” direction of the charge, whereas light takes a bit to catch up.
Note that this apparent FTL effect cannot be used to transmit any information FTL, because, as soon as you try to wiggle the charge to telegraph something, this wiggling will only be sent as EM radiation (light), at light speed.
So, it’s possible for the electrical force to act from a location that never had a charge? If the charge moves at a constant speed long enough and then makes a hard turn away, the charge will act on other objects (at least briefly) as if the charge had continued straight? Does the charge also act on the object as though it had not turned, even after it had, or is the force unilateral?
Yes to the second, not sure how the third is different. That’s why, in part, Newton’s second law does not in general hold for Electromagnetism, but momentum conservation does, if you account for the momentum of the electromagnetic field itself.
That behavior is contrary to naive expectations, right? If I run really fast towards a wall but turn before I reach it, I shouldn’t hit it. It also shouldn’t smash my face in after I make the turn.
The major force involved in billiard balls bouncing off of each other is electric in nature, right?
Those who think they understand General Relativity might warm up to the task of retconning Time-Turners by solving the following conundrum, which takes place in the real world, and so must be solved under the constraint that “magic” is not an allowed answer.
The Earth orbits at 18 miles per second, and causal influences from the Sun travel no faster than light, which takes 8 minutes to reach the Earth from the Sun. Therefore the Earth “sees” the Sun 18x8x60 miles behind the straight line joining the two (using a Sun-centred frame of reference). This is an angle to the radial line of 18x8x60/93000000 = 0.0001 radians, or about 1⁄3 of a minute of angle. The gravitational attraction, being a causal influence transmitted at lightspeed, must be along that line also, thus retarding the Earth in its orbit with an acceleration of 0.0001 times the radial acceleration. This is enough to cancel out the Earth’s orbital velocity in about 1000 years.
This has not happened. Where is the error?
Teach the heliocentric controversy!
The Earth revolves around the point where the sun was 8 minutes ago. In a solar-centric frame of reference, this is where the sun is now. In any other frame of reference, “8 minutes” has changed.
I don’t know the relativistic math, but: Gur Fha vf abg npgvat ba gur Rnegu; vg jbhyq npg vqragvpnyyl ba nal znff gung unccraf gb or naljurer nybat gung beovg. Gb hfr lbhe jbeqf, “Gur tenivgngvbany nggenpgvba, orvat n pnhfny vasyhrapr genafzvggrq ng yvtugfcrrq, zhfg or nybat gung yvar”, ohg vg vf ba rirel bgure yvar vagrefrpgvat gur Fha nf jryy. Gurersber yvtugfcrrq vf abg n pbafgenvag ba jung unccraf.
I’m pretty sure there’s no general relativity in the error:
Qenj gur gevnatyr sebz gur rnegu gb gur fha gb gur cbvag jurer gur rnegu jvyy or jura vg vagreprcgf yvtug sebz gur fha gung vf orvat rzvggrq ‘abj’. Gur shgher Rnegu gung vf frrvat gur yvtug sebz gur fha abj vf frrvat gur yvtug gung gur Rnegu rzvggrq nybat gur ulcbgrahfr, juvpu vf abg ng gur fnzr qverpgvba n gur Rnegu vf ‘abj’, ohg engure vf va gur qverpgvba sebz gur fha gb jurer gur Rnegu jvyy or jura vg vagreprcgf gur yvtug. Va bgure jbeqf, V qvfnterr gung gur Rnegu ‘frrf’ gur fha va gur jebat fcbg.
Jul ner lbh hfvat gur Fha nf gur senzr bs ersrerapr jura nfxvat jung gur Rnegu vf frrvat? (V’z abg fher vs guvf vf gur reebe lbh’er ybbxvat sbe be abg, ohg vg fgevxrf zr nf vaghvgviryl jebat.)
Because it’s an inertial frame, to the accuracy required, but the Earth’s frame isn’t.
Mind you, I can’t claim to know anything about GR, certainly not by this standard. My expectation is that when one does the mathematics, GR does not predict planets spiralling to destruction on thousand-year timescales, and that shminux has located the problematic step. But I do not know the mathematics. My expectation is also that verbal stories to justify why it does or does not happen are only helpful once one has worked out the real mathematical story, and cannot by themselves resolve the matter.
Actually the issue is the statement “Gur tenivgngvbany nggenpgvba, orvat n pnhfny vasyhrapr genafzvggrq ng yvtugfcrrq”.
Umm, what? I studied GR way less than you, but so far as I know, there’s nothing even slightly problematic about that particular phrase. Perhaps you’re referring to its failure to imply the phrase after it? Because the following phrase is definitely wrong.
“The gravitational attraction, being a causal influence transmitted at lightspeed” is wrong here because static fields do not propagate, they are just there. That’s why black hole can retain electric charge and mass, even though “not even light can escape it”. EM and gravitational waves propagate at the speed of light, but that’s not related to gravitational attraction or the coulomb force between bodies. But you know all that. So it’s a wrong premise which motivates a wrong conclusion, which is the next phrase.
I won’t say anything about gravitational waves, but in EM, what you just said is unrelated, totally is related. If you just take the rules for electrostatics and the magnetic field from charges, and apply time-retardation, you recover the electromagnetic waves produced by that charge’s motion. No corrections or adjustments are needed.
The electrical and magnetic fields at a given point is precisely the sum over all charged particles of their momentary electrical and magnetic fields at that point, causally offset… (… plus any background fields that the universe was shipped with, that did not arise from charges. I’m not saying that that’s impossible)
But the point is, it’s totally completely fair to talk about causal influences of a charged particle or mass propagating at lightspeed. If a particle’s sitting there and then gets kicked and settles down, the electrostatic and gravitational fields update to their new values by the particle emitting waves.
That’s true, but that’s waves, not static attraction. In essence, electric attraction is “magically” corrected for the straight-line motion, so the electric field from a moving but non-accelerating charge points exactly in the direction of the charge, not in the direction where the charge was after accounting for time-retardation. This breaks down once you add acceleration, hence EM waves. The original post made this (possibly deliberate) mistake: calculated retarded field for (nearly) uniform motion when calculating the direction of the attractive force. Unless I misunderstood it.
For gravity the corrections resulting in radiation appear even later, its “predictive power” is one order in time derivatives better than that of EM.
Wat. This is so severely counterintuitive I’m going to have to look it up and get a technical explanation. Is there a named effect for this?
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.
Isn’t “the direction of the charge” where you point a telescope to look at it, even if it is moving?
Not necessarily. This is somewhat counter-intuitive, but the light lags the direction of the Coulomb’s law’s attraction if a charge moves past you with a constant velocity (and has been doing so for some time). The attraction force points to the “true” direction of the charge, whereas light takes a bit to catch up.
Note that this apparent FTL effect cannot be used to transmit any information FTL, because, as soon as you try to wiggle the charge to telegraph something, this wiggling will only be sent as EM radiation (light), at light speed.
So, it’s possible for the electrical force to act from a location that never had a charge? If the charge moves at a constant speed long enough and then makes a hard turn away, the charge will act on other objects (at least briefly) as if the charge had continued straight? Does the charge also act on the object as though it had not turned, even after it had, or is the force unilateral?
Yes to the second, not sure how the third is different. That’s why, in part, Newton’s second law does not in general hold for Electromagnetism, but momentum conservation does, if you account for the momentum of the electromagnetic field itself.
That behavior is contrary to naive expectations, right? If I run really fast towards a wall but turn before I reach it, I shouldn’t hit it. It also shouldn’t smash my face in after I make the turn.
The major force involved in billiard balls bouncing off of each other is electric in nature, right?
Evtug, rirelbar xabjf n serr obql whfg zbirf va n fgenvtug yvar.