The Sun, as it is right now, won't have its gravity affect Earth for another 8+ minutes, and the gravity that the Earth feels right now pulling it towards the Sun is actually pulling it towards where the Sun was 8+ minutes ago! (Weird, isn't it?)
I'm almost positive this statement is incorrect.
Relativistic force laws tend to be forced to contain correction factors that ensure that constant velocity motion is "predicted" and the direction of the force is adjusted accordingly - as long as a body is not accelerating, a purely attractive or repulsive force will be pointing at its current position, not its position 8 mins ago.
To see why this must be the case for a repulsive force, at least, imagine two charges riding on frictionless rails that keep them at a constant, finite distance. Suppose they're both moving with some constant velocity (to start, at least) in the same direction - now the place that the force from the other charge appears to be coming from is behind the charge, so if there was no correction factor, each charge would be getting an extra push forwards. This would lead to a runaway "bootstrap" acceleration, and the particles would accelerate to the speed of light. That's a pretty clear violation of the conservation laws, so...
With attractive forces like gravity, there's still a conservation problem, since the charges would slow down to zero speed eventually, but it's always more convincing to cite the runaway solution as a violation of conservation laws than the run-down one, because while the energy could possibly leak out of the charges into the fields, there's nowhere to pull infinite energy from, so it's pretty clear there's a problem if we'd need to.
And this is somewhat different from the runaway self-action solutions that we grudgingly "accept" in E+M, because those tend to involve some limit to infinitesimal size, whereas this is a completely finite situation that we could theoretically set up in the real world with a couple of charged beads or something like that.
I don't think it's incorrect. But for constant velocities it is canceled out because the relative motion causes things to also appear in a different direction. This happens with ordinary light, also. Stars appear in a different place to us because of our relative motion (they appear to be located more "forward" in our direction of motion).
As as example, assume there is a star directly perpendicular to our motion (and that the Earth moves in a straight line). Due to the vector addition of our motion and the light travel direction, it appears to us that the star is located slightly forward of perpendicular (typically by about 1/100 of a degree) Now assume the star were to disappear. During the light travel time the star would have time to move backwards as seen by us (due to our forward motion) so that at the moment it disappears, it appears to be located perpendicular to us.
In the context of gravity, this effect (it's called aberration) exactly cancels and the net effect is that the gravitational attraction is in a direction different from the actual location of the attractor such that it appears that gravity is instantaneous.
This only works for constant velocities, once you have accelerations it becomes more complicated. And it's not a relativistic effect at all, it's present for all waves with finite propagation speeds. You can do this experiment with boats making waves and get the same result.
Edit: And your charge example is not so good. For Galilean invariant theories it's only relative motions that matter. There is no effect if the two are moving with the same velocity. (Plus, once there are relative motions between the charges, there will be induced magnetic fields which affect the dynamics.)
> And your charge example is not so good. For Galilean invariant theories it's only relative motions that matter.
E&M is Lorentz invariant though, not Galilean.
This causes an issue with your starlight aberration example as well. Velocities do not add linearly (though that is a reasonable approximation for low velocities).
> Relativistic force laws tend to be forced to contain correction factors that ensure that constant velocity motion is "predicted" and the direction of the force is adjusted accordingly - as long as a body is not accelerating, a purely attractive or repulsive force will be pointing at its current position, not its position 8 mins ago.
Absolutely correct for E&M. This is because it's a vector field, and the velocity of the source is encoded in how the field moves.
For gravity, it goes one better -- this is a tensor field (hence the notion of gravitons having to be "spin-2" if they were described by a quantum theory), and the acceleration is encoded as well, so you need a non-constant acceleration to notice any difference from the position being instantaneously updated.
Ah, that's interesting, I never realized the rank was what determined the amount of "prediction" that a field does. That's something that should have been obvious on degrees-of-freedom considerations, but I never thought about it in that way before.
That means my bootstrap-runaway argument must be flawed in some way, because a scalar field can't encode any velocity information, and as a classical field equation, it would cause a 1/r^2 force law just like everything else, but aimed at the retarded position.
For Galilean relativity, a sphere expanding at a given speed will remain so after a boost, but will have a net velocity. Densities will remain the same. A moving source shouldn't matter -- any material will itself will set the rest-frame, and must to have a non-infinite propagation velocity. Whereas, emitting particles, these will have the velocity of the source. Ah! if it's a complex wave equation, you can put a varying phase on everything encoding the velocity. But moving to a complex field is in some sense putting more degrees-of-freedom in.
For Einsteinian relativity, a sphere expanding at light speed remains a sphere in any reference frame, but the density along it changes. A moving source should have that same pattern, so it would seem that local differences (i.e. derivatives) could "point out" the velocity of the source. This behaves the same for particles being emitted. Is there a first-order Lorentz invariant scalar wave-equation, or only second-order, using the d'Alembertian? And that seems to introduce another degree-of-freedom as well.
I'm still really confused. How do pions behave? Is it reasonable to consider them as a fundamental scalar field when they're modeled as pairs of quarks? Particularly when they have excited states that are spin-1?
> For gravity, it goes one better -- this is a tensor field (hence the notion of gravitons having to be "spin-2" if they were described by a quantum theory), and the acceleration is encoded as well, ....
Well, that's mind blowing. Where could I go to learn more about this?
Other than "graduate textbooks", I'm not really sure. Jackson does cover the E&M case pretty well (EDIT: in chapters 11 and 14, note especially sections 10.11 and 14.1), but I'd combine it with the treatment in Taylor & Wheeler's highly readable _Spacetime Physics_.
For E&M, the standard way to develop this is to explain magnetism as Lorentz-transformed static electrical attraction/repulsion. Take two wires, and run current through them. Transforming to a frame where the electrons are at rest, but the atoms (and hence protons) are moving, length contraction ends up with the density being different, meaning a net charge in this frame. You then get E&M united as tensor field F, but antisymmetric, meaning the spin-2 components are 0, leaving effectively two spin-1 (vector) components. This lets you do relativistic corrections for a propagating field, and the velocity of the original source gets turned into magnetic effects that act the same as if the source were moving at a constant velocity.
Extremely similar things happen with gravity, if you look at weak-field linearized versions of the Einstein field equations. The actual math ... well, it's rather ugly.
Wait, so you are saying we would feel the effects immediately, but the light would not disappear for eight minutes?
So let's assume once can cause the sun to vanish instantly somehow. If that's the case, would it then be possible to setup a 'gravitational wave' communication system that travels faster than light?
Ie, I would put a whole bunch of stars together, and then have them disappear at certain intervals. My buddy who sits 100 light-minutes away would then detect the changes and decode the message. TADA: Faster-than-light communication.
No, the sun disappearing would not be constant velocity or constant acceleration motion, so the info would indeed only propagate at the speed of light.
Yes, you are correct. The easiest way to think about it is that while the gravity from the earth moves at the speed of light, while it travels toward the sun, it also moves at the same original speed the earth was traveling at.
Te end result is that it appears to travel instantaneously since it ends up at the same place. But this only works for constant velocities - if the earth changes direction suddenly this "correction" will not be correct, since it will still "assume" the earth is still headed in the direction it was heading before.
This is more or less a correct interpretation (the physics are definitely right). Importantly, the behavior is seen in electromagnetism, which is much easier to think about than general relativity, so people should start there.
(But no one will ever read this comment because the topic is 5 days old.)
Yes you are totally right. Furthermore: In relativity there is no "right now" moment. Two simultaneous events in one reference frame can be one after the other in another reference frame. (see: http://en.wikipedia.org/wiki/Relativity_of_simultaneity) This concept is even true in special relativity and has been tested and a basic principle why GPS is working.
I'm not certain your thought experiment applies. Your use of charges implies that we are discussing EM attraction or repulsion, which we are not.
Considering gravity does not have an inverse (a gravitational repulsion) that I am aware of, I'm not certain we can extend your thought experiment to gravity.
Comments
The Sun, as it is right now, won't have its gravity affect Earth for another 8+ minutes, and the gravity that the Earth feels right now pulling it towards the Sun is actually pulling it towards where the Sun was 8+ minutes ago! (Weird, isn't it?)
I'm almost positive this statement is incorrect.
Relativistic force laws tend to be forced to contain correction factors that ensure that constant velocity motion is "predicted" and the direction of the force is adjusted accordingly - as long as a body is not accelerating, a purely attractive or repulsive force will be pointing at its current position, not its position 8 mins ago.
To see why this must be the case for a repulsive force, at least, imagine two charges riding on frictionless rails that keep them at a constant, finite distance. Suppose they're both moving with some constant velocity (to start, at least) in the same direction - now the place that the force from the other charge appears to be coming from is behind the charge, so if there was no correction factor, each charge would be getting an extra push forwards. This would lead to a runaway "bootstrap" acceleration, and the particles would accelerate to the speed of light. That's a pretty clear violation of the conservation laws, so...
With attractive forces like gravity, there's still a conservation problem, since the charges would slow down to zero speed eventually, but it's always more convincing to cite the runaway solution as a violation of conservation laws than the run-down one, because while the energy could possibly leak out of the charges into the fields, there's nowhere to pull infinite energy from, so it's pretty clear there's a problem if we'd need to.
And this is somewhat different from the runaway self-action solutions that we grudgingly "accept" in E+M, because those tend to involve some limit to infinitesimal size, whereas this is a completely finite situation that we could theoretically set up in the real world with a couple of charged beads or something like that.
I don't think it's incorrect. But for constant velocities it is canceled out because the relative motion causes things to also appear in a different direction. This happens with ordinary light, also. Stars appear in a different place to us because of our relative motion (they appear to be located more "forward" in our direction of motion).
As as example, assume there is a star directly perpendicular to our motion (and that the Earth moves in a straight line). Due to the vector addition of our motion and the light travel direction, it appears to us that the star is located slightly forward of perpendicular (typically by about 1/100 of a degree) Now assume the star were to disappear. During the light travel time the star would have time to move backwards as seen by us (due to our forward motion) so that at the moment it disappears, it appears to be located perpendicular to us.
In the context of gravity, this effect (it's called aberration) exactly cancels and the net effect is that the gravitational attraction is in a direction different from the actual location of the attractor such that it appears that gravity is instantaneous.
This only works for constant velocities, once you have accelerations it becomes more complicated. And it's not a relativistic effect at all, it's present for all waves with finite propagation speeds. You can do this experiment with boats making waves and get the same result.
Edit: And your charge example is not so good. For Galilean invariant theories it's only relative motions that matter. There is no effect if the two are moving with the same velocity. (Plus, once there are relative motions between the charges, there will be induced magnetic fields which affect the dynamics.)
> And your charge example is not so good. For Galilean invariant theories it's only relative motions that matter.
E&M is Lorentz invariant though, not Galilean.
This causes an issue with your starlight aberration example as well. Velocities do not add linearly (though that is a reasonable approximation for low velocities).
Yes, I know that it's not exact for high velocities. My point was just that you don't need relativity for this effect to happen.
> Relativistic force laws tend to be forced to contain correction factors that ensure that constant velocity motion is "predicted" and the direction of the force is adjusted accordingly - as long as a body is not accelerating, a purely attractive or repulsive force will be pointing at its current position, not its position 8 mins ago.
Absolutely correct for E&M. This is because it's a vector field, and the velocity of the source is encoded in how the field moves.
For gravity, it goes one better -- this is a tensor field (hence the notion of gravitons having to be "spin-2" if they were described by a quantum theory), and the acceleration is encoded as well, so you need a non-constant acceleration to notice any difference from the position being instantaneously updated.
Ah, that's interesting, I never realized the rank was what determined the amount of "prediction" that a field does. That's something that should have been obvious on degrees-of-freedom considerations, but I never thought about it in that way before.
That means my bootstrap-runaway argument must be flawed in some way, because a scalar field can't encode any velocity information, and as a classical field equation, it would cause a 1/r^2 force law just like everything else, but aimed at the retarded position.
Hmm. Now I need to think about it a bit more.
Suppose we have a scalar field.
For Galilean relativity, a sphere expanding at a given speed will remain so after a boost, but will have a net velocity. Densities will remain the same. A moving source shouldn't matter -- any material will itself will set the rest-frame, and must to have a non-infinite propagation velocity. Whereas, emitting particles, these will have the velocity of the source. Ah! if it's a complex wave equation, you can put a varying phase on everything encoding the velocity. But moving to a complex field is in some sense putting more degrees-of-freedom in.
For Einsteinian relativity, a sphere expanding at light speed remains a sphere in any reference frame, but the density along it changes. A moving source should have that same pattern, so it would seem that local differences (i.e. derivatives) could "point out" the velocity of the source. This behaves the same for particles being emitted. Is there a first-order Lorentz invariant scalar wave-equation, or only second-order, using the d'Alembertian? And that seems to introduce another degree-of-freedom as well.
I'm still really confused. How do pions behave? Is it reasonable to consider them as a fundamental scalar field when they're modeled as pairs of quarks? Particularly when they have excited states that are spin-1?
> For gravity, it goes one better -- this is a tensor field (hence the notion of gravitons having to be "spin-2" if they were described by a quantum theory), and the acceleration is encoded as well, ....
Well, that's mind blowing. Where could I go to learn more about this?
Other than "graduate textbooks", I'm not really sure. Jackson does cover the E&M case pretty well (EDIT: in chapters 11 and 14, note especially sections 10.11 and 14.1), but I'd combine it with the treatment in Taylor & Wheeler's highly readable _Spacetime Physics_.
For E&M, the standard way to develop this is to explain magnetism as Lorentz-transformed static electrical attraction/repulsion. Take two wires, and run current through them. Transforming to a frame where the electrons are at rest, but the atoms (and hence protons) are moving, length contraction ends up with the density being different, meaning a net charge in this frame. You then get E&M united as tensor field F, but antisymmetric, meaning the spin-2 components are 0, leaving effectively two spin-1 (vector) components. This lets you do relativistic corrections for a propagating field, and the velocity of the original source gets turned into magnetic effects that act the same as if the source were moving at a constant velocity.
Extremely similar things happen with gravity, if you look at weak-field linearized versions of the Einstein field equations. The actual math ... well, it's rather ugly.
I think my brain just exploded. Thanks for the references!
Wait, so you are saying we would feel the effects immediately, but the light would not disappear for eight minutes?
So let's assume once can cause the sun to vanish instantly somehow. If that's the case, would it then be possible to setup a 'gravitational wave' communication system that travels faster than light?
Ie, I would put a whole bunch of stars together, and then have them disappear at certain intervals. My buddy who sits 100 light-minutes away would then detect the changes and decode the message. TADA: Faster-than-light communication.
I feel safe to say that's not possible.
No, the sun disappearing would not be constant velocity or constant acceleration motion, so the info would indeed only propagate at the speed of light.
Out of curiosity, why do you think this is not possible?
You can't transmit data faster than the speed of light.
http://cdsweb.cern.ch/record/1122131/?ln=cs
So far as we currently know, classical information (aka bits) cannot be transferred via quantum entanglement (aka spooky action at a distance.)
I love the names they give to things nowadays. In this case, specifically "Spooky action"
I think that when it comes to the term "spooky action", "nowadays" would be around the time Einstein was working on all of this.
Yes, you are correct. The easiest way to think about it is that while the gravity from the earth moves at the speed of light, while it travels toward the sun, it also moves at the same original speed the earth was traveling at.
Te end result is that it appears to travel instantaneously since it ends up at the same place. But this only works for constant velocities - if the earth changes direction suddenly this "correction" will not be correct, since it will still "assume" the earth is still headed in the direction it was heading before.
This is more or less a correct interpretation (the physics are definitely right). Importantly, the behavior is seen in electromagnetism, which is much easier to think about than general relativity, so people should start there.
(But no one will ever read this comment because the topic is 5 days old.)
Yes you are totally right. Furthermore: In relativity there is no "right now" moment. Two simultaneous events in one reference frame can be one after the other in another reference frame. (see: http://en.wikipedia.org/wiki/Relativity_of_simultaneity) This concept is even true in special relativity and has been tested and a basic principle why GPS is working.
I'm not certain your thought experiment applies. Your use of charges implies that we are discussing EM attraction or repulsion, which we are not.
Considering gravity does not have an inverse (a gravitational repulsion) that I am aware of, I'm not certain we can extend your thought experiment to gravity.