> 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.
Comments
> 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!