I consider momentum conservation a “big principle.”, and Newtons 3 laws indeed set out momentum conservation. However, I believe uniform centre of mass motion to be an importantly distinct principle. The drive loop thing would conserve momentum even if it were possible. Indeed momentum conservation is the principle underpinning the assumed reaction forces that make it work in the first place. To take a different example, if you had a pair of portals (like from the game “portal”) on board your spaceship, and ran a train between them, you could drive the train backwards, propelling your ship forwards, and thereby move while conserving total momentum, only to later put the train’s breaks on and stop. I am not asking you to believe in portals, I am just trying to motivate that weird hypotheticals can be cooked up where the principle of momentum conservation decouples from the principle of uniform centre of mass motion. The two are distinct principles.
Abraham supporters do indeed think you can use conservation of momentum to work out which way the glass block moves in that thought experiment, showing that (because the photon momentum goes down) the block must move to the right. Minkowksi supporters also think you can use conservation of momentum to work out how the glass block moves, but because they think the photon momentum goes up the block must move to the left. The thing that is at issue is the question of what expression to use to calculate the momentum, both sides agree that whatever the momentum is it is conserved. As a side point, a photon has nonzero momentum in all reference frames, and that is not an aspect of relativity that is sensibly ignored.
You are actually correct that the photon does have to red-shift very slightly as it enters the glass block. If the glass was initially at rest, then after the photon has entered the photon has either gained or lost momentum (depending on Abraham or Minkowski), in either case imparting the momentum difference onto the glass block. The kinetic energy of the glass block is given by p2/2m where p is the momentum the block has gained, and m is the block’s mass. The photon’s new frequency is then given by ℏω=ℏω0−p2/2m (by conservation of energy) where ω0 was its initial frequency. In practice a glass block will have a very gigantic mass compared to p2, but at least in principle the photon does red shift.
Going into the full gory detail for Abraham.
Abraham: Photon momentum before entering glass ℏω0/c Photon momentum after entering glass ℏω/(nc) (note, new frequency ω, not the old one) Change in photon momentum Δp=ℏω0/c−ℏω/nc The same momentum goes into the glass, so ℏω=ℏω0−(ℏω0/c−ℏω/nc)2/2m
We can re-arrange to put the c^2 in the denominator next to the mass of the glass block. So that the change in the frequency/energy of the photon is scaled by a term that has something to do with the refractive index, along with how the photon energy compares to the rest mass energy of the glass block (mc2). So, as previously said, this is negligible for a glass block that weighs any reasonable amount.
The Minkowski version is almost the same derivation, except the division by refractive index becomes a multiplication, giving: ℏω=ℏω0−(ℏω0/c−nℏω/c)2/2m
Playing with these quadratic equations, to solve for ω, you find that the Abraham version never breaks. In contrast, if you assume it is possible to have a glass block with a reasonably high refractive index, but arbitrarily small mass, then Minkowski eventually breaks and starts giving an imaginary frequency. This maybe says something vaguely negative about Minkowski, but a block of material with a high refractive index but negligible mass is such an unrealistic setup that I don’t think failing in that case is too embarrassing for the Minkowski equation.
Given 12.72, uniform motion of the center of energy is equivalent to conservation of momentum, right? P is const ⇔ dR_e/dt is const.
(I’m guessing 12.72 is in fact correct here, but I guess we can doubt it — I haven’t thought much about how to prove it when fields and relativistic and quantum things are involved. From a cursory look at his comment, Lubos Motl seems to consider it invalid lol ( in https://physics.stackexchange.com/a/3200 ).)
That said, the hypothetical you give is cool and I agree the two principles decouple there! (I intuitively want to save that case by saying the COM is only stationary in a covering space where the train has in fact moved a bunch by the time it stops, but idk how to make this make sense for a different arrangement of portals.) I guess another thing that seems a bit compelling for the two decoupling is that conservation of angular momentum is analogous to conservation of momentum but there’s no angular analogue to the center of mass (that’s rotating uniformly, anyway). I guess another thing that’s a bit compelling is that there’s no nice notion of a center of energy once we view spacetime as being curved ( https://physics.stackexchange.com/a/269273 ). I think I’ve become convinced that conservation of momentum is a significantly bigger principle :). But still, the two seem equivalent to me before one gets to general relativity. (I guess this actually depends a bit on what the proof of 12.72 is like — in particular, if that proof basically uses the conservation of momentum, then I’d be more happy to say that the two aren’t equivalent already for relativity/fields.)
I think the point about angular momentum is a very good way of gesturing at how its possibly different. Angular momentum is conserved, but an isolated system can still rotate itself, by spinning up and then stopping a flywheel (moving the “center of rotation”).
Thank for finding that book and screenshot. Equation 12.72 is directly claiming that momentum is proportional to energy flow (and in the same direction). I am very curious how that intersects with claims common in metamaterials (https://journals.aps.org/pra/abstract/10.1103/PhysRevA.75.053810 ) that the two can flow in opposite directions.
In my post the way I cited Lubos Motl’s comment implicitly rounded it off to “Minkowski is just right” (option [6]), which is indeed his headline and emphasis. But if we are zooming in on him I should admit that his full position is a little more nuanced. My understanding is that he makes 3 points:
(1) - Option [1] is correct. (Abraham gives kinetic momentum, Minkowski the canonical momentum) (2) - In his opinion the kinetic momentum is pointless and gross, and that true physics only concerns itself with the canonical momentum. (3) - As a result of the kinetic momentum being worthless its basically correct to say Minkowski was “just right”(option [6]). This means that the paper proposing option [1] was a waste of time (much ado about nothing), because the difference between believing [1] and believing [6] only matters when doing kinetics, which he doesn’t care about. Finally, having decided that Minkowski was correct in the only way that he thinks matters, he goes off into a nasty side-thing about how Abraham was supposedly incompetent.
So his actual position is sort of [1] and [6] at the same time (because he considers the difference between them inconsequential, as it only applies to kinetics). If he leans more on the [1] side he can consider 12.72 to be valid. But why would he bother? 12.72 is saying something about kinetics, it might as well be invalid. He doesn’t care either way.
He goes on to explicitly say that he thinks 12.72 is invalid. Although I think his logic on this is flawed. He says the glass block breaks the symmetry, which is true for the photon. However, the composite system (photon + glass block) still has translation and boost symmetry, and it is the uniform motion of the center of mass of the composite system that is at stake.
re redshift: Sorry, I should have been clearer, but I meant to talk about redshift (or another kind of energy loss) of the light that comes out of the block on the right compared to the light that went in from the left, which would cause issues with going from there being a uniformly-moving stationary center of mass to the conclusion about the location of the block. (I’m guessing you were right when you assumed in your argument that redshift is 0 for our purposes, but I don’t understand light in materials well enough atm to see this at a glance atm.)
And the loss mechanism I was imagining was more like something linear in the distance traveled, like causing electrons to oscillate but not completely elastically wrt the ‘photon’ inside the material.
Anyway, in your argument for the redshift as the photon enters the block, I worry about the following:
can we really think of 1 photon entering the block becoming 1 photon inside the block, as opposed to needing to think about some wave thing that might translate to photons in some other way or maybe not translate to ordinary photons at all inside the material (this is also my second worry from earlier)?
do we know that this photon-inside-the-material has energy ℏω?
The microscopic picture that Mark Mitchison gives in the comments to this answer seems pretty: https://physics.stackexchange.com/a/44533 — though idk if I trust it. The picture seems to be to think of glass as being sparse, with the photon mostly just moving with its vacuum velocity and momentum, but with a sorta-collision between the photon and an electron happening every once in a while. I guess each collision somehow takes a certain amount of time but leaves the photon unchanged otherwise, and presumably bumps that single electron a tiny bit to the right. (Idk why the collisions happen this way. I’m guessing maybe one needs to think of the photon as some electromagnetic field thing or maybe as a quantum thing to understand that part.)
I presented the redshift calculation in terms of a single photon, but actually, the exact same derivation goes through unchanged if you replace every instance of ℏω0 with E0 and ℏω with E . Where E0 and E are the energy of a light pulse before and after it enters the glass. There is no need to specify whether the light pulse is a single photon a big flash of classical light or anything else.
Something linear in the distance travelled would not be a cumulatively increasing red shift, but instead an increasing loss of amplitude (essentially a higher cumulative probability of being absorbed). This is represented using a complex valued refractive index (or dielectric constant) where the real part is how much the wave slows down and the imaginary part is how much it attenuates per distance. There is no reason in principle why the losses cannot be arbitrarily close to zero at the wavelength we are using. (Interestingly, the losses have to be nonzero at some wavelength due to something called the Kramers Kronig relation, but we can assume they are negligible at our wavelength).
I consider momentum conservation a “big principle.”, and Newtons 3 laws indeed set out momentum conservation. However, I believe uniform centre of mass motion to be an importantly distinct principle. The drive loop thing would conserve momentum even if it were possible. Indeed momentum conservation is the principle underpinning the assumed reaction forces that make it work in the first place. To take a different example, if you had a pair of portals (like from the game “portal”) on board your spaceship, and ran a train between them, you could drive the train backwards, propelling your ship forwards, and thereby move while conserving total momentum, only to later put the train’s breaks on and stop. I am not asking you to believe in portals, I am just trying to motivate that weird hypotheticals can be cooked up where the principle of momentum conservation decouples from the principle of uniform centre of mass motion. The two are distinct principles.
Abraham supporters do indeed think you can use conservation of momentum to work out which way the glass block moves in that thought experiment, showing that (because the photon momentum goes down) the block must move to the right. Minkowksi supporters also think you can use conservation of momentum to work out how the glass block moves, but because they think the photon momentum goes up the block must move to the left. The thing that is at issue is the question of what expression to use to calculate the momentum, both sides agree that whatever the momentum is it is conserved. As a side point, a photon has nonzero momentum in all reference frames, and that is not an aspect of relativity that is sensibly ignored.
You are actually correct that the photon does have to red-shift very slightly as it enters the glass block. If the glass was initially at rest, then after the photon has entered the photon has either gained or lost momentum (depending on Abraham or Minkowski), in either case imparting the momentum difference onto the glass block. The kinetic energy of the glass block is given by p2/2m where p is the momentum the block has gained, and m is the block’s mass. The photon’s new frequency is then given by ℏω=ℏω0−p2/2m (by conservation of energy) where ω0 was its initial frequency. In practice a glass block will have a very gigantic mass compared to p2, but at least in principle the photon does red shift.
Going into the full gory detail for Abraham.
Abraham:
Photon momentum before entering glass ℏω0/c
Photon momentum after entering glass ℏω/(nc) (note, new frequency ω, not the old one)
Change in photon momentum Δp=ℏω0/c−ℏω/nc
The same momentum goes into the glass, so ℏω=ℏω0−(ℏω0/c−ℏω/nc)2/2m
We can re-arrange to put the c^2 in the denominator next to the mass of the glass block. So that the change in the frequency/energy of the photon is scaled by a term that has something to do with the refractive index, along with how the photon energy compares to the rest mass energy of the glass block (mc2). So, as previously said, this is negligible for a glass block that weighs any reasonable amount.
The Minkowski version is almost the same derivation, except the division by refractive index becomes a multiplication, giving: ℏω=ℏω0−(ℏω0/c−nℏω/c)2/2m
Playing with these quadratic equations, to solve for ω, you find that the Abraham version never breaks. In contrast, if you assume it is possible to have a glass block with a reasonably high refractive index, but arbitrarily small mass, then Minkowski eventually breaks and starts giving an imaginary frequency. This maybe says something vaguely negative about Minkowski, but a block of material with a high refractive index but negligible mass is such an unrealistic setup that I don’t think failing in that case is too embarrassing for the Minkowski equation.
here’s a picture from https://hansandcassady.org/David%20J.%20Griffiths-Introduction%20to%20Electrodynamics-Addison-Wesley%20(2012).pdf :
Given 12.72, uniform motion of the center of energy is equivalent to conservation of momentum, right? P is const ⇔ dR_e/dt is const.
(I’m guessing 12.72 is in fact correct here, but I guess we can doubt it — I haven’t thought much about how to prove it when fields and relativistic and quantum things are involved. From a cursory look at his comment, Lubos Motl seems to consider it invalid lol ( in https://physics.stackexchange.com/a/3200 ).)
That said, the hypothetical you give is cool and I agree the two principles decouple there! (I intuitively want to save that case by saying the COM is only stationary in a covering space where the train has in fact moved a bunch by the time it stops, but idk how to make this make sense for a different arrangement of portals.) I guess another thing that seems a bit compelling for the two decoupling is that conservation of angular momentum is analogous to conservation of momentum but there’s no angular analogue to the center of mass (that’s rotating uniformly, anyway). I guess another thing that’s a bit compelling is that there’s no nice notion of a center of energy once we view spacetime as being curved ( https://physics.stackexchange.com/a/269273 ). I think I’ve become convinced that conservation of momentum is a significantly bigger principle :). But still, the two seem equivalent to me before one gets to general relativity. (I guess this actually depends a bit on what the proof of 12.72 is like — in particular, if that proof basically uses the conservation of momentum, then I’d be more happy to say that the two aren’t equivalent already for relativity/fields.)
I think the point about angular momentum is a very good way of gesturing at how its possibly different. Angular momentum is conserved, but an isolated system can still rotate itself, by spinning up and then stopping a flywheel (moving the “center of rotation”).
Thank for finding that book and screenshot. Equation 12.72 is directly claiming that momentum is proportional to energy flow (and in the same direction). I am very curious how that intersects with claims common in metamaterials (https://journals.aps.org/pra/abstract/10.1103/PhysRevA.75.053810 ) that the two can flow in opposite directions.
In my post the way I cited Lubos Motl’s comment implicitly rounded it off to “Minkowski is just right” (option [6]), which is indeed his headline and emphasis. But if we are zooming in on him I should admit that his full position is a little more nuanced. My understanding is that he makes 3 points:
(1) - Option [1] is correct. (Abraham gives kinetic momentum, Minkowski the canonical momentum)
(2) - In his opinion the kinetic momentum is pointless and gross, and that true physics only concerns itself with the canonical momentum.
(3) - As a result of the kinetic momentum being worthless its basically correct to say Minkowski was “just right”(option [6]). This means that the paper proposing option [1] was a waste of time (much ado about nothing), because the difference between believing [1] and believing [6] only matters when doing kinetics, which he doesn’t care about. Finally, having decided that Minkowski was correct in the only way that he thinks matters, he goes off into a nasty side-thing about how Abraham was supposedly incompetent.
So his actual position is sort of [1] and [6] at the same time (because he considers the difference between them inconsequential, as it only applies to kinetics). If he leans more on the [1] side he can consider 12.72 to be valid. But why would he bother? 12.72 is saying something about kinetics, it might as well be invalid. He doesn’t care either way.
He goes on to explicitly say that he thinks 12.72 is invalid. Although I think his logic on this is flawed. He says the glass block breaks the symmetry, which is true for the photon. However, the composite system (photon + glass block) still has translation and boost symmetry, and it is the uniform motion of the center of mass of the composite system that is at stake.
re redshift: Sorry, I should have been clearer, but I meant to talk about redshift (or another kind of energy loss) of the light that comes out of the block on the right compared to the light that went in from the left, which would cause issues with going from there being a uniformly-moving stationary center of mass to the conclusion about the location of the block. (I’m guessing you were right when you assumed in your argument that redshift is 0 for our purposes, but I don’t understand light in materials well enough atm to see this at a glance atm.)
And the loss mechanism I was imagining was more like something linear in the distance traveled, like causing electrons to oscillate but not completely elastically wrt the ‘photon’ inside the material.
Anyway, in your argument for the redshift as the photon enters the block, I worry about the following:
can we really think of 1 photon entering the block becoming 1 photon inside the block, as opposed to needing to think about some wave thing that might translate to photons in some other way or maybe not translate to ordinary photons at all inside the material (this is also my second worry from earlier)?
do we know that this photon-inside-the-material has energy ℏω?
The microscopic picture that Mark Mitchison gives in the comments to this answer seems pretty: https://physics.stackexchange.com/a/44533 — though idk if I trust it. The picture seems to be to think of glass as being sparse, with the photon mostly just moving with its vacuum velocity and momentum, but with a sorta-collision between the photon and an electron happening every once in a while. I guess each collision somehow takes a certain amount of time but leaves the photon unchanged otherwise, and presumably bumps that single electron a tiny bit to the right. (Idk why the collisions happen this way. I’m guessing maybe one needs to think of the photon as some electromagnetic field thing or maybe as a quantum thing to understand that part.)
I presented the redshift calculation in terms of a single photon, but actually, the exact same derivation goes through unchanged if you replace every instance of ℏω0 with E0 and ℏω with E . Where E0 and E are the energy of a light pulse before and after it enters the glass. There is no need to specify whether the light pulse is a single photon a big flash of classical light or anything else.
Something linear in the distance travelled would not be a cumulatively increasing red shift, but instead an increasing loss of amplitude (essentially a higher cumulative probability of being absorbed). This is represented using a complex valued refractive index (or dielectric constant) where the real part is how much the wave slows down and the imaginary part is how much it attenuates per distance. There is no reason in principle why the losses cannot be arbitrarily close to zero at the wavelength we are using. (Interestingly, the losses have to be nonzero at some wavelength due to something called the Kramers Kronig relation, but we can assume they are negligible at our wavelength).