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No, the 4-momentum is conserved as a vector given a reference frame, which means all 4 parts are conserved, which means 3-momentum is conserved. It doesn't "go

No, the 4-momentum is conserved as a vector given a reference frame, which means all 4 parts are conserved, which means 3-momentum is conserved. It doesn't "go somewhere". You seem to be obsessed about conservation of the magnitude of 4-momentum under Lorentz transformations. It's the whole vector I'm talking about, not the magnitude. and change-of-frame is not a necessary part of conservation laws. The claim that you can observe conversation in one frame but not another is absurd. The laws are affine; they hold in any frame or none.

pdonis 10y ago
> the 4-momentum is conserved In standard relativity, yes. But not in your hypothetical "relativistic system where momentum is conserved but energy isn't". You said such a system was easy to imagine. I simply pointed out that, easy to imagine or not, such a system doesn't work. Nothing that I said about 3-momentum being conserved in one frame but not another applies to standard relativity where the 4-momentum is conserved. It only applies to the hypothetical system that you claimed was easy to imagine. I don't understand why you keep talking as though my comments about 3-momentum being conserved in one frame but not another referred to standard relativity. > It doesn't "go somewhere". By "go somewhere" I simply meant energy can get transferred from one part of the system to another. I didn't mean that total energy wasn't conserved (in standard relativity). > change-of-frame is not a necessary part of conservation laws. It is in a relativistic theory, since the principle of relativity requires that the laws of physics, including conservation laws, must hold in all frames. > The claim that you can observe conversation in one frame but not another is absurd. For standard relativity, of course it is. But not for your hypothetical system. > The laws are affine; they hold in any frame or none. This seems to contradict what you said earlier in the same post, that "change of frame is not a necessary part of conservation laws".
- cynicalkane 10y ago
  You seem confused by how conservation works, and intent on misunderstanding it. All I can say is that momentum conservation is Lorentz invariant in my system, and hence cannot depend on choice of reference frame. The conserved quantity is not invariant, and conserved quantities do not need to be Lorentz invariant, and fact almost always are not. If this confuses you, sorry, but I cannot follow your reasoning anymore.
  - pdonis 10y ago
    > All I can say is that momentum conservation is Lorentz invariant in my system If by "my system" you mean your hypothetical theory where energy is not conserved but 3-momentum is, I strongly suspect you have not done the math. See below. > The conserved quantity is not invariant, and conserved quantities do not need to buslye Lorentz invariant, and fact almost always are not. If this confuses you, sorry, but I cannot follow your reasoning anymore. I understand what you are saying here, but it does not refute what I was saying. You are saying that, for example, in standard relativity, energy is conserved--in a given frame, it stays the same through any series of events--but it is not frame invariant; changing frames changes the energy. I agree with that. What I am saying is something different: in your hypothetical relativistic theory in which energy is not conserved, then 3-momentum cannot be conserved in any frame other than one particular one (which in my example was the center of mass frame). I am not saying the numerical value of the 3-momentum has to be the same in all frames. I am saying that in all frames but one, the numerical value of the 3-momentum, as evaluated in that frame, is different after an inelastic collision than before, i.e., 3-momentum is not conserved in that frame. Here is the math backing up that assertion. Suppose 3-momentum is conserved in the center of mass frame. In that frame the 3-momentum is zero. We assume that our system consists of just two billiard balls, each with the same rest mass m. Before the collision, one ball has speed v and the other has speed -v (we can restrict ourselves to one spatial direction). After the collision, one ball has speed w and the other has speed -w, where w < v (because the collision is inelastic). So 3-momentum is zero before and after the collision in this frame, and energy is not conserved--it is smaller after the collision. Now transform to any other frame. Call the relative velocity of the Lorentz transformation u. Then, in this frame, the speeds of the two balls before the collision are (I am using units where c = 1) v+' = (u + v)/(1 + uv) v-' = (u - v)/(1 - uv) and the speeds after the collision are w+' = (u + w)/(1 + uw) w-' = (u - w)/(1 - uw) Now we evaluate the 3-momentum in this frame. Before the collision, it is m ( v+'/sqrt(1 - (v+')^2) + v-'/sqrt(1 - (v-')^2) ) After the collision, it is m ( w+'/sqrt(1 - (w+')^2) + w-'/sqrt(1 - (w-')^2) ) Substituting and straightforward algebra simplifies these to before = 2mu/sqrt[(1-u^2)(1-v^2)] after = 2mu/sqrt[(1-u^2)(1-w^2)] These are obviously not equal, hence 3-momentum is not conserved in this frame. (In standard relativity, where energy is conserved, 3-momentum would be conserved as well; as I mentioned before, the simplest way for that to happen would be for the two billiard balls to heat up, increasing their rest mass. More complicated ways would involve other particles, or the billiard balls emitting radiation, or something like that.)

pdonis 10y ago

> the 4-momentum is conserved In standard relativity, yes. But not in your hypothetical "relativistic system where momentum is conserved but energy isn't". You said such a system was easy to imagine. I simply pointed out that, easy to imagine or not, such a system doesn't work. Nothing that I said about 3-momentum being conserved in one frame but not another applies to standard relativity where the 4-momentum is conserved. It only applies to the hypothetical system that you claimed was easy to imagine. I don't understand why you keep talking as though my comments about 3-momentum being conserved in one frame but not another referred to standard relativity. > It doesn't "go somewhere". By "go somewhere" I simply meant energy can get transferred from one part of the system to another. I didn't mean that total energy wasn't conserved (in standard relativity). > change-of-frame is not a necessary part of conservation laws. It is in a relativistic theory, since the principle of relativity requires that the laws of physics, including conservation laws, must hold in all frames. > The claim that you can observe conversation in one frame but not another is absurd. For standard relativity, of course it is. But not for your hypothetical system. > The laws are affine; they hold in any frame or none. This seems to contradict what you said earlier in the same post, that "change of frame is not a necessary part of conservation laws".

cynicalkane 10y ago
You seem confused by how conservation works, and intent on misunderstanding it. All I can say is that momentum conservation is Lorentz invariant in my system, and hence cannot depend on choice of reference frame. The conserved quantity is not invariant, and conserved quantities do not need to be Lorentz invariant, and fact almost always are not. If this confuses you, sorry, but I cannot follow your reasoning anymore.
- pdonis 10y ago
  > All I can say is that momentum conservation is Lorentz invariant in my system If by "my system" you mean your hypothetical theory where energy is not conserved but 3-momentum is, I strongly suspect you have not done the math. See below. > The conserved quantity is not invariant, and conserved quantities do not need to buslye Lorentz invariant, and fact almost always are not. If this confuses you, sorry, but I cannot follow your reasoning anymore. I understand what you are saying here, but it does not refute what I was saying. You are saying that, for example, in standard relativity, energy is conserved--in a given frame, it stays the same through any series of events--but it is not frame invariant; changing frames changes the energy. I agree with that. What I am saying is something different: in your hypothetical relativistic theory in which energy is not conserved, then 3-momentum cannot be conserved in any frame other than one particular one (which in my example was the center of mass frame). I am not saying the numerical value of the 3-momentum has to be the same in all frames. I am saying that in all frames but one, the numerical value of the 3-momentum, as evaluated in that frame, is different after an inelastic collision than before, i.e., 3-momentum is not conserved in that frame. Here is the math backing up that assertion. Suppose 3-momentum is conserved in the center of mass frame. In that frame the 3-momentum is zero. We assume that our system consists of just two billiard balls, each with the same rest mass m. Before the collision, one ball has speed v and the other has speed -v (we can restrict ourselves to one spatial direction). After the collision, one ball has speed w and the other has speed -w, where w < v (because the collision is inelastic). So 3-momentum is zero before and after the collision in this frame, and energy is not conserved--it is smaller after the collision. Now transform to any other frame. Call the relative velocity of the Lorentz transformation u. Then, in this frame, the speeds of the two balls before the collision are (I am using units where c = 1) v+' = (u + v)/(1 + uv) v-' = (u - v)/(1 - uv) and the speeds after the collision are w+' = (u + w)/(1 + uw) w-' = (u - w)/(1 - uw) Now we evaluate the 3-momentum in this frame. Before the collision, it is m ( v+'/sqrt(1 - (v+')^2) + v-'/sqrt(1 - (v-')^2) ) After the collision, it is m ( w+'/sqrt(1 - (w+')^2) + w-'/sqrt(1 - (w-')^2) ) Substituting and straightforward algebra simplifies these to before = 2mu/sqrt[(1-u^2)(1-v^2)] after = 2mu/sqrt[(1-u^2)(1-w^2)] These are obviously not equal, hence 3-momentum is not conserved in this frame. (In standard relativity, where energy is conserved, 3-momentum would be conserved as well; as I mentioned before, the simplest way for that to happen would be for the two billiard balls to heat up, increasing their rest mass. More complicated ways would involve other particles, or the billiard balls emitting radiation, or something like that.)