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There's a confusion here which you still don't get. Momentum is not 4-momentum. Conservation of latter implies the former but the reverse is not necessarily tru

There's a confusion here which you still don't get. Momentum is not 4-momentum. Conservation of latter implies the former but the reverse is not necessarily true. And nothing in the above is a "subset of conservation of energy"--that's absurd. Actually, I'm not 99% sure, I'm 100% sure. It's very easy to imagine a relativistic system where momentum is conserved but energy isn't. Imagine if the universe was made of inelastic billiard balls, with no internal "heat", so every collision just lost energy. But momentum is still conserved.

pdonis 10y ago
> It's very easy to imagine a relativistic system where momentum is conserved but energy isn't. Imagine if the universe was made of inelastic billiard balls, with no internal "heat", so every collision just lost energy. This won't work. Let's suppose we have just two billiard balls, and that 3-momentum is conserved in the center of mass frame (in which it is by definition zero), but energy in this frame decreases at each collision. (Ignore the fact that this obviously implies that the 4-momentum of the system is not conserved, even though it is undergoing no external interactions.) Now transform into any other inertial frame. You will find that 3-momentum is not conserved either; it has a different nonzero value after a collision than before.
- deleted 10y ago
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- cynicalkane 10y ago
  This is a bizarre comment. If we forget about energy, this is equivalent to saying 3-momentum is not conserved in inelastic collisions in real life. Which is false.
  - pdonis 10y ago
    > this is equivalent to saying 3-momentum is not conserved in inelastic collisions in real life. No, it's not. In real life, energy is conserved, so the energy that is lost from kinetic energy in inelastic collisions needs to go somewhere. And whatever receives that energy will also have momentum (even if it doesn't in the center of mass frame, it will in other frames), so it has to be included in the accounting of momentum conservation as well as energy conservation. A relativistic theory that conserves energy as well as 3-momentum (i.e., standard relativity theory) will still conserve both energy and 3-momentum when you change frames, even in inelastic collisions. (In the simplest case, the lost kinetic energy goes into increasing the rest mass of the two billiard balls, by increasing their temperature.) What you hypothesized was something different: a hypothetical relativistic theory (which obviously does not match our actual world) in which energy is not conserved but 3-momentum is. In such a hypothetical theory, inelastic collisions could take place without the lost kinetic energy going anywhere: it simply disappears. I am simply explaining why such a relativistic theory is not possible: if energy is not conserved, 3-momentum cannot be conserved either (except in one particular frame, the center of mass frame, but that violates the principle of relativity so it is not allowed in a relativistic theory).
    - cynicalkane 10y ago
      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

> It's very easy to imagine a relativistic system where momentum is conserved but energy isn't. Imagine if the universe was made of inelastic billiard balls, with no internal "heat", so every collision just lost energy. This won't work. Let's suppose we have just two billiard balls, and that 3-momentum is conserved in the center of mass frame (in which it is by definition zero), but energy in this frame decreases at each collision. (Ignore the fact that this obviously implies that the 4-momentum of the system is not conserved, even though it is undergoing no external interactions.) Now transform into any other inertial frame. You will find that 3-momentum is not conserved either; it has a different nonzero value after a collision than before.

deleted 10y ago
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cynicalkane 10y ago
This is a bizarre comment. If we forget about energy, this is equivalent to saying 3-momentum is not conserved in inelastic collisions in real life. Which is false.
- pdonis 10y ago
  > this is equivalent to saying 3-momentum is not conserved in inelastic collisions in real life. No, it's not. In real life, energy is conserved, so the energy that is lost from kinetic energy in inelastic collisions needs to go somewhere. And whatever receives that energy will also have momentum (even if it doesn't in the center of mass frame, it will in other frames), so it has to be included in the accounting of momentum conservation as well as energy conservation. A relativistic theory that conserves energy as well as 3-momentum (i.e., standard relativity theory) will still conserve both energy and 3-momentum when you change frames, even in inelastic collisions. (In the simplest case, the lost kinetic energy goes into increasing the rest mass of the two billiard balls, by increasing their temperature.) What you hypothesized was something different: a hypothetical relativistic theory (which obviously does not match our actual world) in which energy is not conserved but 3-momentum is. In such a hypothetical theory, inelastic collisions could take place without the lost kinetic energy going anywhere: it simply disappears. I am simply explaining why such a relativistic theory is not possible: if energy is not conserved, 3-momentum cannot be conserved either (except in one particular frame, the center of mass frame, but that violates the principle of relativity so it is not allowed in a relativistic theory).
  - cynicalkane 10y ago
    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.