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Yes, I know how the 4-dimensional momentum vector in relativity works. I'm not aware of a physical view for which energy and momentum are the same for the purp

Yes, I know how the 4-dimensional momentum vector in relativity works. I'm not aware of a physical view for which energy and momentum are the same for the purposes of something like the EmDrive. In particular, you cannot completely turn one into the other by a choice of coordinates, and you cannot mix them by any physical process. And I could be wrong, but I'm 99% sure you need both energy and momentum conservation to get 4-momentum conservation. No matter your view, conserving the spacelike components of 4-momentum is not "a subset of the conservation of energy". And the truth/falsehood of this view has nothing to do with the "non quantum world". Relativity was an extension of classical electrodynamics before it was anything else.

akiselev 10y ago
> And I could be wrong, but I'm 99% sure you need both energy and momentum conservation to get 4-momentum conservation. I'm almost certain you're wrong. The most mathematically precise definition I remember for conservation of 4-momentum only requires that the interaction is invariant to spacetime translations. I don't see how that requires you to conserve both separately from the whole. > No matter your view, conserving the spacelike components of 4-momentum is not "a subset of the conservation of energy". And the truth/falsehood of this view has nothing to do with the "non quantum world". Relativity was an extension of classical electrodynamics before it was anything else. That was a poor choice of words on my part. I meant that an understanding of conservation of momentum based in classical mechanics is rather meaningless when talking about a drive that might use energy to create particles. What matters is that the total energy of the system is conserved and I stand by my choice of "subset" because momentum is not independently conserved. Calling conservation of momentum a subset of conservation of energy is not even close to calling energy the same thing as momentum and I have absolutely no clue where you got that idea.
- pdonis 10y ago
  > The most mathematically precise definition I remember for conservation of 4-momentum only requires that the interaction is invariant to spacetime translations. I don't see how that requires you to conserve both separately from the whole. "Conservation" of a 4-vector means its components in a given inertial frame do not change. That entails that, from the viewpoint of a given inertial frame, the total energy and each individual component of the momentum are all conserved separately.
- cynicalkane 10y ago
  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
      [deleted]
    - 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).

akiselev 10y ago

> And I could be wrong, but I'm 99% sure you need both energy and momentum conservation to get 4-momentum conservation. I'm almost certain you're wrong. The most mathematically precise definition I remember for conservation of 4-momentum only requires that the interaction is invariant to spacetime translations. I don't see how that requires you to conserve both separately from the whole. > No matter your view, conserving the spacelike components of 4-momentum is not "a subset of the conservation of energy". And the truth/falsehood of this view has nothing to do with the "non quantum world". Relativity was an extension of classical electrodynamics before it was anything else. That was a poor choice of words on my part. I meant that an understanding of conservation of momentum based in classical mechanics is rather meaningless when talking about a drive that might use energy to create particles. What matters is that the total energy of the system is conserved and I stand by my choice of "subset" because momentum is not independently conserved. Calling conservation of momentum a subset of conservation of energy is not even close to calling energy the same thing as momentum and I have absolutely no clue where you got that idea.

pdonis 10y ago
> The most mathematically precise definition I remember for conservation of 4-momentum only requires that the interaction is invariant to spacetime translations. I don't see how that requires you to conserve both separately from the whole. "Conservation" of a 4-vector means its components in a given inertial frame do not change. That entails that, from the viewpoint of a given inertial frame, the total energy and each individual component of the momentum are all conserved separately.
cynicalkane 10y ago
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).