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Why in a collision between particles is the four-momentum conserved within a frame of reference but not invariant between frames of reference?

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    $\begingroup$ I'm not really sure what you're asking. Four-momentum transforms like any four-vector under Lorentz transformations. The magnitude of the four-momentum vector is the invariant mass, which is clearly invariant between frames. $\endgroup$ – Evan Rule Oct 8 '15 at 1:17
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If the 4-momentum were invariant then it would be a scalar. 4-vectors are defined by the way their components mix when we change coordinates. In particular when we apply a lorentz transformation to our coordinates the inverse transformation is applied to the vector.

As a simple example consider what happens to the energy when we boost. If we start in the rest frame of a particle and then boost to a frame in which it is moving we get,

$$E_{\text{rest}}=mc^2 \rightarrow E_{\text{moving}} = \gamma mc^2,$$

if $E$ were invariant then a moving particle would have no kinetic energy.

Its important to keep conserved and invariant seperate in your mind. The total 4-momentum will not change overtime in any given inertial frame, but you can't change those frames and expect it to stay the same. What you can bring with you from frame to frame is the magnitude of the total 4-momentum, i.e., the invariant mass. This is both conserved and invariant.

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Conservation and invariance are fundamentally different things. Conservation means "doesn't change with respect to time". While invariance means "doesn't change with respect to Lorentz transformations". Components of four-momentum transform like vector components and are thus NOT invariant under Lorentz Transformations. But that doesn't prevent them from being conserved. Suppose the four-momentum is conserved in one frame. If you switch to a different frame, the four-momentum components will all be different, but the conservation is preserved.

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