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Noether's Theorem states that if a Lagrangian is symmetric for a certain transformation, this leads to an invariant: Symmetry of translation gives momentum conservation, Symmetry of time gives Energy conservation etc.

The Galilean principle stating that all reference frames that move with constant speed relative to each other are equivalent is also a symmetry principle: Setting up a physical system that is identical to the original except for a constant velocity (boost) added will have the same behaviour.

Shouldn't there be an invariant associated with this symmetry? If yes, what is that invariant?

yippy_yay
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1 Answers1

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The conserved quantity corresponding to boost symmetry is $$ \int d^3 x (P_0 x_i - P_i t) $$ which is the relativistic analogue of $x_{CM} - v_{CM} t$, the position of the center of mass at $t=0$. It is quite a useless conserved quantity, and that is why people don't talk about it.

Prahar
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  • Is this quantity always zero? – yippy_yay Jul 19 '13 at 22:13
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    No. It is the position of the center of mass at $t=0$. – Prahar Jul 19 '13 at 22:15
  • There's a formal similarity to the expression for angular momentum. Not surprising as boosts are a kind of rotation. – Dan Piponi Jul 19 '13 at 22:26
  • But if the center of mass is moving, then the position of center of mass at t = 0 will be moving away, i.e. changing, in the reference frame attached to the center of mass. – yippy_yay Jul 19 '13 at 22:26
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    No. The position at $t=0$ (at a specific time) does not change. The position changes as a function of time though. – Prahar Jul 19 '13 at 22:37
  • If something changes as a function of time, how can it be considered an invariant? By that logic, at t=0, everything would qualify as an invariant (at that specific moment). – yippy_yay Jul 19 '13 at 23:01
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    Right. That's the reason this conserved quantity is useless. Its simply telling us that the center of mass at time $t=0$ is preserved. But that is obviously true! – Prahar Jul 19 '13 at 23:01
  • The conserved quantities under rotations are the 3 components of angular momentum around a point, in space (any point if space is homogeneous). The conserved quantities under Lorentz boosts are the 3 components of "angular momentum" around an instant, in space-time. That instant is any time $t$, not just $t = 0$. – Cham Aug 08 '18 at 20:52
  • I think P0 should be replaced by P0/c^2 since it represents the inertial mass, not the energy. – Sergio Prats May 20 '23 at 07:28
  • It is not a useless quantity, it is simply obvious. It says a particle's position moves in accordance with its linear momentum. Otherwise it would be a mess! – Sergio Prats May 20 '23 at 07:31
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    @SergioPrats we only work in natural units here. – Prahar May 20 '23 at 19:45
  • This equation tells that the overall flow of energy is determined by the overall momentum in the system and for an isolated system it is a conserved quantity. I wonder if there is a similar equation for the "flow of momentum" that is caused by momentum moving itself or by the stresses (see Maxwell tensor), if it exists it should be associated with some symmetry. – Sergio Prats Jul 31 '23 at 07:32