# What convervation law is required by the Lorentz Transformations

#### metrictensor

Time invariance implies conservation of energy. Space invariance implies momentum convervation. What convervation law does the Lorentz invariance imply?

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#### dextercioby

Homework Helper
Angular momentum. Immediate by Noether's theorem for classical fields.

Daniel.

Staff Emeritus
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The Lorentz transformations by definition preserve the four-interval $$c^2t^2 - x^2 - y^2 - z^2$$.

#### pervect

Staff Emeritus
dextercioby said:
Angular momentum. Immediate by Noether's theorem for classical fields.

Daniel.
Conservation of angular momentum is generated by spatatial rotation invariance. Space rotation invariance is indeed part of the Lorentz group. But I suspect the original poster was interested in the symmetries related to the Lorentz boost, not by the spatial rotation part of the Lorentz group.

I seem to recall that this question was discussed before, but I don't recall the conclusion that we came to.

#### metrictensor

The Lorentz transformations by definition preserve the four-interval $$c^2t^2 - x^2 - y^2 - z^2$$.
I was thinking the same thing but there are many 4-vector invariants in SR. Energy-momentum, space-time. The classical conservation laws have one specific quantity conservered not a variety.

#### George Jones

Staff Emeritus
Gold Member
metrictensor said:
Time invariance implies conservation of energy. Space invariance implies momentum convervation. What convervation law does the Lorentz invariance imply?

Regards,
George

#### Berislav

Thinking by analogy, shouldn't it imply conservation of the stress-energy tensor?

#### dextercioby

Homework Helper
Nope, stress- energy tensor is linked to space-time translations.

Daniel.

#### yogi

This should be a straightforward question with an obvious answer - but authors seem to skirt the issue
spatial displacement symmetry - conservation of momentum
temporal displacement symmetry - conservation of energy
isotropic symmetry - conservation of angular momentum

When gauge symmetry is applied to Maxwells's em equations, one consequence is conservation of charge - isn't conservation (invariance) of the spacetime interval also consequent to gauge symmetry?

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#### arivero

Gold Member

Regards,
George
It is a funny answer... the position of the center of mass? Ok, in absence of external forces, the center of mass is a preserved quantity, so it makes sense, or sort of.

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