Spacetime Translational Invariance vs(?) Lorentz Covariance

[tiyusufaly]

Hello,

I have been reviewing some relativity notes, and I am confused over something. I apologize if this seems like a silly or obvious point, but humor me.

When we are talking about Lagrangians in field theory and in regular mechanics, we are often looking at symmetries. Namely, almost all physical theories in the universe have the symmetry of Lorentz covariance, which as I understand it means the Lagrangians transform according to a representation of the Lorentz group. Does that necessarily mean the action is still stationary if we do a Lorentz transformation? If so, what would be the relevant 'Noether's charge'? Would it be the invariant dot-product E^2 - p^2 c^2?

And then, a part of me wonders what the relationship is between Lorentz covariance and spacetime translational invariance, which has as its 'Noether's charge' the relativistic stress-energy tensor, effectively amounting to overall relativistic energy and momentum conservation. Is this relativistic energy and momentum conservation a fundamental property of all physical theories like Lorentz covariance? I had for the longest time thought so, and so I had kind of thought that relativistic energy and momentum was the Noether's charge of the Lorentz covariance symmetry, but now I don't know about that at all.

I ask this question because it occurred to me, the relativistic Navier-stokes equations of fluid dynamics are derived simply by writing the Lorentz covariant Lagrangian, and writing down the equations for relativistic energy and momentum conservation. If relativistic energy and momentum conservation is fundamental among all known physical theories (amounting effectively to the fact that all physical theories have spacetime translational invariance), then we should be able to do a similar writing out of the equations for any such theory. But I don't see people doing that, so I fear I have it all wrong.

Basically, can someone clear up my confusion between Lorentz covariance and spacetime translational invariance? Which ones are more fundamental to all known physical theories? What are the Noether charges/currents? I again am sorry if this seems silly, it is just something to ponder.

Thanks.

 

[George Jones]

The Lorentz group is generated by rotations and boosts. The Poincare group is generated by rotations, boosts, and translations.

Invariance under rotations leads to conservation of angular momentum. Invariance under translations leads to conservation of energy and momentum. Invariance under boosts leads to? See

https://www.physicsforums.com/showthread.php?t=81661.

Representation theory of the full Poincare group leads to relativistic wave equations like, e.g., the Klein-Gordon and Dirac equations. For a little bit more on this, see

https://www.physicsforums.com/showthread.php?p=1314701#post1314701.

 

[Entropia]

Hi there,

Your question is not silly at all, it is actually a very interesting and important one. Let me try to provide some clarification on the concepts of Lorentz covariance and spacetime translational invariance.

Firstly, Lorentz covariance and spacetime translational invariance are both fundamental symmetries in physics, and they are closely related. Lorentz covariance refers to the invariance of physical laws under rotations and boosts (changes in velocity) in spacetime, while spacetime translational invariance refers to the invariance of physical laws under translations in spacetime (changes in position). These two symmetries are connected through the concept of spacetime, which combines space and time into a single four-dimensional entity.

Now, to answer your question about Noether's charge and its relation to these symmetries, let's start with Lorentz covariance. As you mentioned, the Lorentz group is a mathematical representation of these rotations and boosts in spacetime. When a physical theory is Lorentz covariant, it means that the equations of the theory remain unchanged under these transformations. This leads to the conservation of certain quantities, which are known as Noether's charges. In the case of Lorentz covariance, the Noether charge is the conserved quantity associated with the symmetry of rotations and boosts, and it is indeed the invariant dot product of energy and momentum that you mentioned.

Now, for spacetime translational invariance, the Noether charge is the conserved quantity associated with the symmetry of translations in spacetime. This quantity is known as the stress-energy tensor, which represents the energy and momentum density of a system. So, in a way, the Noether charge for spacetime translational invariance is the same as the one for Lorentz covariance, just expressed in a different form.

To summarize, both Lorentz covariance and spacetime translational invariance are fundamental symmetries in physics, and they are both associated with the conservation of certain quantities (Noether charges). The difference is that Lorentz covariance is related to rotations and boosts in spacetime, while spacetime translational invariance is related to translations in spacetime. Both are essential for understanding the behavior of physical systems, and they are closely connected through the concept of spacetime.

I hope this helps to clarify your confusion. Keep exploring and asking questions, as these concepts can be quite complex and