Covariant and noncovariant Lorentz invariants

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Discussion Overview

The discussion revolves around the concept of Lorentz invariants, specifically distinguishing between covariant and noncovariant types. Participants explore theoretical implications, examples from physics, and the transformation properties of various quantities under Lorentz transformations.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants propose that covariant Lorentz invariants are fully contracted Lorentz tensors, which remain invariant under Lorentz transformations.
  • Others argue that noncovariant quantities, such as the product of energy and volume (EV), are numerically identical across Lorentz frames despite not being form invariant.
  • A participant challenges the claim that EV is identical in all frames by providing a specific example involving energy and volume transformations that do not align with the proposed invariance.
  • Another participant asserts that Lorentz invariants are fundamentally invariant and do not change under any representation of the Lorentz group.
  • One participant discusses the distinction between local and global invariant quantities, using the total charge as an example of a global invariant.
  • There is a discussion about manifest and non-manifest Lorentz covariance, with participants debating the implications of these terms on equations like div B = 0.
  • Some participants express uncertainty about whether non-manifest invariants can be reduced to manifest invariant expressions.
  • A later reply suggests that while equations can be rewritten in non-manifest forms, they do not introduce new quantities.
  • Participants explore the possibility of expressing div B = 0 in a manifest covariant way without including Faraday's law, leading to further technical discussion about the equations involved.
  • Corrections are made regarding the proper tensorial forms of the equations discussed, indicating ongoing refinement of ideas.

Areas of Agreement / Disagreement

Participants do not reach consensus on the nature of Lorentz invariants, with multiple competing views on the definitions and implications of covariant versus noncovariant quantities, as well as the distinction between manifest and non-manifest covariance.

Contextual Notes

Some discussions involve unresolved mathematical steps and the dependence on specific definitions of invariance and covariance. The implications of these distinctions are not fully settled within the conversation.

Heirot
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It recently came to my attention that there exists two "kinds" of Lorentz invariants: the covariant and the noncovariant ones.

The covariant ones would be Lorentz scalars e.g. fully contracted Lorentz tensors. If one applies the Lorentz transformation to a covariant Lorentz scalar, one would find it is invariant in form and in numerical value.

On the other hand, quantities like EV (volume times energy) are certainly not Lorentz scalars and they are most certainly not form invariant under Lorentz transformations (energy goes to energy and momentum, volume goes to volume and time), yet they appear to be numerically identical in all Lorentz frames. One can show this by noticing that the volume gets contracted: V = V_0/gamma, while the energy is related to the mass by E = gamma * m. (c=1) Therefore EV=mV_0 which is obviously the same number in every Lorentz frame.

I'm still not quite sure what to make of these quantities. None of the standard textbooks explicitly deals with it. Any thoughts?
 
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EV is not identical in all Lorentz frames. For example, let E be the energy of a certain light wave, and V the volume of George W. Bush's cranium. Then when we transform out of the rest frame of GWB's head, V picks up a factor of 1/gamma, but E changes by the relativistic Doppler shift factor, which is not gamma.
 
Lorentz invariants are not only covariant i.e. change w.r.t. some rep. of the Lorentz group, but they are invariant i.e. they change w.r.t. the trivial rep. of the Lorentz group, i.e. they don't change at all.

So there is only one class of Lorentz invariants, namely invariants.
 
OK, let's try with another example. Consider Maxwell equation div B = 0. It is true in every Lorentz frame. So, one would conclude that the number div B = div' B' = 0 is Lorentz invariant. But if one Lorentz transforms div B = 0 to another frame, they obtain something like a*(div' B') + b*(d/dt' B' + curl' E') = 0, where a and b are some velocity dependent constants. Therefore, div B is not Lorentz invariant but it has the same numerical value in every frame (zero). Would you agree?
 
I believe what Heirot is talking about is the difference between a Lorentz invariant which is a local quantity, i.e. a scalar field, and one which is a global quantity, i.e. the integral of a density.

An example of the latter is the total charge Q, which is obtained by integrating the 4th component of jμ over a hypersurface. Q is invariant in the sense that it doesn't depend on the choice of hypersurface.
 
I think Heirot is talking about manifest and not manifest Lorentz covariance.

\nabla B = 0

is not manifest Lorentz covariant but

dF = 0

is.
 
@Bill K: I'm not sure that's what I'm talking about. Maxwell equation div B should be a local invarinat quantity, although not a scalar field because it doesn't have the transformation properties of a scalar field.
 
tom.stoer said:
I think Heirot is talking about manifest and not manifest Lorentz covariance.

\nabla B = 0

is not manifest Lorentz covariant but

dF = 0

is.

OK, so you agree there exists two types (manifest and non manifest) of invariant quantities. While it's trivial to construct manifest invariants, the question remains whether it is possible to reduce the non manifest invariant (obtained ad hoc) to some manifest invariant expression?
 
Heirot said:
OK, so you agree there exists two types (manifest and non manifest) of invariant quantities.
No. What I am saying is that you can have manifest covariant equations (with covariant quantities like F or even invariant i.e. scalar quantities) which you can re-write in a not manifest covariant form. But that does not introduce any new quantity! B is already contained in F, so the equation div B = 0 does not introduce a new quantity B.
 
  • #10
tom.stoer said:
No. What I am saying is that you can have manifest covariant equations (with covariant quantities like F or even invariant i.e. scalar quantities) which you can re-write in a not manifest covariant form. But that does not introduce any new quantity! B is already contained in F, so the equation div B = 0 does not introduce a new quantity B.

OK, I agree with that. My final question is: can we write div B = 0 in a manifest covariant way which does not include Faraday's law?
 
  • #11
no; using forms the covariant equation reads

dF = 0

using tensors it reads

\partial_\mu F^{\mu\nu} = 0

and these equations always contain both Gauss's law for magnetism and the Maxwell–Faraday equation (= Faraday's law of induction)
 
  • #12
That's what I thought. Thanks!
 
  • #13
I think you mean
##dF=0## (ie, Bianchi) becomes ##d_{[a}F_{bc]}=0## tensorially.
 
  • #14
robphy said:
I think you mean
##dF=0## (ie, Bianchi) becomes ##d_{[a}F_{bc]}=0## tensorially.

oh, of course, sorry, thanks for the correction, it was nonsense! it's the other Maxwell equation, and dF= 0 must read

\epsilon^{\mu\nu\lambda\rho}\,\partial_\lambda \, F_{\mu\nu} = 0
 

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