What Lorentz Covariant Objects Can You Name?

[Phrak]

For starters, there is the covariant vector

(E/c, p).

Dividing by the scalar invariant, h_bar/2∏, where k is the propagation vector, there is

(ω/c, k).

There must be a significant number of covariant objects in electromagnetism...

 

[starthaus]

For starters, there is the covariant vector

(E/c, p).

Dividing by the scalar invariant, h_bar/2∏, where k is the propagation vector, there is

(ω/c, k).

There must be a significant number of covariant objects in electromagnetism...

There is covariant formulation of the laws of electrodynamics
There is an attempt to formulate thermodynamics in a covariant form (not very successful, by R.C.Tolman)
There are a few (failed) attempts to formulate the laws of material resistance (like Hooke law, for example) in covariant form.

As we can see, not all the nature's laws could be formulated in covariant form with the same level of success. Electrodynamics has lent itself to covariant formulation the best.

 

[Phrak]

The failures to identify Lorentz Covariant forms should be an interesting topic, starthaus.

The challenge here, is different. What Lorentz Covariant forms can you identify?

 

[DrGreg]

Well, for a start, look at all the examples listed under the Wikipedia article four-vector:

• four-velocity
• four-acceleration
• four-momentum
• four-force
• four-current
• electromagnetic four-potential
• four-frequency
• wave vector
• dust number-flux

Are you interested only in rank (1,0) or (0,1) tensors, or other ranks too?

 

[pervect]

Obviously important tensors of higher rank (which may not be exhaustive, just the ones that are vital):

The Faraday tensor and its dual. The stress energy tensor (for E&M it's the Maxwell stress-energy tensor, more generally we have the stress-energy tensor itself). These are all rank 2.

For gravity, we have the Ricci and the Einstein at rank 2, and of course the Riemann at rank 4.

I tend to think of volumes in relativity as three-forms (or as their dual vectors, sometimes) - but I'm not sure how standard this practice is. It's implicit in writing a volume element as dx^dy^dz, where dx, dy and dz are one-forms, though (and ^ is the wedge product).

dx, dy and dz are also covariant entities of course.

Thinking of volume as being represented by a vector makes the stress-energy tensor a lot easier to understand, IMO - but I digress.

 

[Phrak]

It's so nice to hear from the both of you DrGreg and pervect!

We should include integrals in the set.

I am sure that for each heuristic Lorentz covariant tensor (pseudotensor) there is an associated Lorentz covariant tensor (or pseudotensor) attached to each spacetime event. This language will sound confusing, but I'll clarifty and attempt to motivate the claim--

For example, the tensor (E/c, p) is heuristic. It tells us about a system and not the values of the system at a space-time event. For these we seem to need length, area, volume, or 4-volume densities. And these are orientable.

pervect, so you are not digressing in bringing up dx,dy,dz wedge products, but spot on.

Exemplary are Maxwell's equations that may be expressed in differential or integral form. The integral form is heuristical--it produces, for example, the total magnetic flux but doesn't say anything about the value of the flux at any given spatial location.

I am certain there is a purely mathematical, and deductive way to obtain heuristic tensors from point-wise tensors and the inverse, within four dimensions, but it evades me without a kick in the right direction.

(E/c, p) might be a good place to start. It is the integral form of something.
It cannot be the stress energy tensor. With directed (orientable) k-volumes (k<=4 dimensions), it should be skew symmetric.

 

[Phrak]

Is anyone familiar with stokes theorem? I see that this is what I've been talking about, some 150 years after the fact.

 

[atyy]

Is anyone familiar with stokes theorem? I see that this is what I've been talking about, some 150 years after the fact.

Yes, that's why the integral forms of Maxwell's equations are not heuristical.

I think this fails in curved spacetime (but that's irrelevant, since you asked about Lorentz covariant objects).

Edit: I'm not sure it fails completely - it does become non-straightforward. What I had in mind is that the covariant conservation of energy in GR does not have a straightforward integral counterpart.

 

[Phrak]

Oh, dear. I used a word I thought I understood one way (heuristic) but means something else entirely, I'm sure I've completely cluttered the issue.

 

[Phrak]

If I take the charge continuity equation which is a one-form, and integrate over the 3 dimensional boundary on the 4 dimensional manifold of spacetime, the other side of Stoke's equation should yield another Lorentz covariant tensor I believe. What is the covariant tensor?