Is mass conserved by mathematical identity?

[Phrak]

In electromagnetism J denotes the oriented charge-current density, J=d*F.

Conservation of charge immediately follows. J=d²*F=0(identically). All exact forms are closed.

We can identify scalar mass as the norm of the one-form, μ=(E/c²,-p/c).

*μ is then spatial mass density. Like charge-current density, it is an oriented 3-form.

Is there anything that tells us that d*μ=0, or that *μ is exact?

 

[Bill_K]

J^μ ≡ δL/δA_μ is locally conserved due to gauge invariance. A_μ → A_μ + λ_,μ implies J^μ;μ ≡ 0.

T^μν ≡ δL/δg_μν is locally conserved due to invariance under coordinate transformations. g_μν → g_μν + ξ_μ;ν + ξ_ν;μ implies T^μν_;ν ≡ 0.

 

[Phrak]

Yes, well, there is more than one formalism. See http://en.wikipedia.org/wiki/Maxwell's_equations#Differential_geometric_formulations

As I understand it, replacing forms with their tensor density equivalents will obtain globally true statements on smooth manifolds.

 

[haushofer]

In quantum mechanics, mass is conserved due to a U(1)-symmetry; algebraically, the Casimir which plays the role of mass in the Bargmann group is a central extension.

 

[tom.stoer]

Yes, well, there is more than one formalism. See http://en.wikipedia.org/wiki/Maxwell's_equations#Differential_geometric_formulations

As I understand it, replacing forms with their tensor density equivalents will obtain globally true statements on smooth manifolds.

The problem is that the rank-2 tensor T does not allow for a well-defined volume integral in general cases; that's why global concepts like "mass" and "energy" are notorious difficult in GR.

 

[henry_m]

An isolated particle lives on a one-diimensional submanifold so forms aren't that interesting; there are only one-forms which are dual to functions.

The pushforward of the metric onto the worldline is just $-d\tau^2$, where $\tau$ is the proper time (here I use the 'mostly +' signature). This is the definition of proper time. If the mass m is regarded as a scalar on the worldline, then its hodge dual *m is the four-momentum form p. By raising the index and pulling back to a vector on the spacetime manifold, we get the ordinary four-momentum vector, defined on the worldline. I guess we then might have something to do with the four-force? If we lower the index, push forward onto the worldline and take the dual I think we should get g(F,U) where F is the four-force and U the four-velocity vectors, and g the metric on the spacetime manifold. But this looks like it'll be the wrong way round to me...

 

[henry_m]

On reflection, my input above is a bit pointless. If we're talking about gravity, the mass there plays an entirely passive role so isn't really analogous to the source in any way.

Here's a better answer: in gravity, the analogous quantity to the source J is the stress energy tensor T. Einstein's equations are $G=8\pi T$. Then T is identically conserved by the Bianchi identities.

 

[Phrak]

The problem is that the rank-2 tensor T does not allow for a well-defined volume integral in general cases; that's why global concepts like "mass" and "energy" are notorious difficult in GR.

Yes, this is why I think dwelling on the stress energy tensor is the wrong approach.

My naive approach is to use parallel arguments to the electric current density where tensors should really be substituted with tensor densities in a curved spacetime, though I didn't think it was necessary to say so at the time.

If *μ is exact this leads automatically to a conservation expression.

Say we have a primative 1-form field, α, and β=dα and *μ=dβ, where μ is as given in the OP. This is what I was attempting to illustrate for the most part, except that perhaps alpha is not necessary but beta is out primative field that is not the exact form of anything.

 

[tom.stoer]

I don't understand; you want to replace tensors with " tensor densities in a curved spacetime". What do you mean? Scaling T with some function of g? But then this new "densitized" object is no longer covariantly conserved.

 

[Phrak]

That's OK, tom.stoer. What I am interested in are cannonically generally covariant expressions (connection free). In this restricted study T doesn't fit, but skew symmetric tensor densities with lower indeces, such as the 3-form of momentum, do.

henry_m gave me a couple ideas. Thanks, henry.