Garth said:
In GR is not the understanding that the 'rest' mass of a particle is taken to be constant and it is energy that is not conserved? i.e. The problem is with the definition of energy rather than mass.
Garth
I did chose a rather poor example - especially if a point particle is used. The mass of a point particle is the invariant of its energy-momentum 4-vector. Probably this should be included as a separate sort of "mass" in GR, along with the other three I mentioned.
Basically I agree that mass in GR is a lot more complicated than one would like, but my POV is not that there is a lack of defintion of the concept, rather that there are too many defintions that are too easy to confuse. Unfortunately this doesn't seem to be able to be avoided. (I suppose it's theoretically possible that someone will come up with some new defintion that's superior to everything else written on the topic, but I'm a bit skeptical.)
Moving on, I will try to see if I can come up with a better example of what I was trying to say.
Let's consider instead a static planet. The Komar mass of the static planet is well defined, and time invariant. It has a couple of formulations. One of the formulations is as an intergal of the stress energy tensor. (It can also be done as a surface intergal).
I am taking the philosophical POV that it makes sense, in this context, to talk about the mass of a piece of the planet, and not just the total mass of the planet. (If the metric isn't static, there's probably no good way to talk about the mass of a piece of the planet.)
The mass of a piece of a planet is then just the appropriate intergal of the stress-energy tensor over a region that is part of the planet rather than the whole planet.
Pehraps this POV is suspect. But that's the way I was thinking about it. I'd be interested in other comments on this point. Is this a non-standard POV?
Using this POV, though, we assign a mass to each small part of the planet, and this mass will depend on the local metric. When we want to find the total mass of an extended object (the whole planet) by adding up the masses of its pieces, we have to include metric corrections to the mass of each piece, depending on where that piece was located.
I'm not familiar at all with bifrucation horizons, unfotunately.
I do agree that the "mass of the universe" isn't well defined, but that's because the universe is neither static nor asymptotically flat.