Particle Energy in Schutz's "A First Course in General Relativity

[pmb_phy]

If a photon just happens to fly through a hole in the box, it's difficult for me to see how one can attribute the energy to the box.

In that case if someone asked you what the mass of a gas was you'd be unable to answer them?

The idea is pretty simple: In the rest frame there is one event, box radiates energy. After the box radiates energy its mass has decreased. In the moving frame there are two events separated in time and the box has three different values of enegy.

A simpler value is if you have a long hot, very thin, rod lying on the x-axis in S in free space. Assume the rod cools uniformly. By symettry the rod will not move. Then at any point in time in S the rod will have one and only one value of energy. In S' it will also me true but the value will be be E = gamma*E_o


[qupte]One can attribute the energy to the volume inside the box, in which case one has shown that the energy in a unit volume is a function of the frame of reference. That dependence of the energy and momentum in a unit volume on the frame of reference of the observer is why the stress-energy tensor is a second rank tensor [/qupte]That tensor is second rank because mass-energy has two gamma factors.

Think more about simultaineity.

Pete

 

[pervect]

In that case if someone asked you what the mass of a gas was you'd be unable to answer them?

If the question is stated unambiguously, there is no problem computing the mass of a gas. When the volume of the gas becomes large enough, though, one will start to run into other more interesting problems in the computation of the mass, where a simple approach of integrating the energy density over the volume will not give the correct answer for the system mass.

The idea is pretty simple: In the rest frame there is one event, box radiates energy. After the box radiates energy its mass has decreased. In the moving frame there are two events separated in time and the box has three different values of enegy.
[/quote

Last I heard the box was gaining energy, now its losing energy? Oy vey! Anyway, so far we've just been arguing about what constitutes the box and what does not constitute the box, and clarifying the issue that if the box gains energy, it gains energy because of some transaction with an entity that is not the box that is losing energy. The rest is accounting details which revolve around what constitutes the box "now", and exactly when we make the bookeeping transaction that deducts the energy from the "non-box" account, and adds it to the "box" account, or vica versa.

Think more about simultaineity.

Pete

In the limit as the size of the box approaces zero, the difference in simultaneity approaches zero, and it becomes a non-issue. This is one of the reasons the approach of describing a system in terms of a stress-energy tensor is very useful. All the trivial accounting details disappear, and one can concentrate on the physics.

 

[pmb_phy]

In the limit as the size of the box approaces zero, the difference in simultaneity approaches zero, and it becomes a non-issue.

Relativity applies to everthing, not just objects of no spatial extent.

Pete

 

[pervect]

I'm not quite sure what your point is supposed to be?

The point I'm making is that energy does not appear from nowhere. Thus, when energy disappears from one set of books, it is moved to another set of books, instantaneously. There is no conceptual problem in making this transaction instantaneous, because the "transaction" of moving the energy from one set of books to another set of books, (possibly involving a physical change in form of the energy) occurs at one specific spatial location, not an extended region.

The specific time and the specific location at which the bookeeping transaction occurs depends on the coordinate system of the bookeeper, which is arbitrary.

If you wish to move the energy of a large object from one set of books to another set of books, you may/will have to subdivide the large transaction into a number of smaller transactions, which occur over an extended time period (in some frames of reference).

Because relativity is a classical theory, one is assumed to be able to divide up space-time into as small a region(s) as are needed to maintain the desired accuracy.

Thus in relativity we have the stress energy tensor, which talks about the density of energy and momentum at a point in space, which is assumed to be a continuous function.

The tricky part in GR of accouting for energy is not in the issue of dividing up energy into small enough chunks, it's another issue entirely.

 

[pmb_phy]

I wasn't trying to make any point whatsoever regarding the conservation of energy. The point I am making is a pretty simple point. If the object is a particle the the term "proper mass" has a well defined meaning. However if the object is not a particle, has spatial extent, and whose mass is changing then the term "proper mass" is not well defined. On can't simply choose any frame they wish to measure the proper mass. In particular one can't always write a meaninful 4-vector for an object nor write E = gamma*E_o

Its all spelled out much clearer here
http://www.geocities.com/physics_world/sr/invariant_mass.htm

To make this much easier to understand consider tube of very tiny cross-section which is laying on the x-axis in S. There is a deviceon each side which will emit flashes of light at regular time intervals in directions perpendicular to the x-axis and in equal amounts. Therefore a series of flashes will be emitted from near each out to the sides with no change in motion of the rod as a whole. If you let "m" = proper mass then the proper mass always has a well-defined quantity in this frame. However if you move to S' which is moving in the +x direction then the proper mass will not awlays have a value found by the rest observer.

Pete

ps - Do you have Ohanian's GR text? Tolman?

Recall why I brought this up. I told hellfire this; hellfire - You asked about the energy of a particle. There is much to be learned about asking about the energy of things which are not particles. Here is an interesting idea I came up with.

I find several of my relativist friends perplexed by this problem for a bit and hence its fun to think about. Plus it instills further that simultaneity is always there waiting to play games with our heads.