All mass is relativistic?

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  • #76
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Photons aren't what you think they are; and you should reread post #39 of this thread.
I understand plenty about photons, I think. I reread post #39. I don't think I have any further questions about this.
 
  • #77
Mister T
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If the solar system is found to have less mass than the individual bodies, is this not due to the gravitational potential energy of their being separated in space?
Not separated, collected.
 
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@1977ub

Mass is just a measure of how much energy a system has in its rest frame: ##E_0 = mc^2##, where ##E_0## is the system's rest energy.

What contributes to a system's rest energy? Answer: all of the energy "internal" to the system.

Say that the system is full of gas molecules. To calculate the system's rest energy (i.e., its mass), we must add up all of the "internal" energy contributions: the kinetic energies of the gas molecules (as measured in the system's rest frame, of course), the potential energy associated with their relative positions, and also the rest energies (masses) of the individual particles.

Then we could "zoom in" on a single molecule and itemize its rest energy as the sum of the kinetic, potential, and rest energies associated with its constituent atoms. We could "zoom in" on a single atom in the molecule and itemize its rest energy as the sum of the kinetic, potential, and rest energies associated with its subatomic particles.

Etc.

The potential-energy contributions associated with the relative positions of a system's constituents can be positive or negative, depending on whether the force in question is repulsive (positive) or attractive (negative). The potential-energy contributions approach zero in the limit that the system's constituents are infinitely far apart.
 
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@1977ub

To calculate the system's rest energy (i.e., its mass), we must add up all of the "internal" energy contributions ...
(or we can just weigh it) :wink:
 
  • #80
Nugatory
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Is this convention really arbitrary in the context of reckoning a system's rest energy?
You're right, it's not arbitrary for that purpose. I was trying/hoping to avoid that subtlety because much of the recent discussion has been about how the rest energy changes as energy enters and leaves the system; and because (as you pointed out while I was writing this) we can jus weigh the system to establish its rest energy.
 
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  • #81
PeterDonis
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What contributes to a system's rest energy? Answer: all of the energy "internal" to the system.
The potential-energy contributions associated with the relative positions of a system's constituents can be positive or negative, depending on whether the force in question is repulsive (positive) or attractive (negative). The potential-energy contributions approach zero in the limit that the system's constituents are infinitely far apart.
(or we can just weigh it)
Since we're having this discussion in the relativity forum (as opposed to the classical physics forum), it's worth pointing out that, in GR, all of these statements have limitations.

First, in relativity, the idea of determining a system's rest energy by adding up all of the "internal" contributions is formalized in what is called the Komar energy (more usually called "Komar mass", because in relativity energy and mass are just different units for the same quantity, and the term "mass" is more usual in the GR literature--as opposed to, say, the QFT literature, where the term "energy" or even "momentum" is more usual). The basic idea is that you integrate the stress-energy tensor over all of space, paying appropriate attention to the fact that spacetime is curved.

However, the Komar energy integral is only well-defined in a limited class of spacetimes, the stationary spacetimes--which are basically the ones in which there is a notion of "space" that is independent of "time" (I use the quotes because for precision these terms should be formalized and made precise, and they can be, but there are a lot of pitfalls lurking for the unwary in doing so). It turns out that these are also the only spacetimes in which there is a well-defined concept of "gravitational potential energy"; and it turns out that, in these spacetimes, the Komar energy integral is what you would expect it to be for a bound system, taking the (negative) contribution of gravitational potential energy into account (basically because taking proper account of spacetime curvature, in such spacetimes, is taking the contribution of gravitational potential energy into account).

Interestingly, the other notion of total energy you mention--just weigh the system--corresponds, if we take "weigh" to mean "determine by measurements purely external to the system", to a different concept of energy in relativity--actually, to two of them, called the ADM energy and the Bondi energy. These are well-defined for a different limited class of spacetimes, the asymptotically flat spacetimes--which are basically the ones which describe an isolated system surrounded by empty space. The difference between them is that the ADM energy never changes--even for a system that emits radiation that escapes to infinity. (The reason is that, at any finite time, the radiation has only traveled some finite distance from the system, because of the finite speed of light, so it is always present somewhere in the spacetime, and the ADM energy will therefore include it.) The Bondi energy was developed in order to make rigorous the idea that systems which emit radiation away to infinity lose energy: basically, the Bondi energy is the ADM energy minus whatever energy is carried away to infinity by radiation. This means that the Bondi energy, unlike the ADM energy, can change with "time" (again, this term needs to be properly formalized and made precise) as a system radiates and becomes more tightly bound.

For the even more limited class of spacetimes which have both properties--stationary and asymptotically flat--the Komar energy and the ADM energy are the same, so everything fits together consistently. But there are also important spacetimes--such as the FRW spacetimes used in cosmology--where none of these concepts of energy are well-defined, and so none of the ideas we have been talking about in this discussion apply.
 
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Thanks, @PeterDonis. Very informative write-up, there.
 

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