High School Does the mass of an object increase in a gravitational field?

[pervect]

I've been thinking my remarks over, and the use of the term "isolated system" is important to my argument. I believe, though, that my thinking is that an isolated system basically IS just one that has an asymptotically flat background metric, at a minimum. Probably it'd be good to add some constraints on gravitational radiation, so that a system that was strongly radiation gravitational radiation wouldn't be isolated because the gravitational radiation was escaping the system.

So it's an issue of semantics, and I'm using the term "isolated system" because I think it's more layman-friendly than "asymptotically flat". But the question arises, are the two notions really the same?

Possibly my thinking is wrong - it's not something I read in a textbook. But currently, I cannot think of any counterexamples. Perhaps someone else can, if so it would be very interesting. I suppose at this point I am proposing that we can think of the idea that the terms "asymptotically flat" and "isolated system" are the same as a conjecture, and try to disprove the idea by finding a counterexample.

Failure to find a counterexample won't necessarily prove anything, but it makes it plausible. And finding a counterexample would be interesting. Part of the issue is semantics - I'm not sure there is a formal definition for "isolated system", in fact, that's sort of what we're trying to figure out.

Birkhoff's theorem is the starting point of my thinking. It says that any spherically symmetric solution of Einstein's field equations in a vacuum must be static and asymptotically flat. Now if we could argue that an isolated system, viewed from a long distance, should be spherical symmetrical, we'd be done.

I don't think this quite works though. Clearly, gravitational radiation won't be spherically symmetrical. But we are already adding some constraints on gravitational radiation in considering the system to be isolated.

Additionally, we can note that the presence of gravitational radiation won't necessarily spoil asymptotic flatness, as long as it dies out fast enough with increasing distance. And I'd expect that to happen.

Anyway, none of this is going to replace a serious study of ADM mass, Bondi mass, and Komar mass as they are currently defined in General relativity. But it might make the discussion more accessible, IF we can accept that an "isolated system" has asymptotic flatness.

 

[PeterDonis]

for motion in SR an object's total energy (which some call its relativistic mass) increases in proportion to its observed time dilation (all per the frame in which the motion is evident). So they ask is there something similar for an object (possibly a radioactive body, so also a clock) experiencing gravitational time dilation on a planet's surface? My answer remains, to first order, the mass measured at a distance will be less by the dilation factor, contrary to the OP expectation.

To me this is a superficial comparison anyway. The analogy between "time dilation due to motion" and "time dilation due to gravity" has so many problems that my preference is to just reject it altogether

.

 

[PeterDonis]

my thinking is that an isolated system basically IS just one that has an asymptotically flat background metric, at a minimum. Probably it'd be good to add some constraints on gravitational radiation, so that a system that was strongly radiation gravitational radiation wouldn't be isolated because the gravitational radiation was escaping the system.

Not just gravitational radiation: any kind of radiation. I think a good working definition of "isolated system" is a system that is asymptotically flat, and whose ADM mass equals its Bondi mass (the latter condition is what excludes any kind of radiation escaping).

the presence of gravitational radiation won't necessarily spoil asymptotic flatness, as long as it dies out fast enough with increasing distance

That's correct (and it goes for any kind of radiation, not just gravitational). The presence of radiation just makes the Bondi mass less than the ADM mass.

 

[pervect]

On further thought, I think I may have a counterexample to my own idea. A closed universe is arguably isolated, I think, but it's not necessarily asymptotically flat. So the issue of the existence or nonexistence of the mass of a closed universe rather spoils my idea. The generally accepted idea (or at least the idea as expressed in MTW's gravitation) is that there isn't a meaningful notion of the mass of a closed universe. So - back to the idea board.

 

[PeterDonis]

A closed universe is arguably isolated

I would say not, because to me, "isolated" implies that there could be an observer "outside" the system somewhere who can measure its properties. That's not true of a closed universe. But the term "isolated" does not have an exact technical meaning, so this is really a matter of personal preference.

 

[nitsuj]

No. Under the conditions you have specified, the reception events will be separated by less than one day of proper time along our worldline (assuming that a "day" is defined to be the number of cesium-clock seconds that will be counted by a clock on Earth during one rotation).

liked that post because "day" was used in quotes and defined as...a day..on Earth...measured in defined seconds; cesium "cycles".

that level of detail "inherits" more ambiguity...