That's food for thought. In a purely Newtonian terms there would be a COM or frames in which the bodies are at rest. In a multi-body GR solution I can see things are not so straightforward although we (presumably) could still calculate the proper acceleration of a test-particle worldline.pervect said:Well, the universe as a whole doesn't have a static observer - the metric doesn't have a timelike killing vector.
A binary star would be another example, again, no timelike killing vector. To demonstrate that the metric is a function of time when you don't have a full GR solution, consider the Newtonian approximation where you have two equal mass stars, and ask if the Newtonian potential U and/or the tidal forces are constant.
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Mentz114 said:That's food for thought. In a purely Newtonian terms there would be a COM or frames in which the bodies are at rest. In a multi-body GR solution I can see things are not so straightforward although we (presumably) could still calculate the proper acceleration of a test-particle worldline.
I understand that a multi-body solution in GR would be time-varying. Does that in principle mean we can't define some coordinate system to write a curve [itex]x^\mu(\tau)[/itex] ?PAllen said:For a binary system, even in COM frame, the potential is time varying. In Newtonian approximation, it would be strictly periodic, but in GR not. Due to GW, the metric would be time varying and aperiodic.
Mentz114 said:I understand that a multi-body solution in GR would be time-varying. Does that in principle mean we can't define some coordinate system to write a curve [itex]x^\mu(\tau)[/itex] ?
I don't know. I was just asking if I could have worldlines. It's been pointed out to me ( three times ) that if stuff is whizzing around then defining 'static' is problematic and I understand that.PAllen said:But how do you distinguish which lines are static?
Mentz114 said:I don't know. I was just asking if I could have worldlines. It's been pointed out to me ( three times ) that if stuff is whizzing around then defining 'static' is problematic and I understand that.
PAllen said:But how could you not have world lines? Even inside an event horizon you have world lines. They may end in finite proper time, but for universe with a big crunch, all world lines end in finite proper time. So, yes, you can always have world lines with varying proper acceleration profiles. You can, in many cases, (IMO) invent various heuristic criteria to consider some pseudo-static (to invent a term)*. But only in very special spacetimes can you pick out a unique family satisfying a geometric criterion for being static.
*This is an idea I've played around with and discussed a few times on these forums. Even for this, I've so far found it necessary to assume asymptotic flatness and 'well behaved' local geometry. And I haven't achieved complete success formalizing these ideas.