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Preamble: we are largely on the same page. I thought I'd draw your (or our mutual readers') attention to a couple details where we may differ a little. > [lin

Preamble: we are largely on the same page. I thought I'd draw your (or our mutual readers') attention to a couple details where we may differ a little. > [link to wikipedia's de Sitter-Schwarzschild page] Let's call it Schwarzschild-de Sitter (SdS, for short, and for ease of literature-searching). More generally there is the McVittie family of metrics of a massive object in a dynamical spacetime. SdS is the limiting case of McVittie where the spacetime is stationary and the central mass is compact, spherically symmetric, 0-angular-momentum, and uncharged. (One could alternatively say that McVittie is a generalized time-dependent SdS.) > black hole solutions assume a perfect vacuum Not quite, but this is mainly a specialist quibble, since the most widely known theoretical black holes are vacuum or electrovac spacetimes. See for example the Kerr-Vaidya black hole solutions, which have either a incoming radiation ("null dust") field or an outgoing one (or both) falling onto resp. shining out of ("roughly Hawking") a spinning black hole. There are many other nonvacuum solutions with a compact central mass (which can look more or less black-hole-like), both exact and non-exact. The important feature of these is asymptotic behaviour, as you touch on in your second-last paragraph, since if the influence of the central mass fades with distance, and the sources are kept distant from each other, that lets us ignore (but see below) the difficulties in combining two or more exact solutions of the Einstein Field Equations into a new exact solution. Linearized gravity is usually applicable and sufficient, and if not one can obtain corrections using post-Newtonian theory. We don't really need numrel unless mass-ratios are small and compactness is extreme. See the handy diagram at <https://en.wikipedia.org/wiki/Post-Newtonian_expansion#/media/File:GR2bodyparameterspace.png https://en.wikipedia.org/wiki/Post-Newtonian_expansion#/medi...>. See also Ellis 2010 (Chapter 2, section 3 on inexact solutions, notably his complaint at the bottom of p. 34 to the top of p. 35) <https://doi.org/10.1017/CBO9780511622724.002 https://doi.org/10.1017/CBO9780511622724.002>, which is handily also at s c y h o b. Re his complaint see also Visser 2014 on horizons: <https://arxiv.org/abs/1407.7295 https://arxiv.org/abs/1407.7295>.

codethief 4y ago
Thanks so much for the references! > > black hole solutions assume a perfect vacuum > Not quite You're right, I should have been more precise here. I was merely trying to say: The spacetimes in the vicinity of celestial bodies can be modeled sufficiently well by vacuum black hole solutions (obviously as long as we don't get too close and don't need to consider accretion disks, radiation and what not).
- raattgift 4y ago
  ... or actual surfaces, which we are allowed in Kerr (or more properly perturbations thereof) or Hartle-Thorne spacetimes where the central body is non-compact, for example. Where modelling an astrophysical body this way tends to fall down is that as far as we can tell the central mass evolves and is in general not uniform in the sense that it has long-lasting multipole moments (that would quickly bald away for a black hole), frustrating the hopes of matching the object's interior with the exterior solution. In comparison, outer space is practically always empty enough of stress-energy that the non-physicality of the vacuum or lambdavac region is rarely the issue (and can be dealt with perturbatively, for example). The other major problem is that binary (and triple) systems are surprisingly commonplace, and the N-body problem will drive one towards approximations of GR for tractability. This is the essence of this thread's cautions on the difficulties in superposing solutions to the EFEs. Worse, the two problems above can feed into one another, like when binary stars raise long-lived bumps on each other. This reminds me to re-read the invigorating <https://link.springer.com/article/10.1007/s00190-016-0927-4 https://link.springer.com/article/10.1007/s00190-016-0927-4> which compares the gravitation of Earth and its messy multibody neighbourhood with several exact vacuum solutions and a couple of formalisms. The first author is the redoubtable Michael Soffel <https://www.iau.org/administration/membership/individual/7332/ https://www.iau.org/administration/membership/individual/733...> A PDF of the paper can also be found at s c y h o b. The kicker is the two paragraphs before section 3.

codethief 4y ago

Thanks so much for the references! > > black hole solutions assume a perfect vacuum > Not quite You're right, I should have been more precise here. I was merely trying to say: The spacetimes in the vicinity of celestial bodies can be modeled sufficiently well by vacuum black hole solutions (obviously as long as we don't get too close and don't need to consider accretion disks, radiation and what not).

raattgift 4y ago
... or actual surfaces, which we are allowed in Kerr (or more properly perturbations thereof) or Hartle-Thorne spacetimes where the central body is non-compact, for example. Where modelling an astrophysical body this way tends to fall down is that as far as we can tell the central mass evolves and is in general not uniform in the sense that it has long-lasting multipole moments (that would quickly bald away for a black hole), frustrating the hopes of matching the object's interior with the exterior solution. In comparison, outer space is practically always empty enough of stress-energy that the non-physicality of the vacuum or lambdavac region is rarely the issue (and can be dealt with perturbatively, for example). The other major problem is that binary (and triple) systems are surprisingly commonplace, and the N-body problem will drive one towards approximations of GR for tractability. This is the essence of this thread's cautions on the difficulties in superposing solutions to the EFEs. Worse, the two problems above can feed into one another, like when binary stars raise long-lived bumps on each other. This reminds me to re-read the invigorating <https://link.springer.com/article/10.1007/s00190-016-0927-4 https://link.springer.com/article/10.1007/s00190-016-0927-4> which compares the gravitation of Earth and its messy multibody neighbourhood with several exact vacuum solutions and a couple of formalisms. The first author is the redoubtable Michael Soffel <https://www.iau.org/administration/membership/individual/7332/ https://www.iau.org/administration/membership/individual/733...> A PDF of the paper can also be found at s c y h o b. The kicker is the two paragraphs before section 3.