Asymptotically Flat Spacetime - Criticism

Criticism

The notion of asympotic flatness in gravitation physics has been criticized on both theoretical and technical grounds.

There is no difficulty whatsoever in obtaining models of static spherically symmetric stellar models, in which a perfect fluid interior is matched across a spherical surface, the surface of the star, to a vacuum exterior which is in fact a region of the Schwarzschild vacuum. In fact, it is possible to write down all these static stellar models in a way which makes clear that they exist in plenitude. Given this success, it may come as a nasty shock that it seems to be very difficult, mathematically speaking, to construct rotating stellar models in which a perfect fluid interior is matched to an asymptotically flat vacuum exterior. This observation is the basis of the most prominent technical objection to the notion of asymptotic flatness in general relativity.

Before explaining this objection in more detail, it seems appropriate to briefly discuss an often overlooked point about physical theories in general.

Asymptotic flatness is an idealization, and a very useful one, both in our current "Gold Standard" theory of gravitation -- General Relativity -- and in the simpler theory it "overthrew", Newtonian gravitation. One might expect that as a (so far mostly hypothetical) sequence of increasingly sophisticated theories of gravitation providing more and more accurate models of fundamental physics, these theories will become monotonically more "powerful". But this hope is probably naive: we should expect a monotonically increasing range of choices in making various theoretical tradeoffs, rather than monotonic "improvement". In particular, as our physical theories become more and more accurate, we should expect that it will become harder and harder to employ idealizations with the same ease with which we can invoke them in more forgiving (that is, less restrictive) theories. This is because more accurate theories necessarily demand setting up more accurate boundary conditions, which can render it difficult to see how to set up some idealization familiar in a simpler theory in a more sophisticated theory. Indeed, we must expect that some idealizations admitted by previous theories may not be admitted at all by succeeding theories.

This phenomenon can be both a blessing and a curse. For example, we have just noted that some physicists hold that more sophisticated theories of gravitation will not admit any notion of an isolated point particle. Indeed, some argue that general relativity does not do so, despite the existence of the Schwarzschild vacuum solution. If these physicists are correct, we would gain a kind of self-abnegating intellectual honesty or realism, but we would pay a hefty price, since few idealizations have proven as fruitful in physics as the notion of a point particle (however troublesome it has been even in simpler theories).

Be this as it may, very few examples of exact solutions modeling isolated and rotating objects in general relativity are presently known. In fact, the list of useful solutions presently consists of the Neugebauer-Meinel dust (which models a rigidly rotating thin (finite radius) disk of dust surrounded by an asymptotically flat vacuum region) and a few variants. In particular, there is no known perfect fluid source which can be matched to a Kerr vacuum exterior, as one would expect in order to create the simplest possible model of a rotating star. This is surprising because of the plenitude of fluid interiors which match to Schwarzschild vacuum exteriors.

Indeed, if some argue that an interior solution which matches to the Kerr vacuum, which has Petrov type D, should also be type D. There is in fact a known perfect fluid solution, the Wahlquist fluid, which is Petrov type D and which has a definite surface across which one can attempt to match to a vacuum exterior. However, it turns out that the Wahlquist fluid cannot be matched to any asymptotically flat vacuum region. In particular, contrary to naive expectation, it cannot be matched to a Kerr vacuum exterior. A tiny minority of physicists (actually, a minority of one) appear to believe that general relativity is unacceptable because it does not allow sufficiently general asymptotically flat solutions (evidently this argument implicitly assumes that we have decisively rejected at least some Machian principles!), but a sequence of increasingly sophisticated and general existence results appears to contradict this assumption.

The mainstream viewpoint among physicists about these matters can probably be summarized by saying as follows:

• while many prominent researchers have tried to invoke Machian principles (including Albert Einstein and John Archibald Wheeler), the status of these principles, in contrast to widely accepted principles like the principle of conservation of momentum, is currently highly equivocal,
• general relativity admits a sufficient variety of solutions to model (in principle) any realistic astrophysical situation, plus (apparently) many highly unrealistic ones.