In article <
cb878d08-ead1-b8e1-de1e-70cac7634527@comcast.net>, I gave a
brief overview of the concept of asymptotic flatness in general
relativity (GR), and I mentioned some very useful concepts (event
horizons, black holes, total energy & angular momentum) which are only
defined if spacetime is asymptotically flat (i.e., which are generally
*not* defined in a non-asymptotically-flat spacetime).
Another very useful concept which is much harder to define without
asymptotic flatness is that of gravitational waves (GWs). The problem
here is that it's rather hard to answer the question "what is a GW?"
unless you have some sort of "background".
That is, our intuitive notion of a GW is that it's a "ripple of
spacetime curvature", i.e., some sort of (propagating) perturbation in
the spacetime curvature. But how do we tell which part of the spacetime curvature is the "ripple" (perturbation) versus which is "just part of
our spacetime"? In an asymptotically flat spacetime, we can go out to
the weak-field region (recall that this is roughly the $r \to \infty$
limit),
[I'm glossing over the fact that there is more than one
$r \to \infty$ limit. E.g., there's the obvious limit
$r \to \infty$, $t \to \text{constant}$ where $t$ is a
"suitable" time coordinate. But in studying GWs it's often
more convenient to instead consider the simultaneous limits
$r \to \infty$, $t \to \infty$, and $t-r \to \text{constant}$;
this part of the far-field region is known as "null infinity".
Roughly speaking, this is "where an outgoing GW or light signal
ends up after an infinite time".]
and precisely because spacetime is almost flat (Minkowski) there (and
because of a bunch of other conditions about derivatives of the metric
being small there), there is a mathematically clean way to separate "perturbation" from "background". Namely, in the weak-field region
there's a *unique* Minkowski spacetime singled out by the physics, up to
Lorenz boosts and rotations
[and some things called "supertranslations" which I
alas have never quite understood].
So, we can say that the difference between the actuasl metric and that
"unique" Minkowski (flat) metric, a.k.a. the "metric perturbation",
*is* the GW.
With this definition we can then make do interesting/useful calculations
(& prove theorems) about GWs, i.e., we can calculate the response of a
detector (located in the far-field region) to incident GWs, we can
calculate the GW energy flux flowing outwards through a Gaussian sphere
at $r = \infty$, and we can even prove that the time derivative of the
total energy of spacetime is equal to the negative of that energy flux.
A famous example: suppose we have a "nearly Newtonian" spacetime which
is close to Minkowski *everywhere*, and that spacetime is asymptotically
flat, and empty except for a binary star system. How much energy does
that binary star system radiate in GWs? For a nearly-Newtonian
asymptotically flat spacetime it's fairly easy to derive an excellent approximate answer, given by the "quadrupole formula"
(see
https://en.wikipedia.org/wiki/Quadrupole_formula for more on this)
which says that the radiated energy flux is proportional to the square
of the 3rd time derivative of the binary star's mass quadrupole tensor.
Again for a nearly-Newtonian-everywhere spacetime, there's a simple
derivation of the quadrupole formulation (see any beginning GR textbook)
which is "rigorous enough for a physicist".
But now suppose our spacetime *isn't* nearly-Newtonian. E.g., what if
our orbiting "stars" are neutron stars or black holes? That simple
proof assumes that the metric is close to Minkowski *everywhere*,
including near to and inside the "stars", so that proof no long holds.
So is the quadrupole formula still valid for such a
binary-relativistic-star system? In the 1980s there was a lot of
controversy about this, with some researchers claiming (based on various approximation methods of uncertain mathematical validity) that in this situation the actual radiated energy might be very different from what
the quadrupole formula would give, possibly even different in sign!
After a lot of hard work by a whole bunch of very talented mathematical relativists, this controversy was eventually resolved (by the mid to
late 1990s, I think) and a mathematically rigorous proof of the
quadrupole formula was constructed which applies even to binary black
holes.
All (or at least a great deal) of the mathematical machinery which lets
us prove these nice theorems is only defined if the spacetime is
asymptotically flat. I suspect that there may be ways of defining these
things for *some* non-asymptotically-flat spacetimes, but we would
certainly loose a big part of the nice mathematical framework we have
for an asymptotically flat spacetime. (I suppose an analogy might be
trying to do calculus with functions which aren't Riemann-integrable.)
So, in conclusion, asymptotic flatness is a very nice property for a
spacetime to have, and if it doesn't hold then we loose a bunch of
useful and important concepts and mathematical tools.
[Moderator's note: Those interested in the history of the concept of gravtational waves might enjoy arXiv:2111.00330:
@INPROCEEDINGS{ MDiMauroEN21a ,
AUTHOR = "Marco Di Mauro and S. Esposito and A.
Naddeo",
TITLE = "Towards detecting gravitational waves:~a
contribution by Richard Feynman",
BOOKTITLE = "{T}he {S}ixteenth {M}arcel {G}rossmann
{M}eeting",
YEAR = "2022",
EDITOR = "Remo Ruffini and Gregory Vereshchagin",
ADDRESS = "Singapore",
PUBLISHER = "World Scientific"
}
(The above is preliminary publication information; probably at most the
title might not be the final one.) -P.H.]
-- "Jonathan Thornburg [remove -color to reply]" <
jthorn4242@gmail-pink.com>
Dept of Astronomy & IUCSS, Indiana University, Bloomington, Indiana, USA
currently on the west coast of Canada
"C++ is to programming as sex is to reproduction. Better ways might
technically exist but they're not nearly as much fun." -- Nikolai Irgens
"that applies to Perl, too!" -- me
[Moderator's note: As a Fortran man, I pity your sex life. :-) -P.H.]
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