Natural Science Forum / Physics / Research / July 2008

In principle, if you use real rulers and clocks to implement your
coordinate charts, then you must include them in your energy-momentum
tensor. Note, however, there is no necessity to use such instruments to
construct coordinates, and the equations and formalism work just as well
if you just arbitrarily assign labels to the points of the manifold,
being careful to obey the rules (uniqueness within a chart, using
4-tuples of real numbers for the labels, continuity and
differentiability of the labels commensurate with the topology of the
manifold, etc.). Such labels, of course, do not carry any mass or
energy, and are not entered into the stress-energy tensor. Of course you
do have the problem of defining a metric in terms of such coordinates,
but if you plan ahead for this, it can be done independently of the test
of GR for which the coordinates to be used....

In practice, of course, the rulers and clocks a real experimenter might
use are VASTLY smaller than the astronomical objects being observed and
used to test GR, so the approximation that neglects their contributions
to the stress-energy tensor is an extremely good one. For instance, the
satellite of Gravity Probe B is so much smaller than the earth and sun
that one can consider it to follow a geodesic path to much better
accuracy than one can measure its trajectory (the orientations of its
gyroscopes are another matter). The clocks and telescopes on earth
clearly do not significantly affect the contribution of the earth to the
geometry of the manifold. No rulers or clocks are erected near the
satellite's orbit, and coordinates out there are constructed via optical
and radio observations of the satellite (and other objects) from earth.

Tom Roberts

I would say it's not a bug, it's a mathematical model (idealization
and abstraction) of reality.  Within the formal system of mathematics,
we can use coordinates (and other such concepts) to reason about the
properties of solutions of the Einstein equations.

For example, we can show that the Schwarzschild metric (written in
whatever coordinates you like) is a solution of the vacuum Einstein
equations, we can determine the physical interpretation of (say) an
areal radial cooridinate, and we can integrate the geodesic equation
to find the coordinate trajectories of geodesics.

So far this is all perfectly good *mathematics*, but it doesn't yet
have anything to do with *physics*.  If we want it to be physics,
then we need to make a connection to the "real world".  This means we
need to consider realizations of our various abstractions/idealizations.
For example, to investigate the behavior of the GPS system we might
idealize (approximate) spacetime near the Earth as part of Schwarzschild
spacetime, i.e. we might approximate the universe as containing a
spherically symmetric Earth and nothing else.

In this case the particular approximation you mentioned in your question
(ignoring the stress-energy tensor of the coordinate-measuring apparatus)
is *very* good: the GPS satellites have a mass that's 21 or so orders
of magnitude smaller than that of the Earth, so their stress-energy
perturbations to the Einstein equations in the vicinity of the Earth
are going to be on the order of one part in 10^21.

Of course, there are lots of other approximations here, e.g. the
Earth isn't really spherically symmetric, there are gravitational tidal
forces from the Moon, Sun, and other massive bodies in the universe,
and the (cosmological) structure of the universe may well differ from
the "asymptotically flat" which is implicit in the Schwarzschild
solution.

Much of the "art of physics" consists of correctly judging which
approximations are good ones, and which are dubious.  For example,
the Schwarzschild-spacetime approximation turns out to be excellent
for calculating (say) the effects of gravitational redshift on the
GPS signals.  But if we want to investigate the GPS satellites'
orbits around the Earth, then we'd better take into account the
non-sphericity of the Earth and the gravitational tidal forces of
the Moon and Sun.

ciao,

That's not the "Hawking-Ellis" presentation of GR. It's the standard
treatment of manifolds and differential geometry that's been prevalent
since the turn of the 20th century. Hawking and Ellis have nothing,
per se, to do with any of that. In all likelihood, you're only pinning
their names to it, because it's the first place you saw it.

If you set everything up by echo-location signalling (with light) and
internal atomic clocks, you can subject the sources, themselves to the
equations governing the dynamics, express the solutions (e.g. the
number of send-receive echoes or the number of atomic oscillations) in
terms of whatever underlying coordinate system you wish to write out
the solutions in, and then invert to express the coordinate system in
terms of these. After inverting, you have the coordinates expressed in
terms of the material events, WITH the solutions already taken into
account.

But, in practice, tiny satellites in geosynchronous orbit are going to
have an utterly negligible back-reaction effect on the field that will
almost certainly be well beyond the ability of its technology (or any
other human-developed) technology to detect -- hence making the issue
of back-reaction effects irrelevant. Just think of all of the twisting
and contorsion that has to go into merely designing and constructing a
pendulum balance to detect the near-field for ordinary matter.

In any case, you can write it out as an iterative series: (1)
situation without back-reaction + invert to the material coordinates;
(2) include back-reaction as a minor modification (+ invert to
material coordinates and reverberate this onto everything else); (3)
include back-reaction of back-reaction; etc. It's a rapidly converging
series, since the back-reaction is much less than the field being
measured.

After all, you're not using giant celestial objects as clocks or
rulers.

It's certainly true that the idea of a manifold is an idealization, but
not quite in the sense you mean.  The starting manifold has no "clocks"
or "rulers" even in an idealized sense -- there is no concept of
distance or time until the metric is introduced as a superstructure.  A
manifold does assume some qualitative sense of closeness, in the sense
of having open sets, and some sort of global topology; but the more
important assumption is that one can somehow label events, giving them
(abstract) coordinates.

It's important, though, that this labelling is almost arbitrary.  If the
final theory depended on the actual existence of such labels, you would
be quite right to worry.  But the only observables in GR are actually
coordinate-independent objects, things that don't depend on a choice of
how to label points or a "background" spacetime with identifiable
individual points.  So in that sense, it is bootstrapped away in the
end. That's the real meaning of general covariance.

(This causes tremendous problems for quantum gravity, because there are
some nice theorems that coordinate-independent objects are necessarily
nonlocal, and quantum field theory is not at all good at dealing with
nonlocal operators.)

You might want to look at the discussion of Einstein's "hole argument"
in the Stanford Encyclopedia of Philosophy, which is not bad -- see
http://plato.stanford.edu/entries/spacetime-holearg/

Steve Carlip