stargene wrote:
I would like to know if there is a straightforward way of calculating
the sum of two different gravitational potentials U and V at some point
x.

To simplify as much as possible, I imagine, say, that two neutron stars
n1 and n2 are in circular orbit around their mutual center of gravity
(*) and that the point x is always outside their orbit but on a line
joining their centers...Something roughly like:

x _ _ _ _ _ _(n1) _ _ _(*) _ _ _(n2)

and r1 is distance between x and n1; r2 is the distance between x and n2.

Separately and classically, U, for n1 might be:
U = -G M(n1)/r1 and V, for n2 might be: V = -GM(n2)/r2 . But
what would GR say about this?

So, assuming that r1 and r2 are easy to define, would the sum U+V be
analogous to SR's addition of velocities:

w=(u+v)/(1+uv) ,

..where U=-u^2 , V=-v^2 and their sum W = -w^2 ?

Or am I wildly off base? I just realized an added wrinkle he n1 and
n2 are possibly in relativistic motion and my `straight line' thinking
might be naive.

In Newtonian mechanics, it is indeed correct to add the two potentials,
so the potential at x due to the two neutron stars is
-G M(n1)/r1 - GM(n2)/r2

In general relativity (GR) things are a lot more complicated, but
the short answer is that there's no simple formula for combining the
gravitational effects of multiple bodies (here the two neutron stars)
analogous to special relativity's addition of velocities.

And, in GR r1 and r2 aren't easy to define any more. More precisely,
there are several different plausible definitions. For example
(a) we could (gedanken) measure r1 and r2 by putting a radar set at x
and bouncing radar signals off the two neutron stars (and somehow
correcting for the neutron stars' finite radia, but we won't worry
about that here), or
(b) we could (gedanken) measure r1 and r2 by laying down a bunch of meter
sticks end-to-end between x and n1 and n2 and counting how many meter
sticks it takes to span each of these distances, or
(c) we could (gedankey) measure r1 and r2 by putting theodolites at two
different places off to the sides of the x-n1-n2 line, and using
standard surveying (basically trigonometry based on measured angles)
In GR, (a), (b), and (c) will typically all give slightly *different*
results. (They'll differ by amounts on the order of the neutron stars'
Schwarzschild radia, i.e., on the order of 10% of their physical radia.)
Another way to state this is that (a), (b), and (c) must necessarily
agree *if* we're in a flat spacetime (one where, e.g., Euclid's axioms
of geometry hold). But in GR, spacetime is generally non-flat (Euclid's
axioms are violated, so (a) differs from (b) differs from (c).

r1 and r2 here are examples of coordinates. GR deals with the
non-uniqueness of coordinates by basically saying we can use any
coordinates we want (so long as we do so consistently). The GR object
which describes the gravitational field (the spacetime metric) encodes
not just the gravitational field but also the choice of coordinates,
[note to experts: I'm deliberately fudging the
distinction between a tensor as an abstract geometric
object, and as a matrix of its coordinate components]
and the GR analog of Newton's 1st and 2nd law (the geodesic equation)
is defined in such a way that it gives the same physical results
regardless of what coordinates we choose.

But what this means in practice is that in order to solve for the
spacetime metric, we have to *choose* a coordinate system. This
amounts to specifying 4 free functions everywhere in spacetime.
And we have to do this carefully, because if we do it wrong, it's
very easy to get coordinates which are singular (like latitude/longitude
near the Earth's poles) even in a perfectly well-behaved spacetime,
and that causes no end of confusion.

[[
In Newtonian mechanics the gravitational potential is a
scalar field, i.e., it's a single number at each point
in spacetime. But in GR the spacetime metric is a
*tensor* field, which has 10 independent components at
each point in spacetime. So in a sense there are 10
gravitational potentials in GR.

It turns out there are really only 2 dynamical degrees
of freedom in the spacetime metric, but in general
there's no easy way to separate those out -- we have to
choose the coordinates, then solve for the full
spacetime metric. This is complicated, both
conceptually, mathematically, and computationally.
]]

Another complication worth noting is that in GR the two orbiting
bodies necessarily radiatiate gravitational waves (GWs), and those
GWs themselves carry energy and thus gravitate. So to really determine
the gravitational effects at point x we need to calculate all the GWs
that are around as well.

Fortunately, in many practical situations (including the one we
asked about) there are useful approximations which greatly simplify
the problem:

For example, we can expand all the GR equations in powers of the
velocities of the two neutron stars. More precisely, we expand in
powers of v/c, where v is the velocity of one of the neutron stars.
If we assume that the neutron stars haven't (yet) physically collided,
and that they were slowly moving when they were far apart, then their
velocities now are at most a few tenths of the speed of light, i.e.,
v/c is no more than 0.2 or so, so a power series in v/c should converge
pretty quickly. Doing such a "post-Newtonian approximation" consistently
and correctly is still a big job, but it's easier than solving the
full Einstein equations. The result of such a calculation is a set
of series expansions in v/c for all the interesing dynamical quantities.

If we take just the leading-order terms in the post-Newtonian series,
the result is "just" Newtonian mechanics and gravitation.

Another approximation is to assume that GWs are negligible. There
are a couple of different mathematical ways to make this assumption
(the most common way is to assume that *space* (not spacetime) is
"conformally flat"), but the result is to greatly simplify the
mathematical structure of the Einstein equations (from 10 hyperbolic
PDEs down to 4 elliptic PDEs). This approximation turns out to be
pretty good for orbiting neutron stars, or even for orbiting black
holes until they get very close to each other.

So, to summarize, in GR the original poster's nice simple question
becomes harder to pose (we have to specify precisely how we're defining
r1 and r2), and *much* harder to answer.

ciao,
--
-- "Jonathan Thornburg [remove -animal to reply]"
Dept of Astronomy & IUCSS, Indiana University, Bloomington, Indiana, USA
currently on the west coast of Canada
"There was of course no way of knowing whether you were being watched
at any given moment. How often, or on what system, the Thought Police
plugged in on any individual wire was guesswork. It was even conceivable
that they watched everybody all the time." -- George Orwell, "1984"