Mass of a Schwarzschild object?
Hi, Logarythmic,
Logarythmic said:
Well, the context is like this: the equations of orbital motion of a two-body problem in GR only depend on two parameters, the Schwarzschild masses. How can I explain what this is just by using a few words? Are there any relations to the Newtonian masses?
Where are you reading or hearing this? I think there is some confusion.
Can you take a look at the discussion of test particle motion in the Schwarzschild vacuum soltuion in a standard gtr textbook, such as the ones listed on this page? http://www.math.ucr.edu/home/baez/RelWWW/reading.html#gtr Barring that, can you look at this old post by myself? http://www.math.ucr.edu/home/baez/PUB/effpot Is this what you are asking about?
If so, note that only one mass parameter appears, because the Schwarzschild vacuum solution describes a spherically symmetric static gravitational field (according to gtr) outside an isolated nonrotating object, and the "test particles" are assumed to have a mass so small thay they do not appreciably disturb this ambient gravitational field. As we sometimes say, the Schwarzschild vacuum solution solves the
one-body problem in gtr; the
two-body problem is much more difficult and remains the subject of current research.
There is no notion of "Schwarzschild mass" (at least, no such notion is known to me), although I sometimes see mention of "Schwarzschild objects" or "Schwarzschild masses" (but that is just shorthand for "an isolated nonrotating object producing a static spherically symmetric gravitational field", which, it must be admitted, is quite a mouthful.)
Without context, I can only assume that you are asking about the mass parameter m which appears in the standard line element expressing the Schwarzschild solution
ds^2 = -(1-2m/r) \, dt^2 + 1/(1-2m/r) \, dr^ 2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),
-\infty < t < \infty, \; 2 m < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi
(this defines the metric tensor in terms of the Schwarzschild chart in the exterior region), should be related to the notion of mass familiar from Newtonian physics.
If so, it is fair to ask: how is this parameter related to the mass of the aforementioned nonrotating object (generating a static spherically symmetric gravitational field), as treated in Newtonian gravitation?
The answer comes in part from considering the "Newtonian limit of general relativity" (weak ambient fields or equivalently spacetime models with small curvatures, plus slowly moving test particles in said spacetime models), which gives a rather general way to find the relationship between various Newtonian concepts and analogous concepts in gtr, and in part from considering the "far field region" of the Schwarzschild vacuum, i.e. studying this particular spacetime model. Far from the object, the curvatures are weak, and orbiting test particles are also slowly moving, so we can compare a Keplerian analysis (Newtonian gravitation) with the result of the analysis of test particle motion in the Schwarzschild vacuum. This allows us to identify the parameter m above with the Kepler mass, as deduced from observations of test particles in distant circular orbits, as measured by distant observers.
In addition to these elementary considerations (which are discussed in detail in almost every gtr textbook), there are some further considerations which are quite a bit trickier.
http://en.wikipedia.org/w/index.php?title=Mass_in_general_relativity&oldid=83547460 should give you some idea of some of these issues.
