Schwarzchild Mass Explained - What is it?

[Logarythmic]

Can someone briefly explain to me what the schwarzchild mass is?

 

[pervect]

Could you give us some context as to where the phrase appears?

 

[Chris Hillman]

Hi, pervect, congragulations on your award!

And Logarythmic, I was also about to ask for context when I saw that p beat me to the punch, as it were! The obvious guess is that you are asking about the parameter "m" in the standard form of the line element for the Schwarzschild vacuum solution, but if so, "Schwarzschild mass" is not a standard term in the literature. Mass is tricky in gtr. By the way, I think p did a fine job with the Wikipedia article http://en.wikipedia.org/w/index.php?title=Mass_in_general_relativity&oldid=83547460, although the usual caveats apply for anyone reading a later version.

 

[Logarythmic]

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?

 

[Chris Hillman]

Mass of a Schwarzschild object?

Hi, Logarythmic,

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.

 

[Logarythmic]

That's a lot of info. ;) I was reading in http://luth2.obspm.fr/IHP06/lectures/damour/DamourDeruelleAIHP85.pdf , page four, row 2..

 

[Chris Hillman]

Hi, Logarythmic,

Damour and Deruelle are discussing the two-body problem in gtr. As I stated, this is much too hard to find useful exact solutions in closed form (such as the Kerr solution, which solves the one-body problem), so they are studying a standard method of approximation which has been very highly developed over many decades (Damour is one of the leaders in this work).

Hope this clarifies the situation!