Schwarzchild Mass Explained - What is it?

In summary, "Schwarzschild mass" is not a standard term in the literature of general relativity. It refers to the mass parameter "m" in the standard form of the line element for the Schwarzschild vacuum solution, which describes a spherically symmetric static gravitational field outside an isolated nonrotating object. This parameter is related to the mass of the object as treated in Newtonian gravitation through the "Newtonian limit of general relativity" and analysis of test particle motion in the far field region of the Schwarzschild vacuum. However, the two-body problem in gtr remains a difficult subject of current research.
  • #1
Logarythmic
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Can someone briefly explain to me what the schwarzchild mass is?
 
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  • #2
Could you give us some context as to where the phrase appears?
 
  • #3
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.
 
  • #4
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?
 
  • #5
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 [Broken] Barring that, can you look at this old post by myself? http://www.math.ucr.edu/home/baez/PUB/effpot [Broken] 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 [tex]m[/tex] which appears in the standard line element expressing the Schwarzschild solution
[itex] 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), [/itex]
[itex] -\infty < t < \infty, \; 2 m < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi [/itex]
(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.
 
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  • #7
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!
 

1. What is the Schwarzchild mass?

The Schwarzchild mass is a concept in physics that describes the mass of a non-rotating, uncharged black hole. It is named after the German physicist Karl Schwarzchild, who first calculated the mathematical solution for a non-rotating black hole in 1916.

2. How is the Schwarzchild mass calculated?

The Schwarzchild mass is calculated using the formula M = r_s / 2G, where M is the mass of the black hole, r_s is the Schwarzchild radius, and G is the gravitational constant. The Schwarzchild radius is the radius at which the escape velocity exceeds the speed of light, and it is directly proportional to the mass of the black hole.

3. What does the Schwarzchild mass represent?

The Schwarzchild mass represents the amount of matter that has collapsed into a singularity at the center of a black hole. It is also used to describe the strength of the gravitational field of a black hole.

4. How does the Schwarzchild mass affect spacetime?

The Schwarzchild mass causes a distortion in the fabric of spacetime, which is the cause of the strong gravitational field around a black hole. This distortion is what causes objects to accelerate towards the black hole and makes it impossible for anything, including light, to escape from within the event horizon.

5. Can the Schwarzchild mass change?

According to the theory of general relativity, the Schwarzchild mass is a constant value that does not change. However, in the real world, black holes can grow by consuming matter and merging with other black holes, which would increase their mass. So while the Schwarzchild mass is a theoretical concept, the mass of a black hole in reality can change over time.

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