The Schwarzschild Metric: Obtaining Equation M=Gm/c^2 & Newton Law at Infinity

In summary, the equation M=Gm/c^2 is used to determine the mass of a black hole in units where G and c are equal to one. M stands for the mass of the black hole, and it is a conversion factor from mass units to length units. The Schwarzschild radius is equal to twice the equivalent length determined by this equation. The derivation of this equation can be found by searching online for the Schwarzschild radius.
  • #1
TimeRip496
254
5
How do you obtain this equation M=Gm/c^2. What does M stand for? Is is Newton law at infinity? Again what is this Newton law at infinity?
 
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  • #2
TimeRip496 said:
How do you obtain this equation M=Gm/c^2. What does M stand for?

M is just the mass of the black hole using units in which G and c are both equal to one.

We don't have to make this substitution but if we don't we'll be schlepping factors of G and c around everywhere in our equations, and they're complicated enough already.
 
  • #3
Nugatory said:
M is just the mass of the black hole using units in which G and c are both equal to one.

We don't have to make this substitution but if we don't we'll be schlepping factors of G and c around everywhere in our equations, and they're complicated enough already.
Do you mind telling me a source for such derivation? Cause all the Internet gives is just the derivation of the schwarzschild radius.
 
  • #4
TimeRip496 said:
Do you mind telling me a source for such derivation?

It's just a conversion factor from mass units to length units; ##Gm / c^2## converts the mass ##m## to an equivalent length. The Schwarzschild radius corresponding to ##m## is just twice that equivalent length.
 
  • #5


The Schwarzschild Metric is a mathematical equation that describes the curvature of space-time around a massive, non-rotating object. It is derived from Einstein's theory of general relativity and is often used to study the behavior of objects in extreme gravitational fields, such as black holes.

To obtain the equation M=Gm/c^2, one must solve the Schwarzschild Metric for the parameter M, which represents the mass of the massive object. This is done by setting the metric equal to zero and solving for M, which results in the equation M=Gm/c^2.

The letter M in this equation stands for the mass of the object, while G is the gravitational constant and c is the speed of light. This equation is important because it relates the mass of an object to its gravitational field, and it helps us understand the effects of gravity on space and time.

The concept of "Newton's law at infinity" is a bit misleading. It refers to the behavior of objects in the limit of large distances from the massive object, where the effects of gravity become weaker and weaker. In this limit, the equation M=Gm/c^2 reduces to Newton's law of universal gravitation, which describes the gravitational force between two objects based on their masses and the distance between them. So, in a sense, the equation M=Gm/c^2 can be seen as a generalization of Newton's law of gravitation that takes into account the curvature of space-time.
 

1. What is the Schwarzschild metric?

The Schwarzschild metric is a mathematical formula in the field of general relativity that describes the curvature of spacetime around a non-rotating, spherically symmetric mass. It is named after the German physicist Karl Schwarzschild who first derived the metric in 1916.

2. How is the Schwarzschild metric obtained?

The Schwarzschild metric is obtained by solving Einstein's field equations, which describe how matter and energy affect the curvature of spacetime. This involves using techniques from differential geometry and tensor calculus, and can be quite complex. The final result is the Schwarzschild metric, which is a solution to these equations for a static, spherically symmetric mass.

3. What is the equation M=Gm/c^2 in relation to the Schwarzschild metric?

The equation M=Gm/c^2 is known as the Schwarzschild radius, and it is a key component of the Schwarzschild metric. It represents the distance from the center of a massive object at which the escape velocity would equal the speed of light. This means that anything within this radius, including light, would be unable to escape the strong gravitational pull of the object.

4. How does the Schwarzschild metric relate to Newton's law of gravitation at infinity?

At large distances from a massive object, the Schwarzschild metric reduces to Newton's law of gravitation, which describes the gravitational force between two objects. This is known as the Newtonian limit of the Schwarzschild metric, and it shows that general relativity is consistent with Newton's theory of gravity in this regime.

5. What are some real-world applications of the Schwarzschild metric?

The Schwarzschild metric has been used in various real-world applications, such as in the GPS system for accurately measuring time and position on Earth. It is also used in simulations of black holes and other astronomical objects, as well as in the study of gravitational lensing, which is the bending of light by massive objects in space.

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