# Newtonian derivation of Scwarzchild radius

• RK1992
In summary, the derivation of the Schwarzschild radius works because it is a limit of a relativistic theory. However, the Newtonian mass is not the same in GR as it is in classical mechanics.
RK1992
can anyone say why the derivation works? my teacher went through it in class and sort of said "don't question it" (which i hate) but it's still annoying me now even though it's a few weeks since i finished college.

KE = GPE
0.5mv² = GMm/r
r=2GM/v²

and then if the escape velocity is the speed of light, then the radius of the region of space where you can't escape is given by r=2GM/c²

but the equation KE=0.5mv² is just valid for small values of v and can be obtained from the expansion of m(γ-1)c²... m(γ-1)c² is not defined for v=c, though, so why does the maths turn out nicely when its so clearly wrong to use KE=0.5mv²?

thanks :)

Even though you should be using relativity to figure this out, which I never learned, it just so happens that if you do use classical mechanics you do get the right answer. I derived the same answer when I was in college. I knew the methodology was wrong and I was surprised when I got the correct answer. So I see two possibilities, either your professor, like me, found the correct answer but did not know why it was correct, in other words he was not a physicist, or he did not want to go through the relativistic derivation that would have been above the class' ability to understand.

The derivation of what the Schwarzschild radius is that I have seen basically compared the time-time components of the metric in a Schwarzschild background with the classical Newtonian potential in a metric theory with a small perturbation from flat spacetime. So this assumed the test particle was non-relativistic and a fair distance away so the Newtonian potential was the correct limit to compare to.

Pengwuino said:
The derivation of what the Schwarzschild radius is that I have seen basically compared the time-time components of the metric in a Schwarzschild background with the classical Newtonian potential in a metric theory with a small perturbation from flat spacetime. So this assumed the test particle was non-relativistic and a fair distance away so the Newtonian potential was the correct limit to compare to.

yeah, this seems much too complicated to explain to a level students ^^

RK1992 said:
yeah, this seems much too complicated to explain to a level students ^^

Basically, you setup a relativistic theory and you look at how that relativistic theory looks in the non-relativistic limit. The small perturbation above flat spacetime just means that you're looking at something like the earth-sun where the distances are so great and the center objects mass is so small that relativistic effects can be ignored.

Another thing to realize is that mass is not so simple in general relativity. The M in $R=2GM/c^2$ is not the same mass as in $F = GmM/r^2$. I haven't gone into GR enough to actually speak of how you approximate the Newtonian mass from the GR mass though.

Pengwuino said:
Basically, you setup a relativistic theory and you look at how that relativistic theory looks in the non-relativistic limit. The small perturbation above flat spacetime just means that you're looking at something like the earth-sun where the distances are so great and the center objects mass is so small that relativistic effects can be ignored.

Another thing to realize is that mass is not so simple in general relativity. The M in $R=2GM/c^2$ is not the same mass as in $F = GmM/r^2$. I haven't gone into GR enough to actually speak of how you approximate the Newtonian mass from the GR mass though.

heh, can't wait to study it all properly. thanks :)

## 1. What is the Newtonian derivation of Schwarzchild radius?

The Newtonian derivation of Schwarzchild radius is a mathematical equation that predicts the radius of a non-rotating, spherically symmetric object at which its escape velocity would equal the speed of light. This is based on the Newtonian theory of gravity and is used to calculate the event horizon of a black hole.

## 2. How is the Schwarzchild radius calculated using Newtonian theory?

The Schwarzchild radius is calculated by equating the escape velocity from the surface of an object to the speed of light. This results in an equation where the mass of the object is divided by the square of the speed of light to give the radius at which the escape velocity equals the speed of light.

## 3. What assumptions are made in the Newtonian derivation of Schwarzchild radius?

The Newtonian derivation of Schwarzchild radius assumes that the object is non-rotating, spherically symmetric, and has a constant mass throughout its volume. It also assumes that the object is not influenced by other external forces besides gravity.

## 4. How does the Newtonian derivation of Schwarzchild radius differ from the relativistic derivation?

The Newtonian derivation of Schwarzchild radius is based on Newtonian gravity and does not take into account the effects of relativity. The relativistic derivation uses Einstein's theory of general relativity to calculate the Schwarzchild radius, which takes into account the effects of gravity on the curvature of space-time.

## 5. Is the Newtonian derivation of Schwarzchild radius accurate?

The Newtonian derivation of Schwarzchild radius is only an approximation and is not entirely accurate. It does not take into account the effects of relativity, which are significant at high velocities and strong gravitational fields. The relativistic derivation is a more accurate and precise method for calculating the Schwarzchild radius.

Replies
11
Views
1K
Replies
2
Views
1K
Replies
11
Views
2K
Replies
7
Views
461
Replies
75
Views
6K
Replies
4
Views
3K
Replies
1
Views
1K
Replies
21
Views
1K
Replies
7
Views
6K
Replies
20
Views
1K