Understanding Black Hole Equations Conceptually

[Ryan Jackson]

I get how to derive black hole equations mathematically. But conceptually, how does it make sense that the radius of a black hole is 2MG/c^2, for example?

 

[Mgcini Keith Phuthi]

A conceptual, albeit incomplete way to understand why the radius of a black hole can be considered as what you stated is by considering that if nothing can escape a black hole (not really!), then the escape velocity of any particle from a black hole must be the speed of light which is the limiting speed. With this in mind the calculation is not very difficult to derive and can be done using familiar concepts in Newtonian mechanics.

 

[fresh_42]

We have a good Insights article series about it. Maybe you should have a read:
https://www.physicsforums.com/insights/schwarzschild-geometry-part-1/#toggle-id-1

I haven't done the search there, but I'm confident there are even more contributions among the "Insights".

 

[Drakkith]

I get how to derive black hole equations mathematically.

Do you? The actual field equations are incredibly difficult to solve even for the simplest examples. Just look at this derivation of the Schwarzschild solution: https://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

But conceptually, how does it make sense that the radius of a black hole is 2MG/c^2, for example?

As far as I understand, Schwarzschild found a solution to Einstein's field equations that gave the gravitational field surrounding a non-rotating, spherically symmetric body. One of the terms of this solution has the form $\frac{1}{2m-r}$ where $m=\frac{GM}{c^2}$ and $r$ is the distance from the center of the body.
The Schwarzschild radius is the value for $r$ such that this term becomes singular (undefined). This occurs when $r=\frac{2GM}{c^2}$ and the terms on the bottom of the fraction subtract to zero.

That's about the best I can do to explain why the Schwarzschild radius is what it is.

 

[Mister T]

I get how to derive black hole equations mathematically. But conceptually, how does it make sense that the radius of a black hole is 2MG/c^2, for example?

The derivation of any equation is the way you make sense of that equation. In other words, if you claim you understand the derivation that implies you understand the result. If you don't understand the result it's not possible to have understood the derivation because the result is part of the derivation!

 

[pervect]

I get how to derive black hole equations mathematically. But conceptually, how does it make sense that the radius of a black hole is 2MG/c^2, for example?

R=2GM/c^2 makes perfect conceptual sense. For instance, if you work out the units GM/c^2 has the proper units to be a length. (Type G*kg/c^2= into Google, for instance, which brings up Google Calculator and computes the units for you). I suppose you're saying you don't understand the equation - fair enough, but your question isn't specific enough for me to figure out what you don't understand, what your concerns are, or even what your background is, so I don't see how I can give a more helpful answer.

Interpreting R as a "radius" isn't quite right though, unfortunately. 2*pi*R can be interpreted as a circumference, though.