Answer to Question #131735 in Astronomy | Astrophysics for Mason Anderson

Question #131735
The Schwarzschild radius of a black hole depends on its mass m, the speed of light
c, and the gravitational constant G, which has units of m3/(kg · s2). Use dimensional analysis
to write down an expression for the Schwarzschild radius in terms of these three quantities.
Explain your reasoning. (Hint: The Schwarzschild radius has dimension of length)
1
Expert's answer
2020-09-08T09:14:10-0400

Let's take the gravitational constant "[G] = \\dfrac{m^3}{kg\\cdot s^2}" and try to get rid of "m^2" in the numerator and "kg\\cdot s^2" in the denominator.

First, let's notice that "[c^2] = m^2\/s^2", thus, we can exclude these units by putting "c^2" into the denominator of the final expression. Then, only "kg" has left. To get rid of it we should put "[m] = kg" into the numerator of the final expression.

Finally, obtain:


"R = G\\dfrac{m}{c^2}"

Checking dimension, get:

"[R] = \\dfrac{m^3}{kg\\cdot s^2}\\cdot \\dfrac{kg\\cdot s^2}{m^2} = m"

Answer. "R = G\\dfrac{m}{c^2}".


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