Derive Formula for Surface Area of Black Hole

[Werg22]

Homework Statement

Einstein's theory of general relativity is relativistic model of gravity. "Relativistic" means it involves the speed of light, c; "gravity" means it involves Newton's constant, G. Use dimensional analysis to derive a formula for the surface area, A, of a black hole of mass M, up to a dimensionless proportionality constant.

Homework Equations

N/A

The Attempt at a Solution

I have little idea of how to tackle this question. Any sort of help would be appreciable.

 

[mgb_phys]

Not sure I follow the question.
The only meaningful surface area I can think of for a black hole is the area of a sphere with the Schwarzschild radius.

 

[m2003]

I would approach this problem by first understanding the basic concepts and equations involved. In this case, we are dealing with Einstein's theory of general relativity, which involves the speed of light (c) and Newton's constant (G). We also need to find a formula for the surface area (A) of a black hole of mass (M).

One key concept to keep in mind is that black holes have an event horizon, which is the point of no return for anything that enters it. The event horizon is also the surface area of the black hole, which is what we are trying to find.

Using dimensional analysis, we can start by looking at the units of each variable. The speed of light, c, has units of distance divided by time (m/s). Newton's constant, G, has units of distance cubed divided by mass multiplied by time squared (m^3/kg*s^2). The mass, M, has units of mass (kg). And the surface area, A, has units of distance squared (m^2).

From these units, we can see that the formula for the surface area must contain a combination of distance, time, and mass. We can also see that the units of A must be proportional to the units of G and M, and inversely proportional to the units of c. This gives us the following equation:

A ∝ G^α M^β / c^γ

where α, β, and γ are exponents that we need to determine. To do this, we can use the principle of dimensional homogeneity, which states that all terms in an equation must have the same units. In other words, the units on the left side of the equation (A) must be equal to the units on the right side of the equation.

Using this principle, we can set up the following equation:

m^2 = (m^3/kg*s^2)^α * kg^β / (m/s)^γ

Simplifying this equation, we get:

m^2 = m^(3α) * kg^(β-γ) * s^(-2α-γ)

Comparing the powers of each unit, we get the following equations:

2 = 3α
0 = β - γ
0 = -2α - γ

Solving these equations, we get:

α = 2/3
β = γ
γ = -4/3

Sub