# How does one determine black hole mass based on the curvature of a photon?

1. How much mass would a black hole need to create a Schwarzschild radius that would trap a photon, whereby the photon would (to an outside observer) be continually curved 0.004km/s at the horizon?

(assume a non-spinning black hole, then contrast that with a photon traveling in either direction along the equator of a spinning black hole)

You're presumably thinking of the last stable orbit for photons. The radius of this orbit is given by:

$$r = \frac{3GM}{c^2}$$

so:

$$M = \frac{rc^2}{3G}$$

for $r = 4$m I make this a mass of about $1.8 \times 10^{27}$kg.

• Very close to mass of Jupiter in fact, though you'll have fun trying to squash Jupiter into within a 2.7m radius. Oct 4 '13 at 14:36
• If photon is traveling at c, with respect to an outside observer, wouldn't one distinguish the radius from the angular velocity of 4m/s around a much larger radius? Don't you have to turn the angular velocity of 4m/s to a circumference and radius based on c? Oct 4 '13 at 15:18
• Oops, I misread your question as wanting the radius of the orbit to be 0.004km. I don't understand what you mean by continually curved 0.004km/s at the horizon. Oct 4 '13 at 15:28
• I updated the post with a graphic example. I guess I'm trying to derive the circumference/radius based on c and the assumed angular velocity of an orbiting photon. Oct 4 '13 at 15:42

It seems that the first step is to approximate the radius using Newton's Method.

I think this comes out to a ~1.12344e13 km radius (1.187 light year) using a photon traveling c with an angular momentum of 4m/s. This would equate to a diameter of 150,193.78AU (2.375 light years). The black hole needed to create this horizon would be 7.5646E42 kg, which is a bit bigger than the mass of the Milky Way at 6E42 kg.

The observable universe has a Schwarzschild radius of approximately 10 billion light years, and the Milky Way would have a ~.9419 ly radius, which means that such a black hole would be larger than the supermasssive black hole at the center of the Milky way plus all the matter (~6E42 kg).