# Black Holes and Schwarzschild Radius

FROM THE LECTURE SERIES: INTRODUCTION TO ASTROPHYSICS

## A black hole has no surface. It’s a real-life point mass. It is an infinitely deep gravitational pit. And the pit has slippery sides: if you get too close, you inevitably fall in. The mass really is concentrated into a single point in space, so as you approach, the gravitational force gets stronger, without bound all the way down to r equals zero, where it becomes infinite.

### Can We Escape a Black Hole?

Let’s say we’re cruising around the galaxy, and through some careless navigation error, we find ourselves a distance r-naught from a black hole. In a panic, we turn away from the hole and fire our thrusters, giving the spaceship a velocity v-naught directed away from the hole. Will we escape?

We can figure that out based on the conservation of energy. The initial energy of the spaceship is the kinetic energy, 1/2 mv-naught-squared, plus the gravitational potential energy, minus-G times big M little m over r-naught. Here big M is the mass of the black hole, and little m is the mass of the spaceship.

At some later time, we manage to make it out to some larger distance, r-one, with a slower speed, v-one, since the black hole’s gravity has been slowing us down. So, the energy is 1/2 mv-one-squared minus GMm over r-one, which we can set equal to the initial energy.

### Escape Velocity

Let’s think about the case in which we just barely escape. In that case r-one approaches infinity, and the potential energy approaches zero. In addition, v-one approaches zero, because we had just enough energy to make it out, with no leftover kinetic energy. So, the total energy in this case must be zero, giving us a simple equation, which we can solve for v-naught, giving square root of 2G big M over r-naught.

That’s the formula for the escape velocity. If we have enough fuel to go at least that fast, we escape. Otherwise, we fall in. But here’s a thing about a black hole. Since there’s no surface, there’s no minimum value for r-naught. There’s nothing equivalent to the Earth’s surface, within which the gravitational force starts getting weaker.

So, the escape velocity grows without bound as r-naught approaches zero. And at some point, the escape velocity exceeds the speed of light—the fastest speed it’s possible for anything to attain—and in that case, no amount of fuel would have been enough. To find the value of r-naught where that happens, we set the escape velocity equal to c, the speed of light, and solve for r-naught, giving 2GM over c-squared.

This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.

So, even though the black hole itself has zero size, 2GM over c-squared is a sort-of radius: it’s the radius of no return. It’s called the Schwarzschild radius after Karl Schwarzschild, the first person to solve Einstein’s equations of general relativity exactly, for the case of a point mass. Another, more evocative name is the event horizon. From beyond that horizon, no light can reach you, so you can’t see any events that might be going on there.

Speaking of general relativity it is not used in our calculation. Our calculation used Newton’s law of gravity instead, and Newtonian equations for the kinetic energy and potential energy. In Einstein’s theory of gravity, which is more accurate than Newton’s, those expressions don’t hold.

So, does that mean our calculation of the Schwarzschild radius was bogus? Well, in a sense, yes. Any time we’re dealing with strong gravitational fields, and speeds approaching c, we shouldn’t expect to get the right answer with Newtonian physics.

Let’s find the Schwarzschild radius of a black hole with the mass of the Earth. In that case M is 6 times 10 to the 24 kilograms, giving a Schwarzschild radius of 9 millimeters. What that means is that if we could somehow compress the entire Earth down to the size of a marble, we would have ourselves a black hole. For the mass of the Sun, the Schwarzschild radius comes out to be 3 kilometers, so we can write the formula as a scaling relation: Rs equals 3 kilometers times M over M sun.

Q: What is Schwarzschild Radius? Who is it named after?

Even though the black hole itself has zero size, 2GM over c-squared is a sort-of radius: it’s the radius of no return. It’s called the Schwarzschild radius after Karl Schwarzschild, the first person to solve Einstein’s equations of general relativity exactly, for the case of a point mass. Another, more evocative name is the event horizon. From beyond that horizon, no light can reach you, so you can’t see any events that might be going on there.

Q: What can be the approximate Schwarzschild Radius of a black hole with the mass of the Earth?

To calculate the Schwarzschild radius of a black hole with the mass of the Earth, M is 6 times 10 to the 24 kilograms, giving a Schwarzschild radius of 9 millimeters. What that means is that if we could somehow compress the entire Earth down to the size of a marble, we would have ourselves a black hole.

Q: What is a black hole?

A black hole has no surface. It’s a real-life point mass. It is an infinitely deep gravitational pit. And the pit has slippery sides: if you get too close, you inevitably fall in. The mass really is concentrated into a single point in space, so as you approach, the gravitational force gets stronger, without bound all the way down to r equals zero, where it becomes infinite.