Black Holes and the Arrow of Time

By Shawn Radcliffe

In our everyday life, time flows in one direction—forward. When you put a frozen pizza in the hot oven, the pizza heats up. When you hit a baseball, it flies over the wall (if you’re having a good day at bat). When you knock a coffee mug off a table, the mug shatters on the floor.

These actions don’t happen the other way around. The “arrow of time” is unidirectional. You wouldn’t expect to put a hot pizza in the oven and take it out twenty minutes later frozen. Or to have a baseball return over the wall and hit your bat. And you can wait for years, but that smashed coffee mug is never going to reassemble itself and hop back up on the table.

Things are a little different, though, when it comes to the laws of physics. In general, these laws don’t care which way time is flowing for us. Imagine taking a video of a moving object—such as a swinging pendulum or an accelerating train—and playing the video forward. And then backward. The equations used to describe the movement of the object are the same regardless of which way the video is playing—even when we know that the train is going backwards.

So how can the arrow of time be described as unidirectional? The answer is entropy. Entropy is a measure of the state of disorder within a system. For example, a shattered mug on the floor is more disordered than a whole mug sitting on a table. A room full of gas is more disordered than the same amount of gas squeezed into a tiny steel cylinder. More disordered equals higher entropy.

In a post on Nautilus, physics graduate students Andrew Turner and Alex Tinguely describe entropy as “our lack of knowledge about the underlying state of a system.” For example, we may know the temperature and pressure of a cylinder filled with gas, but we don’t know the position and velocity of each gas particle inside the cylinder.

If you opened the cylinder, the gas would flow out into the room. Because the room is larger than the cylinder, there are many more possible states for the gas particles to be in. We can still measure the temperature and pressure of the room, but we know even less about the positions and velocities of the particles. Our “lack of knowledge”—and the entropy—of the system has increased.

The second law of thermodynamics states that entropy can only increase, never decrease. Gas particles in a room will never spontaneously all move inside the cylinder. Likewise, a broken coffee mug (higher entropy) will never spontaneously reassemble itself into a whole mug (lower entropy).

This is where the unidirectional arrow of time comes from. Intuitively, we understand that entropy can only increase, although we don’t usually think of it in those terms. You could, of course, force the gas back into the cylinder, but overcoming entropy like this requires the addition of energy to the system.

So what does this have to do with black holes? Black holes have the most clear dividing line between “inside” and “outside” of any object in the universe. This line is known as the event horizon.

When an object passes a black hole’s event horizon, it never comes back. Even the information about the object is gone. It’s as if the object has been erased entirely from the part of the universe outside the black hole. Think of it as a perpetually shattered coffee mug … that you will never see again … ever.

If you think about a black hole without taking quantum physics into account, you get what Turner and Tinguely call a “classical black hole.” These have only a few properties, such as mass, angular momentum (the speed of its spinning) and electric charge.

Classical black holes also have no entropy. So when an object crosses the event horizon, the object’s entropy is basically erased from the universe. If this were true, it would mean that the overall entropy of the universe decreased, which contradicts the second law of thermodynamics.

But if this law were null and void (terms that could also be used to describe black holes), then you might come home one day to find that your broken coffee mug is whole again. But this never happens, so something else must be going on.

It turns out that black holes have other properties as well, including temperature. This is also known as its Hawking temperature, named after the late physicist Stephen Hawking, who discovered this property of black holes.

This discovery led to the Bekenstein–Hawking formula which shows that black holes do actually have entropy—and it depends on the surface area of the black hole. Turner and Tinguely go into much more detail about the math behind this formula, but for our purposes we just need to know that black holes have entropy.

So when an object falls into a black hole, the entropy of the part of the universe outside the black hole decreases, but the entropy of the black hole itself increases, as does its surface area. The total entropy of the universe, though, doesn’t decrease.

Thanks to a quantum modification, the second law of thermodynamics still holds. And our arrow of time is unidirectionally functioning. But sadly, our poor coffee mug is still broken.