How Time Becomes Space Inside a Black Hole | Space Time

Matt O'Dowd (0:03)

This episode is sponsored by Crunchyroll.

Today on Space Time, we're going to talk about time-space, or the strange switching in the roles of space and time that occurs in the mathematics when we drop below the event horizon of a black hole.

What does this bizarre statement, space and time switching roles even mean? Is this space-time dyslexia purely a mathematical quirk, or does it correspond to real timey-wimey weirdness? We've been working up to this one, so you might want to hit pause and check out these episodes if you think you need some more background.

First, we'll think about what the flow of time looks like without black holes or even space-time curvature. When we talked about the geometry of causality, we saw that this quantity that we call the space-time interval governs the flow of cause and effect, the only reliable ordering of events in a relative universe. I'm going to show you the math one more time, and then we'll get back to doing all of this graphically. The space-time interval is defined like this, for boring old flat or Minkowski space. Different observers may report that two events are separated by different distances, delta x, and by different amounts of time, delta t. However, all observers record the same space-time interval. If one event causes a second event, the space-time interval must be or negative. That just means that a light-speed causal link may have traveled between them. You could say that an object at a given space-time instant is caused by whatever version of itself existed an instant earlier. So world lines of objects have decreasing space-time intervals. In fact, forward temporal evolution requires a negative space-time interval.

In flat space-time, that negative sign in front of the delta t drives that forward evolution. This makes t the time-like coordinate, while x is the space-like coordinate. For causality to be maintained, the time-like coordinate must always increase. Reversing causality means flipping the sign of the space-time interval. In our episode on superluminal time travel, we saw that in flat space, this means traveling faster than light. Which is, of course, impossible. But if we introduce a black hole, we now have a second way to flip the sign of the space-time interval. We're going to see how this changes the behavior of time in very strange ways. Add a non-rotating, uncharged black hole, and the space-time interval becomes this. This comes from Karl Schwarzschild's solution to the Einstein field equations, the very first accurate description of a black hole. I've left out a few terms. This equation assumes no orbital motion, only motion towards or away from the center of the black hole, which is a distance r away. That rs is the Schwarzschild radius, the radius of the event horizon. Very far from the event horizon, the Schwarzschild interval becomes the good old Minkowski interval, and time and space are nicely separated.

But if an object gets close to the event horizon, so r just a little bit bigger than rs, that stuff in the two brackets describes extreme warping of spacetime. But as long as you're outside the event horizon, time behaves itself mostly. A negative spacetime interval still means causal movement, and the only way to break causality is still with faster-than-light travel. Things change radically below the event horizon when r gets smaller than rs. Then both of these brackets become negative. The entire delta r stuff is now negative, and the delta t stuff is positive. Below the event horizon, there is only one way to maintain the respectable causal progression, expected of a well-mannered temporal entity. That's to fall inwards, to have a non-zero delta r. As it happens, you don't have a choice. Space itself is falling inwards faster than the speed of light towards the central singularity. It carries you with it and drives your personal clock forward as it does so. In the mathematics, the coordinate r, which once represented distance, now grants the negative sign needed to maintain your causal flow.

It becomes timelike. It's uni-directional. Meanwhile, the coordinate previously known as time, t, lost its negative sign and becomes spacelike. So it can be traversed in any direction or not traversed at all. But what does all of this time-space switching actually look like?

Let's fall into the black hole one more time. Now graphically, instead of mathematically. Back out here in the regular universe, it's pretty obvious where the past and the future are. On our ever-popular space-time diagram, we see a sharp division between the two. Our past light cone encompasses all of space-time that could have influenced us, while our future light cone shows us the parts of the universe that we might ever hope to encounter or influence.