How Time Becomes Space Inside a Black Hole - Script (Patreon)

Today on space time we’re going to talk about timespace. Or the strange switching in the roles of space and time that occurs in the mathematics when we drop below the vent horizon of a black hole. 

Intro Graphic

So, what does that bizarre statement – space and time switching roles - even mean? Is this spacetime dyslexia purely a mathematical quirk, or does it correspond to real timey-whimey 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. OK?

Let’s get started. First, we’ll think about what the flow of time looks like without black holes or even spacetime curvature. When we talked about the geometry of causality, we saw that this quantity that we call the spacetime 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 - but then I’ll get back to doing all of this graphically. The spacetime 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 spacetime interval. 

If one event causes a second event, the spacetime interval must be zero or negative. That just means that light-speed causal link may have traveled between them. You could say that object at a given spacetime instant is “caused” by whatever version of itself existed an instant earlier. So worldlines of objects have decreasing spacetime intervals. In fact forward temporal evolution requires a negative spacetime interval. In flat spacetime that negative sign in front of the Delta-t drives that forward evolution. This makes T the time-like coordinate, while x is the spacelike coordinate. For causality to be maintained, the time-like coordinate must always increase.

Reversing causality means flipping the sign of the spacetime 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 spacetime 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 spacetime interval becomes this. This comes from Karl Schwarzschild solution to Einstein’s 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, a distance r away. That r_s 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. If an object gets close to the event horizon – so r just a bit bigger than r_s – 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 FTL 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’s 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 time-like. 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 spacetime diagram we see a sharp division between the two. Our past light cone encompasses the all of spacetime that could have influenced us, while our future light cone shows us the universe that we might ever hope to encounter and influence. Which direction is the future? “Ahead” along our time axis and at right angles to all of our space axes. Our future light cone stares fixedly forwards, encompassing all spatial directions equally. 

This is no longer true if we introduce gravity. Close to a massive object, your future is no longer at right angles to space. It becomes slightly tilted in the direction of that mass. Send out a burst of future-defining light rays and they won’t spread out evenly because they bend towards the gravitational field. As you approach the event horizon of a black hole, more and more light rays are turned towards the event horizon. Your future light cone and your time axis begin to blur together with the inward radial axis of the black hole. 

At this point it’s time we switched diagrams. Close and within a black hole the Penrose diagram is much more useful. It deals with the extreme stretching of space and time by compactifying lines of constant space or time close to its boundaries. We talked about these diagrams previously, but an important thing to remember is that the lines of constant space and time are bent so that light cones remain upright, and light always travels at a 45 degree angle, even inside the black hole..

This entire diagonal line represents the event horizon. Watch what happens to our view of the past and future universe as we approach the event horizon. Our entire future encompasses more and more of the event horizon. That last, tiny sliver is a narrowing window directly above that you could escape to at close to the speed of light. Meanwhile, our past lightcone now encompasses light that has been struggling to escape from just above the event horizon since the distant past, but we see nothing from below that horizon.

Yet as soon as we pass the horizon everything changes. The outside universe exits our future light cone, which now just contains the singularity. We also begin to encounter a new set of photons from the past. At the moment of crossing, light rays from the event horizon itself are suddenly visible. In fact you plummet through a sea of light that is eternally climbing outwards, but getting nowhere. After that you have access to the history of the interior of the black hole. As you fall with the faster-than-light flow of spacetime you overtake light that is outward-pointing. That light it isn’t actually making headway outwards; it’s trying to swim upstream and failing against the faster-than-light inwards cascade of spacetime. Some of this might be light from the collapsing surface of the star that first formed the black hole - emitted long before entered the event horizon. It appears to come from below you because its trying to climb upwards. In fact it was emitted at larger radii than wherever you encounter it.

Also in our past light cone are light rays that are pointed inwards - some of them coming from the outside universe. This light overtakes us as we fall. This is light that entered the event horizon after we did, and appears to reach us from above. We can try to move towards either source of light - down towards light from the black hole's past or up towards light from the black hole's future. Those directions are now described by what was once the time coordinate. But it’s no longer time-like - you can traverse it in either direction, making it space-like. 

Doing so isn’t actually traveling in time - even though there’s a sense of past events in one direction - the collapsing star, and future events in the other - everything that fell into the black hole after us. But remember that our future light cone actually just points forward to the singularity. If we try to accelerate in either direction - up or down - we just quicken our demise. Best just to fall - it’s the last mercy granted by the black hole; it transports us to our doom by the slowest path. Unless we resist.

There’s still a sense of spatial up-ness and down-ness, however the true radial dimension isn’t space-like - it’s time-like. Every photon that reaches you was emitted at some larger radius than when you encounter it, even if it’s old light struggling outwards. The past is radially outwards. And all possible future directions lead radially inwards, in the same way that all worldlines move towards the future in the outside universe. Time IS layered radially and R is time-like, unidirectional. The singularity becomes a future time, not a central place.

In fact the Schwarzschild metric really gives two separate spacetime maps in a single equation – one for above and one for below the event horizon. The coordinates r and t play different roles in those regions. There are other coordinate systems in which that switch never happens. But this mysterious dimensional flip does give us some fascinating insight into how time and space blend together in what is perhaps the strangest place in all of spacetime.