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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 matemathical(sic) 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.

OK? Let's get started.

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 zero or negative.

That just means that a lightspeed causal link may have traveled between them.

You could say that an object at a given spacetime instant is caused by way of a 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" is just a little bit bigger than "rₛ"; that stuff in the two brackets

describes extreme warping of space-time.

But as long as you're outside the event horizon

time behaves itself,

mostly.

A negative space time 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 Δr stuff is now negative

and the Δt 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 Δr.

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 unidirectional.

Meanwhile, the coordinate previously known as time, "t"

lost its negative sign and becomes space-like. 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.

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 switch diagrams.

Close to, and within the 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 curved

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 universe as we approach it.

Our entire future light cone

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 light cone now encompasses light that has been struggling to escape from just above the event horizon

since the distant past.

But we still see nothing from below the 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, we plummet through a sea of light that is eternally climbing outwards, but getting nowhere.

After that, we have access to the history of the interior of the black hole. As we fall with the faster-than-light flow of space-time,

we overtake light that is outward pointing. That light isn't actually making headway outwards,

it's trying to swim upstream and failing against the faster-than-light cascade of space-time. Some of this light

might be from the collapsing surface of the star that first formed the black hole emitted long before we entered the event horizon.

It appears to come from below us because it's trying to climb upwards. In fact though,

it was emitted at larger radii than wherever we 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 holes past; or up, towards light from the black hole's future.

Those directions — those spatial freedoms — 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 towards 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. Below the event horizon

there's still a sense of spatial up-ness and down-ness,

however, the old radial dimension isn't space-like; it's time-like.

Every photon that reaches us was emitted at some larger radius than wherever we 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 world lines 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 Scwarzschild metric

really gives two separate space-time 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 space-time.

Thanks to Crunchyroll for sponsoring this episode.

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Again that's crunchyroll.com/spacetime

Hey guys. In our recent Space Time Journal Club we talked about a paper describing a recipe for making these weird things

called time crystals and about how some researchers have now successfully created them.

Let's continue the discussion.

A few of you are unsure of what constitutes a time crystal. At this point the use of the term is pretty loose.

It refers to any quantum system

whose internal interactions result in a periodic change from one state to another and then back again.

The systems that were tested use electron spins, which pull on each other to cause a

cascading flow of flipping spins that cycles back and forth.

Really though, the term time crystal is just used to refer to anything that has a pattern of internal states that repeats over time.

It doesn't also have to be a regular crystal that has a repeating spatial pattern. In fact it typically won't be.

Colin Brown asks if the spin flip oscillation is only dependent on the electromagnetic field oscillation

and he also asks why that's so special.

Firstly, yeah these time crystals

oscillate at an integer multiple of the electromagnetic field frequency, so the time crystal oscillation

and the EM field oscillation are in resonance. For every 1, 2, 3, et cetera

cycles of the time crystal, the EM field gives a little push.

It has to be an integer factor because if the EM field were pushing

halfway through the time crystal period it would be pushing in the wrong direction.

It's like when you're pushing a swing. You don't need to push every time to keep it going,

but you do need to push at the right time or you'll slow it down.

Colin and others also ask why this is so special. After all, lots of things oscillate,

especially when you push them. So in Frank Wilczek's initial idea his hypothetical time crystals didn't require any input energy

to keep them oscillating. For Wilczek's time crystals the oscillating state is an equilibrium state and the

oscillations were supposed to go on forever without any energy input.

That would be special and weird even if the original period was defined by an external EM field frequency.

It's now been proved mathematically that time crystals can't exist in equilibrium.

To keep them oscillating you need to keep putting in energy.

However the experimental results are still exciting because these systems did develop their own internal oscillations

that resisted changes from the outside, forcing EM field oscillation. So the oscillations were in a sense

fundamental, just not sustainable.

Is this more interesting than, say, a swinging pendulum?

Well pendulums are pretty cool, so I don't know.

They're a new type of oscillating system that could have their own uses.

I'd like to thank Dankulous Memelord for what for us

spurred a useful discussion on the comprehensibility of this episode.

We're always monitoring feedback to improve the clarity of the show

but it's worth commenting here that our goal on Space Time is a little bit different to most science media.

We're interested in providing a bridge to understanding the real science that goes a bit further than an introductory level.

And that's especially true of Space Time Journal Club. If you're finding an episode a bit much on first viewing

it's often a good idea to also check out some other sources which might give a better intro.

But please persist because understanding our universe is well worth all of the work.