If you travel faster than light, you can travel backwards in time.

In fact, breaking the cosmic speed limit doesn't just allow time to be reversed.

It demands it.

Any FTL journey will appear to someone, somewhere in spacetime, as time travel.

In fact any FTL ship set on the right trajectory can find its way back in time to a point before its journey began.

In today's challenge question, I propose a race-- a race to save a nearby exoplanet.

You're going to show me how you win that race.

And in the process, you'll prove to yourself that FTL travel is time travel.

This question will draw a lot on our recent episode on the geometry of causality.

Check it out.

You're going to need the tools I give you there.

In that episode, we talked about the spacetime diagram and how it transforms between observers traveling at different speeds.

We talked about how the constant or invariant nature of the speed of light governs that transformation.

In fact, there's only one possible way to map between different observers' reference frames.

That's using the Lorentz transformation.

It lets us figure out what spacetime looks like for every observer, no matter what his or her velocity is.

If two events happen in spacetime, observers with different velocities will report different separations between them, in both space and time.

We can combine those space and time intervals into the spacetime interval.

Everyone agrees on its value.

If we represent lines of constant spacetime interval with respect to x and to equal zero as contours, then we can transform the spacetime diagram into a 3D graph, in which causality must always flow downhill.

To increase your spacetime interval, to cross these contours backwards uphill, you must travel faster than light.

But why does this have to mean traveling back in time?

Let's have a race to find out.

A race to a newly found habitable planet 100 light years away.

Whoever first reaches this temperate terrestrial world gets to claim it for a king country, or their own selves.

An evil interplanetary corporation is intent on strip mining the planet to oblivion.

They launched the fastest ship that exists, the Annihilator, powered by an anti-matter drive capable of reaching 50% light speed.

Eager to save the world, or perhaps to expand your own galactic holdings, you resolve to win the race.

You don't have a faster ship, but you do have the wait equation.

It tells you that sometimes it's better not to launch immediately, but to wait until tech advancements allow a faster ship to be built.

So you start work on an Alcubierre warp drive.

It takes you 100 years to build a ship capable of traveling at twice the speed of light.

Happily, medical technology developed faster, and you live long enough to paint the name Paradox on its gleaming hull.

With the Annihilator still only halfway to its destination, the Paradox will easily overtake it and win the race.

My challenge to you.

What does the captain of the Annihilator see at the moment the Paradox overtakes?

To answer this, you'll need to draw a spacetime diagram showing the world lines of the two ships.

I recommend you start from the perspective of someone waiting around back at Earth.

When you transform the diagram to the perspective of the Annihilator, the paradox should suddenly appear to act like a time machine.

It will be helpful to draw the hyperbolic spacetime interval contours on your diagram.

That'll let you transform between reference frames.

For example, if this is the world line from Earth's perspective of the first part of the Annihilator's journey during the construction of the Paradox, then they transform to the Annihilator's perspective like this.

Note that the endpoints of the world lines stay on the same contours, because their spacetime intervals don't change.

I also have an extra credit question.

Find a spacetime trajectory that allows you to fly the Paradox all the way back to the beginning of the race, the beginning in both space and time at the moment the Annihilator is launched.

Show that the time travel implied by FTL isn't just an illusion.

It should actually allow travel into the past.

Submit your answers with full explanations and spacetime diagrams within two weeks of the release of this video to be in the running.

A selection of correct submissions will receive a Space Time T-shirt and a habitable exoplanet.

Email responses to pbsspacetime@gmail.com, and make sure you use the subject line time warp challenge, because we filter by subject line.

See you next week for a new episode of Space Time.