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.