Race to a Habitable Exoplanet - Time Warp Challenge Script (Patreon)

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 FLT 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 from 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 observer’s reference frames – that’s using the Lorentz transformation. It lets us figure out what spacetime looks like for every observer, no matter his or her velocity. If two events happen in spacetime, observers with different velocities will report different separations between them, in both space and time. But we can combine those space and time intervals into the spacetime interval. EVERYONE agrees on ITS value.

Written this way, the spacetime interval is always negative for slower-than-light movement between two events. In fact, every step along the worldline must decrease the spacetime interval. If we represent lines of constant spacetime interval  with respect x and t=0 as contours, then we can transform the spacetime diagram into a 3-D 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 king, country, or their own selves. An evil interplanetary corporation is intent on strip-mining the planet to oblivion. They launch the fastest ship that exists – the Annihilator – powered by an anti-matter drive capable of reaching 50% light speed. Eager to save this world – or perhaps to expand your own galactic holdings, you resolve win the race. You don’t have a faster ship, but you do have the “wait equation”. It tells us that sometimes it’s better not to launch immediately, but wait until tech advancements allow a faster this 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 paint the name Paradox on its gleaming hull. With the Annihilator still only half way 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 worldliness 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’ll be helpful to draw the hyperbolic spacetime interval contours on your diagram. That’ll let you transform between those reference frames. For example, if this is the worldline from Earth’s perspective of the first part of the Annihilator’s journey during the construction of the Paradox, then they’d transform to the Annihilator’s perspective like this. Note that the end-points of the worldlines 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 me to fly the Paradox all the way back the beginning of the race – the beginning in both space and time, at the moment the Annihilator is launched. Show that the time travel that FTL allows isn’t just an illusion – it should allow travel into the past.

Submit your answers with full explanations and spacetime diagrams within two weeks of release of this video to be in the running. A selection of correct submissions will receive a spacetime t-shirt and a habitable exoplanet. Email responses to psbspacetime@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.