Superluminal Time Travel + Time Warp Challenge Answer Script (Patreon)

Traveling faster than light and traveling backwards in time are the same thing. Today I’m going to prove that to you.

Faster than light travel is, understandably, a staple of science fiction. The reality of the vast scale of our universe – even of our galaxy – is … inconvenient for tales of star-hopping adventures or warring galactic empires. Enter the warp drive, and hyperspace, and stargates, and the infinite improbability drive. Plenty of ways to make traveling across impossible distances as challenging as a weekend roadtrip. And sci-fi makes instantaneous communication a breeze with ansibles, sophons, subspace relays, and tachyonic antitelephones. Actually, that last one is a thought experiment that demonstrates that if you communicate using tachyons – hypothetical FTL particles – it’s possible to receive a reply to your message before your message is sent.

By choosing the right path and the right reference frames, any superluminal motion can lead to information or objects returning to their origin before they depart. Today I want to show you how to navigate such a path. To do that, we’re going to need a map. We’ll do this in flat space, so we need a flat or Minkowski spacetime diagram. We’re going to add spacetime interval contours. We’ve spent some time talking about how these contours define the flow of causality. If you aren’t familiar, you should probably watch that episode first. In addition, today’s episode is going to answer the recent time warp challenge question, so spoiler alert.

In the olden days – before Albert showed us the way – time was thought of as universal – it was assumed that the entire universe exists simultaneously in a state of now, and that all points move forward in time at a constant rate for all observers governed by one global clock. In the olden days, the same time axis of a spacetime diagram would apply to everyone. But no longer! Einstein showed us that there’s no universal clock. Instead, every spacetime traveler carries their own clock. The tick rate of your clock and your perception of simultaneity depends on your velocity. There’s no absolute notion of velocity, so everyone can be considered motionless from their perspective. Everyone draws their spacetime diagram time axis parallel to their direction of motion because that’s their experience of stillness. The ticks marks on that time axis also depend on velocity and represent the speed of everyone’s personal clock – their proper time. And everyone also has a different space axis, representing chains of simultaneous events according to their perspective. Those axes reflect symmetrically around this 45-degree path, representing the unvarying speed of light.

Connect the ticks of all possible time axes and you get these nested hyperbolae. These are contours of constant spacetime interval. A straight-line journey to any location on one of these contours seems to take the same amount of time for every traveler. The spacetime interval is special because every traveler will agree on which contours a set of different events lie, even if they don’t agree on temporal ordering of those events. If we write the spacetime interval for flat space with a negative sign in front of the time part, then changes in your spacetime interval have to be negative as long as you travel at less than light speed – greater than 45 degrees on the diagram. Each contour is smaller than the one before and so forward temporal evolution means rolling down the causality hill. On the other hand, superluminal travel – paths at less than 45 degrees - means revisiting previous contours; traveling uphill. That uphill journey is equivalent to time travel. To prove it, let’s think about the scenario that I proposed in the recent challenge question.

You are in a race to claim a newly discovered exoplanet 100 light years away. Your competitor immediately launches a 50%-light-speed ship – the anti-matter powered Annihilator. You decide to wait, taking a century developing an Alcubierre warp drive. Your ship, the Paradox, can travel at twice the speed of light. Let’s see what that looks like on the spacetime diagram. We’ll plot the worldlines of these ships as recorded by someone waiting back on Earth. Earth doesn’t move from its own perspective. It just hangs out at x=0 and rolls upwards in time.

The Annihilator races off towards the exoplanet, 100LYs this way. At 50% light speed the Annihilator makes a 67.5 degree angle. At that speed it’ll take 200 years to reach its destination. Meanwhile, your own worldline remains on Earth as you build the Paradox. However when you launch you take a 22.5 degree path – twice as much space covered per time tick compared to light. You overtake the Annihilator at around the 67LY mark and reach your destination 150 years after the race began. Congrats, you win. I just hope your rejuvenation tank is still working.

But winning was never the question. I’m really curious about what the crew of the Annihilator see at that moment you pass them. Do they perceive you as traveling in time? And now that you’ve mastered FTL, can you pilot the Paradox back to a point before the race even started?

To see what the Annihilator sees, let’s transform the spacetime diagram to their perspective - in fact we need to do a Lorentz transformation. Their time axis is their own worldline, and their space axis is symmetrically reflected around the path of light. Now add hyperbolic spacetime interval contours. These are the spacetime intervals as calculated from the zero-point in space and time – the beginning of the race. Because the spacetime interval is invariant to Lorentz transformations, when we shift to the velocity frame of the Annihilator, we just make sure events stay on the contours they started on. The Annihilator perceives itself as stationary, and sees Earth racing away in the opposite direction at half light-speed, while its destination races towards it at the same speed.

We can figure out the Paradox worldline because we know which spacetime interval contours it’s on when it departs from Earth and arrives at its destination. It actually ends with the same spacetime interval that it started with – that’s already suspicious. So transforming this we get that the Paradox appears to travel its entire course in zero time according to the Annihilator’s local clock.  

What does this look like to the crew? Just trace photon paths, assuming for a moment that an FTL ship doesn’t produce infinitely red- or blue-shifted photons.  The Paradox outraces its own photons to the Annihilator, and then continues to emit photons backwards after it passes. The Annihilator sees a series of photons from both directions that arrive simultaneously. To its crew, the Paradox appears to materialize out of nowhere, and then proceeds split in two; one races onwards to its destination, the other travels in reverse towards Earth.

That looks a bit like time travel, but actually a physicist on board the Annihilator would infer that all points in the Paradox’s journey happened simultaneously. Let’s look at the perspective of a different spacetime traveler---one traveling at very near the speed of light. When we transform the diagram to their perspective, we see that the Paradox really does appear to travel backward in time according to this time axis. But this is just perspective, right? Well no, not if we can find a way to bring to Paradox back to a point in space before it was built. 

To do that we first need to outrace photons that were emitted at the spacetime point we want to precede: in this case the start of the race. Let’s return to the reference frame of the Earth to do this. We can keep flying the Paradox until we cross this ominous 45-degree boundary. Let’s fill in the spacetime diagram with all four quadrants. These ones represent the regions inaccessible for sub-light-speed travelers starting at the origin. Now we transform back to the near-light-speed reference frame. In that frame the Paradox has moved into a region that appears to be prior to the start of the race. If we assume that this trajectory is valid then there’s no limit to how far into the past we can travel. If we travel far enough, then when we finally turn the Paradox around its twice-light-speed movement will take it back to the beginning of the race, long before it was even built. 

This seems like a trick, and it sort of is. We constructed this time-traveling path using two different reference frames – in this case Earth’s and then a near-lightspeed frame. Normally that would be fine because spacetime events marking the different stages of a sub-light-speed journey transform consistently between these frames. However when we introduce FTL, things get messed up. Superluminal paths aren’t real worldlines. Real worldlines don’t point backwards in time under Lorentz transformations. While we can define a chain of events that looks like an FTL journey, these aren’t paths that objects can take. And that includes us. Remember, we are temporal creatures. Our experience of the universe is a thing that emerges from the forward causal evolution of the matter we are composed of. Reverse the flow of time and you reverse the flow of … you. Even our fantasies of time travel are just another pattern emerging from our one-way path through spacetime.