[Music] traveling faster than light and travelling 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 adventure 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 road trip and sci-fi makes instantaneous communication a breeze we are answerable so fons subspace relays and tachyonic anti telephones actually that last one is a thought experiment that demonstrates if you communicate using tachyons hypothetical faster than light or superluminal particles it's possible to receive a reply to your message before the message is sent by choosing the right path and the right reference frame 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 will do this in flat space so we need a flat or minkovski space-time diagram we're going to add space-time 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 timeless 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 at the same time access to the space-time diagram would apply to everyone but no longer Einstein showed us that there is no universal clock instead every space time 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 space-time diagram time axis parallel to their direction of motion because that's their experience of stillness the tick marks on that time axis also depends 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 hyperbola these are contours of constant space-time interval a straight-line journey to any location on one of these contours seems to take the same amount of proper time for every traveller the space-time interval is special because every traveller will agree on which contours a set of different events lie even if they don't agree on the temporal ordering of those events if we write the space-time interval for flat space with a negative sign in front of the time path then changes in your space-time interval has to be negative as long as you travel at less than the speed of light so 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 paths powers at less than 45 degrees mean revisiting previous contours traveling uphill that uphill journey is equivalent to time travel to prove it let's think about the scenario I proposed in a recent challenge question there went something like this you're in a race to claim a newly discovered exoplanet 100 light-years away your competitor immediately launches a 50% Lightspeed ship the antimatter powered Annihilator you decide to wait taking a century developing and alcubierre's 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 will plug the world lines 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 equals zero and rolls upward in time the Annihilator races off towards the exoplanet a hundred light-years this way at 50% light speed the Annihilator would take 200 years to reach its destination from Earth's perspective meanwhile your own world line remains on earth as you build the paradox however when you launch you travel at twice the speed of light and so you reach the exoplanet in 50 years from launch day also from Earth's perspective you overtake the Annihilator at around the 67 light year mark and finish the race 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 sees at that moment you pass them do they perceive you as traveling in time and now that you've mastered faster-than-light travel can you pilot the paradox back to a point before the race even started to see what the Annihilator sees let's transform the space-time diagram to their perspective in fact we need to do and Lorentz transformation their time axis is their own world line and their space axis is symmetrically reflected around the path of light now ad hyperbolic space-time into a contours these are the space-time intervals as calculated from the zero point in space in the beginning of the race because the space-time 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 that they started on the united perceives itself as stationary and seize earth racing away in the opposite direction at half light speed while its destination races towards you at the same speed we can figure out the paradox worldlines because we know which space-time interval contours is on when it departs from Earth and arrives at its destination the paradox still appears to be traveling forward in time with respect to the Annihilator even though it's traveling faster than light but what does this look like to the captain of the Annihilator well you just trace the 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 as it catches up to the Annihilator and then it continues to emit light backwards behind it after it passes so the Annihilator sees a series of photons coming from both directions that arrive simultaneously the paradox appears to materialize out of nowhere and then proceeds to split in two one paradox seems to race onwards towards its destination while the other travels in Reverse back towards Earth now that looks a bit like time travel but a physicist on the annihilator would still infer that the paradox is moving forward in time upwards according to the annihilators own time axis however there are perspectives where time travel seems real let's look at the perspective of a different space time 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 backwards in time according to this new time axis but this is just perspective right well no not if we can find a way to bring the 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 space-time point that we want to proceed in this case is the start of the race let's return to the reference frame of the earth students we can keep finding paradox until we cross this is ominous 45-degree boundary let's fill in a spacetime diagram with all four quadrants these ones represent the regions inaccessible for sublight 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 how far in the past we can travel if we travel far enough then when we finally turn the paradox around it's twice light speed movement will take us back to the beginning of the race long before the paradox was ever 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 light speed frame normally that will be fine because space-time events marking the different stages of the sublight speed journey transform consistently between these frames however when we introduce faster than light travel things get messed up superluminal paths aren't real world lines real world lines don't point backwards in time under Lorentz transformations well we can define a chain of events that looks like an FTL journey these aren't paths that real 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 that we're 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 trajectory through the temporal part of space-time Thanks to