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 with our galaxy, is inconvenient for tales
of star-hopping adventure or warring galactic empires.
Enter the warp drive, and hyperspace,
and star gates, 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 have ansibles, sophons, subspace relays,
and tachyonic antitelephones.
Actually, that last one is a thought experiment
that demonstrates that if you communicate using tachyons,
hypothetical faster-than-light or superluminal particles,
it's possible to receive and reply to a 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.
We'll do this in flat space, so we need a flat or Minkowski
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 add to 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 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 depend on velocity
and represent the speed of everyone's personal clock
in 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 proper 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 the 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 space interval
have 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,
paths 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
that I proposed in the recent challenge question.
It 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 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 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 0 and rolls upward in time.
The Annihilator races off towards the exoplanet,
100 light years this way.
At 50% lightspeed, 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've reached the exoplanet
in 50 years from launch date, 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 a Lorentz transformation.
Their time axis is their own world line,
and their space axis is symmetrically reflected
around the path of light.
Now, add hyperbolic spacetime into our 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
that 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 world line
because we know which spacetime interval contours
it's 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
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 Annihilator's own time
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 admitted at the space time point
that we want to perceive.
In this case, it's 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
Let's fill in the space time 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 lightspeed reference
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, it's twice lightspeed 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-lightspeed frame.
Normally, that would be fine, because spacetime events
marking the different stages of a sub-lightspeed journey
transform consistently between these frames.
However, when we introduce faster-than-light travel,
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 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 spacetime.
Thanks to everyone who submitted answers
to the timewarp challenge.
If you see your name below, we randomly
selected your correct answer to win a spacetime t-shirt.
Shoot us an email at PBSspacetime@gmail.com with
your mailing address, US t-shirt size--
small, medium, large, et cetera--
and let us know which of these T's you'd like.
Also, it's that time of year again,
time for the annual PBS Digital Studios survey.
This is not a graded quiz, nor will it
tell you which Disney princess you are,
but it will really help both Spacetime and PBS figure out
what you guys are into and what you want for the future.
Click on the link in the description
to fill out the survey.
It should only take about 10 minutes,
and we'll be giving away PBS shirts to 25 randomly selected