 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
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 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
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 Annihilator's 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 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
degree boundary.
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
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, 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
participants.
