I started this series saying that relativity is about understanding how things look from
different perspectives, and in particular, understanding what does and doesn't look different
from different perspectives.
And at this point you'd be justified to feel like we've kind of just trashed a bunch of
the foundational concepts of physical reality: we've shown how our perceptions of lengths
and spatial distances, time intervals, the notion of simultaneous events, and so on,
are not absolute: they're different when viewed from different moving perspectives, and so
aren't universal truths.
And if we can't agree on the length of something, what can we agree on?
Anyway, the point is, this relativity thing so far kind of just feels like it's leaving
us hanging.
I mean, all we've really got is the fact that the speed of light in a vacuum is constant
from all perspectives – which, while it's true, doesn't feel nearly as helpful in describing
objects and events the way that lengths and times are.
Luckily, there is a version of length and time intervals that's the same from all moving
perspectives, the way the speed of light is.
You know how if you have a stick that's 10 meters long and you rotate it slightly and
measure its length, it won't be 10 meters long in the x direction any more - it'll be
shorter?
Now, if you know some math you'll tell me it's not actually shorter, and you can still
calculate its true length using the pythagorean theorem as the square root of its horizontal
length squared plus its vertical length squared.
And yes, this is the case.
You can use the pythagorean theorem to calculate the true length of the stick regardless of
how it's rotated.
But you don't need to use the pythagorean theorem at all - if you just rotate the stick
back so that it's a hundred percent lying in the x direction, then you just measure
it as 10 meters long and that's that.
No pythagorean theorem necessary.
In some sense, this is what gives us justification to use the pythagorean theorem to calculate
the length of rotated things – sure, it's important that the pythagorean theorem always
gives the same answer regardless of the rotation, but it's critical that it agrees with the
actual length we measure when the object isn't rotated.
And it turns out there's a version of the pythagorean theorem for lengths and times
in spacetime that allows us to measure the true lengths and durations of things - the
lengths and durations they have when they're not rotated.
Except, as you know from Lorentz transformations, rotations of spacetime correspond to changes
between moving perspectives, so true length and true duration in spacetime correspond
to the length and duration measured when the object in question isn't moving - that is,
true length and true time are those measured from the perspective of the object in question.
For example, suppose I'm not moving and I have a lightbulb with me which I turn off
after four seconds.
As we know, any perspective moving relative to me will say I left my lightbulb on for
more than four seconds – like, you, moving a third the speed of light to my left, will
say I left it on for 4.24 seconds – that's time dilation.
However, this is where the spacetime pythagorean theorem comes in – it's like the regular
pythagorean theorem, but where instead of taking the square root of the sum of the squares
of the space and time intervals, you take the square root of their difference (\sqrt{\Delta
t^{2}-\Delta x^{2}}).
Now we need a quick aside here to talk about how to add and subtract space and time intervals
from each other – I mean, one is in meters and the other seconds, so at first it seems
impossible to compare them to each other.
But in our daily lives we directly compare distances and times all the time – we say
that the grocery store is five minutes away, even though what we actually mean is that
it's 1 km away; it just takes us 5 minutes to bike 1 km, so we use that speed to convert
distance to time.
In special relativity, however, we convert not with bike speed but with light speed - that
is, how long it would take light to go a given distance.
For example, light goes roughly 300 million meters in one second, so a light-second is
a way to compare one second of time with one meter (and second is WAAAAAAY bigger!).
So, back in our example situation, where from my not-moving perspective I had my lightbulb
on for 4 seconds - from your perspective it was on for 4.24 seconds before I turned it
off, in which time I had traveled 1.4 light-seconds to your right.
And the spacetime version of the pythagorean theorem simply tells you to square the time,
subtract the square of the distance (measured using light-seconds), and take the square
root of the whole thing.
Voilá - 4 seconds!
We used observations from your perspective to successfully calculate the true duration
I had my light on - the duration that I, not moving, experienced.
And it works for any moving reference frame.
Here, from a perspective in which I'm moving 60\% the speed of light to the right, I left
my lightbulb on for 5 seconds, during which time I moved 3 light seconds to the right.
Square the time, subtract the distance squared, take the square root, and again, we've got
4 seconds: the true, proper duration of time for which my lightbulb was on.
This all works similarly for true, proper lengths, too: here are two boxes that spontaneously
combust 1200 million meters apart – at least, it's 1200 million meters from my perspective,
in which the boxes aren't moving.
From your perspective, in which the boxes and I are moving a third the speed of light
to the right, the distance between the combusting boxes is now 1273 million meters, and the
time between when they spontaneously combust is now 1.41 seconds, which converts, using
the speed of light, to 425 million meters.
We're again ready for the spacetime pythagorean theorem: square the distance, subtract the
square of the time (measured in light-meters), and take the square root of the whole thing
to get...
you guessed it, 1200 million meters.
Specifically, what we just did was use Lorentz-transformed observations from your perspective to calculate
the true distance between the boxes from their (and my) perspective.
And it would work from any other moving perspective, too.
The bottom line is that in special relativity, while distances and time intervals are different
from different perspectives, there is still an absolute sense of the true length and true
duration of things that's the same from everyone's perspective: anyone can take the distances
and times as measured from their perspective and use the spacetime pythagorean theorem
to calculate the distance and time experienced by the thing whose distance or time you're
talking about.
Perhaps it should be called “egalitarian distance” and “egalitarian time”.
But sadly no, these true distances and times are typically called “proper length” and
“proper time”.
And the spacetime pythagorean theorem, because it combines intervals in space and time together,
has the incredibly creative name “spacetime interval”.
But don't let that get you down: spacetime intervals allow us to be self-centered and
lazy!
Spacetime intervals allow fast-moving people to understand what life is like from our own,
non-moving perspectives.
The astute among you may have noticed that there was some funny business going on regarding
whether or not we subtracted distance from time or time from distance - the short story
is that it just depends on whether you're dealing with a proper length or a proper time.
The long story is an age-old debate about what's called “the signature of the metric”.
And if you want practice using proper time and spacetime intervals to understand real-world
problems, I highly recommend Brilliant.org's course on special relativity.
There, you can apply the ideas from this video to scenarios in the natural world where special
relativity really affects outcomes, like the apparently paradoxical survival of cosmic
ray muons streaming through Earth's atmosphere.
The special relativity questions on Brilliant.org are specifically designed to help you go deeper
on the topics I'm including in this series, and you can get 20% off of a Brilliant subscription
by going to Brilliant.org/minutephysics.
Again, that's Brilliant.org/minutephysics which gets you 20% off premium access to all
of Brilliant's courses and puzzles, and lets Brilliant know you came from here.