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. Video source: https://www.youtube.com/watch?v=WFAEHKAR5hU
