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  • 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 perspectiveswhich, 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 thingssure, 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 secondslike, you, moving a third the speed of light to my left, will

  • say I left it on for 4.24 secondsthat's time dilation.

  • However, this is where the spacetime pythagorean theorem comes init'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 timewe 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 apartat 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 calledegalitarian distanceandegalitarian time”.

  • But sadly no, these true distances and times are typically calledproper lengthand

  • proper time”.

  • And the spacetime pythagorean theorem, because it combines intervals in space and time together,

  • has the incredibly creative namespacetime 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 calledthe 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.

I started this series saying that relativity is about understanding how things look from

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B1 pythagorean pythagorean theorem theorem spacetime length relativity

Spacetime Intervals: Not EVERYTHING is Relative | Special Relativity Ch. 7

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    林宜悉 posted on 2020/03/28
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