Name: 時空間。すべてのものが相対的ではない｜特殊相対性理論 第7章 (Spacetime Intervals: Not EVERYTHING is Relative | Special Relativity Ch. 7)
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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

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

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

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

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.

You can use the pythagorean theorem to calculate the true length of the stick regardless of

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

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

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

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

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

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,

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

Perhaps it should be called “egalitarian distance” and “egalitarian time”.

But sadly no, these true distances and times are typically called “proper length” and

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

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時空間。すべてのものが相対的ではない｜特殊相対性理論第7章 (Spacetime Intervals: Not EVERYTHING is Relative | Special Relativity Ch. 7)

perspective

stick

critical

debate

時空間。すべてのものが相対的ではない｜特殊相対性理論 第7章 (Spacetime Intervals: Not EVERYTHING is Relative | Special Relativity Ch. 7)

perspective

stick

critical

debate

時空間。すべてのものが相対的ではない｜特殊相対性理論第7章 (Spacetime Intervals: Not EVERYTHING is Relative | Special Relativity Ch. 7)