Name: 時空図｜特殊相対性理論 第2章 (Spacetime Diagrams | Special Relativity Ch. 2)
Uploaded: 2021-01-14T10:19:33.000Z
Duration: 14 min 31 s
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Physics is all about the motion of things – how planets and stars move, how electrons

and protons move, how the movement of molecules results in emergent properties like temperature,

The role of relativity, in physics, is to study how that motion looks from different

Here I'm using “relativity” in a general sense, to mean from any different possible

perspective, moving, accelerating, or otherwise; special relativity in particular is concerned

with how motion looks just from a limited, or “special,” set of perspectives.

But either way, relativity (special or not) is about how the motions of things look from

Like, if you were looking at the earth and the moon, depending on where you were and

how you were moving, it might look like the moon is moving around the earth in a giant

circle, or back and forth on a straight line, or that the earth and moon together are tracing

But if the motion of the earth and moon can be described in such different ways, what

does any one of these descriptions actually tell us about the earth and moon?

Is one of them “right” and the others “wrong”?

Is there some preferred perspective for observing the earth and moon that gets closer to the

It's the goal of relativity to answer these kinds of questions.

In fact, relativity can essentially be summed up as two basic ideas:

To figure out how objects and their motion look from different perspectives, and

To notice which properties of objects and motion don't look different from different

We've already given an example of number 1, with different ways the motion of the earth

and moon can look from different perspectives.

Number 2, the idea of finding things that don't look different from these perspectives

In the earth and moon case, for example, all three perspectives appear quite different.

But after a while, you might notice that regardless of the perspective, the maximum physical distance

between the earth and the moon appears to be the same.

There's something that's independent of perspective!

Maybe it's a fundamental property of the earth-moon system, and not just an artifact of my particular

And this is why relativity is so important in physics: by studying what changes and what

doesn't about a physical system when you change your perspective, you are zeroing in on universal

Facts that remain true from many perspectives throughout the universe (like, perhaps that

the distance between the earth and the moon is 384399 kilometers), are literally more

universal than a fact that only holds true at a single place and time (like, that the

angle between the moon and earth is 150 degrees).

Relativity is a way of thinking that helps you to evaluate how universal a given truth

To make all this tangible, we need a rigorous way of describing moving things and of describing

changes to how that motion looks when you change your perspective.

We're ultimately going to build up to special relativity, which has to do with motion over

time, but we'll start with non-moving things just to get a sense of how intuitive relativity

You're probably familiar with specifying the position of a cat on a plane using xy coordinates:

this cat is three tick marks to the right of our point of reference, and two tick marks

up, so we say it's at position x=3 and y=2, which typically gets written as just a pair

However, (3,2) is not a universal truth – I mean, it's just based on where I'm standing,

But over here, where you are, maybe you're rotated by 30 degrees, and you made the tick

marks closer together, and suddenly the cat is at a different position: x=9, y=9.

In fact, it's possible to specify the cat's position using any x and y values we want,

depending on our point of reference: which corresponds mathematically to where we put

our axes and how we orient and scale them.

So clearly, specifying the position of something is not a universal truth.

Or, in relativity parlance, “position is relative.”

A more universal, or absolute, truth can be found if you have two cats: let's say they're

And I'm going to stop drawing a person at the origin point of the axes from now on,

but you should remember that the axes we use represent a particular perspective and orientation

Ok, so The distance between these two cats is clearly 5 – they're at the same y value

If we move and rotate our point of reference now, the cats are at positions... uh, x=1,

So they differ by 4 in the new x direction and 3 in the new y direction.

But the overall distance between the cats, which we can find using the pythagorean theorem,

is the square root of 4 squared plus 3 squared, which is the square root of 25 which is 5.

Which is the same distance we calculated with the original axes!

This turns out to be a general truth: on a plane, the distance between two things doesn't

change if you change your perspective just by shifting your point of reference or your

I like to think about this as similar to how if I take a piece of paper, and slide it around

and rotate it, I haven't actually changed anything on the piece of paper.

Or, in relativity parlance, “distances are absolute.”

The geometric intuition for this is that you can move your axes around, slide them up and

down, and rigidly rotate them, without affecting your description of the distance between two

If you like, we can make this mathematically precise by calling the original coordinates

x and y, and the new coordinates x new and y new; then when we've slid the x axis an

amount Delta x (technically called a “translation by Delta x”), we say that x new=x-Delta

x, and when we slide the y axis by an amount Delta y (technically called a “translation

by Delta y”) we say that y new=y-Delta y.

The minus sign is there because if you slide your origin point closer to something, its

Changes of orientation are a little fancier, but it's really just some geometry: if you

re-orient the x and y axes counterclockwise by an angle theta, the new coordinates look

like x new=x times cos theta minus y times sin theta and y new = y times cos theta plus

If you want a fun algebra exercise, you can use these equations (or even their 3D counterparts!)

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時空図｜特殊相対性理論第2章 (Spacetime Diagrams | Special Relativity Ch. 2)

perspective

ultimately

stick

assume

時空図｜特殊相対性理論 第2章 (Spacetime Diagrams | Special Relativity Ch. 2)

perspective

ultimately

stick

assume

時空図｜特殊相対性理論第2章 (Spacetime Diagrams | Special Relativity Ch. 2)