Name: ローレンツ変換｜特殊相対性理論 第3章 (Lorentz Transformations | Special Relativity Ch. 3)
Uploaded: 2021-01-14T10:19:26.000Z
Duration: 12 min 18 s
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The goal of relativity is to explain and understand how motion looks from different perspectives,

and in particular, from different moving perspectives.

程度の副詞

It's easy enough to describe motion itself – if something is moving relative to me,

that means it has different positions at different times, which I can plot on a spacetime diagram.

This straight line corresponds to motion at a constant velocity of say, v units to the

And the question we're interested in is what do things look like from the moving perspective?

Of course, the answer to this question is a physical one, and is determined by experimental

And that evidence will come into play, but first we need to understand what it means,

in terms of spacetime diagrams, to view something from a moving perspective.

We'll start with a key property of spacetime diagrams: when someone draws a spacetime diagram

from their own perspective, on that diagram they're always, for all time, located at position

x=0, since they're always a distance of 0 away from where they are.

Or in other words: a spacetime diagram like this represents your perspective only if your

worldline is a straight vertical line that passes through x=0.

If, on a spacetime diagram, the worldline describing your motion leaves x=0 and goes

anywhere else, that means you're moving relative to the perspective of that particular diagram,

With this in mind, to describe how things look from the perspective of a moving object,

like this cat, we simply need some way to transform spacetime diagrams that makes the

worldline of the cat into a straight vertical line through x=0; or in other words, we want

to make the spacetime diagram where the cat is moving into one where the cat's worldline

That's not something we can do just by sliding the whole plot left or right or up or down,

like we've done for perspectives from different locations.

No, changes of velocity require some sort of rotationy thing to change the angle of

the worldline, and importantly, whatever this rotationy thing does should be generalizable

to a world line at pretty much any angle, since there was nothing special about the

particular speed the cat happened to be going.

There are also two important pieces of experimental evidence that we'll need to take into account:

first, if I measure the cat as moving at a speed v away from me, then the cat will measure

me as moving at that same speed v away from it, and likewise if we're moving towards each

Which means we not only want to transform the spacetime diagram in a way that the cat's

angled line becomes vertical, but we also want the angle between our two lines to stay

the same after the transformation – that is, from the cat's perspective, I should be

The second piece of evidence we'll come to later.

Let's focus just on the section of the cat's worldline from time t=0, where it's at x=0,

This section is a straight line between those two points, and we want it to end up as a

straight vertical line, so we can simply leave the t=0,x=0 point unchanged while moving the

t=4,x=2 point onto the time axis (where x=0).

And there are really only three general possibilities for how to do this: either this point gets

moved onto the time axis while keeping it at the same point in time, t=4, or it gets

moved onto the time axis at an earlier time (say, t=3), or a later time (like t=5).

There's a very nice geometric way to picture these possibilities.

If we think again of motion on a spacetime diagram as a series of snapshots, like, at

time t=0 the cat is at position 0, at time t=1 the cat is at position 0.5, at time t=2

the cat is at position 1, etc, then the transformation where points move to the time axis and keep

the same time just looks like sliding each snapshot over a corresponding amount; the

possibility where points move to the time axis at a later time looks kind of like some

sort of rotation around the origin; and the possibility where points move to the time

axis at an earlier time looks kind of like some sort of squeezy rotation.

The reason these last two involve rotating the snapshots rather than just sliding is

to make sure that the angle between the cat's worldline and my worldline stays the same

before and after the transformation – it's a fun little geometry puzzle to understand

Now, among these three, the option that makes the most intuitive sense based on our everyday

experiences of the passage of time, is that a given point in time should stay at the same

point in time, and just slide over to the time axis.

I mean, we don't noticeably experience time travel every time we hop on a train or bike

And this sliding does mathematically work – if we move things at time t=1 a half meter

to the left, and things at time t=2 one meter to the left, and so on, then we'll have a

description from the cat's perspective – the cat's not moving, and I'm moving to the left

If we want the perspective of somebody who's going a meter per second to the right relative

to the cat, we can slide the snapshots over even farther, and now the cat's going a meter

per second to the left, and I'm going a meter and a half per second to the left.

And of course we can slide back to my perspective from which the newcomer is going a meter and

This kind of sliding change of perspective is normally called a “shear transformation,”

but that's when both dimensions are space dimensions: since one of our dimensions is

time, a shear transformation represents a change in the velocities of things, so in

As in, rocket boosters boosting you to a higher speed.

However, it turns out that boosts in the physical universe are not actually described by shear

This is where the second and most famous piece of experimental evidence comes in: the speed

As you've probably heard, starting in the late 1800s, physicists built up mountains

of experimental and theoretical evidence that the speed of light in a vacuum is always the

same, even if you measure it from a moving perspective.

This is, of course, entirely unintuitive from our everyday experiences with velocities,

where if you throw a ball from a standstill and then from a moving vehicle, the ball thrown

from the vehicle will be moving faster relative to the ground.

And yet, experimental results show that light does not behave like everyday objects: shine

light from a standstill, or from a moving vehicle, and its measured speed relative to

Shear transformations simply can't accomodate this feature of light's behavior: they change

all velocities equally by sliding each snapshot an amount proportional to its time.

No velocity remains unchanged – if you draw the worldline of a light ray and then change

to a moving perspective using a shear transformation, the speed of that light ray will change, which

Luckily, one of the other two options for boosting to a moving perspective can accomodate

a constant speed of light: remember the transformation where the snapshots do a kind of squeeze rotation,

and points move to the time axis at earlier times?

字幕リスト動画再生

ローレンツ変換｜特殊相対性理論第3章 (Lorentz Transformations | Special Relativity Ch. 3)

perspective

sort

perceive

constant

ローレンツ変換｜特殊相対性理論 第3章 (Lorentz Transformations | Special Relativity Ch. 3)

perspective

sort

perceive

constant

ローレンツ変換｜特殊相対性理論第3章 (Lorentz Transformations | Special Relativity Ch. 3)