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#### 狭义相对论#3：洛伦兹变换

Lorentz Transformations | Special Relativity Ch. 3

The goal of relativity is to explain and

understand how motion looks from different perspectives,

and in particular, from different moving perspectives.

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

right every second.

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

evidence gathered by actually moving.

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 spacetimediagrams: 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 thatpasses 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,

and thus it’s not your perspective.

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

coincides with the time axis.

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 differentlocations. 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 other.

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 moving.

So that’s the first piece of experimental evidence.

The second piece of evidence we’ll come tolater.

Let’s focus just on the section of the cat’s worldline from time t=0,

where it’s at x=0,

to t=4, where it’s at x=2.

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 possibilitiesfor 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 picturethese 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 why.

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 thetime axis.

I mean, we don’t noticeably experience

time travel every time we hop on a train or bike

or plane.

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

half a meter every second.

It works for other speeds, too.

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

a half per second to the right.

This kind of sliding change

of perspective is normally called a “ shear transformation, ”

but that’s when both dimensions are spacedimensions: since one of our dimensions is time,

a shear transformation represents a change

in the velocities of things, so in

physics it’s called a “boost.”

As in, rocket boosters boosting you to a higherspeed. However,

it turns out

that boosts in the physical universe are not actually described by shear

transformations.

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

of light.

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 movingperspective.

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 relativeto the ground.

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

light from a standstill, or from a moving vehicle,

and its measured speed relative to

the ground will be the same.

Shear transformations simply can’t accomodatethis 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

is wrong. 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 earliertimes?

This kind of transformation can amazingly leave one speed unchanged,

even while it changes

all other speeds.

More amazingly, the unchanged speed is leftunchanged in all directions.

Let’s do an example.

Here’s a set of snapshots from my perspective

with a slow-moving sheep and two fast-moving cats,

and let’s suppose that we have experimental evidence

that cats always move at the same

speed regardless of perspective.

If we want to describe this situation from the perspective

of the sheep, we can’t simply

slide the snapshots

over so the sheep isn’t moving and its worldline coincides with the

time axis, since that would change the speedof the cats. But,

if we slide and rotate and stretch the snapshots

like this, then look – we’ve transformed

the diagram to both describe things

from the sheep’s perspective and keep the cats moving

at the same speed they were before.

You might note that the various cats appear

to be spaced out differently along their worldlines,

but that just means that the constant-time snapshots

from my perspective aren’t constant-time

snapshots from the sheep’s perspective.

The important thing is

that the angle of the cats’ worldlines – which represents their

speed – has remained unchanged.

It’s kind of amazing to me that this works at all;

that it’s mathematically and physically

possible for all speeds except one to change!

But it is possible with these squeeze rotationy things,

and they’re the answer to our question

of how to describe motion from a moving perspective. Well,

not by keeping the speed of cats constant,

but by keeping the speed of light constant:

by doing squeeze rotations so

that a moving perspective’s angled worldline becomes vertical

without changing the speed of light – that is,

without changing the slope of the worldlines

for light rays.

These squeeze rotationy things are called Lorentz Transformations,

named after one of

the first people to derive the correct mathematical expression

for them – it looks kind of like

the equation for rotations that we saw in the last video,

and I’ll post a followup video

showing how to derive this using just a few simple assumptions and experimental facts.

Lorentz Transformations are at the heart

of special relativity – they’re the thing that

Lorentz and Einstein and Minkowski and

others figured out was the correct description of

how motion looks from moving perspectives in our universe,

and they’ll be the foundation

of the rest of this series, too. Now,

Normally, physicists draw their spacetimediagram tickmarks such that if every vertical

tickmark represents one second, a horizontaltickmark represents 299,792,458 meters, which

means that the speed of light,

which is 299,792,458 meters per second, is drawn as a 45° line

– to the right for right-moving light,

and to the left for left-moving light.

With this scaling,

a Lorentz Transformation that leaves the speed of light constant simply

consists of squeezing everything

along one 45° line and stretching along the other in

a particular, proportional way.

You can see immediately how this changes the angles

of all of the other worldlines, that is,

changes how we perceive their speeds,

and yet doesn’t change any of the light rays.

And it turns out

that it’s possible to actually build a mechanical device that does Lorentz

Transformations for you: here it is!

Just like how a globe has the structure

of rotations built into it in a fundamental way,

and you can simply turn the globe to see how rotations work,

rather than doing a lot of

complicated math, this spacetime globe hasLorentz Transformations built in: it does

the math of special relativity for you,

allowing you to focus on understanding the physics

of motion from different perspectives!

Here’s a quick example:

from my perspective, I’m always at the same position as time passes,

while the cat is moving away from me to the right

at a third the speed of light, and the

light rays from my lightbulb are moving out to the right and left.

Using the time globe,

I can do a Lorentz transformation to boost into the cat’s perspective.

And from the cat’s perspective,

the cat – naturally – stays at the same position as time passes,

while the cat views me as moving away from it

at a third the speed of light to the left,

and the speeds of the light rays from my lightbulb are still the same,

still at 45° angles.

I just love how tangible and hands-on this

is – normally when people are first introduced

to special relativity and how motion looks

from different perspectives, it’s done with

a bunch of messy, incomplete,

algebraic equations – but you don’t need the equations to understand

the ideas of special relativity and how motion looks from different perspectives.

You just need an understanding of spacetime diagrams, and a time globe.

And so in the rest of this series,

I’m going to be using the time globe extensively to

dive into all of the normally confusing things you’ve heard

time dilation, length contraction, the twins paradox,

relativity of simultaneity, why you

can’t break the speed of light, and so on.

I have to say a huge thank you

to my friend Mark Rober for helping actually make the time

globe a reality ( you may be familiar

with his youtube channel where he does incredible

feats of engineering,

like this dartboard that moves so you always hit the bullseye ).

He devoted a huge amount of time, effort,

and engineering expertise to turn my crazy

idea into this beautiful, precision,

hands-on representation of special relativity and I’m

supremely indebted to him – this serieswouldn’t be possible otherwise.

And if you’re eager for more details,

I’m planning another whole video about the time

globe itself.

In the meanwhile, to get more hands-on

with the math of special relativity, or economics,

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