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 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, 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 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 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. 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, 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 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 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 the

time 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 space

dimensions: 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 higher

speed. 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 piece

of 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 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 the ground will be the same. 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 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 earlier

times? This kind of transformation can amazingly

leave one speed unchanged, even while it changes all other speeds. More amazingly, the unchanged speed is left

unchanged 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 speed

of 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, as we’ve seen, Lorentz transformations

look different depending on what speed you’re trying to keep constant, or how you’ve scaled

your axes. Normally, physicists draw their spacetime

diagram tickmarks such that if every vertical tickmark represents one second, a horizontal

tickmark 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 has

Lorentz 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 about in Special relativity: 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 series

wouldn’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

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