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Spacetime Diagrams | Special Relativity Ch. 2

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 property like temperature and so on.

The role of relativity in physics is to study

how that motion looks from different perspectives.

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

which will get to later.

But either way, relativity special or not

is about how the motions of things look from different perspectives.

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 out a spiralling path through space.

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 true description of what’s happening?

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

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

1. To figure out how objects and their motion look from different perspectives.
1. 从不同的角度了解物体和物体的运动
and two, to notice which properties of objects and motion
2. 研究不同的视角下 物体本身和运动过程
don’t look different from different perspectives.

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,

that’s a little trickier.

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 appeals to be the same

So you might say: Aha, 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 point of view.

And this is why relativity is so important in physics.

By studying what changes and what doesn’t about a physical system

you are zeroing in on universal truths.

Literally, facts that remain true from many perspectives throughout the universe

like perhaps that 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 is.

OK, enough philosophizing.

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,

just to get a sense of how intuitive relativity can be.

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 of numbers like (3,2).

However, (3,2) is not a universal truth.

I mean, it’s just based on where I’m standing, and how I’m oriented.

But over here, where you are, maybe you’re rotated by 30 degrees,

and you made the tick marks close together,

and suddenly the cat is at a different position, x=9, y=9,

even though the cat hasn’t moved.

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 at x=0, y=0, and x=5, y=0.

I’m not gonna draw a person at the origin of the axes from now on,

but you should remember that the axes we use

represent a particular perspective and orientation from which we measure things.

The distance between these two cats is clearly 5,

they’re at the same y value and their x values differ by 5.

If we move and rotate our point of reference now,

the cats are at positions, x=1, y=1 and x=5, y=4.

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 the perspective just by shifting your point of reference or your orientation.

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 things.

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 slide the x axis an amount ΔX,

technically called a “translation by x”,

we say that Xnew=X-ΔX,

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 here because if you slide your origin point closer to something

its new x and y coordinates will be smaller.

Changes of orientation are a little fancier,

but it’s really just some geometry:

if you reorient the x and y axes counterclockwise by an angle theta,

the new coordinates look like x new=x cosθ-y sinθ

and y new = y cosθ+ x sinθ.

And if you want a fun algebra exercise,

you can use these equations or even their 3D counterparts

to check that indeed that the distance between two points doesn’t change

when you slide or rotate your axes.

But the messiness of all the details here

really clouds the simplicity of what’s going on.

The important geometric idea I want you to remember

is that rotating and sliding axes

doesn’t change the distance between two points.

However, the distance between two points does change

if we’re allowed to change the spacing of the tick marks.

If when we change our axes,

we also double the tick marks,

then the distance between the cats becomes 10, not 5.

Turns out, distance measured in numbers,

is not so universal.

But there is a more universal truth.

Suppose we have a stick that’s 1 tick mark long,

according to the original axes,

conventionally this thing might be called a meter stick.

And now we can say that the two cats are five sticks apart.

When we again move and rotate our axes

and change the spacing of the tick marks,

the cats are again 10 tick marks apart,

but the stick is also now 2 tick marks long,

so the distance between the cats is still 5 sticks.

This is an example of an even more general physical truth:

The distance between two things, measured in terms of another physical thing,

doesn’t change when you change your perspective by shifting your point of reference

or orientation, or the spacing of your tick marks.

In relativity parlance, we’d say that

the ratio of two distances is absolute.

Or basically, if you want to actually describe a distance,

you can’t just specify a number,

like I’m five away from you.

You have to say what you’re measuring distance in terms of,

and what number of those things your distance is equal to.

This is kind of a subtle point and is very important

if you’re interested in metrology, the study of measurement and units.

But because it doesn’t really play a major role in special relativity,

from now on I’m going to be a bit sloppy and just assume that

whenever we’re talking about distances, we’re talking about distances not as numbers

but in terms of some reference distance, like meters, or cats, or whatever.

And the same will apply to times.

when we talk about a time interval, we’ll assume it’s a time interval

in terms of some reference time, like the second,

which brings us to the motion of objects over time.

To describe a moving object,

it’s customary to use a horizontal coordinate axis for the left-right x position.

But instead of using the vertical axis to represent height y, we use it to represent time t.

So for something not moving,

something that stays at the same position x at time t=0, t=1, t=2 and so on,

we draw a straight vertical line through x.

For something moving one meter per second to the right,

we draw a line that goes one meter to the right

for every second that transpires vertically.

It’s important to note that we’re not saying

that the object is moving through 2D space along a 45 degree line.

The object is moving purely one-dimensionally along the x axis,

and we’re just showing those different one-dimensional positions as time passes.

This whole “time on the vertical axis” thing

can also be a bit weird at first since in most other situations

you’ve probably encountered time plotted on a horizontal axis.

But vertical time has its merits, and more importantly,

it’s convention at this point.

So it’s worth getting used to.

I like to think of each horizontal line

as representing a different snapshot of a scene.

We could show the snapshots one after another as time actually passes, of course,

but it’s useful to be able to see all of the snapshots at once.

So if we display each snapshot at a consecutive vertical position,

we get a nice representation in a single static image

of motion that normally takes place over time.

This geometric way of representing motion over time

is called a “space-time” diagram.

And it’s so central to intuitively understanding relativity

that it’s worth doing a few more examples.

Say we have a cat attached to a spring,

bouncing back and forth, left and right.

If we plot this motion on a spacetime diagram,

as time passes we see the cat move left and right,

leaving behind a trace in the shape of a sine wave.

On the other hand,

if we’re given a spacetime diagram and want to recover the motion of the cat

we simply slide the diagram downwards at a constant rate

and move the cat left and right so that it follows along the traced-out path.

This is important: a traced-out path in a spacetime diagram

is a faithful recording of an object’s motion.

And these paths are called “world-lines”, presumably because they show

where in the world the object has gone

though by “world” we often mean

“room” or “solar system”, or “universe”.

Any particular point on a worldline has coordinates t and x,

which we write as a pair telling us

for time t what position x the object was located.

So far we’ve just been representing one-dimensional motion on our spacetime diagrams,

just one spatial direction the object is moving in,

and then time as the vertical axis.

If we want to use a spacetime diagram to represent motion in two dimensions,

like the moon orbiting the earth,

we actually need three dimensions to do so.

The two horizontal directions for the moon and earth to move in,

and the vertical direction to trace out the snapshots as time passes.

It’s pretty cool,

but if you have multiple particles moving complicatedly,

this can get really messy on a 2D screen.

And it’s physically impossible to make a full spacetime diagram

for three dimensional motion,

because you would need four spatial dimensions to do so.

Three horizontal directions for the motion

and a vertical direction for time,

which of course is impossible in our universe

with its measly three spatial dimensions.

So physicists normally restrict their spacetime diagrams

to just one or two spatial dimensions,

and time going vertically.

So how does relativity work with spacetime diagrams?

That is, now that we know

how to describe motion geometrically,

how do changes in perspective affect that description?

Let’s take as an example me staying put right at x=0,

and a cat moving one meter per second to the right away from me

starting at time t=0.

It may not surprise you to notice

that when you slide the x axis to the left or right,

the particular x positions of the cat and I have

at any particular time change,

but the distance between us doesn’t change.

At time t=0 we’re still 0 meters apart,

no matter how much use like the x axis back and forth,

at time t=2 we’re 2 meters apart, and so on.

So you can slide the x axis back and forth however you like

and spacial distances don’t change.

Similarly, if you slide the time axis up and down,

the absolute time like when the cat starts moving away from me changed.

But time intervals don’t change:

the cat still takes 2 seconds to get 2 meters away from me.

So you can slide the t axis up and down,

and distances in time are left unchanged.

If we have 2-dimensional motion,

then changes in the orientation of the two spatial axes

also don’t change the distances between objects at any particular time.

Essentially, you can re-orient the xy axes however you like.

So the relativity we applied to purely spatial diagrams

applies pretty well to space-time diagrams, too.

To summarize the major takeaways:

how changes in perspective do or don’t affect motion

We can describe motion over time geometrically

using spacetime diagrams.

And spacetime diagrams can help us see

how changes of perspective affect how the motion of objects looks.

and orientation correspond to sliding the axes around

and rigidly reorientating the spatial axes,

without changing the spatial distance between two points at the same time

and without changing the temporal distance between two points at the same location in space.

However, all of this relativity is static.

and by that I mean that we haven’t yet talked about how motion looks

from a perspective that is itself moving.

That’s ultimately the key to special relativity,

and will be the subject of the next video.

If you’d like to play around with some spacetime diagrams yourself,

I highly recommend checking out the “propagation of light” interactive quiz

This quiz is seriously cool-

it uses spacetime diagrams to guide you through

how astronomer Ole Rømer deduced that the speed of light must be finite

just by observing the orbit of Jupiter’s moon Io.

It’s a super clever idea,

and the quiz does a great job

of using spacetime diagrams to help visualize the situation

and guide you through calculating the speed of light yourself.

In fact, this quiz is part of a whole course on Special Relativity

that Brilliant has available at brilliant.org/StaathofMinutePhysicsSpecialRelativity.

And doing problems like theirs after watching videos like mine

is a great way to practice

and really internalize the ideas of special relativity.

you can get 20% off by going to Brilliant.org/minutephysics,

or even better, go to brilliant.org/MinutePhysicsSpecialRelativity,

which lets Brilliant know you came from here

and takes you straight to their relativity course.