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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 properties 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.
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, and
2.
To notice which properties of objects and motion 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 appears 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 when you change your perspective, you are zeroing in on universal
truths – literally.
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
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, but we'll start with non-moving things 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 closer 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.
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
from which we measure things.
Ok, so 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... uh, 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 your 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 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
new x and y coordinates will be smaller.
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
x times sin theta.
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 “the two cats are 5 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 10/2, or 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.”
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 is based off
of how long it takes a certain electromagnetic wave to vibrate once).
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 1-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 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.
Though if you feel comfortable with space-time diagrams, world-lines, and so on, by all means,
skip ahead.
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 “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.
Pretty cool, huh!
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 spatial 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 me and the cat at any particular time change, but the distance
between us doesn't.
At time t=0 we're still 0 meters apart, 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.
Similarly, if you slide the time axis up and down, the absolute time when the cat starts
moving away from me changes, 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.
And if we have 2-dimensional motion, then changes in the orientation of the two spatial
axes 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: relativity is about understanding how changes in perspective
do or don't affect motion.
For objects in space, 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.
Like how changing your position and orientation correspond to sliding the axes around and
rigidly changing the orientation of the spatial axes, and yet these transformations don't
change the spatial distance between two points at the same time or 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 on Brilliant.org, this video's sponsor.
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/MinutePhysicsSpecialRelativity, and doing problems like theirs after watching
videos like mine is a great way to practice and really internalize the ideas of special
relativity.
If you decide to sign up for premium access to all of their courses and quizzes, 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.