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 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
the math of special relativity, or economics, or machine learning, I highly recommend Brilliant.org,
this video’s sponsor. In conjunction with my video series, Brilliant
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