- [Voiceover] So we've already been able to explore
a lot with our little thought experiment
where I am floating in space.
I'm at the center of my frame of reference,
and right at time equals zero
in my frame of reference,
a friend comes in a spaceship passing me by
with a velocity v,
I'll say the magnitude is v,
and it's going in the positive x direction.
We're just going to focus,
and we have been focusing,
just on the x dimension for simplicity.
And we've thought about reconciling space
and time in my frame of reference
relative to her frame of reference.
The conundrum that we've faced
in previous videos is how do we reconcile
that the speed of light is always
going to be the same in every frame of reference?
And to reconcile them, we had to essentially come
up with the idea of spacetime.
I should say it even faster, spacetime.
Spacetime, let me write it out, spacetime.
And the first time I heard about spacetime,
I assumed that people were just talking
about space and time as independent things
and just plotting your point in space and time.
But when people talk about spacetime,
they're really talking about this continuum of one thing,
and we're just talking about different directions
in spacetime.
They could have called this something else.
They could have called this spime,
or tace, or stace,
or a lot of different things.
But this is spacetime,
and it's this idea that it's this continuum.
And when we started to make spacetime diagrams,
we realized in order for the speed of light
to be absolute, that time and space
weren't as independent of each other
as we thought, and they weren't as absolute
as we thought.
And we constructed these Minkowsky
spacetime diagrams for each
of our frames of reference.
So my frame of reference,
the spacetime diagram,
is here in white,
and for my friend's frame of reference,
her spacetime diagram is here
in this blue color.
The angle formed between these axes,
between the x and the x prime axis
and the ct and the ct prime axis,
this angle alpha here,
this is going to be dependent on her
relative velocity in my frame of reference.
So if her velocity is v,
or the magnitude of her velocity is v
in my frame of reference,
this angle we've already seen
is going to be the inverse tangent
or the arctangent of the ratio
between her relative velocity
and the speed of light.
So this is going to be equal to the inverse tangent
of v over c.
So the faster she goes,
these two things are going to start squeezing together,
and if somehow she were to approach the speed
of light, they would both approach a 45 degree angle
and actually start to coincide
if they were actually able to approach
the speed of light.
And that's all interesting already,
this idea that space and time are not
as independent, that it's all a continuum
called spacetime, but some of you have probably said,
well, I want to deal with some more
tangible numbers here.
For example, if this event right over here
in spacetime, we can think about it
from my frame of reference,
and we can think about it
from her frame of reference.
In my frame of reference,
I would view the coordinates here.
This coordinate would be x,
and this coordinate right over here
would be ct.
We had a whole video on why we think
of time in terms of ct.
The units here literally would be meters.
We could think of it as light-meters if we like.
So that would be the coordinates
in my frame of reference.
Well, what would be the coordinates
in her frame of reference?
Well, we've already thought about how to read
these Minkowski spacetime diagrams.
To find her x prime coordinate,
we would just go parallel to the ct prime axis.
So that would be the x prime coordinate
in her frame of reference.
And to figure out the ct prime coordinate,
we would just go parallel to the x prime axis.
So this would be the ct prime coordinate.
Now how do you actually go in between,
transform, from x to x prime
and from ct to ct prime?
And to do that, we're going to introduce
in this video the Lorentz transformations.
And what they do is they allow us to do
exactly what we just needed to do.
They'll allow us to go from x, ct
to x prime
and ct prime.
And to help us think about it,
I'm going to introduce some variables,
and hopefully it will show the symmetry
of the Lorentz transformations.
You might see them written in other ways
in other sources, and we'll reconcile all
of those in the future.
But the Lorentz transformations,
we'll start with what we call the Lorentz factor
because this shows up a lot
in the transformation.
So I'll just define this ahead of time.
So the Lorentz factor, denoted by the Greek letter
gamma, lowercase gamma,
it is equal to one over the square root
of one minus v squared over c squared.
Now sometimes you might even see it written like,
well, I'll write it another way.
Sometimes you might see it written as gamma,
let me do it in that same color,
same reddish color,
gamma is equal to one over the square root
of one minus beta squared.
You might say, well, what is beta?
Well, beta is another variable that shows up
a lot when we're thinking about special relativity.
And beta is just the ratio
between the relative velocity,
her relative velocity in my frame of reference,
and the speed of light.
It shows up a lot,
even this angle alpha here,
we could have said this is the inverse tangent
of beta.
This becomes a little bit simpler when you write
the Lorentz factor.
And when we look at the actual transformation
between the coordinates, we'll see that beta
becomes useful again, at least the way
I like to write it.
So if we want to think about what x prime
is going to be, so we can write
x prime is going to be equal
to the Lorentz factor,
let me do it in that red color,
is going to be equal to the Lorentz factor
times x
minus, and now we're going to say
beta times ct,
and ct prime
is going to be equal to the Lorentz factor gamma,
let me do that same color again,
switching colors is sometimes difficult,
gamma times, and we'll see it's just
the other way around.
It is going to be c,
let me do it in that green color,
ct
minus, you might even guess what I'm
about to write based on the symmetry
that we see here,
ct minus beta times x.
And I really want you to appreciate this
because it really shows that space and time
are really just different directions
in the spacetime and there's a nice symmetry
to them right over here.
Notice we have an x and we have an x.
We have a ct and we have a ct.
So when we're thinking about x prime,
it's x minus beta times ct,
and we're scaling it by the Lorentz factor,
and then when we're thinking about time,
well we do it the other way around.
We're still scaled by the Lorentz factor,
but now it is ct minus beta times x.
Now this might all seem like Greek to you,
and we actually are using Greek letters,
but in the next video,
I'll actually use some sample numbers here,
and you'll see that evaluating these
is just a little bit of straightforward algebra.