- [Voiceover] We have been doing some interesting things
in the last few videos.
We let go of our Newtonian assumptions
that the passage of time is the same
in all initial frame of reference
that time is absolute that one second
in my frame of reference is the same
as one second passing in your frame of reference.
We even let go of the idea that space is absolute
that one meter in a certain direction
in my frame of reference is going to be the same
as one meter in your frame of reference
if we're are moving at relative velocities
with respect to each other.
And what that allows us to do, it allowed us to reconcile
what is actually observed in the universe.
And that's the idea that speed of light is an absolute.
That regardless of what inertial frame of reference we're in
regardless of the speed of the source of the light
that we will always measure light travelling at three times
or roughly three times 10 to the eighth meters per second.
And when we let go of our assumptions
about absolute time and absolute space
and we did assume absolute velocity for light.
It gave us a very interesting diagram
because it took our x prime axis from being
on top of the x axis and put it an angle relative
to the x-axis.
Now, one question that you might have
is well what is this angle, the way I've drawn it,
it looks like the angle between x prime and x
is the same as the angle between t prime and t.
Or as we did in the last videos, we turned our units in time
where the units along this axis, the ct-axis,
instead of saying seconds,
we're now going to measure it in meters,
and once again watch the last video
if you're having an issue with that.
But it looks like this angle and this angle is the same.
What I want to do in this video is feel good about the fact
that they really are the same.
So, one of the convenient things when a couple videos ago,
we went through this thought experiment
of in my friend's frame of reference,
her admitting a photon of light
bouncing off of the spaceship
that's three times 10 to the eighth meters
in front of her, and then coming back to her.
As we assumed that as we mark off the meters,
and as we mark off the seconds
that the distance from the origin
for three times 10 to the eighth meters is the same
as the distance from the origin is one second.
And now it makes even more sense 'cause now we call this
three times 10 to the eighth meters
and this times three times 10 to the eighth meters as well
and so the speed...
When we depict the path of light of a photon
in either of these space time diagrams,
it's always going to be at a either
a positive 45 degree angle or a negative 45 degree angle
depending on the direction that light is traveling in.
In fact, that was what we used to establish
that this x prime axis is going to be,
it's not going to sit on top of the x-axis,
it's going to be at an angle.
But I don't actually think about that angle.
So, if we say that this angle right over here is alpha
and if we were to continue this 45-degree line
that we had to remember, any path of a photon
is gonna be at 45 degree angle, so let me continue that.
So, this is going to look like that
and then we could go back
and continue it right like that.
Well, we know that this going to be 45 degrees.
45 degrees right over there this is going to be...
Whoops, this is going to be 45 degrees right over there
and we know, we that this green triangle that I'm...
It's not green, it's not different enough,
let me get a better color.
This??, no that's still green.
We know that this, this purplish triangle
that I'm setting up right here.
This is an isosceles triangle.
How do I know that this is an isosceles triangle?
Well, this hash mark right over here, on the ct prime axis
that is three times 10 to the eighth meters
and this hash mark on the x prime axis is also
three times 10 to the eighth meters.
And, so, this side is equal to that side
and so we know that the base angles of an isosceles triangle
are going to congruent.
So, this angle is going to congruent to that angle.
Well, if those two angles are congruent,
then this angle is going to congruent to that angle.
Because they are supplementary to those two base angles.
And so notice, if we look at...
If we look at this triangle...
Let me do this in a new color...
If we look at this triangle right over here
and compare it to this triangle right over here
notice they both have a 45 degree angle,
they both have this blue angle which is congruent
so the third angle has to be congruent.
They always have to add up to 180 degrees
if you have two angles of two different triangles
of the same, then the third angle has to be the same
and so, if this is alpha, I shall do...
Let's see, I've already used...
I'll do four arcs there, if that angle is alpha
then this angle is going to be alpha as well.
And, so, this is really interesting,
it's a beautiful kind of symmetry,
and it really comes out of the fact
that the speed of light should always be measured
at three times 10 to the eighth meters per second
in any inertial frame of reference.
Notice, if we go back to my frame of reference
that photon that I emit from my, my flashlight
it will look like this.
It will look like this.
It's path in this Minkowski space time
from my point of view will look like that
and notice, from my friend's point of view
the one that's travelling on a ship for every amount
let's say, let's pick a certain point in space time
right over there.
Well, her x time coordinate is going to be right over there
and her ct prime coordinate is going to be right over there
so that makes sense.
In one, in one I guess we could say light meter
that thing travels a meter.
So, it is still travelling at the speed of light
from her point of view.
Even though, she is moving at half the speed of light
relative to me.
And you can think about what would happen
if she was moving even faster.
Someone moving even faster.
Their ct prime axis would be yet at an even,
at an even more severe angle
so it might look something like that.
And then, what would their x prime angle look like?
Well then, their x prime angle is going to symmetric
around that line that shows the path of light,
so it would look like this.
So, this is someone, let's say who's moving
even faster relative to me, let's call that ct prime prime
and this would be their x prime prime.
As you can imagine, you get close
as that second frame of reference
gets closer and closer to speed of light
from my frame of reference, their coordinate axis
are just going to get more and more squanched up
around, around this 45 degree angle, so that's really neat.
Now, another thing I wanted to think about
and now, I won't draw it in this video
just 'cause I think I've given you enough to digest is
there's a symmetry here.
If my friend is moving at half the speed of light
moving at half the speed of light
in the positive extraction according to me,
then from her point of view,
I'm moving at half the speed of light
in the negative x prime direction, so you should think about
well what would if I were to,
if we were to look at this right over here
what would my frame of reference look like
if I were to project it on top of that.