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- [Voiceover] We have been
doing some interesting things

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in the last few videos.

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We let go of our Newtonian assumptions

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that the passage of time is the same

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in all initial frame of reference

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that time is absolute that one second

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in my frame of reference is the same

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as one second passing in
your frame of reference.

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We even let go of the idea
that space is absolute

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that one meter in a certain direction

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in my frame of reference
is going to be the same

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as one meter in your frame of reference

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if we're are moving at relative velocities

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with respect to each other.

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And what that allows us to
do, it allowed us to reconcile

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what is actually observed in the universe.

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And that's the idea that
speed of light is an absolute.

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That regardless of what inertial
frame of reference we're in

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regardless of the speed
of the source of the light

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that we will always measure
light travelling at three times

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or roughly three times 10 to
the eighth meters per second.

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And when we let go of our assumptions

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about absolute time and absolute space

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and we did assume absolute
velocity for light.

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It gave us a very interesting diagram

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because it took our x
prime axis from being

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on top of the x axis and
put it an angle relative

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to the x-axis.

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Now, one question that you might have

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is well what is this angle,
the way I've drawn it,

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it looks like the angle
between x prime and x

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is the same as the angle
between t prime and t.

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Or as we did in the last videos,
we turned our units in time

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where the units along
this axis, the ct-axis,

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instead of saying seconds,

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we're now going to measure it in meters,

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and once again watch the last video

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if you're having an issue with that.

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But it looks like this angle
and this angle is the same.

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What I want to do in this video
is feel good about the fact

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that they really are the same.

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So, one of the convenient
things when a couple videos ago,

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we went through this thought experiment

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of in my friend's frame of reference,

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her admitting a photon of light

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bouncing off of the spaceship

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that's three times 10 to the eighth meters

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in front of her, and
then coming back to her.

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As we assumed that as
we mark off the meters,

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and as we mark off the seconds

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that the distance from the origin

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for three times 10 to the
eighth meters is the same

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as the distance from the
origin is one second.

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And now it makes even more
sense 'cause now we call this

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three times 10 to the eighth meters

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and this times three times 10
to the eighth meters as well

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and so the speed...

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When we depict the path
of light of a photon

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in either of these space time diagrams,

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it's always going to be at a either

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a positive 45 degree angle
or a negative 45 degree angle

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depending on the direction
that light is traveling in.

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In fact, that was what
we used to establish

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that this x prime axis is going to be,

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it's not going to sit
on top of the x-axis,

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it's going to be at an angle.

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But I don't actually
think about that angle.

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So, if we say that this angle
right over here is alpha

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and if we were to continue
this 45-degree line

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that we had to remember,
any path of a photon

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is gonna be at 45 degree
angle, so let me continue that.

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So, this is going to look like that

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and then we could go back

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and continue it right like that.

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Well, we know that this
going to be 45 degrees.

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45 degrees right over there
this is going to be...

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Whoops, this is going to be
45 degrees right over there

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and we know, we that this
green triangle that I'm...

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It's not green, it's not different enough,

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let me get a better color.

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This??, no that's still green.

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We know that this, this purplish triangle

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that I'm setting up right here.

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This is an isosceles triangle.

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How do I know that this
is an isosceles triangle?

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Well, this hash mark right
over here, on the ct prime axis

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that is three times 10
to the eighth meters

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and this hash mark on
the x prime axis is also

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three times 10 to the eighth meters.

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And, so, this side is equal to that side

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and so we know that the base
angles of an isosceles triangle

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are going to congruent.

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So, this angle is going to
congruent to that angle.

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Well, if those two angles are congruent,

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then this angle is going
to congruent to that angle.

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Because they are supplementary
to those two base angles.

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And so notice, if we look at...

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If we look at this triangle...

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Let me do this in a new color...

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If we look at this
triangle right over here

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and compare it to this
triangle right over here

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notice they both have a 45 degree angle,

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they both have this blue
angle which is congruent

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so the third angle has to be congruent.

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They always have to add up to 180 degrees

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if you have two angles of
two different triangles

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of the same, then the third
angle has to be the same

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and so, if this is alpha, I shall do...

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Let's see, I've already used...

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I'll do four arcs there,
if that angle is alpha

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then this angle is going
to be alpha as well.

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And, so, this is really interesting,

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it's a beautiful kind of symmetry,

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and it really comes out of the fact

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that the speed of light
should always be measured

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at three times 10 to the
eighth meters per second

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in any inertial frame of reference.

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Notice, if we go back
to my frame of reference

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that photon that I emit
from my, my flashlight

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it will look like this.

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It will look like this.

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It's path in this Minkowski space time

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from my point of view will look like that

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and notice, from my friend's point of view

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the one that's travelling
on a ship for every amount

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let's say, let's pick a
certain point in space time

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right over there.

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Well, her x time coordinate is
going to be right over there

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and her ct prime coordinate is
going to be right over there

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so that makes sense.

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In one, in one I guess
we could say light meter

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that thing travels a meter.

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So, it is still travelling
at the speed of light

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from her point of view.

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Even though, she is moving
at half the speed of light

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relative to me.

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And you can think about what would happen

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if she was moving even faster.

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Someone moving even faster.

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Their ct prime axis
would be yet at an even,

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at an even more severe angle

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so it might look something like that.

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And then, what would their
x prime angle look like?

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Well then, their x prime
angle is going to symmetric

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around that line that
shows the path of light,

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so it would look like this.

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So, this is someone,
let's say who's moving

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even faster relative to me,
let's call that ct prime prime

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and this would be their x prime prime.

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As you can imagine, you get close

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as that second frame of reference

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gets closer and closer to speed of light

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from my frame of reference,
their coordinate axis

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are just going to get
more and more squanched up

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around, around this 45 degree
angle, so that's really neat.

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Now, another thing I wanted to think about

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and now, I won't draw it in this video

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just 'cause I think I've
given you enough to digest is

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there's a symmetry here.

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If my friend is moving at
half the speed of light

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moving at half the speed of light

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in the positive extraction
according to me,

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then from her point of view,

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I'm moving at half the speed of light

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in the negative x prime direction,
so you should think about

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well what would if I were to,

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if we were to look at this right over here

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what would my frame of reference look like

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if I were to project it on top of that.