You may have loved the algebraic derivation of the Lorentz
transformations. You may not have enjoyed the algebra.
Either way, let's remember the logic. What we did is we assumed four facts.
One, that if a blue observer is moving with speed V relative to a black
observer. Then, the black observer is observed by
the blue one to be moving with velocity minus v.
And so, relativity works in that way. That was assumption number one.
Assumption number two was that motion that is observed by the black observer to
be inertial. In other words, to have a straight world
line is observed by the blue observer to have a straight world line.
In other words to be inertial. Newton's laws hold in both frames in the
sense that if an object is observed to be experiencing no force in one frame.
It's observed in the other frame to also be experiencing no force.
That was assumption number two. Assumption number three was the statement
that the velocity of light is the same in all frames.
And putting all those assumptions together, we were ineluctably led to
these Lorentz transformations. Which remember, relate the position and
time of any event. Something happening, a birthday party
starting. As measured by one observer to the
position and time of the same event as measured by another observer.
Two observations about this before we go on, to derive the consequence of this,
which are the meat of special relativity. two things to note right off the bat.
observation number one is if you look at this denominator.
You notice that if v^22 over c^22, is bigger or equal than to one.
Then this denominator is either zero, or you get an infinity or it's negative and
the radical is imaginary. What we conclude, is that if you put
together are, moderest assumptions which we're going to assume as part of physics,
along with the idea of two observers moving relative to each other with a
speed larger than the speed of light you get nonsense.
Conclusion, since we're not going to abandon relativaty, two observers can not
move relative to each other with speeds larger than the speed of light.
The speed V in all of are Lorentz transformations will forever be less than
the speed of light. Therefore, when we talk about
superluminal motion, you cannot accelerate yourself in any way, and we
will see how physics stops you from accelerating yourself to speeds faster
than light. So,
comment one, v^22 is always less than c^2.2.
Comment number two is I worked in a one-dimensional universe.
We don't really live in a one dimensional universe.
we live in a three-dimensional universe so you can ask what happens to the y and
z directions. It's the motions.
I can always pick my x direction. The point along the direction of motion,
what happens in the other directions? Well, in the other directions,
life is much simpler. y prime and z prime are simply given by y
and z. Nothing happens that takes not too much
calculation, so all of the action is indeed in one dimension, the direct,
along the direction of motion. Let's see what we, what happens.
Let's follow through, go to the consequences of adopting these
Lorentz transformations. By far the most confusing one is the idea
of simultaneity. Notice,
we are not shocked that when we draw the now and the here of the black observer
and then we draw the here of the blue observer.
This describes a tilted world line. Well, good.
That's not shocking. We expect this describes the fact, that the blue
observer is moving. What is shocking and very confusing, is
that if you look at the Lorentz transformations and we look for the
collection of all events that, such that they are characterized by t prime equal
to zero, you find that, that is not equivalent to t equal to zero, but rather
to t equal vx / c^2 so that the, blue observers now differs from the black
observers now. They agree and when I say now, these are
all of the events in the history of the universe that each observer imagines to
have occurred simultaneously with in this case, their meeting.
Because that's how he said, t as t prime is zero.
Is, they both started their watches at the moment when they met.
And so this violation of simultaneity is the cause of much, much, much confusion,
and many paradoxes. And to, to, to, get a sense of it for,
for first how much should we worry. Let's measure the magnitude of this effect.
So for example, when I am driving in my car.
Things that I consider simultaneously are not the same as things that a bystander
on the sidewalk considers simultaneous if you move far enough out in the direction
of my motion. It's a happy coincidence that if I'm
moving, say, at ten meters per second, a reasonably fast clip then I can
estimate that our time our, our, our space axes or our definitions of
simultaneity will differ by about a second for objects that are a light year
away. So if something is happening a light year
away, and I measure when I think it happened.
And I'm driving in my car towards it. And someone standing on the sidewalk
measures that time. We will disagree by a second over the
distance of a light year. But how are you going to know what's
going on right now, at a distance of a light year?
That's a subject of our next question. So we'll, we'll address that in a second.
Nonetheless when you are moving not at ten meters per second but at relativistic
velocities, this idea of simultaneity meaning something different, is
significant and understanding it correctly, or learning to wrap your mind
around the weirdness that it is, is the, the, the, the main obstacle to
feel comfortable with relativity. And that comes from working problems.
We'll probably have an optional clip where I will work paradoxes in relativity
to give you a sense for how that works. you'll note that when I've drawn these
diagrams, I am often drawing the red lines which are the world lines. Remember
of the two light beams at 45 degree angles.
Now, the tilt of the world line from the vertical should correspond to the
velocity or the, the speed with which something is moving.
So essentially I'm saying that light moves with speed one,
the way that's typically implemented is that you plot not t, but ct, on your
diagram. So you're effectively working in units
such that the speed of light is one. In my field we do that all the time
anyway and so I have to remind myself to put the c's in.
The c's will be in all my equations. I do not promise to always remember the
to plot ct on this plot. But I am indeed plotting ct and if you
remember that what you are plotting is ct then you can see that the geometry of the
situation is as depicted here the 45 degree
world line of a light beam disects the angle between the time and space axis for
any observer so if you have a green observer who is moving faster than the
blue observer that means their t double prime access will be tilted farther away
from the vertical. Then, so well, there x double prime axis
be tilted farther from the horizontal. This is the geometry of the Lorentz
transformations that we wrote up. So, how do we know what's going on?
A light year away right now? What do we actually mean by right now?
Well now that we understand that the speed of light is a constant we can give
a technical measurable operative meaning to what is going on right now.
if for example 6,000 light years away on the crab nebula something is happening
now. Well, I don't know it yet but in exactly
6,000 years the light from there will reach Earth.
So I define what's going on now at the crab nebula to be what we will see on
Earth in 6,000 years. If I make that definition that now I have
an operational meaning of what the present is at any distance things
happening now are the things from which we will hear at a time related to its
distance from us by this expression. And so if you use that operational
definition of what you mean by the present, then.
Following through the behavior of light beams.
Drawing them onto that diagram that I drew before.
You find that this relativity of simultaneity is obviously a fact is a, a
property of the fact that light moves at the same speed in all frames.
Okay, we'll swallow simultaneity, will work
through it. it's only, it only matters at large
distances or high speed, so the fact that it conflicts with our everyday
experience, well, we don't have everyday experience of moving at the speed of
light. Well what other weird things do the
Lorentz transformations teach us? Well, for starters,
that teaches about something called length contraction which is a very
puzzling a property and we need to understand it so let's see what it does
how do you measure the length of something?
Well, here's the idea. suppose that the moving blue observer is
holding a ruler this ruler has nominal length l prime.
Bought it at a store, it's a one meter ruler, so L prime may be one meter.
He's holding it and so it's left end is in his hand at X prime equals zero, it's
left end is a distance one meter, L prime, to the right of that.
And of course it's moving with him and he measures its distance and he says the
distance is L prime, very good. We want to know what happens if the black
observer measures the length of the same ruler?
So how do we do that we turn our friends the Lorentz transformations which contain
all of the information and we use these transformations to understand where the
black observer sees the ends of the ruler so not too difficult let's see what we
get. the left end of the ruler is at all time
at x prime equal to zero. It's clear from setting this that x prime
equals 0 corresponds to x equals vt. So, good.
The left end of the ruler is in the hands of the blue observer, and as seen by the
black observer, it's moving to the left, to the right with speed v.
Good. What about the right end of the ruler?
I set x prime equal to l prime, and I get an equation that says l prime.
is x2 - vt divide it, at any time by the square root of 1 - v^2 / c^2.
So I can solve this. I multiply through by the square root.
Erase it here, put it here. And I then move the vt over to the other
side and I find that x2 is vt + l prime route 1 - v^2 / c^2.
Okay, let's write that out cleanly.
x1 is this, x2 is this.
Well, good. Both ends of the ruler are moving to the
right at speed v because it's at rest in the moving frame, and the distance
between them is constant. Good.
The ruler is not wiggling. What is the distance between the two ends
at any given time? At any given time, this is the distance
between the two ends of the ruler, so this is what you'd call the length of the
ruler, and so I conclude that the length of the ruler is this number, and since
this multiplier here this square root is smaller than one, the ruler shrunk.
I see a shorter ruler than the person moving with the ruler sees.
In other words when you move something fast its length shrinks.
Along the direction of motion. This is a real effect.
Now, of course, relativity says that if I were holding, if the black observer is
holding a ruler of length L, and you ask what length does the blue observer
measure for that same ruler? Well, same expression, flip the sign of v
makes no difference. The blue observer thinks the black
observer's ruler is shrinking and of course these equations are inconsistent
and so what you have to keep in mind is that they mean two very different things.
This is the length as measured by the black observer of a ruler of length L
prime at rest in the blue frame, so moving with the blue observer.
This is the length as measured by the blue observer of a ruler of length L that
is moving with the black observer. Anything when it moves relative to us and
I don't need an observer for it, the ruler is an observer all by itself.
When a ruler moves along its length, it appears shorter.
Again, unless the speeds are relativistic, this is not important.
When the speeds are relativistic, this is important, and it does happen.
So, moving fast messes with your idea of length.
It also messes with your idea of time. We know about rulers.
Let's think about clocks. So imagine that the moving observer is
holding a clock in their hand. And that clock ticks.
And we'll select two specific ticks. One at the time of the historic meeting
between the two observers at x prime = t prime = 0x= = x = t = 0.
And then on the one sometime later say a second.
So the moving observer measures the tick of the clock at t0, = 0, t prime = 0/ And
then at t prime = 1 or capital T prime. I want to know, what does the black
observer think about this moving clock? In this case, it's easier to use the
inverse Lorentz transformations. The ones that give me x and t and
functions of x prime t prime. Remember they're no harder to write down
than the other ones. You just flip the sign of v and so we're
working at x prime = to zero in this case.
So that's easy. Both terms with x prime are irrelevant
for all of this process because the clock is always at x prime = to 0.
And so I set t prime equal to 0. And I find directly that t1 is zero.
Okay, that's good. We agree that the clock was ticking at
the same event, which was the meeting of the observers.
Excellent, we agree about that. Events are unambiguous things.
But then, what about t2? Well t2 is equal,
t prime is now blue t prime. Pardon me for not respecting the color
scheme. But so it goes divided by 12.
- v^2 / c^22. and indeed, if you ask the black
observer. When did the clock tick?
This is the time when it ticked for the second time.
This when it ticked for the first time. Write those down.
It is clear that the time interval between those two is a lot longer than t
prime. It is t prime divided this quantity less
than one. The black observer observes the blue
clock to be running slow. Of course, by the same relativistic
argument we're used to by now, if the blue observer looks at a clock that is
stationary in the black frame, if you observe that the black blocks are moving
slow. And again, the fact that these two are
consistent just like the fact that the other two length contraction equation
expressions are consistent is associated with being confused about simultaneity.
What you mean by simultaneity in the two frames is different.
And that makes these, it possible for each observer to observe the other's
clocks moving slow. We'll deal with that in the paradoxes
section and we'll soon discuss about, something about why we know this is true.
First, let's follow one important astronomical consequence of this.
We have Doppler shifts a we describe Doppler shifts in the traditional sense
as having to do with radial motion if an object is moving away from me then each
subsequent wave front that it emits has a longer distance to go so I get a red
shift that's moving at negative blue shift.
If it's moving transversely there is no Doppler shift but now there is see, the
reason for their, their there's a Doppler shift that's imagined that an object that
is moving has some atom in it that is vibrating the period in which that atom
vibrates is determined by the physics of the atom and that physics is invariant
because of relativity. And so that atom addressed in its own
frame is vibrating at exactly the same frequency that the same atom would
vibrate when I held it in my hand here addressed in my frame.
But when moving, it vibrates with the same rate relative
to the clock that I observed to run slow. Thusly, the vibrations of the atom are
slowed down by any motion so there's a redshift.
There's always a redshift. The frequency is decreased.
the speed of light is constant. The frequency times the speed of light is
always. the frequency times the wavelength is
always the speed of light. So there is a relativistic redshift that
occurs even if in the case when there is transverse motion.
There's no radial motion whatsoever. And this has to do with time dilation.
So there's always a red shift. what happens if the object happens to
actually be moving radially? Well, then you have this time delation
effect but in, that happens independent of the direction.
But in, in addition to this, you have the geometric fact of how far each light
frame has to go. So you basically multiply the two effects
and you find the relativistic formula for the Doppler shift.
notice I'm using the astronomical convention here where the positive sign
of v means the object is receding from us so we've now converted completely to the
astronomical radial velocity component so this is for a radial or longitudinal I
perhaps should have called it radial because that's what we have called it.
And transverse is tangential when looking at a star.
So all observations are have Doppler shift.
When an object is moving at us sufficiently rapidly.
the the, the blue shift from negative v here can overcome the red shift from
this. But this is the relativistic expression
for a Doppler shift. Okay, that's a lot to swallow.
Is this for real? What is going on?
The answer is yes. We have a lot, a ton of experimental
evidence for each and every one of these phenomenon.
we observe for example, length contraction, when we collide large ions
in accelerator experiments, we accelerate a gold nuclei to high energies, to
velocities close to the speed of light and these are large objects remember they
are hundreds of ferometer across and when they collide the collision happens much
too fast. For relative to the size of these nuclei
and the reason is that as they observe each other well because of their large
relative velocity, these things are colliding as essentially pancakes.
And therefore they pass through each other in a much faster time than would
the nuclei given their size and the speeds that we know because we know how
fast we've accelerated them. time delation is something that's
measured all the time if you want to give we, we take unstable particles.
Unstable particles decay within some time.
We put them in an accelerator, speed them up to a large fraction, a substantial
fraction of the speed of light and they last forever.
Why? Because their clocks are retarded.
Our time their time is dilated relative to ours and we can keep you unstable for
hours by moving them fast enough. there are more low speed measurement of
this. In fact, the lowest speed, I think at
which this has been successfully measured has been obtained by taking really high
precision atomic clocks, putting one of them on the runway, putting another one
on a jet that circumnavigated the globe a couple of times and then noticing when
they landed that the one that was moving fast was in fact, behind the atomic clock
that had been left on the ground. The effects of jet speeds are tiny.
So, you need very high precision atomic clocks.
There is no doubt that these are all facts.
They're counter intuitive, too bad. We have to change our intuition.
now how does it happen? And the other question is also how you
take this ruler. What happened to the rest of the ruler?
You got a one meter ruler, you're now moving at some large fraction of the
speed of, you move it at some large fraction of the speed of light.
The ruler is 90 centimeters length, in length, what happened to the other ten
centimeters? Well the answer is what determined the
length of the ruler in the first place. Well the ruler is held together by
electromagnetic forces. Electromagnetic interactions, remember
these are Maxwell equations, are invariant in the Lorentz transformations,
which means that they give the same physics.
If you do Lorentz transformations electromagnetic interactions if you were
able to make the count detailed calculation of what determines the size
of this ruler would actually predict that it shortens when you move it fast.
In fact Lorentz, when he first derived the Lorentz contraction, the length
contraction expression was not trying to do relativistic physics.
He had not adopted Einstein's principle of relativity he was doing calculations
with electromagnetic fields and he saw that the electromagnetic field of a
moving charge appears compressed along the direction of its motion he was
observing that Maxwell's equations know about Lorentz contraction.
Which is not surprising since Lorentz contraction follows from the invariance
of the speed of light which is a property of Maxwell's equations.
Woo-hoo. Similarly, build a clock of any kind and
it will run slow when you move it to high speed.
in fact we just did that calculation for one clock construction, the light clock.
The light clock moving transversely where we assumed that the speed of light was a
constant. We found slowed down by one over the
square root of 1 - v^2 / c^2. What about the longitudinal time clock?
You'll do that in the homework. one last thing before we let go of these
weird and bizarre consequences of Lorentz transformations, we talked about velocity
addition, what happens when you add to velocity, so back to our old picture.
We have the blue observer, we have the black observer.
The blue observer note sees an object and the object is moving at a velocity which
I should properly call u prime, and I ask at what
I will call it u I think, I guess. And I ask, at what velocity will the
black observer observe this object to be moving?
So again, I draw the world line of the moving object as x prime = blue u * t
prime. I use my Lorentz transformations, plug in
this information, and solve for x. So, for example, I can do the calculation
x = ut prime + vt prime / the square root of 1 - v^2 / c^2.2.
On the other hand t = t prime + vx prime, x prime is ut prime / c^2 / square root
of 1 - v^2 / c^2. And, what does that tell me?
Well this is equal to u plus v, just as it was in the Galillean case except for
the funny square root there. This is equal to t prime * 1UV + uv / c^2
divided by the same square root. So indeed x is some number times t,
inertial motion translates to inertial motion and the number is obtained by
dividing this by that. The reason I didn't write out the square
roots is because they happily cancel. I see that x is in fact not uV + v * t as
it was before but uV + / 1UV + uv / c^2. This is the new velocity addition
formula. Let me write it cleanly.
Here's the new velocity addition formula and
the great here I've remembered to put u prime and what is the great property of
this? Well if u and v are both small compared
to the speed of light if you are dealing with slow moving objects then you can
erase this u2 / c * v over c and you're back to Galileo.
Logic makes sense at the small velocities at which we're used to applying it.
The other nice property of this is that if you plug in vC = c.
If you ask, if I add some speed, if I observe something moving at a speed C,
what will you observe? Well if vC = c, then it's an easy
experiment to show that if, sorry not v, v isn't always necessary, if u prime is
equal to c, then uC. = c.
So this has a property that on one hand at low velocity, this is Galileo.
On the other hand, light travels at the speed of light no matter who is measuring
it. These are the consequences of Lorentz.
we can play with, if you'd like, you can go on from here to
play with some simultaneity puzzles and if not, we'll move on to constructing
relativistic physics.