Okay. So we found some, at the, at the cost of
a lot of time. We found some nice property of Newtonian
mechanics. Well, invariant, under these
transformations of Galilean relativity, as observed by one observer and as
observed by another observer, moving relative to the first with a constant
velocity. Physics is the same.
What makes that a principle is what you do when it doesn't work.
And when it doesn't work is when we introduce Maxwell's equations.
Electrodynamics causes a problem. Remember, Maxwell wrote down his
equations as a description of the interactions of electric currents and
electric charges, and magnetic fields and so on. And he found when he wrote the
right equations, that his equations had emitted wave solutions, and he computed
from known measured quantities the speed of those waves, and discovered that it
agreed with the measured speed of light, and therefore, had discovered that light
was an electromagnetic phenomenon. But he measured the speed of light,
and what that means is, now that you remember
our velocity edition formula, was that the speed of light as measured by the
black observer, or the speed of light as measured by the blue observer, they
should get different results depending upon which way they're moving. If they
both time the motion of a pulse of light, they should obtain different results.
So Maxwell's equations can at most hold for one of those observers and the other
observer must detect different electric and magnetic phenomena so that when they
compute the speed of light they will get an answer consistent with their
measurements. A technical way of saying this is that
the electromagnetism, the equations, Maxwell's equations are not invariant, do
not retain their form when you do the change of variables that we call Galilean
relativity transformations, what we computed in the previous clip.
So there are two possible solutions to this.
I mean look, Newton, albeit with some error, computed the speed of sound
already and sound moves with a speed that you can compute and it moves with that
speed through air, and if you are moving through the air or if there is a brisk
wind blowing, then of course sound will move at two different speeds, upwind and
downwind, because it moves at a constant speed through the air and then in
addition is carried on by the air's motion.
So, the solution that Maxwell, in fact, adopt, suggested,
is that there is something called, he, he called the luminiferous ether,
some medium through which light propagates.
And it's the properties of this, this ether that cause light to propagate at
speed c, much as it is the properties of air that cause sound to propagate at the
speed at which it propagates. And Maxwell's equations would then be
valid for observers at rest relative to this ether.
Just like the speed of sound would be measured correctly for an observer.
If someone is, there is a brisk wind blowing, you would measure different
properties of air, compared to what you would measure in a,
steady, stationary air, and when you are moving relative to the ether, the
velocity edition formula will tell you, you will measure a different speed of
propagation of light. The other, much more radical solution,
which will turn out to be experimentally, verifiably correct in fact, is that light
propagates with a uniform speed C through space.
there is no real ether, no material implementation of what the medium through
which light propagates. The medium if you want is space.
Maxwell's equations, in fact, hold for all inertial observers whatever their
constant velocity. Remember an inertial observer is one for
who Newton's first law holds. That means that all inertial observers
relative to each other, are moving at constant velocities.
We don't know who the rest observer is, but we know what the collection of all
inertial observers are all those for whom Newton's First Law works.
If Newton's First Law works in your, measurements.
Then, when you measure the properties of electricity and magnetism,
you will find that they obey Maxwell's equations,
and therefore if you measured the motion of the path of light, you would find that
it moves at the speed c, and the speed c will not depend on your velocity relative
to another observer. This contradicts common sense, so let's
see why it is that we believe it. Indeed the initial idea was that Maxel
was right. There is an ether through which light
propagates at speed C, and then your task is to measure your velocity relative to
the ether. And you can do that by measuring the
properties of light. And you do this with the following
experimental apparatus. in Einsteinian terms it's called a light clock, in
others an interferometer. What you're doing is you will set up two
mirrors separated by distance L, as drawn over here on the right.
And you will set up a beam of light to bounce between them.
We know how to do this, this is technology that was well developed in the
nineteenth century. And you can imagine that over here on the
mirror on the right there is a counter that ticks over.
We have a pulse of light bouncing back and forth, every time the pulse of light
hits the mirror on the right, counter ticks over.
You now have a clock and it's a good clock because the speed of light being a
constant. The time that it takes light to bounce
back and forth through this interferometer is twice the length
divided by C because that is how far light has to travel.
So this is true if light moves at a speed C, in other words, if you're at rest with
respect to the ether. What happens if you take this whole clock
and, assuming that the transparency is addressed with respect to the ether, you
set this whole clock moving at a speed V to the right.
Well, what that means is that when the light beam starts out here and is moving
to the left, the mirror on the left is itself moving to the right, towards the
light beam. The relative velocity between, the pulse
with which the pulse of light is approaching the mirror is, of course, C
plus V, when moving in this direction. Let me use red pen to indicate the motion
of light relative to this moving, interferometer.
The speed of light in this direction would be c plus v.
This is exactly the addition, velocity addition formula we computed.
On the other hand, when light is moving from the left mirror back towards the
right mirror, then the right mirror is receding from it, and the light pulse
will be closing on the mirror at a speed C minus V.
And so, it'll take a shorter time for light to move from here to there than it
does for light to move from here to there, but you can add the two.
And you can obtain the, time that it will take light to move to the left and then
back to the right by taking the time it takes to move to the right.
plus the time it took to move to the left.
And doing some algebra, you can write it this way.
You see that the faster. The clock is moving at the end of the
day, the longer it will take for a light pulse to move back and forth.
if you think about it that's because it spends more time moving at the slower
velocity because this is a larger number with a smaller denominator.
Okay so moving the clock should slow it down so you can measure.
yeah, but how do you know what the speed of the clock is when it's addressed, if
you don't know what addressed is? Remember, it's hard to detect when you're
addressed. well the trick is, that was what the
clock will measure when it's moving along its axis.
What happens if it moves perpendicular to its axis?
So now we're taking the same light clock, except now the motion is with a speed v
in the up direction. So we have those two mirrors and they're
moving up. Of course, now if a light pulse bounces
off the right mirror and moves directly to the left, it'll just go off into space
because the left hand mirror will have moved up in the time that it took the
light to transverse the, to traverse the clock. And so the light beam that we're
interested in is a light beam that bounces back and forth between the moving
mirrors. So it bounced off the red mirror here, hit the mirror on the left here,
will bounce off and hit the right mirror back there, when the right mirror has
moved up to this position. Right?
So, and what we want to know is how long it
takes light to transverse, traverse the clock, when the clock is moving in this
direction. This is a geometric calculation.
We're going to assume that despite the fact that there is a small motion
relative to the, ether of the light beam, that this angle is small enough, or V is
small compared to C, and to leading order, we're going to assume that light
actually moves with a speed C. What that tells us is that if little T is
the time it takes a light pulse to go from here to here, and by symmetry little
T is also the time it takes it to get back so bit T's going to be just little T
here is half the period of this light clock.
Well we can do a calculation over the time little t.
the light beam has traversed the distance ct.
times t, in the same time the clock has moved up by a distance vt, times t.
So that's this vertical blue segment over here.
The distance between the mirrors is L. The, an application of the Pythagorean
theorem shows us that l squared plus v squared t squared is the length square,
of the length of a hypotenuse. We can solve this for t, and we find that
t is given by this. Remember that the period of the clock is
twice t so we write it down, and we see that moving gensferously also slows down
the clock, but by a different factor. Notice the square root here.
So the light clock slows down by one factor when its moving longitudinally by
another factor when its moving transversely.
If we spin the light clock around and assume that the Earth is not at rest with
respect to the either we should find that we measure one
That we measure a periodic change in the period of a light clock as its relative
motion with respect to the either changes in its direction.
And this measurement was essentially to a very high precision by Michaelson and
Morely in 1887 and their result is that there is no change.
There is no periodic change in the period of a light clock and so there are various
theories advanced how to understand this perhaps just as if you measure the speed
of sound on average north south and east west you get the same answer, despite the
fact that the Earth is rotated to the east perhaps the either is being dragged
along by the Earth. Lots of complicated ideas.
the simplifying Gordian knot resolving solution of Einstein proposed in 1905
because there is no such things as, as the ether.
Maxwell's equations hold at any inertial frame.
Consequences that you have to change your rules for velocity addition.
Galilean relativity needs modification. Deep changes in what we think of as the
obvious properties of space and time are necessary, but remember, Newtonian
physics needs to still be valid, and hence, Galilean relativity, in some
appropriate limit. So we're going to take Einstein's idea
and see what it tells us, and then show why we believe it.