Main content

## 2015 Challenge — Physics

Current time:0:00Total duration:9:36

# A visualization of special relativity

## Video transcript

The theory of Special Relativity, which is
commonly associated with Albert Einstein, has proven to be one of the most successful
theories in all of physics, surviving any test we can think of to try to prove it wrong. However, it has a reputation for only being
possible to learn through headache inducing thought experiments where the student has
to keep up with pulses of light, multiple clocks, and moving observers all while trying to adapt to a new concept of space and time. While all of this is fascinating, and I really
encourage you to look into the thought-experiment development of special relativity, my goal
in this video is to present you with a more direct intuition for how special relativity
works without using as much math. Let’s start with explaining Galilean Relativity,
where the outcome of any experiment will be the same regardless of the frame of reference—that
is to say, if all relevant parts of the experiment are given some boost with the same speed and
direction, the experiment will still come out the same. The concept of a boost will be important later
on. For this discussion, think of a boost as a push into a new frame of reference. Imaging throwing a ball straight up in your
driveway. It will fall straight down. Now imagine that you and the ball are in a car that is moving at a constant 60 mph west. From the frame of reference of the ground,
if you throw the ball up, it will move upward, but also westward at 60 mph to form an arc. If I asked what the ball looks like to you,
you would quickly tell me that it looks like it is moving straight up and straight down,
just like in your driveway. While this seems obvious, it is only because of symmetry that
we can make this assumption. The idea behind Galilean Relativity is that
you can skip the step where you consider the frame of reference from the ground because
of the symmetry between all frames of reference. In fact, the ground is just the Earth’s
surface, which is rotating at some thousand miles per hour relative to the axis. But even
the axis orbits the Sun, which in turn orbits the Galaxy. It therefore follows that all frames of reference are equally valid, or in other words, there is no way to determine if you are moving or at rest in any absolute sense, only the relative motion between different objects is physically
meaningful. This is called the principle of relativity. Okay. Now we need to switch gears for a minute
to talk a little bit about electromagnetism and waves. Electromagnetism is the theory describing
the one unified interaction between charges, currents, magnets and light. In the nineteenth century, it was realized
that the equations for electromagnetism had a class of solutions that corresponded to
physical waves in the electromagnetic field. It did not take long for experiments to produce
these “electromagnetic waves” in the laboratory, and confirm that light is just a small subset
of these waves with a particular range of frequencies. The electromagnetic field was thought to exist
in the æther, with its own state of motion, through which electromagnetic waves of light
travelled at a relative speed of 300 million meters per second. This, however, poses a problem for Galilean
Relativity. To see why, we are going to use an analogy—water waves. Imagine you are floating in a lake, and you
drop a baseball in the water. The disturbanc e created in the height and
motion of the water will propagate outward as an expanding ring. This ring will let us
determine if we are rest with respect to the water. As long as you are in this lake, you can determine
if you are at rest by dropping a baseball in the water and checking to see if you are
remain in the center of the expanding ring-shaped wave around you. If you are moving, you will be able to see
this as the center of the expanding wave moves away from you–you don’t stay in the center
of the expanding ring. Giving such a test to determine if you are
moving or at rest may seem to contradict the principle of relativity, but remember, you
are only measuring your motion relative to the water and its associated frame of reference. So, according to the æther idea, we should
be able to do the same thing with an electromagnetic or light wave as we did in the lake. Which
would mean that we should be able to see our movement in the æther by measuring lightwaves. Physicists Albert Michelson and Edward Morley
tried an experiment to measure this. What they found is that it does not appear
that there is any motion at all through the æther, even at times of year when the earth
is moving at a velocity up to 60 km/sec different. This experiment showed that the æther theory
was wrong, and that the electromagnetic field did not behave like some cosmic lake with
a fixed background frame. Instead, the electromagnetic field treats
all frames of reference equally, which means that the speed of light is 300 million meters
per second in every frame of reference. This is called the principle of constancy of the
speed of light. But wait! How can this be consistent with
Galilean Relativity? Galilean Relativity says that if a Galileo
Galilei is moving toward you at 1 million meters per second and he shines a light at
you that leaves him at 300 million meters per second, then the light will arrive to
you moving at 301 million meters per second, breaking the principle of constancy of the
speed of light. This inconsistency led theoretical physicists
Henri Poincaré, Albert Einstein, and Hendrik Lorentz, as well as mathematician Hermann Minkowski to develop a new theory of space and time. Somehow it had to be true that there is no
preferred frame of reference and the speed of light is the same in all frames of reference. Instead of following the original thinking
of Einstein or Lorentz, I am going to use the fact that everyone thinks they are in
the center of the universe. No, really — all observers see themselves as staying in the
center of a pulse of light emitted from them, regardless of their state of motion. The resolution found by the fathers of Special
Relativity required replacing Galilean Relativity. So we need symmetry that respects the principle
of relativity and also the principle of constancy of the speed of light as well. Enter Lorentzian
Relativity which was discovered by Lorentz and Einstein. A visualization of the principle of Relativity
will require a geometry in which every point and every direction is essentially the same.
There is a theorem in mathematics that there are exactly three such geometries. To begin with, let’s look at a geometry
that does not satisfy this property. On the cube, not any two points are the same type.
For example, while these two points are geometrically equivalent, these two are not. The geometry that does fit our criteria is
the plane. Since the plane is infinite any point could call itself the center. The next one is the sphere. Since the sphere
is perfectly round, any point could call itself the center or the pole. In both of these geometries each point is
basically the same. The third and final geometry is one that is
less familiar, since there is no perfect way to represent it in three dimensional space.
It is called the hyperbolic plane, and it is what we will use to understand relativity. In our visualization, the regular plane will
represent Galilean relativity and the hyperbolic plain will represent Lorentzian relativity. So here we go… Imagine a car is moving at some speed through
a large parking lot. According to Newtonian physics, the car can have any speed in any
direction. The speed of the car can be represented as a point on this grid of tiles. To move from one tile to an adjacent one would
require a standard boost, say 10 mph. Remember that a boost is just a push into
a new frame of reference. Each point on this circle represents a velocity
of 10 miles per hour in a different direction, and each point on this circle represents a
velocity of 20 miles per hour. This grid represents Galilean relativity. Notice that no particular
tile is special: we can easily shift the grid around and put any tile in the center. Now I am going to state something that will
become clear later: nothing can move faster than the speed of light. Therefore the velocities
that the car can move at are represented inside of this circle. For example, this point represents
half the speed of light north. If you imagine this circle is the size of a dinner plate,
any speeds that would practically be obtained by a vehicle on earth stay in a tiny circle
only about a thousand atoms in diameter. So, the new grid of tiles looks like this.
The most important thing to understand is that the tiles become more and more compressed
near the boundaries. Each tile is still the same amount of boost, but they shrink in size
as you approach the edge. This explains why you can never get to the
speed of light. To increase your speed, you need to undergo a certain boost, moving to
the next tile. But as you get near the speed of light, each tile takes you less and less
distance with the same boost. So while the speed of light is only a finite number of
“meters per second” away, it is still an infinite number of tiles that you must
cross to get there. So you can see when you compare our two grids
that in the first one (which represents Galilean relativity), the velocities are additive — each
tile adds the same amount of boost or change in velocity regardless of which frame of reference
you are observing it from. In the second grid (which represents Lorentzian relativity),
each tile adds the same amount of boost, but it looks completely different depending on
where you are viewing it from. If you viewing a change that is far away, the boost or tile
appears very small, but if you are in the center or close by, the tile appears very
large. With a bit of visualization, you can imagine
that the grid could be shifted so that this arc moves to this arc, with all of these tiles
getting compressed over here, and these tiles expanding to fill up all of this space. So, you can see that any tile could reasonably
call itself the center tile, and so it follows, each tile sees the speed of light as a circle
around it. Now maybe you can see why, with Lorentzian
relativity, each observer sees light emitted from them moving out in a perfect sphere regardless
of their state of motion which in our visualization is represented by which tile they are on which. One last thing to note is that Galilean Relativity
is an approximation to Lorentzian relativity as long as the speeds involved are small.
For example, the 1000 atom diameter circe representing typical speeds of human vehicles
is hardly affected by the distorting near the speed of light. The theory of special relativity found by
Einstein gave rise to a whole new picture of space and time, which became incorporated
into some of the most profound and successful theories in all of physics. I wish I could go on to talk about these ideas
of spacetime and mass-energy, but unfortunately I am out of time. I hope I’ve given you a taste of Special
Relativity, and why you can never reach the speed of light without using all of this. I want to encourage anyone interested to find
material on the topic of Special Relativity and to look at the woodcut Circle Limit III
by M.C. Escher which inspired this video. My next video will be on Schrödinger.