[MUSIC PLAYING] Does the speed of light actually have anything to do with light?

[MUSIC PLAYING] So what is it about the speed of light that's so special?

Why does the universe seem to conspire to, one, keep photons from traveling at any speed but 300,000 kilometers per second in a vacuum-- according to any observer-- and, two, keep anything from traveling faster than that speed?

The answer this statement is false, or at least backwards.

The universe doesn't arrange itself to keep the speed of light constant.

In fact, spacetime couldn't care less about light.

The cosmic speed limit is about something much deeper.

This universal constant is, perhaps, more accurately, the speed of causality.

In a previous episode, we talked about causality by way of the spacetime interval.

Causal connections give us the only ordering of events that all observers will agree on.

But why must causality have a maximum speed?

And why is that speed the same as the speed of light?

To understand this, let's first get our heads around two of the most important insights in physics ever.

Ready?

First, 1632-- frilly collars, pilgrims in Plymouth, and in Italy, Galileo was about to be dragged off by the Inquisition for his book supporting Copernicus and the whole, Earth is not the center of the universe, thing.

But in his book, there's another, less well known idea-- his Principle of Relativity.

This is not Einstein's Relativity, but, instead, the brilliant precursor.

Not only is Earth or, indeed, any other location not special, but Galileo posits that no velocity is special, either.

To put it another way, all experiments should give the same results regardless of the velocity of your non-accelerating, or inertial, frame of reference.

This Galilean Relativity is an incredible insight that Isaac Newton would later codify into his Laws of Motion.

Fast forward to the 1800s-- top hats, steam trains, and mad experiments to uncover the laws of electricity and magnetism.

Enter James Clerk Maxwell, scientific maestro, who weaves these laws into his eponymous equations, describing the entire electromagnetic phenomenon with such elegance.

By the late 1800s, we have Maxwell's equations, Newton's mechanics, various other awesome theories.

And there's this sense that physics might be done, except there are hints of something horribly wrong lurking in the math-- actually, two hints.

The first clues to the bizarre quantum nature of reality had emerged.

And more importantly for this episode, Maxwell's equations had cast confusion on the sacred Galilean Relativity.

In fact, we now know that even Newton's mechanics were using assumptions that implied an infinite speed of light, which is really bad.

It would imply that space and time and matter don't exist.

But I'm getting ahead of myself.

First, let me explain the issue with Maxwell's equations.

Imagine a pony on roller blades with a monkey skateboarding along its back.

And make it an electric monkey.

Why?

Well, magnetism comes from moving electric charges.

So an electric skater monkey on a rollerblading pony generates a magnetic field, obviously.

And I can figure out the field strength from Maxwell's equations based on what I see is the monkey's total velocity.

But what is that velocity?

Galileo and Newton tell us that total monkey speed equals pony blade speed plus monkey skate speed.

But what if this very clever pony also solves Maxwell's equations?

She sees the monkey moving at only monkey skate speed, and so gets a totally different magnetic field.

So who's right, me or the pony?

The key lies in what we actually measure.

We don't measure magnetic field.

We measure its effect.

We measure force.

And the pony measures the same force that I do.

See, there's a velocity-dependent trade-off between the electric and magnetic fields.

The two work together to give you the same electromagnetic, the Lorentz, force, regardless of reference frame.

This tells us that the electromagnetic force holds clues to the fundamental interplay between space, time, and velocity.

How do we unravel that connection?

It's going to be encoded in the transformation that will allow Maxwell's equations to jump seamlessly between reference frames-- the transformation that represents space and time in our reality.

This transformation thing, it's like a mathy magic wand that you wave at your description of spacetime or your physical laws.

And it'll bump you between reference frames Harry Potter style.

Wingardium leviosa.

An example is the Galilean transformation, which basically says that velocities add together and space and time don't depend on velocity.

Newton's mechanics use it.

And we just applied it to Maxwell's equations to get total monkey speed.

But it turns out that there's no way to write out Maxwell's equations so that they give consistent results under the Galilean transformation.

They aren't invariant to that transformation.

They sort of give the right force at low speeds.

But the fields are a mess.

And at high speeds-- forget about it.

So does this mean Maxwell was wrong?

No.

It means that Galilean transformation is wrong.

The transformation underpinning Newton's mechanics is wrong.

The only transformation that works is called the Lorentz transformation.

And it was discovered even before Einstein's Relativity.

But it was Einstein who realized that the Lorentz transformation tells us how space and time are connected and that it also predicts the speed of causality.

Now, you can get to this transformation the way Lorentz and Einstein did by requiring a constant speed of light.

As an example, there's a link to the derivation via the spacetime interval in the description.

But forget about the speed of light.

This transformation is so profound that it is inevitable based on a few simple statements about the nature of space and time.

Let me show you how.

First, we're not going to pretend that we know how velocities add.

We don't know that total monkey speed equals pony blade speed plus monkey skate speed.

Why would you assume such as a thing?

Next, no preferred inertial reference frame.

Under our new transformation, the laws of physics will work the same regardless of position, orientation, or velocity.

It doesn't matter where the pony is, how fast it's going, or in what direction it's skating.

This must be true.

The Earth is whizzing around the sun, the sun around the Milky Way.

Position, orientation, and velocity are changing massively.

Yet our experiments don't seem to care about that.

Finally, assume that the universe make sense.

Require that we can consistently transform between reference frames.

I should be able to use the same transformation to get to the monkey's frame as I use to get to the pony's frame just by using the different velocities.

I should also be able to jump consistently through multiple frames of reference and back again.

I can go to the monkey's frame by first going to the pony's frame and then going from pony to monkey.

And I can also get back to my frame by putting a minus sign on the velocities.

Essentially, we're just requiring basic consistency in how the dimensions work.

Finally, finally-- use these axioms to do a teensy bit of algebra.

See the link in the description.

The result is the Lorentz transformation.

It's the only one that satisfies all of these pretty fundamental statements about the relativity, symmetry, and consistency of our universe.

It must describe our reality.

And therefore, there must be a cosmic speed limit.

Why?

This absolute speed limit-- let's call it c-- is the one parameter defining the Lorentz transformation.

Through this parameter, the Lorentz transformation predicts the cosmic speed limit.

Now, the Galilean transformation turns out just to be a special case of the Lorentz transformation where c equals infinity.

And honestly, just from the symmetry and relativity arguments that we made, c could be infinity.

But for other reasons-- still unrelated to light-- we know that it cannot be.

See, the Lorentz transformation finally allows us to write down a version of Maxwell's equation that is invariant to transformation.

With it, we can write down one law for electromagnetism that works in all frames of reference.

This is further evidence that our new transformation accurately describes our reality.

But it only works for a very specific value of c. That value has to be a combination of the fundamental constants of Maxwell's equations.

For the laws of electricity and magnetism to work, we need a finite maximum cosmic speed, even without considering light.

But check this out.

The exact same combination that gives us the cosmic speed limit also happens to define the speed of propagation of electromagnetic waves-- the speed of light.

c is the speed of light.

But it's the speed of causality first.

It's the maximum speed at which any two parts of the universe can talk to each other.

In fact, it's the maximum speed at which any observers can see two parts of the universe talk to each other.

Because of this, it's the only speed that any massless particle can travel.

So lights or photons, also gravitational waves and gluons all have no mass.

And so they travel at the maximum possible speed.

Mass is an impediment to motion.

No mass, no impediment.

So massless things go as fast as it's possible to go.

In fact, the very existence of mass and space and time tells us that the universal speed limit is finite.

Einstein's interpretation of the meaning of the Lorentz transformation gives us the special theory of relativity-- time dilation, length contraction, and, of course, mass to energy equivalence, as described by the famous equation, E equals mc squared.

Awesome episode on that one here.

These are unavoidable conclusions once we have the basic relationship between space and time as described by the Lorentz transformation and we accept Einstein's interpretation of it.

So what happens without a universal speed limit and we pretend c equals infinity?

There is no matter.

Because it would take infinite energy to make any mass.

There is only massless particles traveling at infinite speed.

Time dilation and length contraction are infinite.

There is no time and space, no cause or effect, because all locations and times communicate with each other instantly.

The universe is an infinitesimal here-and-now This is all pretty paradoxical.

And so there are, by definition, inconsistencies in this picture.

However, the paradox itself tells us that an infinite speed limit is impossible.

The finite speed of causality is fundamental to us having a universe in the first place.

And we want a universe so I can see you back here on the next episode of "SpaceTime."