Практика Shadowing: Electromagnetism Derived From Geometry! - Изучайте разговорный английский по видео

You might know that gravity is geometry.

Mass curves spacetime, and that curvature is what we experience as gravity.

But what if I told you that electromagnetism, force responsible for the light,

electricity, magnetism, chemistry, and basically everything else you experience every day, that is also geometry.

In modern physics, electric and magnetic fields can be thought as a kind of curvature.

Just not the curvature of space-time, but the curvature of a strange geometric object attached to every single point in space.

If that sounds bizarre, it gets even stranger.

Following this idea will eventually lead us to the possibility of extra dimensions, string theory, and one of the deepest questions in all of physics.

Could all of the fundamental forces of nature be unified by geometry?

But first, let's see how electromagnetism emerges from from nothing more than symmetry and geometry.

Let's start with something more familiar, electric potential.

It's the same thing you see on batteries, except for batteries, it measures the potential difference from one end of battery to the other.

The electric potential is a scalar function, and for a battery where we only consider one dimension along the length of the battery, it might look like this.

If I put a positive charge on one end, it's going to move down to the lower potential, like a ball rolling down the hill.

So it moves that way, producing a current.

The rate of change of this electric potential gives us a vector field, which we know as the electric field.

You may have seen this rolling ball analogy before, and it's incomplete, but it does show us something useful, what we physicists call a gauge symmetry.

What is the physical reality here?

What do we actually care about?

Is it the electric field, or is it the electric potential?

The force we experience is directly proportional to the electric field, but remember, the electric field only cares about changes of the potential, or the slope of these hills,

not the exact value of the potential itself.

That means any constant scalar addition to my potential, like raising the absolute elevation of these hills, does not affect the physical reality.

This is called a gauge symmetry.

The physical reality didn't change, just the way that you described it did.

Electromagnetism hinges on this particular idea.

Now, I want you to visualize a more complicated geometry.

At every point in spacetime, you have a little circle and an arrow.

The arrow winds around the circle, and as I move through spacetime, the arrow may end up in a different place than when it was before.

This is hard to visualize, so let's use a lower-dimensional analogy.

Consider a string of finite radius.

Spacetime is projected to the length of the string.

Imagine a little ant walking along the string.

Not only can it walk along the length, but there exists a little circular dimension along the circumference of the string.

As the ant moves through space, it might also move along this circular part, which we track by its phase.

Now, here's the catch.

The physical reality does not care about the particular value of the phase at any point.

We only observe the movement along the length of the string.

That means any math we have to describe the ant moving along the string must be gauged symmetric to the phase.

To see this more clearly, we need some math.

All of the necessary physics can be derived from the action.

What is an action?

The action describes an object's Lagrangian density integrated through spacetime as it moves along a path.

The Lagrangian density is built upon different notions of energy density, but it itself is not energy.

I'll do a brief overview here, but if you're interested in learning more about action in Lagrangian mechanics, I highly recommend Veritasium's video.

But back to the action.

When a particle moves through spacetime, depending on the specific path it takes, the action takes different values.

We can thus think of the action as a number that labels every possible path an object can take.

Nature is very efficient and stable, so classically it chooses the path that is insensitive to small variations, or stationary.

Before we actually perform this variation to find the correct path, we need to consider all kinds of energy density that the Lagrangian density contains.

I'm going to hand wave a little bit here, sorry mathematicians, because I don't want to delve into the full quantum mechanics.

If you are interested, I highly recommend the videos on Richard Bellio's channel.

Here's the gist.

In quantum field theory, rather than describing the motion of individual particles, we consider a field of the probability to find the particle at any particular region.

The Lagrangian density of this field is given by this equation here.

The exact nature of this psi and gamma isn't too important for our discussion now.

What matters is that psi, which describes the probability density, depends on the specific phase at any point in spacetime, but the magnitude, or probability amplitude,

found by taking the the complex conjugate and multiplying doesn't.

This is the symmetry we were talking about.

However, this is a partial derivative, and the index is a placeholder that represents individual coordinates, in our case x, y, z, and t.

If I take a regular partial derivative of the wave function, I get something dependent on derivatives of the phase.

For some intuition, consider our string again and draw a straight line along the string.

This describes a path in which the object is subject to no forces.

It just moves in a nice straight line at a constant velocity.

Now if we add a twist, the line twists as well, even though it should just be straight.

The line curving is an artifact of the space it lies on, rather than it actually being a curved line.

As an example, if you've ever seen a flight map on a plane, it looks like they've taken a curved path rather than a straight one.

They are taking a straight path, more or less, it's just that the background we live on, that being Earth, is curved and not flat, regardless of what people on line say.

want to quantify the actual derivative of the line, not due to the background.

So rather than doing a regular derivative, we need to take a covariant derivative.

Covariant means it measures the physical change, while accounting for this background geometry.

In our case, the covariant derivative is defined as the regular derivative with this new contribution that cancels out the unwanted background.

This effectively unwinds our string so that we can measure the actual change.

To make this derivative of the phase and this covariant derivative pop out, we have to give it some structure.

So we postulate that there exists a field A, which we'll call the vector potential for reasons we'll see later.

This field transforms in a particular way to local phase changes, such that our covariant derivative remains gauged symmetric.

This field essentially tells us how spacetime connects to this circular geometry, which is why in math this is often called a connection.

Now plugging in my covariant derivative into the Lagrangian density, I find a new interaction term between the particle and the vector potential.

The quantity multiplying the vector potential is called the forecurrent, and again we'll see why later.

But we're still missing something.

Remember, you can roughly think of these terms in the action corresponding to different types of energy densities.

We have energy of the charge existing at all, the energy of the charge interacting with the vector potential, but what about the energy of the vector potential itself?

Remember our analogy, the string can't just be twisted at a small localized point, but it instead propagates throughout the length of the string.

Since the default state is an untwisted string, it took energy to get the string all twisted, and thus there exists an energy density associated not with a particle, but with the twisting of the string.

How do we quantify this background twisting energy density?

Think of a 2D surface.

At each point, we track its phase.

Consider a tiny loop in the space, such that we move a little bit in x, a little bit in y, and then back in x and back in y.

At each step, the phase can change.

If we consider only very tiny changes, we only need to consider up to linear order.

So first, we have the the x component of a along a tiny piece of x.

Then we have to move up a little piece in y.

We also have to consider how the y component of a had changed as we moved along the x direction.

If we repeat this throughout the loop, we find that everything cancels out except for this term.

Normally, you would think that if I move a phase across the space in a loop, it would turn to the same value, but this is not the case.

As an analogy to why this term may not vanish, consider a vector on a sphere.

Let's consider a loop defined by moving along on the equator 90 degrees, then moving up to the North Pole, then moving back down, keeping the vector parallel to the surface at all times.

We find that the vector has rotated 90 degrees, despite returning to the same point.

This happened because the surface that the vector has moved on is curved.

100

So this term measures curvature, and due to the nature of twisting, it may not actually vanish.

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You may have heard of this concept of curvature before in physics, for example in general relativity, where matter curves spacetime.

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In fact, this is a direct analogy.

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Charge acts as a source for the geometric structure of the phase, just as mass acts as a source for spacetime curvature and general relativity.

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This curvature gives a kind of energy density at that point.

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But why might curvature be related to energy?

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For a mechanical interpretation, consider a spring.

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A spring's energy, in the ideal first-order case, is proportional to how much it's stretched squared.

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The square assures positive energy, such that the energy of displacement stays the same regardless of whether it's stretched or compressed.

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This stretching or compressing is somewhat like curvature, in the sense that curvature is displacement from the geometric object's flat state.

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Thus, if our geometric object were something physical with a kind of tension, like a spring, the energy, at least to first order, is proportional to quadratic deviations from flat, or given by this expression.

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This is also the simplest gauge invariant term we could choose.

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variance means that applying the covariant derivative instead does not affect this quantity.

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In fact, physicists today motivate this not by thinking that this geometry is physical, but by Occam's razor.

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It's simply the simplest.

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So ignoring the quantum part, our total action is now given by this interaction term, which roughly corresponds to the energy of the four-current interacting with the phase,

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and the geometry itself which the four-current creates.

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This means if there exists a current, which is like a little localized twist, that twist, due to the nature of the geometry must propagate outward to the entire rest of space-time,

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just like how a charge induces electric field in the rest of space.

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Let's now actually solve this.

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The forecurrent is an initial condition, so we can't change or vary the forecurrent.

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So what we need to solve is the vector potential that nature will choose, such that the action is stationary.

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I'll spare you from the variational calculus and simplifications, which you can find in my notes, linked below.

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But what you are left with is that the derivative of the curvature is proportional to the four current.

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After all that work, something remarkable happens.

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The equation that tells the curvature how to respond to charge is not a new law of physics at all.

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It's Maxwell's equations, which describe all of electromagnetism.

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Or rather, only two of Maxwell's equations, but the other two will come later.

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I guess it doesn't really look like electromagnetism though.

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Where's the electric field or magnetic fields?

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They're hiding in there, don't worry.

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I'm going to label this curvature with an antisymmetric matrix, or more accurately, tensor.

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Antisymmetry means that if I swap the indices, it turns negative.

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We'll eventually discover what each of these terms in the tensor mean.

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Let's first consider the derivatives along the time component.

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Then, if I treat this as a vector, I get that the divergence of the time component of the curvature equals the time component of the forward current density.

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The time component of the forward current density is just the charge density, the amount of charge that is only moving through time.

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So the divergence of this vector is proportional to the charge density.

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This is Gauss's law.

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The same thing that gives the famous inverse square law of a point charge.

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So these components of the curvature tensor are actually electric field components in disguise.

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Now let's consider the derivative along the spatial components.

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On the source side, I get the three current density, so the charge is moving through space.

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Taking all the derivatives on this side, I can group the electric field terms.

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This has a derivative of time.

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The rest of this is a mess, but if I label the terms in the tensor by magnetic field components in this specific way, I get that this is just the curl of the magnetic field.

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This is Ampere's law.

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If we relate each component of the vector potential back to the curvature tensor, we find that the time component of the vector potential is the electric potential.

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So the electric field is defined by the gradient of the time component of the vector potential

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and the time derivative of the spatial components of the vector potential.

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Similarly, we find that the magnetic field is defined by the curl of the spatial components of the vector potential.

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Since the divergence of a curl is always zero, which you can see here by the symmetry of mixed partial derivatives,

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this means the divergence of the magnetic field is always zero, which is Gauss's law for magnetism.

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Now, if we calculate the curl of the electric field, the curl of a gradient is also zero, for the same reason as above.

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We can switch the order here and do the curl first, and then the time derivative.

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But the curl of the spatial components of the vector potential is just the magnetic field.

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So we find that the curl of the electric field is related to the time derivative of the magnetic field.

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This is Faraday's law of induction.

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There you have it!

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All of Maxwell's laws have popped out of the geometry, specifically the curvature of this geometric object which has the symmetry of a circle.

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Everything we've called electric and magnetic fields is really part of this one object and its curvature.

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Electric and magnetic fields are not fundamental objects at all.

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They're different faces of the same underlying geometry, which is why they are unified into electromagnetism.

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We can also derive the Lorentz-Force Law in local charge conservation.

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For the sake of time, I won't go through the derivation.

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You can find it on Richard Bellio's video.

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I recommend giving it a try.

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That gives all of electromagnetism.

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It's all geometry, just the geometry of a very weird object that twists and curves in ways that obey this circular symmetry.

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I'm being a little fast and loose with my terminology here though.

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In modern physics, this is not an actual, tangible geometric object.

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In quantum field theory, this geometric object is identified with an electromagnetic gauge field, which is then quantized to support the existence of photons.

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You can convince yourself

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that this has something to do with light by solving the variation with only the curvature term

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and no charges to source it.

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It doesn't necessarily mean that the curvature tensor is zero.

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There can be oscillations of magnetic and electric fields, or light.

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So that's how quantum field theory is understood.

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But what if this curvature is not a separate field, but a part of the spacetime geometry that particles can also move along?

179

Remember our string example?

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The circle around the string was a mathematical object for visualization.

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But what if it were real?

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If the radius of the string is small enough, as we zoom out, it looks like a one-dimensional object.

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Thus, our four-dimensional spacetime could actually be five-dimensional, but rather than flat, it has a tiny curled-up or compactified dimension,

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in which the geometry of this dimension is what causes electromagnetism.

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This is what Calusa and Klein proposed 100 years ago, and it works to a certain degree.

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There are some additional pieces, and it's not clear what the radius of the Curled-up dimension should be, and we certainly haven't been able to detect it.

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Tiny Curled-up dimensions we haven't been able to detect?

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Sounds a lot like string theory, and indeed when extra dimensions later appeared naturally in string theory, Calusa and Klein's old idea became relevant again.

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Technically, we haven't said anything about particles being strings, but suppose there really is a tiny circular extra dimension.

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To an observer small enough to explore it, a particle's position around that circle would be a genuine physical coordinate.

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In electromagnetism, however, the phase is not observable.

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Only differences in phase matter.

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One way to picture this is to imagine replacing a particle with a tiny loop of string

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that wraps all the way around the circular dimension.

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Rotating the string around the circle doesn't change the string itself, so there's no way to distinguish between one orientation from another.

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That's exactly the kind of redundancy that gauge symmetry describes.

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There's another reason why this is so appealing and why there's so much hype around string theory.

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Recall that the Lagrangian density of electromagnetism is given by its curvature.

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And in gravity, the Lagrangian density is also given by its curvature.

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We typically think of these curvatures as being of different things, but what if there were a single geometric object where these are just different facets of curvature on that object?

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This is the question that all these unified field theories or unified geometric theories, like string theory, ask.

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Although, now, we know there exist other forces, with their own symmetries and coveratures.

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The question is then, what is the geometric object that describes our universe?

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What does it look like?

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Fortunately for string theory, and why it was so exciting, is that we've been able to find such objects in which gravity and electromagnetism,

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and even sometimes other gauge interactions, pop out.

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Unfortunately for string theory, we've not found just one, but an infinite number of them that might work, and no experimental evidence to support which one is correct.

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Whether or not string theory turns out to be correct, one of the deepest lessons of modern physics remains.

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Gravity, electromagnetism, and even the existence of forces themselves.

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They all appear to be different reflections of the same underlying mathematical structure.

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Symmetry, geometry, and curvature.

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At least to me it's really beautiful how an astonishing amount of modern physics appears to arise from these mathematical abstractions.

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Thanks for watching.

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If you'd like to see more detailed notes, check out the link in the description.

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This video is also a submission to the 2026 Summer of Math Exposition.

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If you'd like to support me in the competition by ranking my video, you can check out how below.

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