シャドーイング練習: Electromagnetism Derived From Geometry! - 動画で英語スピーキングを学ぶ

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.

101

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

102

In fact, this is a direct analogy.

103

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.

104

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

105

But why might curvature be related to energy?

106

For a mechanical interpretation, consider a spring.

107

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

108

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

109

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

110

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.

111

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

112

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

113

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

114

It's simply the simplest.

115

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,

116

and the geometry itself which the four-current creates.

117

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,

118

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

119

Let's now actually solve this.

120

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

121

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

122

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

123

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

124

After all that work, something remarkable happens.

125

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

126

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

127

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

128

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

129

Where's the electric field or magnetic fields?

130

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

131

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

132

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

133

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

134

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

135

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.

136

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

137

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

138

This is Gauss's law.

139

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

140

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

141

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

142

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

143

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

144

This has a derivative of time.

145

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.

146

This is Ampere's law.

147

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.

148

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

149

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

150

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

151

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

152

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

153

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

154

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

155

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

156

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

157

This is Faraday's law of induction.

158

There you have it!

159

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.

160

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

161

Electric and magnetic fields are not fundamental objects at all.

162

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

163

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

164

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

165

You can find it on Richard Bellio's video.

166

I recommend giving it a try.

167

That gives all of electromagnetism.

168

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

169

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

170

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

171

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

172

You can convince yourself

173

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

174

and no charges to source it.

175

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

176

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

177

So that's how quantum field theory is understood.

178

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?

180

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

181

But what if it were real?

182

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

183

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

184

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

185

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

186

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.

187

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

188

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.

189

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

190

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

191

In electromagnetism, however, the phase is not observable.

192

Only differences in phase matter.

193

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

194

that wraps all the way around the circular dimension.

195

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

196

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

197

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

198

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

199

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

200

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?

201

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

202

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

203

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

204

What does it look like?

205

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,

206

and even sometimes other gauge interactions, pop out.

207

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.

208

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

209

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

210

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

211

Symmetry, geometry, and curvature.

212

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

213

Thanks for watching.

214

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

215

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

216

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

✨ おすすめ動画

279 文

電磁気学と幾何学の関係を学びながら英語力を向上！

このビデオでは、重力が幾何学であることから始まり、電磁気学も同じく幾何学的な曲率から生まれるという驚くべき考え方が説明されています。難しい物理学の概念ですが、話し手の明快な語り口で理解しやすくなっています。このような学術的な内容のビデオは、英語シャドーイングの良い教材です。専門用語も含めてリスニングとスピーキングのスキルを磨くことができます。

日常会話に使える5つのフレーズ

"What if I told you that..."（もし～だと言ったらどう思いますか？） - 相手の興味を引く時に使える表現です。
"It's the same thing you see on..."（～で見るのと同じものです） - 身近な例で説明する時に便利です。
"That means..."（つまり～です） - 考えをまとめる時の定番フレーズです。
"To see this more clearly..."（これをもっとはっきり見るために～） - 説明を深める時に使います。
"Nature is very efficient..."（自然は非常に効率的です） - 普遍的な事実を述べる時の表現です。

ビデオを使ったシャドーイングのステップバイステップガイド

このビデオは少し難しいですが、shadow speakの技法を使えば効果的に学べます。まず、1分程度のセクションを選び、字幕なしで聞いて全体的な流れを把握しましょう。次に、字幕を見ながら1文ずつ繰り返し聞き、発音とイントネーションを覚えます。そして、話し手の後ろについてshadowspeaksを行い、リズムを合わせます。最後に、字幕を隠して自分で話してみて、どこが難しかったかをチェックします。この方法を続けると、動画で英語学習の成果がすぐに現れます。

物理学の面白さを楽しみながら、英語のスキルも上達させましょう。shadow speechは継続が大切ですが、少しずつ進めれば必ず成果があります！

この動画の文法

話し手がよく使っている文型を、動画の実際の表現とともに紹介します。

文型	動画での表現
現在完了形 have/has + 過去分詞 — 過去の出来事が今も関係している	you've ever seen · has rotated · has moved
受動態 be + 過去分詞 — 誰がするかより、何が起きるかに焦点を当てる	can be thought · be unified · is called
関係詞節 who / which + 節 — 人や物について情報を加える	geometry, which is · current, which is · zero, which is

シャドーイングとは？英語上達に効果的な理由

シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。

シャドーイングのやり方: ステップ別の完全ガイドを読む →

シャドーイング練習: Electromagnetism Derived From Geometry! - 動画で英語スピーキングを学ぶ

LET THERE BE... Voltage? | Maxwell's Equation #2 Explained for Beginners

Your Skeleton Is Electric

A mind-blowing explanation of symmetry | Frank Wilczek for Big Think

Superfluid Dark Matter

Vectors and 2D Motion: Crash Course Physics #4

Has quantum mechanics proved that reality does not exist?

Mathematics and Physics | Lesson 2 | Understanding the Connection

What Actually Drives Attraction (According to Psychology)

1 6 What Illusions Teach Us

How Long Can You Balance a Pencil?

What science can’t answer, according to physicist Jim Al-Khalili | Big Think

Introduction to Optoelectronics and Photonics

電磁気学と幾何学の関係を学びながら英語力を向上！

日常会話に使える5つのフレーズ

ビデオを使ったシャドーイングのステップバイステップガイド

この動画の文法

シャドーイングとは？英語上達に効果的な理由