Spacetime in Geometric Algebra

The right way to look at
Spacetime and Maxwell's Equations

(or should I say "equation")

Maxwell's equations Maxwell's equations

Not so much taught at school, but there is a 4-dimensional version

The 4-vectors:
Potential \(A_\mu=(\phi, -\overrightarrow{A})\)
Charge \(J^\mu=(\rho, \overrightarrow{J})\)
Differential \(\partial_\mu=\frac{\partial}{\partial x^\nu}=(\frac{\partial}{\partial t}, \overrightarrow{\nabla})= (\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})\)

Define the rank-2 4-form
Faraday field \(F_{\mu\nu}= \partial_\mu A_\nu - \partial_\nu A_\mu=\) \[= \begin{bmatrix} 0 & \frac{\partial \phi}{\partial x} + \frac{\partial A_x}{\partial t} & \frac{\partial \phi}{\partial y} + \frac{\partial A_y}{\partial t} & \frac{\partial \phi}{\partial z} + \frac{\partial A_z}{\partial t} \\ -\frac{\partial \phi}{\partial x} - \frac{\partial A_x}{\partial t} & 0 & -\frac{\partial A_x}{\partial y} + \frac{\partial A_y}{\partial x} & -\frac{\partial A_z}{\partial x} + \frac{\partial A_x}{\partial z} \\ -\frac{\partial \phi}{\partial y} - \frac{\partial A_y}{\partial t} & \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} & 0 & -\frac{\partial A_z}{\partial y} + \frac{\partial A_y}{\partial z}\\ -\frac{\partial \phi}{\partial z} - \frac{\partial A_z}{\partial t} & \frac{\partial A_z}{\partial x} - \frac{\partial A_x}{\partial z} & \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} & 0 \end{bmatrix}= \begin{bmatrix} 0 & -E_x & -E_y & -E_y \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & =B_x\\ E_z & -B_y & B_x & 0 \end{bmatrix} \]

And the Dual field \(F^*_{\alpha\beta}=\frac{1}{2}\epsilon_{\alpha\beta\gamma\delta} F^{\gamma\delta}\)

The final 2 Maxwell's equations: \[DF=J\]\[DF^*=0\]

Introduction

The original Maxwell's equations are often seen as complex and unintuitive.

Compare with the equations that we currently teach in school. More modern formulations that are more compact and elegant.

\[\overrightarrow{\nabla} \cdot \overrightarrow{E} = \frac{\rho}{\epsilon_0}\]

\[\overrightarrow{\nabla} \cdot \overrightarrow{B} = 0\]

\[\overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{\partial \overrightarrow{B}}{\partial t}\]

\[\overrightarrow{\nabla} \times \overrightarrow{B} = \mu_0 \overrightarrow{J} + \mu_0 \epsilon_0 \frac{\partial \overrightarrow{E}}{\partial t}\]

and some more.

\[\frac{\partial \rho}{\partial t} + \overrightarrow{\nabla} \cdot \overrightarrow{J} = 0\]

\[\overrightarrow{B} = \overrightarrow{\nabla} \times \overrightarrow{A} \qquad \overrightarrow{E} = -\overrightarrow{\nabla} \phi - \frac{\partial \overrightarrow{A}}{\partial t}\]

The final 2 Maxwell's equations: \[DF=J\]\[DF^*=0\]

This is not just a mathematical trick. It reveals the underlying geometric structure of electromagnetism and spacetime.

We use Vectors, but we also multiply them. So this is more than a vector space, it is an Algebra, a Geometric Algebra.

\[\overrightarrow{\nabla} \cdot \overrightarrow{E} = \frac{\rho}{\epsilon_0}\]

\[\overrightarrow{\nabla} \times \overrightarrow{B} - \mu_0 \epsilon_0 \frac{\partial \overrightarrow{E}}{\partial t}= \mu_0 \overrightarrow{J}\]

\[\overrightarrow{\nabla} \cdot \overrightarrow{B} = 0\]

\[\overrightarrow{\nabla} \times \overrightarrow{E} + \frac{\partial \overrightarrow{B}}{\partial t} = 0\]

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\[\alpha 1=(e1+e2)(e_1+e_2) =\]\[e_1e_1 + e_1e_2 + e_2e_1 + e_2e_2 = \]\[2 + e_1e_2 + e_2e_1\]

Geometric Algebra

Space Algebra

An algebra is a vector space with a multiplication \(W=VU\)

Associative? \((ab)c=a(bc)\), Commutative? \(ab=ba\), Distributive? \((a+b)c=ac+bc\)

Add multiplication to a vector space \(e_1,e_2, e_3\)

We want \(e_ie_i\) to be a multiplication scalar (marked as 1), and a new member of the algebra

We actually want any combination VV to ba a scalar

This leads to \(e_1e_2=-e_2e_1\)

But \(e_1e_2\) is a new member of the algebra,
define \(e_{12}\equiv e_1e_2\)

Also from \(e_1 e_{23}\) we get \(e_{123}\)

\[1\]
\[e_1\]
\[e_2\]
\[e_3\]
\[e_{23}\]
\[e_{31}\]
\[e_{12}\]
\[e_{123} = I\]

Mechanism of Space Algebra Multiplication

Let's practice some simple multiplications

\[e_{12}e_{23} = e_{13} = - e_{31}\]

\[e_{23}e_{13} = -e_{23}e_{31} = - e_{21} = e_{12}\]

\[e_{31}e_{123} = e_{3}e_{23} = - e_{2}\]

And also complex multiplications

\[(\alpha e_{1} + \beta e_{12})(\gamma e_{3} + \delta e_{23}) =\]

\[= \alpha \gamma e_{1}e_{3} + \alpha \delta e_{1}e_{23} + \beta \gamma e_{12}e_{3} + \beta \delta e_{12}e_{23} = \]

\[= (\alpha \gamma + \beta \delta) e_{13} + (\alpha \delta + \beta \gamma) e_{123}\]

The product of two vectors A and B

\[AB =(A_1 e_1 + A_2 e_2 + A_3 e_3)(B_1 e_1 + B_2 e_2 + B_3 e_3) = \]

\[= (A_1 B_1 + A_2 B_2 + A_3 B_3) 1 + \]

\[+ (A_1 B_2 - A_2 B_1) e_{12} + (A_2 B_3 - A_3 B_2) e_{23} + (A_3 B_1 - A_1 B_3) e_{31}\]

Symmetric and antisymmetric parts

\[\overrightarrow{A} \overrightarrow{B}=\overrightarrow{A} \cdot \overrightarrow{B} + \overrightarrow{A} \wedge \overrightarrow{B}\]

\[(\overrightarrow{A} \times \overrightarrow{B} = - I (\overrightarrow{A} \wedge \overrightarrow{B}))\]

What is multiplied here?

Look again at a complex multiplication \((\alpha e_{1} + \beta e_{12})(\gamma e_{3} + \delta e_{23}) =\)
what can the coefficients \(\alpha, \beta, \gamma, \delta\) be?
what will \(\alpha \gamma e_{1}e_{3}\) be?
If they are just numbers, then the multiplication coefficient is just a number \(\alpha \gamma\).
If they are functions of some variables, then the multiplication is a function of those variables.
If they are operators, then they operate on the other element (like \(\partial_x sin(x)\)).
So... \((\partial_x e_1 + \partial_y e_2 + \partial_z e_3)(E_x e_1 + E_y e_2 + E_z e_3)\) is really...
\((\partial_x E_x + \partial_y E_y + \partial_z E_z)1\) which is a scalar! We can call it \(\overrightarrow{\nabla} \cdot \overrightarrow{E}\)! or \(Div(\overrightarrow{E})1\)
But there is also the Anti-symmetric part...
So \(\overrightarrow{\nabla} \overrightarrow{E} = \overrightarrow{\nabla} \cdot \overrightarrow{E} + \overrightarrow{\nabla} \wedge \overrightarrow{E} = Div(\overrightarrow{E})1 - Curl(\overrightarrow{E})I \)

Space Algebra Representation

Blade 0 - Scalar
Pressure, density, Energy, Charge
Blade 1 - Vectors
Momentum, Velocity, Electric field, Magnetic potential, Nabla
Blade 2 - Bivectors (pseudovectors)
Angular momentum\((L=r\times p)\), Magnetic field\((B=\nabla\times A)\), Vorticity\((W=\nabla\times v)\)
Blade 3 - Trivectors (pseudo-scalars)
Helicity\((H=v\cdot W=v \cdot (\nabla \times v))\)
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\[e_{31}\]
\[e_{12}\]
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Spacetime Algebra

Good for representing spacetime physics, including Maxwell's equations

\[1\]
\[\gamma_0\]
\[\gamma_1\]
\[\gamma_2\]
\[\gamma_3\]
\[\gamma_{10}\]
\[\gamma_{20}\]
\[\gamma_{30}\]
\[\gamma_{23}\]
\[\gamma_{31}\]
\[\gamma_{12}\]
\[\gamma_{123}\]
\[\gamma_{230}\]
\[\gamma_{310}\]
\[\gamma_{120}\]
\[\gamma_{0123} =I\]
10
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i0
0-i
0iσx
-iσx0
0iσy
-iσy0
0iσz
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0σx
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0σy
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Electromagnetic Field Equation

The simplest spacetime theory

Start with a vector \(A\)

Differenciate it \(\nabla A=F\) to get the electromagnetic field bivector

Differentiate that \(\nabla F = J\) to get the source vector

\(\nabla F = J\) is the one and only Maxwell's equation

The Electromagnetic Field

\(A=(A_0, \overrightarrow{A}) = e_0 A_0 + e_1 A_1 + e_2 A_2 + e_3 A_3\)

\(\nabla=(e_0 \frac{\partial}{\partial t}, \overrightarrow{\nabla}) = e_0 \frac{\partial}{\partial t} + e_1 \frac{\partial}{\partial x} + e_2 \frac{\partial}{\partial y} + e_3 \frac{\partial}{\partial z}\)

\[ \definecolor{lRed}{RGB}{255, 120, 120} \definecolor{lBlue}{RGB}{100, 180, 255} F = \nabla A = \begin{matrix} \color{green} e_0 e_0 \frac{\partial A_0}{\partial t} & + \color{lRed} e_0 e_1 \frac{\partial A_1}{\partial t} & + \color{lRed} e_0 e_2 \frac{\partial A_2}{\partial t} & + \color{lRed} e_0 e_3 \frac{\partial A_3}{\partial t} \\ + \color{lBlue} e_1 e_0 \frac{\partial A_0}{\partial x} & + \color{green} e_1 e_1 \frac{\partial A_1}{\partial x} & +\color{lRed} e_1 e_2 \frac{\partial A_2}{\partial x} & + \color{lBlue} e_1 e_3 \frac{\partial A_3}{\partial x} \\ + \color{lBlue} e_2 e_0 \frac{\partial A_0}{\partial y} & + \color{lBlue} e_2 e_1 \frac{\partial A_1}{\partial y} & + \color{green} e_2 e_2 \frac{\partial A_2}{\partial y} & + \color{lRed} e_2 e_3 \frac{\partial A_3}{\partial y} \\ + \color{lBlue} e_3 e_0 \frac{\partial A_0}{\partial z} & + \color{lRed}e_3 e_1 \frac{\partial A_1}{\partial z} & + \color{lBlue} e_3 e_2 \frac{\partial A_2}{\partial z} & + \color{green} e_3 e_3 \frac{\partial A_3}{\partial z} \end{matrix}\]

\( -{\color{green}\frac{\partial A_0}{\partial t}} + {\color{green}\frac{\partial A_1}{\partial x}} + {\color{green}\frac{\partial A_2}{\partial y}} + {\color{green}\frac{\partial A_3}{\partial z}} = -\dot{A_0} + \nabla \cdot A = 0\) (Lorentz condition)

\( e_0 e_1 \left({\color{lRed}\frac{\partial A_1}{\partial t}} - \color{lBlue}\frac{\partial A_0}{\partial x}\right) + e_0 e_2 \left({\color{lRed}\frac{\partial A_2}{\partial t}} - \color{lBlue}\frac{\partial A_0}{\partial y}\right) + e_0 e_3 \left({\color{lRed}\frac{\partial A_3}{\partial t}} - \color{lBlue}\frac{\partial A_0}{\partial z}\right) + e_1 e_2 \left({\color{lRed}\frac{\partial A_2}{\partial x}} - \color{lBlue}\frac{\partial A_1}{\partial y}\right) + e_1 e_3 \left({\color{lRed}\frac{\partial A_3}{\partial x}} - \color{lBlue}\frac{\partial A_1}{\partial z}\right) + e_2 e_3 \left({\color{lRed}\frac{\partial A_3}{\partial y}} - \color{lBlue}\frac{\partial A_2}{\partial z}\right) \)

\[F = e_0 e_1 E_x + e_0 e_2 E_y + e_0 e_3 E_z + e_1 e_2 B_x + e_1 e_3 B_y + e_2 e_3 B_z\]

\[\overrightarrow{E}= \frac{\partial \overrightarrow{A}}{\partial t} - \overrightarrow{\nabla}A_0\quad \overrightarrow{B} = \overrightarrow{\nabla} \times \overrightarrow{A}\]

The Field Equation

\[F = e_0 e_1 E_x + e_0 e_2 E_y + e_0 e_3 E_z + e_1 e_2 B_x + e_1 e_3 B_y + e_2 e_3 B_z\]

\[ \definecolor{lRed}{RGB}{255, 120, 120} \definecolor{lBlue}{RGB}{100, 180, 255} \nabla F = \begin{matrix} e_0 e_0 e_1 \frac{\partial E_x}{\partial t} & e_0 e_0 e_2 \frac{\partial E_y}{\partial t} & e_0 e_0 e_3 \frac{\partial E_z}{\partial t} & e_0 e_2 e_3 \frac{\partial B_x}{\partial t} & e_0 e_3 e_1 \frac{\partial B_y}{\partial t} & e_0 e_1 e_2 \frac{\partial B_z}{\partial t} \\ \color{yellow} e_1 e_0 e_1 \frac{\partial E_x}{\partial x} & e_1 e_0 e_2 \frac{\partial E_y}{\partial x} & e_1 e_0 e_3 \frac{\partial E_z}{\partial x} & \color{red} e_1 e_2 e_3 \frac{\partial B_x}{\partial x} & e_1 e_3 e_1 \frac{\partial B_y}{\partial x} & e_1 e_1 e_2 \frac{\partial B_z}{\partial x} \\ e_2 e_0 e_1 \frac{\partial E_x}{\partial y} & \color{yellow} e_2 e_0 e_2 \frac{\partial E_y}{\partial y} & e_2 e_0 e_3 \frac{\partial E_z}{\partial y} & e_2 e_2 e_3 \frac{\partial B_x}{\partial y} & \color{red} e_2 e_3 e_1 \frac{\partial B_y}{\partial y} & e_2 e_1 e_2 \frac{\partial B_z}{\partial y} \\ e_3 e_0 e_1 \frac{\partial E_x}{\partial z} & e_3 e_0 e_2 \frac{\partial E_y}{\partial z} & \color{yellow} e_3 e_0 e_3 \frac{\partial E_z}{\partial z} & e_3 e_2 e_3 \frac{\partial B_x}{\partial z} & e_3 e_3 e_1 \frac{\partial B_y}{\partial z} & \color{red} e_3 e_1 e_2 \frac{\partial B_z}{\partial z} \\ \end{matrix}\]

\[ e_0\left(\color{yellow}\frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z}\color{black}\right) + e_1\left(-\frac{\partial E_x}{\partial t} + \frac{\partial B_z}{\partial y} - \frac{\partial B_y}{\partial z}\right) + e_2\left(-\frac{\partial E_y}{\partial t} + \frac{\partial B_x}{\partial z} - \frac{\partial B_z}{\partial x}\right) + e_3\left(-\frac{\partial E_z}{\partial t} + \frac{\partial B_y}{\partial x} - \frac{\partial B_x}{\partial y}\right) \] \[\overrightarrow{\nabla}\cdot\overrightarrow{E} + \left(\overrightarrow{\nabla}\times\overrightarrow{B}- \frac{\partial \overrightarrow{E}}{\partial t}\right)= J = \left(\rho, \overrightarrow{J}\right)\]

\[ e_{123}\left(\color{red}\frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z}\color{black}\right) + e_{230}\left(-\frac{\partial B_x}{\partial t} + \frac{\partial E_y}{\partial z} - \frac{\partial E_z}{\partial y}\right) + e_{301}\left(-\frac{\partial B_y}{\partial t} + \frac{\partial E_z}{\partial z} - \frac{\partial E_x}{\partial x}\right) + e_{012}\left(-\frac{\partial B_z}{\partial t} + \frac{\partial E_x}{\partial x} - \frac{\partial E_y}{\partial y}\right) \] \[\overrightarrow{\nabla}\cdot\overrightarrow{B} + \left(\overrightarrow{\nabla}\times\overrightarrow{E}+ \frac{\partial \overrightarrow{B}}{\partial t}\right)= 0\]

The Lorentz Condition

\( \nabla A = \nabla \cdot A + \nabla \wedge A\)

\( \nabla \cdot A = -{\color{green}\frac{\partial A_0}{\partial t}} + {\color{green}\frac{\partial A_1}{\partial x}} + {\color{green}\frac{\partial A_2}{\partial y}} + {\color{green}\frac{\partial A_3}{\partial z}}\)

Assume \(\nabla \cdot A = \theta \neq 0\).

Solve \(\nabla^2 \chi = \theta\). Solution always exists.

A new potential \(A' = A - \nabla \chi\). This doesn't change \(F\) since \(\nabla \wedge \nabla \chi = 0\).

It does change the divergence: \(\nabla \cdot A' = \nabla \cdot A - \nabla^2 \chi = \theta - \theta = 0\).

Thank you

Questions?