Spacetime in Geometric Algebra
The right way to look at
Spacetime and Maxwell's Equations
(or should I say "equation")
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.
\[\nabla \cdot \overrightarrow{E} = \frac{\rho}{\epsilon_0}\]
\[\nabla \cdot \overrightarrow{B} = 0\]
\[\nabla \times \overrightarrow{E} = - \frac{\partial \overrightarrow{B}}{\partial t}\]
\[\nabla \times \overrightarrow{B} = \mu_0 \overrightarrow{J} + \mu_0 \epsilon_0 \frac{\partial \overrightarrow{E}}{\partial t}\]
and some more.
\[\frac{\partial \rho}{\partial t} + \nabla \cdot \overrightarrow{J} = 0\]
\[\overrightarrow{B} = \nabla \times \overrightarrow{A} \qquad \overrightarrow{E} = -\nabla \phi - \frac{\partial \overrightarrow{A}}{\partial t}\]
The new form is very compact and elegant, but it is not just a mathematical trick. It reveals the underlying geometric structure of electromagnetism and spacetime.
Note that we use Vectors, but we also multiply them. So this is more than a vector space, it is an Algebra, a Geometric Algebra.
Space Algebra
Add multiplication to a vector space , ,
We want to be a multiplication scalar (marked as 1)
So tha "scalar" symbol 1 becomes a member of the algebra
We actually want any combination VV to ba a scalar
This leads to = -
But is a new member of the algebra
Mechanism of Space Algebra Multiplication
Let's practice some multiplications
\[e_{12}e_{23} = e_{13} = - e_{31}\]
\[e_{23}e_{13} = -e_{23}e_{31} = - e_{21} = e_{12}\]
\[e_{31}e_{123} = e_{3}e_{23} = - e_{2}\]
\[(\alpha e_{1} + \beta e_{12})(\gamma e_{3} + \delta e_{23}) =\]
\[= \alpha \gamma e_{1}e_{3} + \alpha \delta e_{1}e_{23} + \beta \gamma e_{12}e_{3} + \beta \delta e_{12}e_{23} = \]
\[= (\alpha \gamma + \beta \delta) e_{13} + (\alpha \delta + \beta \gamma) e_{123}\]
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
\[=\overrightarrow{A} \cdot \overrightarrow{B} + \overrightarrow{A} \wedge \overrightarrow{B}\]
\[(\overrightarrow{A} \times \overrightarrow{B} = I (\overrightarrow{A} \wedge \overrightarrow{B}))\]
\[\overrightarrow{\nabla}= \partial_x e_1 + \partial_y e_2 + \partial_z e_3 \quad \overrightarrow{\nabla} \overrightarrow{A} = \overrightarrow{\nabla} \cdot \overrightarrow{A} + \overrightarrow{\nabla} \wedge \overrightarrow{A}\]
Space Algebra Representation
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Spacetime Algebra
Good for representing spacetime physics, including Maxwell's equations
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Electromagnetic Field Representation
The simplest spacetime theory - Start with a vector A and differentiate it
\[F = \nabla A = \nabla\cdot A + \nabla\wedge A\]
If we assume \(\nabla\cdot A = 0\) then F is a pure blade 2 bivector
\[F = E_x e_{01} + E_y e_{02} + E_z e_{03} + B_x e_{23} + B_y e_{31} + B_z e_{12}\]
\[\nabla F = J \quad \text{where } J = \nabla\cdot F \quad\text{since} \quad \nabla\wedge F = 0\]
\[\nabla F = \nabla \nabla A = \nabla^2 A = \text{□} A \]
Maxwell's Equation
\[\nabla F = J\]
Thank you