- #1
AndrejR
- 1
- 2
- TL;DR
- I'm trying to go from 4D to 3D notation easily, instead of writing out all the elements.
Hi, so the four-dimensional generalization of
$$\vec{B}=\mu\vec{H}$$
is
$$F_{\lambda \mu}u_{\nu} + F_{\mu \nu}u_{\lambda} + F_{\nu \lambda}u_{\mu} = \mu (H_{\lambda \mu}u_{\nu} + H_{\mu \nu}u_{\lambda} + H_{\nu \lambda}u_{\mu})$$
From these four-tensors and four-vector I should be able to derive equation
$$\vec{B} + \vec{E} \times \vec{v}/c = \mu (\vec{H} + \vec{D} \times \vec{v} / c) $$
which I am, but only by writing out all the 64 elements (combinations of lambda = 0, 1, 2, 3 nu = 0, 1, 2, 3 mu = 0, 1, 2, 3).
Surely there must be a simpler way? Perhaps I can somehow take advantage of the fact that F is antisymmetric? Or is there some trick in this 4D notation which I am missing?
Btw. it's all in CGS and those four-tensors and four-vector are
$$
F_{\mu \nu} =
\begin{pmatrix}
0&E_x&E_y&E_z\\
-E_x&0&-B_z&B_y\\
-E_y&B_z&0&-B_x\\
-E_z&-B_y&B_x&0\\
\end{pmatrix}
$$
$$
H_{\mu \nu} =
\begin{pmatrix}
0&D_x&D_y&D_z\\
-D_x&0&-H_z&H_y\\
-D_y&H_z&0&-H_x\\
-D_z&-H_y&H_x&0\\
\end{pmatrix}
$$
$$
u_{\mu} =
\begin{pmatrix}
\frac{1}{\sqrt{1-v^2/c^2}}\\
\frac{-\vec{v}}{c\sqrt{1-v^2/c^2}}
\end{pmatrix}
$$
$$\vec{B}=\mu\vec{H}$$
is
$$F_{\lambda \mu}u_{\nu} + F_{\mu \nu}u_{\lambda} + F_{\nu \lambda}u_{\mu} = \mu (H_{\lambda \mu}u_{\nu} + H_{\mu \nu}u_{\lambda} + H_{\nu \lambda}u_{\mu})$$
From these four-tensors and four-vector I should be able to derive equation
$$\vec{B} + \vec{E} \times \vec{v}/c = \mu (\vec{H} + \vec{D} \times \vec{v} / c) $$
which I am, but only by writing out all the 64 elements (combinations of lambda = 0, 1, 2, 3 nu = 0, 1, 2, 3 mu = 0, 1, 2, 3).
Surely there must be a simpler way? Perhaps I can somehow take advantage of the fact that F is antisymmetric? Or is there some trick in this 4D notation which I am missing?
Btw. it's all in CGS and those four-tensors and four-vector are
$$
F_{\mu \nu} =
\begin{pmatrix}
0&E_x&E_y&E_z\\
-E_x&0&-B_z&B_y\\
-E_y&B_z&0&-B_x\\
-E_z&-B_y&B_x&0\\
\end{pmatrix}
$$
$$
H_{\mu \nu} =
\begin{pmatrix}
0&D_x&D_y&D_z\\
-D_x&0&-H_z&H_y\\
-D_y&H_z&0&-H_x\\
-D_z&-H_y&H_x&0\\
\end{pmatrix}
$$
$$
u_{\mu} =
\begin{pmatrix}
\frac{1}{\sqrt{1-v^2/c^2}}\\
\frac{-\vec{v}}{c\sqrt{1-v^2/c^2}}
\end{pmatrix}
$$