Tensors for Physics

388 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations

18.3 Exercise: Derivation of the 4D Stress Tensor

18.4 Exercise: Flatlanders Invent the Third Dimension and Formulate their
Maxwell Equations
The flatlanders of Exercise7.3noticed, they can introduce contra- and co-variant
vectors

xi=(r 1 ,r 2 ,ct), xi=(−r 1 ,−r 2 ,ct).

With the Einstein summation convention for the three Roman indices, the scalar

product of their 3-vectors is

xixi=−(r 12 +r 22 )+c^2 t^2 =−r^2 +c^2 t^2.

They use the differential operator

∂i=

(

∂

∂r 1

,

∂

∂r 2

,

∂

∂ct

)

,

and form the 3-vectors

JI=(j 1 ,j 2 ,cρ), Φi=(A 1 ,A 2 ,φ/c)

from their current and charge densities and their vector and scalar potentials.

How are the relations

B=∂ 1 A 2 −∂ 2 A 1 , Ei=−∂iφ−∂A/∂t, i= 1 , 2 ,

which are equivalent to the homogeneous Maxwell equations, cast into the pertaining

three-dimensional form? Introduce a three-by-three field tensorFijand formulate

their homogeneous Maxwell equations.

How about the inhomogeneous Maxwell equations in flatland?