criticisms of the einstein field equation - Alpha Institute for Advanced ...

CHAPTER 2. A REVIEW OF EINSTEIN CARTAN EVANS (ECE) . . .
2.13 Appendix 4: Standard Tensorial Formulation
of the Homogeneous Maxwell Heaviside
Field Equations
The standard tensorial formulation developed in this appendix is:
∂µ F µν = ∂ µ Fµν = 0 (2.1)
and is needed as a baseline for the development of ECE theory. The field tensor
is defined as:
⎡
⎤
F µν =
⎢
⎣
0 cB 1 cB 2 cB 3
−cB 1 0 −E 3 E 2
−cB 2 E 3 0 −E 1
−cB 3 −E 2 E 1 0
⎥
⎦ . (2.2)
where, in standard covariant - contravariant notation and in S.I. units:
1 ∂ ∂ ∂ ∂
∂µ = , , , ,
c ∂t ∂X ∂Y ∂Z
(2.3)
∂ µ
1 ∂ ∂ ∂ ∂
= , − , − , − ,
c ∂t ∂X ∂Y ∂Z
(2.4)
x µ = (ct, X, Y, Z), (2.5)
xµ = (ct, −X, −Y, −Z). (2.6)
The metric and inverse metric tensors in Minkowski space-time are equal, and
are given by:
gµν = g µν =
⎡
⎢
⎣
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
⎤
⎥
⎦ . (2.7)
Indices are raised and lowered with the metric, for example:
where
Fµν = gµρ gνσ F ρσ
(2.8)
g00 = 1, g11 = g22 = g33 = −1 (2.9)
and so on. Therefore:
F01 = g00 g11F 01 = − F 01 , F02 = − F 02 , F03 = − F 03
65
(2.10)