Part 3.3
Curvature in the Single Polarization ECE Vacuum
To be a Beltrami field, the vector 𝒖 must satisfy the following [11]
𝒖×∇×𝒖= 𝟎
(175)
This is equivalent to [12]
∇×𝒖 =𝑘𝒖
(176)
A Beltrami field of the first kind [12] is a potential field where
𝒖 = ∇𝜓
(177)
with
𝑘=0.
(178)
A Beltrami field of the second kind [12] occurs when
𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
(179)
Meaning that
∇∙𝒖=0.
(180)
A Beltrami field of the third kind [12] occurs when 𝑘 a function of the coordinate variables and
equations (175) or (176) apply.
The ECE vacuum for a single state of polarization is given by
𝑩= ∇×𝑨−𝝎×𝑨 = 𝟎
𝑬
𝟐
(181)
𝜕𝑨
= −∇𝜙 + 𝝎𝜙 = −𝜔0 𝑨 − 𝜕𝑡 = 0
(182)
For the sake of simplicity, the vacuum state label has been omitted from the variables.
For any scalar potential 𝜙
∇ × ∇𝜙 = 0 .
(183)
Therefore the field given by ∇𝜙 is a Beltrami field of the first kind.
From (182) therefore, since 𝝎𝜙 = ∇𝜙 , it is a Beltrami field of the first kind, i.e.
∇ × 𝝎𝜙 = 0
(184)
Using the standard vector expansion, equation (184) can be written
∇ × 𝝎𝜙 = 𝜙∇ × 𝝎 + ∇𝜙 × 𝝎 = 𝟎
(185)
But from equation (182)
∇𝜙 × 𝝎 = ∇𝜙 ×
∇𝜙
=0
𝜙
Thus equation (185) becomes
∇ × 𝝎𝜙 = 𝜙∇ × 𝝎 = 𝟎
(186)
making 𝝎 a Beltrami field of the first kind for the single polarization vacuum, as shown earlier.
Since, the single polarization curvature is given by
𝑹𝑠𝑝𝑖𝑛 = ∇ × 𝝎
(187)
We note that the single polarization vacuum requires from equation (186)
𝑹𝑠𝑝𝑖𝑛 = 0 .
(188)
Also for a single polarization, from equations (8) and (15) we have that
𝑹𝑜𝑟𝑏 × 𝑨 = 𝟎
(189)
𝑹𝑜𝑟𝑏 ∙ 𝑨 = 0
(190)
This leads to the conclusion that
𝑹𝑜𝑟𝑏 = 0
(191)
From equation (182)
𝜕𝑨
−𝜔0 𝑨 =
(192)
𝜕𝑡
so that
1 𝜕𝐴𝑖
𝜔0 = − 𝐴
𝑖 𝜕𝑡
𝜕
= − 𝜕𝑡 𝐿𝑜𝑔(𝐴𝑖 ) .
(193)
Using
𝑹𝑜𝑟𝑏 = −∇𝜔𝑜 −
𝜕𝝎
𝜕𝑡
=0
(194)
We see that
𝜕
𝝎 = ∇ ∫ 𝜕𝑡 𝐿𝑜𝑔(𝐴𝑖 ) 𝑑𝑡 = ∇𝐿𝑜𝑔(𝐴𝑖 ) .
Therefore
(195)
𝝎 × 𝑨 = ∇𝐿𝑜𝑔(𝐴𝑖 ) × 𝑨
(196)
A typical term for this cross product can be written
1
𝐴𝑖
𝜕𝐴
𝜕𝐴
(𝐴𝑗 𝜕𝑘 𝑖 − 𝐴𝑘 𝜕𝑘 𝑖 )
𝑘
𝑗
which in [13] was shown to be zero for the single polarization vacuum.
Therefore
𝝎×𝑨=∇×𝑨=𝟎 .
(197)
Thus 𝑨 and by equation (192), 𝜔0 𝑨 are Beltrami fields of the first kind.
In summary, we have for the single polarization vacuum, that 𝑹𝑠𝑝𝑖𝑛 and 𝑹𝑜𝑟𝑏 are zero and that
𝑨 , 𝝎, ∇𝜙, 𝜔0 𝑨 , and 𝝎𝜙 are Beltrami fields of the first kind.
References
[11] http://mathworld.wolfram.com/BeltramiField.html
[12] Thahar Amari,et. al., “Computing Beltrami Fields”,
https://www.ljll.math.upmc.fr/~boulmezaoud/PDFS/ARTIC/Amari_Boulbe_Boulmezaoud.pdf
[13] Myron W. Evans, Horst Eckardt, Douglas W. Lindstrom, “Generally Covariant Unified
Field Theory”, Ch. 16,Vol. VII, Abramis Publishing, UK , 2011