From Osher Doctorow
I have referred in the last few Sections to the lack of symmetry
between gravitation and electromagnetism in GR, but the lack of
symmetry in Maxwell's equations between the Electric and Magnetic
Fields and between Electric Field time change and spatial rotation
with Magnetic time change is much easier to establish via Probable
Influence/Causation.
Using the express Dt for the time partial derivative, H for Magnetic
Field, E for Electric Field, J for current density, we obtain by
adding Maxwell's Equations:
1) Curl(H + E) + Dt(H) - Dt(E) - J = 0
and adding 1 to both sides of (1) yields:
2) [Curl(H + E) + Dt(H)] - [Dt(E) + J] + 1 = 1
which in terms of Probable Causation by A of B, P(A-->B), says:
3) P{(Dt(E) + J) --> (Curl(H + E) + Dt(H))} = 1
Namely, the time rate of change of the Electric Field E plus the
Current Density which is the density of the time rate of change of
Charge Causes maximal Probable Causation (1 on the right hand side of
(3)) on the Rotation of H + E and on the time rate of change of the
Magnetic Field Dt(H).
The converse is not true, even though the Electric and Magnetic Fields
turn out to Cause each other's rotations separately maximally by Time
change according to Maxwell's Equations. The clue that this latter
"bi-Causation" is less fundamental than the one above is the lack of
symmetry between the (Probable) Causations of the Electric and
Magnetic Fields separately upon each other. Thus, we have from
Maxwell's Equations:
4) Dt(H) = -Curl(E)
5) Dt(E) = Curl(H) - J
The signs - vs + are asymmetric above, but so is J which as a time
change density of Charge should really belong on the left hand side of
(5), further aggravating the asymmetric forms.
An interesting exercise for readers is arguably to convert (4) and (5)
to Probable Influence/Causation Equations similarly to the method that
led to (3).
Osher Doctorow
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