I was hoping
Farsight would take advantage of this opportunity to demonstrate his knowledge of vector calculus, but I guess he's been too busy
promoting crackpottery in other threads.
Vorpal answered first:
Therefore, the transcribed equation (9) should probably read
(9) -∇·(e×h) = F∙u - ∂U/∂t,
which would be correct. The "reversal of direction" that Heaviside refers to would be that in the electromagnetic version has a different sign:
-∇·(E×H) = J∙E + ∂u/∂t.
So Heaviside was not quite that careless after all. The transcribers of his paper, however, are somewhat more disappointing.
I knew equation (9) was wrong and strongly suspected it was missing a minus on its left hand side (becoming certain of that within ten minutes of asking
Farsight's help), but I still wasn't sure whether the mistake was Heaviside's or Jefimenko's. I thank you for answering that question.
Farsight, as evidence for his mastery of electromagnetism and vector calculus, has often referred to his many quotations of a single sentence from page 558 of John David Jackson's
Classical Electrodynamics, third edition. To return the favor he has done me by not responding to my request for the education he offered, I'll direct his attention to section 6.7 and equation (6.108), which is Jackson's statement of the electromagnetic equation
Vorpal wrote above.
With that background, let's return to the out-of-context sentence fragment
Farsight quoted to demonstrate his appreciation of
ctamblyn's remark that gravitomagnetism "is a sometimes-useful approximation and analogy, and definitely not something equivalent to GR in its full gory detail." In this expanded quotation from Heaviside's paper, I have highlighted the missing minus sign
in red, highlighted
Farsight's quote-mine
in brown, and highlighted the more important parts of that paragraph
in blue:
Heaviside said:
Now if we multiply (5) by
e, we obtain
e∙(∇×h) = e∙(ρu) - e∙c ∂te, (8)
or, which is the same,
= - ∇∙(e×h) = F∙u - ∂tU, (9)
if
U = c
e2/2. But ∂
tU represents the rate of exhaustion of potential energy, so - ∂
tU represents its rate of increase, whilst
F∙
u represents the activity of the force on ρ, increasing its kinetic energy.
Consequently, the vector e×h expresses the flux of gravitational energy. More strictly, any circuital flux whatever may be added.
This e×h is analogous to the electromagnetic E×H found by Poynting and myself. But there is a reversal of direction. Thus, comparing a single moving particle of matter with a similarly-moving electric charge, describe a sphere round each. Let the direction of motion be the axis, the positive pole being at the forward end.
Then in the electrical case the magnetic force follows the lines of latitude with positive rotation about the axis, and the flux of energy coincides with the lines of longitude from the negative pole to the positive.
But in the gravitational case, although h still follows the lines of latitude positively, yet since the radial e is directed to instead of from the centre, the flux of energy is along the lines of longitude from the positive pole to the negative. This reversal arises from all matter being alike and attractive, whereas like electrifications repel one another.
In Heaviside's inexact (pre-relativistic) analogy, the vector field
e is analogous to the electric field
E, and the vector field
h is analogous to the magnetic field
H.
Farsight continues to reject the legitimacy of
E and
H, even to the point of saying they are not fields. It is therefore extremely difficult for me to understand how
Farsight could accept Heaviside's analogy.
I hope
Farsight will explain whether he rejects the legitimacy of Heaviside's
e and
h in the same way he rejects
E and
H. If he does reject
e and
h, I hope he will explain why he accepts an analogy based upon fields he rejects. If he does not reject
e and
h, then I hope he will explain why Heaviside regards
e and
h as analogous to the vector fields
Farsight rejects.
I doubt whether
Farsight is any more willing to provide that requested explanation than he has been to demonstrate his mastery of vector calculus, but we can hope. This is, after all, an opportunity for him to teach us something about physics.