Matters of Gravity

Episode #23

It had been another beautiful morning waking up to their view of the lake. After a relaxed breakfast and more commiseration to shore up their position on the details on the application of the Poisson equation to gravitation and electrostatics, they decided they were ready to call Julie.

The introductory conversations were cheerful as among the close friends they had become. Preliminaries included an update on how the dean was performing and whether Julie had yet to regret not taking that job herself, how the 'love birds' were doing, would Julie attend Margie and Tommy's upcoming wedding, and other I'm okay, you're okay chatting with a finale of weather being fine at both ends of the conversation and that they would meet with her face-to-face in a mere day or so.

Then somewhat apprehensively Maria broached the ostensible subject of their call. "We need to discuss the status of our joint investigation so we'll all be in synch when we meet," she said. "Roger has convinced me that his derivation of the inverted exponential charge distribution is correct. The proof of its validity is based squarely on existing theory associated with the broader Poisson boundary value problem."

"Oh yeah?" Julie said somewhat skeptically. "I thought we had concluded that it wasn't."

"I know we had, but I wasn't yet up to speed or concentrating very well because of... well, you know... and I hadn't heard all the arguments. We need to look at it again with Roger. The unusual nature of his conjecture for the potential seems necessary to me now in as much as it results naturally from the solution of the Poisson equation when the correct boundary conditions are in place. I think that's what we were missing. Those boundary conditions needed to be revised to meet realistic expectations at the origin. That's what we didn't look at when we went over it earlier Julie." She smiled at Roger in acquiescence.

"Why do we have to alter boundary conditions that no one saw fit to alter for centuries?" Julie asked. "Isn't that just changing the question to fit the answer we want? It sounds a bit hocus pocus to me."

"No. Singularities are anathema," Roger interjected. "always have been.  No one should have posited point particles and ignored the associated non-realistic nature of such a proposition. It's long since been inappropriate to say, 'Well... they're very tiny'; maybe there isn't a complete singularity and leave it at that. Don't you think?"

"So you're eliminating a particular point in space in order to get rid of point particles. What does that buy us?"

"Yeah, I guess you could say it that way. I wouldn't, but yeah. Getting rid of singularities is one hell of a purchase at any price though don't you think."

"Once one accepts a realistic boundary condition at the origin," Maria chided in, "the conjectured potential as proportional to the total amount of symmetrical charge beyond the distance of a test location instead of the traditional total charge within that distance divided by the distance itself, is the result. It's like the inversion of Gauss's law, so, sure, it's very nonintuitive. But... the gradient of that gives us the inverted exponential instead of the direct inverse square without having to ignore another term that results when taking the gradient. It seems crazy, but both conjectures work virtually identically at a large enough remove. In the limit they are identical but this one works everywhere. What's not to like?"

"As I recall, you did have a disagreement with Gauss, so that's it then. How does that 'prove' that the charge distribution of an indivisible particle must be of the form we agreed on with no proof?" Julie asked.

Roger responded, "According to the uniqueness theorem, there is ambiguity in the solutions if there is an incomplete set of boundary conditions. The boundary condition at infinity allows the conventional form but not uniquely. Completing the boundary conditions by including the origin – the center of the distribution – enforces that the potential expression you take exception to is, in fact, the only valid solution. The traditional one does not satisfy those completed boundary conditions. It was a solution to the homogeneous Laplace equation that assumed empty space between charged points. The Poisson equation is the more general (inhomogeneous) equation, but you know all that. I'm going to tell you this anyway since you doubted some of it. We know field strength is the gradient of the potential E(r) = grad(V(r)) which results in the inverse square law q(r)/r 2 deriving directly from the Laplace equation. We're going to share a file we put together of this derivation. Did you get it shared, Maria?"