singularity

Let me first make it clear that I have limitted knowledge on the subject and, for that matter, in any subject. I consider the philosophical aspects of it more as I believe scientific progress emerges only after a firm philosophical foundation is established for a given quest.

It is my limitted understanding that what creates and transports electromagnetic forces is more or less understood and verified experimentally. That's not the case with gravity. Very little is know of how it propagates and virtually nothing about the creation mechanism. Except, if gravity is a ghost creation of our minds and it's essentially another form of electromagnetism. In this case, 300 or so years of intense research and debate will go down the drain as the biggest deraillement in the history of human thinking. Another explanation has to do with a consiratory behavior to derail research for "obvious" reasons.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>[Makis]: Maybe Dr. Van Flandern would like to assist here and through some photons into these uestions.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

Sorry for the delay. I was away on travel.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Newton proved that given an elliptical orbit, the force law is an inverse square of the distance. The inverse problem, i.e. given the nature of the law, what will be the resulting trajectory, has not been answered in my opinion yet, since all proofs are circular arguments based on geometrical constructions, like for instance the one given by R. Feynman.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

The field of celestial mechanics is all about determining the totality of possible trajectories for any given force law. An inverse square force does not always produce an elliptical orbit, but does always produce a conic section (circle, ellipse, parabola, or hyperbola). This can be demonstrated both analytically and numerically. The latter is done by numerical integration of the force law using any starting positions and velocities one chooses.

To understand the analytical proof, one must be familiar with the equation for conic sections. Then it is simply a matter of double integration of the equation relating the acceleration of a point mass to the inverse-square force law, then eliminating time from the resulting expressions. What remains is an equation for a conic section.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Even when using differential calculus, the knowledge about the orbits is implicit in the models used. Furthermore, implicit use of kepler's laws is made without proof. (As an example, Newton used kepler's 3rd law to derive the force law and most Physics books Newton's Universal Gravitation Lwa to derive Kepler's law.....no comment....)<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

These points may be correct, but are irrelevant to a proper analytical or numerical demonstration.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>The second question is simply about the nature of the mechanism that creates gravity, or universal gravity, and propagates it, whether that's an infinite or finite speed. Nothing is known with certaintly (experimental proof) about that either.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

See my chapter “Gravity” in the new book <i>Pushing Gravity: New Perspectives on Le Sage's Theory of Gravitation</i>, M. Edwards, ed., Apeiron Press, Montreal, 93-122 (2002). I doubt you could want much more in terms of a simple, easily visualized and understood mechanism for gravity. -|Tom|-

TVF

Thank you for the answers. I will definetively get the book and studied it to the best of my ability.

However,

Isn't a fact that kepler's third law remains an empirical observation and given this fact, how do we justify analytical proofs that contain empirical laws (observer dependent)? I guess one of the questions is that in lieu of Kepler's law what would be the results of all celestial mechanics calculations we now know?
(Obviously, the conic section is a direct consequense of kepler's law and not Newton's law of motion.)

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Isn't a fact that kepler's third law remains an empirical observation<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

No. Kepler's third law was empirical when first formulated, but was subsequently derived analytically from the inverse square law. Again, see any book on beginning celestial mechanics for details. I use Danby's <i>Fundamentals of Celestial Mechanics</i> myself, but it is heavy on vector mechanics, which not everyone has the background to read easily. But once you have mastered the derivation, you can easily see how the period-distance law would change for any other force law. -|Tom|-

I may be dead wrong here but it is my view that without Kepler's empirical law, Newton couldn't have derived his inverse law. As I mentioned earlier, Physics books use the inverse law to derive Kepler's law. There is a circularity here, in my view of course, but again I might be wrong and would appreciate any feedback.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>I may be dead wrong here but it is my view that without Kepler's empirical law, Newton couldn't have derived his inverse law. As I mentioned earlier, Physics books use the inverse law to derive Kepler's law. There is a circularity here, in my view of course, but again I might be wrong and would appreciate any feedback.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

You are quite right from the perspective of history. Newton had no idea why an inverse square law ought to exist, and derived it empirically from the observed ellipses. However, we now know why inverse-square behavior is intrinsic to nature. [Any unbound phenomenon must spread in two dimensions as it propagates through a third spatial dimension and time.] And we can take that law (common to all forces within their primary range of applicability) and derive with rigorous mathematics that orbits must be conic sections and Kepler's third law must hold in the two-body problem.

The circularity of the reasoning is interrupted by our present-day understanding (which Newton lacked) that inverse-square laws have a fundamental reason to exist that does not rely on empiricism. The Meta Model development in <i>Dark Matter...</i>, for example, invokes this principle. -|Tom|-