Tuesday, July 31, 2012

Newton's Gravity Waves (Part I) Dumping Center of Mass

 Many are under the impression that 'Gravity Waves'  are a prediction of Einstein's General Relativity (GR), and their discovery would be a proof of GR against Newton's Gravity Theory (NGT).

This is actually completely false.  A quick review shows this is not the case at all.

Newton vs. Einstein

On the one hand, the confusion was mainly caused by Newton's state of knowledge at the time.  Newton had no suspicions concerning an absolute limit to speed, and so conceived of gravity as an 'instantaneous' force, which took no time to propagate between locations.
Newton had no need of an elaborate mechanism such as a 'field', a wave, or a spray of particles to transmit gravitational force between objects.  It was simply an axiom that gravity was 'instantaneous'.  This force was treated as a 'static' or unchanging, unmoving force, based on distance and mass alone, independent of time.

Forces between masses

Einstein on the other hand, had already interpreted time geometrically as a 'special' 4th dimension.  This went beyond a mere charting or diagramming technique, but became fundamental, with the suggestion of a 4-dimensional geometric 'manifold' known as Spacetime, combining both space and time.

Minkowski Space

Einstein then created "Special Relativity", a theory about electromagnetic waves, which posed that light was made of massless particles (photons) that always traveled at "lightspeed".  This theory seemed to suggest that no object with mass could travel as fast or faster than light.

Light Speed Barrier

Einstein now incorporated "Special Relativity" into his new Gravity theory, his "General Relativity".   If gravity was to be treated as a 'field', like other forces, it would also be limited in its speed of propagation, and might be mediated by 'particles', now called "gravitons".  One could expect to detect 'gravity waves' traveling through Spacetime at the Speed of Light (or slower).

Basic  Newtonian Gravity

Newton's Gravity theory however also predicts Gravity waves, when stripped of its naive baggage.  To do this we have to keep some notions, and reject others.

Fundamental to Newton, are the following Axioms:
(1)  Space is three-dimensional and Euclidean.  This is one of the features that distinguishes Newton from Einstein, and modern formulations of Newton use a Euclidean Spacetime Manifold.

(2)   Gravity is fundamentally a static force, based on mass and distance alone, and independent of time.  This is how Newton formulates it at a fundamental level.  All dynamic effects of Gravity via motion, are based on treating Gravity as an instantaneous force, and applying it using Newton's equation, in combination with Newton's other laws of motion.

(3)  The Gravitational force falls off according to the Inverse-Square Law.   We can accept this as fundamental,  because its apparent failure at galaxy-distances may be accounted for in a variety of ways.   The same Inverse-Square Law in regard to electric charges, and appears to hold universally:
"...experimental results reveal that the validity of its inverse square nature can be unassailable almost to a
certainty at the macroscopic level, the length scale of which
has been shown to be of the order of 10^13 cm by laboratory and geophysical tests reviewed above. As for the microcosmic scale, the well-known Rutherford experiments on the scattering of alpha particles [indicates] that
Coulomb’s Law would be valid at least down to distances
of about 10^−11 cm, which is roughly the size of the nucleus.  Modern high energy experiments on the scattering of electrons and protons proved that Coulomb’s inverse square law was successful even down to the fermi range. Thus, the evidence from experimental results reveals that the inverse square Coulomb’s Law is valid not only over the classical range, but deep into the quantum domain also, a total length scale spanning 26 orders of magnitude or more: this range is impressive but still finite.
- Experimental tests of Coulomb’s Law,
Liang-Cheng Tu and Jun Luo (2004)
 Thus scientists tend to prefer to keep the Inverse Square Law as a fundamental concept for such forces, look for other explanations for anomalies in star-orbits around the galaxy.

Rejecting the Fixed Center of Mass Concept

Next, we remind the reader that there is a key problem with Newton: The Center of Mass concept (CM) contradicts the Sphere Theorem (ST).

This is not an experimental issue, but a problem of self-contradiction between two ideas that have been naively combined in popular expositions of Newtonian gravity. 

Newton's Sphere Theorem (ST)

Newton claimed that a Uniform Solid Sphere (uniform in mass density) acts just as a point-mass of the same mass, located at its center.  Thus the force between two spheres (e.g. celestial bodies) can be calculated by simply using the distance between their geometric centers:
 (There is a minor qualifier, namely that the distance 'd' is greater than that given by adding up the radius of each object, ensuring that the objects don't overlap, i.e., share mass.  Rigid objects should be able to approach to touching without a failure of the formula).

In Newtonian Gravitational theory this is not considered a mere approximation (i.e., true for distant objects only), but a fundamental theorem.   Any anomaly caused by say, equatorial bulge (from spin) or shearing (from proximity effects) would be accounted for by deformation of the spheres and displacement of mass, not any kind of failure of the concept or of the formula's predictive power.

The Sphere Theorem is "proved" by extrapolating from calculations for hollow spheres and combining the results.  Thus comes the importance of Newton's claim and 'proof' of the force exerted by a Hollow Uniform Sphere of Negligible Thickness (HST).   Newton's original 'proof' depends upon a result of infinitesimal Calculus, (the so-called mathematical "Sphere Theorem").  We will look again at this in a moment.

Gauss' Law for Gravity

Meanwhile, Newton's original proof was neglected, in favour of a mathematical result by Gauss, applied also to the Electric Field.  Gauss' Law is built from a different approach, that of classical 'field theory'.

 The Center of Mass Concept (CM)

 The Center of Mass concept (CM) builds upon and generalizes Newton's original idea with the Sphere (ST).    The idea behind a Center of Mass is intuitively attractive.  In a static case (no motion), every massive object must exert a fixed force upon another nearby object.

The CM for a Half-Sphere for instance, is defined and calculated as being fixed at 1/3 the distance along the 'radius' from the flat side. 

The Equivalent Point-Mass (EPM)

 The force an object exerts on a test-mass will be calculated from Newton's equation, by treating it as a point-mass, with distance and direction measured from the CM of the object to the CM of the test-mass.
This force will be a fixed size and direction, and so becomes a vector.  Any such force-vector can be represented by an equivalent point-mass (EPM) at a specific location.  Note that this mass could be anywhere along the direction-axis until we specify the actual mass of the replacement.  By definition, the EPM is formally given the same mass as the object it is to replace, fixing its location. 

 Using this definition and usage, the EPM is both a hypothetical object (a point mass located so that it exerts the same force as an object under discussion), and also a local position in space, relative to the geometry of the object being replaced.

With a sphere, the EPM is placed at the Geometric Center (GC) by fiat, and should in any case be placed somewhere along the axis between the two objects, due to symmetry.  The direction of the force vector (along this axis) is not in dispute. 

It is also assumed by Newton that the distribution of the mass (near and far halves) in a uniform sphere balances out as well, and results in nailing the EPM at the GC. 
Note that this second result doesn't simply follow from the symmetry of the sphere, since the viewpoint of the test-mass is from outside the sphere, not from the center, and the only symmetry from that viewpoint is radial  around the center-to-center axis.   By inspection, we can see the following embarrassing facts:
(1)  If the Center of Mass (CM) of each half-sphere is fixed relative to the half-sphere's geometry, (e.g., 1/3 along its axis of symmetry), then the CM of each half is an equal distance from the CM of the whole sphere (by symmetry and inspection).

(2)  The force exerted from each half cannot be equal.  The nearer half exerts a greater force.   Suppose equivalent point-masses (EPMs) are placed at each location (each half the mass), to replace the half-spheres. 

(3)  Suppose each EPM is moved toward the CG of the original sphere:  The far EPM does not increase in force the same amount as the near one decreases, because each is a different absolute distance, and the force must follow the Inverse Square Law, meaning it is non-linear with distance.  Equal adjustments in distance at different locations cannot have the same value. That is, the two EPMs for the half-spheres cannot equal in force the EPM of the whole sphere.
 Moreover, calculating total force by addition of vectors using the CM gives contradictory results, because the force is less if we divide the sphere into halves through the axis, and more if we divide through the plane perpendicular to the axis through the sphere and test-mass. 

 For a fuller treatment of the definition and method of calculating the Center of Mass and EPM, see our Article on the CM here.

 Since the Center of Mass concept is in fact self-contradictory  with extended objects and close distance, we must reject it as only an approximation, which fails when the radius/extension is in the same order of magnitude as the distance.

Thus physicists naturally reject the CM in favour of the Sphere Theorem. 

But if the naive and oversimplified method of calculating the CM and EPM is dangerously false, what is really going on?  The force must be deterministic, and the EPM must be defined somehow.  Far worse, we've really proved too much.  If the CM concept has problems, what kind of confidence can we have in the Sphere Theorem? 

The problem is more slippery than we might think.   For instance, suppose that Newton's claim about spherical objects is also an "approximation"?   How could we test it?  A theorist might suggest that the choice to use the Geometrical Center of a sphere is possibly arbitrary, meaning that the correction might only mean an adjustment of the Gravitational Constant.  Or perhaps the error is 'self-correcting' in that the error for one sphere-size balances the other.  The pragmatic experimentalist might simply say, "Measure the Gravitational Constant, and use it.  Don't worry about esoterical questions."

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