More Detail — Einstein’s Great... Oversights — Science

Approximating a “Uniform Gravitational Field”

The “Equivalence Principle”

“Gravitational Lensing”

What Will The Failure of Relativity Mean?!

The “Equivalence Principle”

Most popular introductions to Einstein’s Theory of Relativity give an account of Einstein discovering the “equivalence principle” and how it formed the basis of general relativity (distinct from special relativity since it includes gravity). Somewhere between 1907 and 1911 (biographers differ), Einstein introduced the key concept, the “equivalence principle”, in which gravitational acceleration was held to be indistinguishable from acceleration caused by mechanical forces. Gravitational mass and inertial mass were — and still are — therefore held to be identical.

The usual picture is an elevator: in “part 1a” of Einstein’s gedanken experiment the elevator is stationary in a uniform gravitational field, in part 1b the elevator is undergoing uniform acceleration in a space with no gravity, in part 2a the elevator is free-falling in a gravitational field, and in part 2b the elevator is undergoing zero acceleration in a space with no gravity. It was Einstein’s blindingly brilliant idea that parts 1a and 1b were actually “equivalent”, likewise for 2a and 2b, which idea he made perhaps the most fundamental concept of his general relativity, the “equivalence principle”:

“uniform gravitational field (with no acceleration)”
is “equivalent” to
“uniform acceleration (in a space with no gravity)”
“free fall” in a “uniform gravitational field”
is “equivalent” to
“uniform zero acceleration (in a space with no gravity)”
in all cases we abstract out all the other usuals, as well, electromagnetic fields, etc.; this might be described as “the laws of physics (sometimes except for gravity) look the same in each local Lorentz frame, i.e. in each local inertial frame of reference with Lorentz-ness as a special case”

From the foundational “equivalence principle” derive the business of “mass creating the curvature of space, in turn creating gravity”, that most famous equation in the world: E = mc², and so on.

Every scientist in the world — and almost every non-scientist, also fascinated by The Man and The Idea — has poured over it, well, usually popular renderings of it, looking for inspiration in the simplicity of the ideas and how much brilliant physics derives from them. Perhaps some even look for hints of... oversights.

It is surprising that more people haven’t openly objected to the “equivalence principle”, because there are so many serious inter-synergizing... oversights that it is difficult to decide where to start to unravel the spaghetti-yarn.

But suffice it to say, if the “equivalence principle” falls, general relativity will fall comparably. It falls... freely.

SECTIONS

Impatient?! A Quick Look at 3 Potentially Fatal Flaws

What Will The Failure of Relativity Mean?!

The “Equivalence Principle”

Approximating a “Uniform Gravitational Field”

“Gravitational Lensing”

Approximating a “Uniform Gravitational Field”

A “uniform gravitational field” is a Gedanken Convenience Concept. (See also comments with regard to a possible “constant gravitational field” concept in the Summary; it is actually almost the same concept, but described in a way that points out its... oversight.) A “uniform gravitational field” is an approximation (just as a “constant gravitational field” would be), and we may soon wish to have other approximations to smoke in our Gedanken Pipe, as well. As an approximation it requires strange things, e.g. a truly uniform field would have all partial derivatives (of the field) identically equal to zero. It turns out that this requirement is part of the downfall of the concept, or rather of inappropriately making it a theoretical concept as opposed to what it should have remained, a Gedanken Convenience Concept.

If we were to approximate a cosine curve with a straight line we might find it to be adequate to many purposes, but a cosine curve has qualities, e.g. of variation, that suggest that there are many scientific purposes for which a straight line approximation would not be adequate. In that same way, obviously, we can have much better approximations to gravitational fields than approximating them as “uniform”, but they can be highly context dependent.

We will next look at what we would need to do to reality to get... uh... to get a gedanken real world approximation to the gedanken “uniform gravitational field” approximation of a real gravitational field. Make sense?!

A Preposterous Approximation: The best possibility for approximating a uniform gravitational field is also the one that initially sounds the most preposterous, or at least the most satirical. If we set up a “mass” in accordance with the equations for force, mass, distance, and “gravitational constant”, so that we have the field we want at a given “point” a certain distance away from that mass, only a circle of points with that mass at the center and passing through that given point will have the field we want. We can move the mass away from that given point while at the same time increasing the mass to yield that same desired field-acceleration at that point, at the same time increasing the radius of curvature for that circle/arc of points, until, in the limit, we have an “infinite mass” that is an “infinite distance” away from our original point, and the field we want will pertain “approximately” (i.e. deviate only “infinitesimally”) in a “finite” (i.e. much larger than merely “infinitesimal”) “4-dimensional neighborhood” of that point. All the quotes are to remind us that these are very arguable terms, especially in view of the successes of quantum theory.

Although this approach to creating a “uniform gravitational field” works, no one ever really seems to consider it, let alone mention it, in discussions of the “equivalence principle”. This is too bad, because it actually gives the most theoretically workable approximation to a uniformly non-zero gravitational field, especially when compared with what has so far been considered acceptable along these lines.

The Usual Approximation (and its... oversights): The usual approach is to restrict ourselves to a region so small that the change in the acceleration “due to gravity” over the region can be considered to be “infinitesimal”, thus, in that region we have an adequate approximation to a “uniform field”. If we wish to be satirical, we can describe the problems that arise from this approach as “manifold”.

The main problem with the usual approach is that, although we can reduce the size of Δg (i.e. the change in the acceleration “due to gravity” over the region) to as close to zero as we might like, we cannot make the value of Δg/Δr as small as we would like. In fact, as Δg/Δr becomes the first (partial) derivative of g (with respect to the distance r in space) as the region becomes “infinitesimal”.

For a gravitational field, g, to be truly uniform, all partial derivatives of g with respect to anything (∂g/∂?), of any order (e.g. ∂³g/∂?₁∂?₂∂?₃), must be identically equal to zero (i.e., not merely “infinitesimal”), even “locally”:
∂g/∂? 4 0
without question.

But there is no possible way we can even approximate a zero partial derivative for g just by making the region “infinitesimal”, since mathematically all derivatives are derived from making the region “infinitesimal” to best approximate the limit of the ratios of the changes. E.g. the limit of Δg/Δr as Δr D 0 4 ∂g/∂r (a partial derivative since we assume that g is a function of more than just r). The “Preposterous Approximation”, given above, however, does allow these partial derivatives of g to be made arbitrarily close to zero, but it does so... rather preposterously.

This first (partial) derivative of g with respect to distance r, ∂g/∂r, is what underpins the well-known Roche Limit of astronomy. When astronomical bodies-particles are inside a planet’s or other body’s Roche Limit (still not nearly as well studied as it deserves), they tend to be torn apart by “tidal forces” (questionably named) that are a mostly a function of the densities of the bodies-particles involved (this last has still not attained community awareness) and of the first partial derivative of the gravitational field strength, which is a function of the distance from the center of mass of the planet (e.g. Saturn, whose rings are almost all within what is now estimated to be that planet’s Roche Limit). If we have 2 infinitesimal regions — local Lorentz frames, free-falling toward the planet — with 2 (of necessity “infinitesimal”) particles of the proper density in each (it’s not a function of the size — within reason — of the particles, just their density, ∂g/∂r and their distance apart), the particles will accelerate apart if ∂g/∂r (times their separation distance, assuming that expression is an adequate approximation) has a value that overwhelms the gravitational attraction of the particles that would otherwise accelerate or keep them together (i.e. the “tidal forces” win out).

For completeness, we can note in passing that, in fact, there is no need for the (necessary) Δg ≠ 0 within e.g. our elevator to derive from a 1/r² gravitational field with its (on the order of) 1/r³ first derivative; any non-zero Δg will give the “infinitesimal density” test particles a non-zero relative acceleration (after all, g is an acceleration, by definition, and therefore Δg is a relative acceleration), and this is all that is needed. (Note that if we take Δg to be only “infinitesimally” different from 0 — as when we take the limit of Δg/Δr as Δr goes to 0 in calculus — then we must accept that there will only be an “infinitesimal” relative acceleration. That's how calculus works.)

This acceleration apart will usually take place very slowly in reality (for example if the absolute value of ∂g/∂r or the distance between the centers of mass of the particles is very small), but it does occur, in reality. We need not concern ourselves with this here since what we are after is the gedanken acceleration apart, which need only be gedanken observable, even if only gedanken “infinitesimal” (since it is clearly gedanken distinguishable from absolute zero). I.e., all we need to be able to gedanken distinguish are the 2-3 cases of > 0, < 0, and perhaps = 0. The reference to “Lorentz frame” here reminds us that even if the 2 particles are free-falling toward the planet, they will accelerate apart if they are within the Roche Limit (actually a variant of it), and accelerate together or stick together if not; this result does not depend on other forces, accelerations, or any non-Lorentz factors, even though it might be overwhelmed by them.

But, as already noted above, the acceleration apart or not of the particles is partly a function, it turns out, pretty much only of the density of the particles, and not their size or mass per se. (See APPENDIX.) By reducing the density to “infinitesimal”, we have a gedanken experiment that distinguishes a gravitational field from a uniformly accelerated frame.

This difference of the acceleration apart or together of 2 test particles, relating to ∂g/∂r ≠ 0 (or even just to Δg ≠ 0) and to the densities and separation distances of the particles, in the phenomenological physics of 2 general relativity-wise equivalent regions — one an ostensibly “uniform gravitational field” and the other a “uniformly accelerated gravity-free region”, each with the same “constant” value of “g” — can be considered one of Einstein’s Great... Oversights.

To reason from a false premise — e.g. the “theoretical (implies the) real” existence of a “uniform gravitational field”, or the “theoretical (implies the) real” existence of a “local Lorentz frame”, or even just “this approximation is good enough” — does not guarantee false results, in fact if one is aware that one is doing it, it can be good for detecting errors, but otherwise there is a very good reason why it has a bad reputation.

Plus, the smaller the region, the closer we must get to any mass that is within the region. Even though, depending on the mass-density distribution, we don’t necessarily approach a singularity within the region, we are, however, forced into an ever closer proximity to a maximum of non-uniformity in the “uniform gravitational field”.

It is a rather serious condition to place on general relativity that it can only be made to apply successfully to local or global regions of spacetime that are not only totally devoid of mass themselves, but also not within a finite distance of any mass, and therefore of any matter-energy in the sense of Einstein. This rather limits the scope of application, especially if one takes into account that one of the great accomplishments of general relativity is — ostensibly — its ability to describe what happens in the cases of the extremely strong — “infinite” — gravitational “fields” (curved spacetimes) of black holes.
AND, these same considerations apply to Einstein’s requirement that the speed of light remain constant in all directions in all inertial frames of reference, since matter-energy disturbs the vacuum to the extent that the speed of light can even be quite a bit lower in matter that is not very dense , such as leaded crystal, diamond, or the atmosphere of Earth or even of the Sun (which latter must yield at least some heretofore unacknowledged refractive bending-lensing of light which is mistaken for gravitational lensing), which are all far from even neutron star density let alone black hole density.
We can look for catastrophic failure of the theory of relativity in precisely the place where it has achieved its greatest popularity: black holes.

There is also the further perennial problem that whenever we are dealing with a manifold-like substance that e.g. any sufficiently smooth curve (a differentiable curve), begins to look like a “globally” “straight line” when we look at “infinitesimal”, “locally Euclidean” subsections of it, and extrapolate or “reintegrate” to the global. (This generalizes to the logic that a ≈ b and b ≈ c, therefore a ≈ b ≈ c ≈ d... ≈ x ≈ y ≈ z... We are supposed to avoid using such logic, but in practice we don’t.) This means that the reasoning that goes into finding that a curved spacetime path looks straight to a particle moving in it (ostensibly the great insight that general relativity offers) is in “manifold” danger of being itself a combination of circular and-or merely artifactual, since the mathematics (of mathematical “manifolds”) we use so commonly has that characteristic as an artifact.

The concepts of 1) “locally Euclidean” — which artifactually does not permit curvature to be extrapolated from its “local Euclidean-ness” by linking the “locally Euclidean” regions — and 2) the “curvature of space (-time)” with its “straight line” “geodesics in curved spacetime” are strongly theoretically at odds with each other as they are used in relativity.

SECTIONS

Impatient?! A Quick Look at 3 Potentially Fatal Flaws

Approximating a “Uniform Gravitational Field”

The “Equivalence Principle”

“Gravitational Lensing”

What Will The Failure of Relativity Mean?!

“Gravitational Lensing”

There is a problem with the concept of “gravitational lensing” that may very well correspond to the... oversights noted above. It has to do with atmospheric refraction, which is not only known to exist, but has been studied since ancient times.

The empirical support for gravitational lensing comes from the pictures of stars surrounding the Sun made during some full solar eclipses. When compared to the positions that are known for the stars, i.e. without the Sun present to distort those positions (apparently through some kind of lensing effect), the positions during the solar eclipse were shifted by an average amount that corresponded reasonably well to the amount predicted by Einstein.

So far so good? Maybe... but, then again, maybe not. Statistics rears its ugly head...

When one looks at all the star displacements derived from the actual published photographs of the total eclipses of 1919 and 1922 , one of the characteristics that stands out is that the apparent positions of the stars are not... “uniformly lensed”. Instead of always appearing further from the Sun as if by a highly regular lensing effect, in the amount and direction of the change in apparent position they appear to be rather RANDOMLY distorted from their known positions.

See Coles, Einstein and the Total Eclipse, p. 55, for a hand-drawn representation of ~90 stars and their displacements as found in the photo by William Wallace Campbell (1862-1938) and Robert Julius Trumpler [misspelled as “Trumper” in Coles] (1886-1956) of the 1922 total eclipse in Wallal, Australia. (The 1922 photos were considered to be of much better quality than those of Eddington in 1919 in Africa, but by then Einstein had already become world famous in a world that wished his idea of relativity to be true.)
Many of the stars appear closer to the Sun as opposed to further away as they would appear if “uniformly lensed”, many are displaced with a large tangential component, and many appear not to have been displaced at all. The distances that they are displaced bears no regular relation to the distance from the Sun. It's as if...

Have you ever been out driving on the highway on one of those days that is so hot that those roiling, watery, mirage-like images form off in the distance just at the horizon, just above the hot ground or hot highway asphalt? This is a refractive effect that comes from the roiling density changes in the atmosphere caused by the heating effect of the ground as it is in turn heated by the Sun. This is what the star displacements actually look like in the photographs, as if they are being refractively distorted by the roiling solar atmosphere.

The question that comes next is:
What about quantitative aspects of the star displacements?

Well, we have an interesting problem. The Earth’s atmosphere is known to distort the apparent position of the Sun by ~1.67 degrees (if we look at the maximum bending of the Sun’s rays as they travel into and back out of the Earth’s atmosphere, “tangenting” the surface). Astronomers (as of the late 1990s) estimate the density of the Sun’s atmosphere as roughly 1/10,000th that of the Earth’s atmosphere (with no estimate of “experimental error”). If we assume linearity (because of the low densities involved), we would expect the Sun’s atmosphere to distort the apparent positions of the stars by ~1.67/10,000 degrees or ~0.599 arc seconds. (See http://aa.usno.navy.mil/faq/docs/RST_defs.html for detailed info regarding sunrise-sunset, which takes into account the magnification of the Sun’s disk at sunrise-sunset and from which the amount of bending of sunlight in the Earth’s atmosphere at sunrise-sunset can be calculated.) The problem is that this value is VERY CLOSE, as these things go, to the value predicted by Einstein (~1.74 arc seconds), especially if we add in the value Einstein calculated for Newtonian theory (~0.87 arc seconds, due to the effective mass equivalence of the photons; the sum being ~ 1.47 arc seconds), and it lies very crudely “in the middle” of the rather random displacement values (especially if one takes all of them into account, instead of just a carefully chosen few; see Note just below) actually found for the eclipse of 1922. (The original publications of these photos are hard to find; one must delve into the stacks of a university library, which is worth the trouble if you live near a university like Stanford, like I did; so instead see the easily obtainable Coles, Einstein and the Total Eclipse, p. 55.)

Note on Statistics: the values calculated from the total eclipse photos of 1919 and 1922 for star displacements were criticized at the time by many in the scientific community for the questionable statistical significance of the rather small samples used to evaluate star displacements and the bending of light around the Sun. E.g. only 7 stars in one case and 5 stars in another case were used to calculate the values. This criticism of statistical insignificance takes on much more meaning if one looks at many more of the actually rather random star displacements. See Coles, Einstein and the Total Eclipse, p. 55, for a hand-drawn representation of ~90 stars and their displacements as found in the photo by Campbell and Trumpler of the 1922 eclipse. (I found it acceptably accurate when comparing it with my memory of an original that I found in the Stanford stacks, Astronomical Society of the Pacific, but since I moved to Brasil I have not been able to find the photo copy I made for the precise citation.) Then compare that with the illustration of 15 stars and their displacements found in Misner, Thorne, Wheeler, Gravitation, p. 11; when compared with the ~90 stars in the Coles illustration, one can guess that these samples of 7, 5, and 15 stars were very carefully selected from a rather ragamuffin population, indeed. I, for one, am reminded of Mark Twain’s insightful comment about statistics. (He later claimed to have quoted Disraeli.) It has been instated as the primordial law of statistics:
The Zeroth Law of Statistics
   “There are lies;
     there are damn lies;
     and then there are... statistics.”

Another problem is that no one (I checked fairly carefully in the late 1990s, but you never know) has ever calculated and subtracted out atmospheric refraction effects, whose existence is infinitely less disputable than that of “gravitational lensing”, from the data, “all of it”. Doing so wouldn’t leave much to support Einstein, unless general relativity were substantially revised.

If the star displacements were due to “gravitational lensing”, one would expect the displacements to be purely radial, i.e. to be VERY regularly directed away from the Sun, by VERY regular amounts as a function of the angular distance from the Sun, with no observable tangential displacement(s). One would expect only an extremely small standard deviation from the radial displacement values predicted by “gravitational lensing”.
If the star displacements were due to by “atmospheric lensing”, i.e. to the optical refractive effects of the solar atmosphere, we would expect to find... what we in fact find: rather random radial displacements by an average amount that — by a somewhat unfortunate coincidence — also happens to be VERY close to the amount predicted by Einstein for “gravitational lensing”; a large standard deviation of the radial displacements from the average radial displacement value predicted by “atmospheric lensing”; sizeable random tangential displacements, also with a large standard deviation. Both the tangential displacements and the standard deviation of the radial displacements are rather too sizeable to put off to “observational error” when trying to make them consistent with “gravitational lensing”.

How do we interpret all the photos of galaxies and other large astronomical objects “gravitationally lensing” the galaxies etc. behind them? It should be no great surprise that all gravitational bodies — stars, galaxies, galactic clusters — all have atmospheres. It comes with the territory. Empty space doesn’t exist if we demand zero partial pressures for atmospheric components. The larger the astronomical entity, the larger the mass and the larger-denser-etc the atmosphere, and the less roiling we would expect compared with that which would naturally be found near solar surfaces. Densities would have a much greater chance to “average-even out” over vaster distances the further they got from those extreme temperature gradients, huge solar flares, etc.

The density distributions of (usually low density) astronomical entity atmospheres will largely conform to the mass distribution, especially that of high density astronomical globs of matter. One will find astronomical entity atmospheres everywhere one would also think to look for matter curving space enough to exhibit gravitational lensing. So what we find in all cases is what we would expect from lensing due to atmospheric refraction, all the more because the patterns of regularity and irregularity match expected atmospheric type variations much more than what should be extremely regular gravitational lensing variations (i.e. an almost complete lack thereof).
Another support for gravitational lensing is “the increase of the time taken by electromagnetic radiation along a path close to a massive body” (from Encyclopedia Britannica 2002, DVD). But... atmospheres also cause this same effect, as a function of density, distance, etc.

So, again, atmospheric refraction stands out as a primary explanation of what is probably not Einstein’s “gravitational lensing”. Perhaps “gravitational lensing” can be found, but at least an order of magnitude less than we now conceive it.

As of the late 1990s, no one has (published) carefully studied star displacements around Jupiter or the other planets. This should have been a first-order-of-business when the solar eclipse photos of star displacements were first raising interest in Einstein’s theory. They should rather quickly give a nod in some direction concerning the “relative” contributions of gravity and atmospheric refraction. They should be considered a next-order-of-business now, as should a careful mapping of the density distribution of the solar atmosphere as it varies over time, and very careful estimates of how this would effect apparent star position displacements, and speed of propagation.

SECTIONS

Impatient?! A Quick Look at 3 Potentially Fatal Flaws