“AT A DISTANCE” PART I (mentioned above)

Einstein denounced Newton’s — and Maxwell’s, and just about everybody’s — “action at a distance”, but he replaced it with “curved spacetime at a distance”...
I.e., for Einstein, “matter curves spacetime” that is “at a distance” from the matter that curves it.

It doesn’t sink the ship, but it is embarrassing when pointed out.
RATING AS AN... OVERSIGHT: ?

NO ABSOLUTES?! “AT A DISTANCE” PART II

Einstein objected to “action at a distance” (a la Newton), and he wanted everything (e.g. movements of particles) to look “linear” “locally”; but “locally” is still “at a (non-zero) distance”...

But in fact, “local” is a subjective concept. It really depends on a human “sense” of what is “nearby”. Rates of change tend not to change as one makes regions (neighborhoods, in the parlance of real variable analysis) infinitesimally small, in fact that is what calculus is all about, and just the opposite of what is assumed in relativity. I.e. relativity assumes that by making regions arbitrarily small, we can make all deviations from Euclidean-linear effectively go to zero. But even though we can choose a spacetime region so small that the change in a given velocity (Δv) will be arbitrarily small, this does not make the rate of change of the velocity (dv/dt) small, nor quantities like (∂g/∂r), where g is field strength and r is distance.

Accelerations — classic rates of change — must go to zero (as must all higher order derivatives with respect to time) if we are to have “uniform motion”. As we shall see, in reality no region can be made small enough to actually give a “uniform gravitational field”, especially one that has the (partial derivative) rate of change of the field with respect to distance identically equal to 0 over the whole region, and especially if the region has within itself even a small amount of mass (matter-energy). (In fact, all the field’s partial derivatives of any order must be identically equal to zero, including any “radius of curvature”, as in the “curvature of spacetime”.) This allows any size regions with real world-style gravitational fields to be gedanken distinguished from uniformly accelerated gravity-free regions. (See APPENDIX.)

In establishing his concept of “local”, Einstein actually replaced “absolute space” and “absolute time” with an invisible, unacknowledged “absolute magnitude (of “locality” in spacetime)”. “Relativity” theory  does not allow for e.g. the “local” being “relative”, for its being itself “global” with its own “local”.

Think of a sheet of log paper, concerning ourselves only with position or movement in the logged dimension. We can be between the line for 10 thousand and the line for 10 million, or we can be between the line for 10^-33 and 10^-30. The amount of space we have in each case — on paper; isn’t that always the way it goes?! — is the same, in a reverse bandwidth sort of sense: 3 log chunks, each 1 power of 10 wide. There might be problems with quantum thresholds, but what the heck. The too large and the too small, both tend to not resonate enough with our in-between to be noticeable. We can eventually start looking for cosms microing and macroing up and down this no longer absolute but relative spectrum of “local” and “global”. (This is not intended as any kind of reference to quantum mechanics, which has its own set of problems that have concerned scientists for decades.)

Again, it doesn’t sink the ship, but it is embarrassing when pointed out, and it is food for thought.
RATING AS AN... OVERSIGHT: ?

COMMON GENERAL... OVERSIGHTS: “Convenience Entities”

A type of fundamental... oversight common to pretty much all of science:

              “Convenience Entities” and their abuses.

The convenience can be e.g. a “Calculational Convenience Entity” (CCE), like the ever-useful  — but also ever-abuseful — “infinitesimals”, which can essentially never exist in reality, but can be very convenient for simplifying and-or shortening calculations of approximations (of increasingly dubious theoretical accuracy). Another kind of convenience is a gedanken convenience, like “Gedanken Convenience Concepts” (GCC) which are useful for gedankening, but again can’t exist in reality, (again) like “infinitesimals”, or like “uniform gravitational fields” or “local Lorentz frames”. “Singularities” should be recognized as Gedanken Convenience Concepts.

These “Convenience Entities”, which are not theoretical entities per se but rather support entities, are often treated as if they in fact can exist in reality, which they cannot do, or as if they can exist in theory, which they cannot do either, making any analysis that uses those assumptions of their existence in reality-theory reasoning from false premises. Reasoning from false premises does not guarantee false results, but has a deservedly bad reputation. This is not the most confidence inspiring way to proceed, even if the mathematics is inspiringly, even blindingly, brilliant. Strange things can happen when one reasons from the properties of members of the empty set. This relates to the common fallacy — not just common in science — of the map, artifacts of the map, map-making and mapping processes, etc. becoming the territory in the minds of the map-makers and map-users. We should look for system-wide and-or systemic failure that is a combination of systematic and asystematic errors.

A set of fundamental... oversights suggesting systemic heedlessness, and thus suggesting systemic failure that is a synergistic compounding of systematic and asystematic errors.
RATING AS AN... OVERSIGHT: ????

“INFINITESIMALS”

The Convenience Entity, the Gedanken Concept of “infinitesimal” was first introduced long ago as (implicitly) a Calculational Convenience Entity, but Einstein’s relativity almost seems to require the real existence of “infinitesimals”, e.g. “infinitesimal masses” (not just “test masses”, a “gedanken concept”), which cannot truly exist. (See a short discussion of “infinitesimals” in Newton’s Great... Oversight. See also why an “infinitesimal” can never truly be an absolute zero.)

This is not intended to be an objection relating to quantum mechanics. It relates to Newton’s law of gravity that any mass, no matter how small, has a non-zero gravitational effect on all other masses at any distance. We find this to be true in reality, so far, and so Einstein’s relativity would seem to need to accord with this. But, does it?! If lighter and heavier particles-masses, (literally) in reality are supposed to accelerate at the same rate (or better, an “infinitesimally different rate”), then those particles must have not only a theoretical but a real  (both as opposed to merely Gedanken-Convenience Concept) “infinitesimal” mass. (See the RELATIVITY REQUIRES ABSOLUTE SPACETIME sub-section below. There is a further serious theoretical problem relating to the difference in masses; see the Mass Difference sub-sub-section of the sub-section just mentioned, below.)

There are also “manifold problems” that result from assuming that errors are “infinitesimal” (and remain so, especially when integrated throughout-over the whole or any sizeable section of the manifold), either in theory or in reality. (See next sub-section.)

A combination of fundamental general and fundamental particular... oversights, this is rather strongly a synergistic set of such... oversights. It is intimately involved with probably fatal fundamental.. oversights in Einstein’s relativity.
RATING AS AN... OVERSIGHT: ????

“MANIFOLD FLAWS”

A manifold is a mathematical entity (used as a theoretical concept that in actuality should have remained a Gedanken Convenience Concept) which has regions (everywhere, i.e. infinitesimal neighborhoods of every point) that are “locally Euclidean”. This concept is also fundamental to relativity, whose “locally Euclidean” regions are “local Lorentz frames”.

But there are general problems with the concept of manifolds that are not generally acknowledged. In practice, the regions are “infinitesimal” in order to get values of this-or-that to approximate constants, or zero, or whatever. But, the error terms (for example, those concerned with derivatives), considered “infinitesimal”, are generally ignored when gedankening the extrapolation or re-integration back to the global scale. I.e. we forget that when we “integrate” the “approximations” in a “local” “region” of a “manifold” back up to a “global” xyz in-of that “manifold”, we are also implicitly integrating the (ostensibly) “infinitesimal” errors of those “approximations”. There is no general guarantee whatsoever that such integrals will remain un-embarrassingly small. But as physicists we have been implicitly assuming in our use of “manifolds”, which started as a Gedanken Convenience Concept, that the integrals of those infinitesimal errors will remain “infinitesimal” (or at least refrain from embarrassing us publicly), as bad an assumption for Einstein as it would be for Newton... or Leibniz.

After all, integrating infinitesimals is what integration is all about, and one can get any end result, even infinite results. In science in general as well as in relativity in particular, everyone usually assumes that re-integration will be kind as far as these things go, but in fact...

In fact, it is no mere coincidence that the usual logic involved in manifolds is the kind that can “prove” that any “sufficiently smooth curve” is a “straight line”, and that Einstein conceives of particle motion to be in “straight line” “geodesics in curved spacetime”.

This is a classic example of an artifact of the synergistic interaction between our concept of manifolds (which should have remained a Gedanken Convenience Concept instead of becoming a theoretical concept) and our psychology of ignoring “infinitesimals” as being “too small to make a difference” when effectively extrapolating-integrating from “infinitesimal” regions back to the global level in a manifold.

• A truly “locally Euclidean” region (i.e. the curvature of the spacetime — or whatever other property must be “locally uniform”, such as its gravitational field — and all its derivatives, of any order, must be identically equal to absolute 0) cannot take part in a curved space, unless it does so discontinuously. Truly “locally Euclidean” regions cannot gradually give way to non-Euclideanly curved regions. At the very least the “locally Euclidean” regions must be interspersed with sufficient numbers of sufficiently non-Euclideanly curved regions to give the curvature desired.

A general example relating to the Gedanken Convenience Concept of “locally Euclidean”: given a straight line and a point on that line, consider the class of sufficiently smooth (derivable) curves that pass through that point that are tangent to that line. If we take a small enough neighborhood of that point, all of the curves seem more or less the same, and we might be tempted to ignore the essential differences in the curves when we try to extrapolate from this neighborhood back to a larger region of the manifold in which the line and point are embedded. We do this in our use of manifolds.

This all points at a generally unacceptably flawed set of basic assumptions that is-are not generally acknowledged.
RATING AS AN... OVERSIGHT: ????

THE “EQUIVALENCE PRINCIPLE”
“Uniform Gravitational Fields” PART I

If someone made a scientific pronouncement of the concept of the “constant gravitational field”, that gravity was a fixed constant, i.e. the same everywhere throughout the cosmos that it was at sea level, and based a scientific theory on it... well, we would question his sanity, not to mention the quality of his education.  In any case we would not allow it as competent science, at least not since the time of Newton, with his (to us) scientifically verified inverse square law of gravity. But for some reason we accept a similar pronouncement by Einstein, that of the “uniform gravitational field”, even though we know from Newton and repeatable scientific observation that no such thing is likely to exist anywhere, even “locally” (see more on this above and below, e.g. with regard to derivatives). We can see that any theory based on this “constant gravitational field” would start out with fundamental... oversights, and would be likely to have serious... problems that derived from them. (For example, we can note that science based on the above concept of a “constant gravitational field” should not get very far away from sea level, but Einstein gets very far away from where a “uniform gravitational field” can be said to hold, taking us with his relativity all the way to black hole singularities and their essentially “infinite” mass densities and gravitational fields or curvature (depending); see More Detail on Approximating a “Uniform Gravitational Field”.)

Let us go out on a scientific limb and say:

No such thing as a “uniform gravitational field” (in fact this is a Gedanken Convenience Concept) is actually “possible” in the reality of our spacetime “quantinuum” (at least given the fact that any mass — and therefore any matter-energy — one way or another has a gravitational field, and given the inverse square with distance nature of that gravitational field, all worded as necessary in the appropriate Einstein-ese), even in an “infinitesimal” region.

This may sound like we are “‘proving’ that something is impossible”, an act that is itself supposed to be impossible, but it is the quibble about the contingency of inverse square with distance that makes it conceivably a competent statement to proffer. If one looks at the various ways that “fields” that approximate Newtonian inverse-square-with-distance gravity can be combined to give a “uniform gravitational field”, even over a “relatively small volume of spacetime”, we... well, we just don’t get a lot of possibilities, especially since we need a non-zero uniform field with all of its (partial) derivatives identically equal to zero over the entire region. Let’s emphasize that last:

• For a gravitational field, g, to be truly uniform, all partial derivatives of g with respect to anything (∂g/∂?), of any order, including any “curvature of spacetime”, must be identically equal to absolute zero (i.e., not merely “infinitesimal”), even “locally” (i.e. in an “infinitesimal” region):
        ∂g/∂? 4 0
  without exception. Physicists desperately need to ask themselves “if in each local region we have ∂g/∂? 4 0 (where ? is e.g. distance or time), just where, when and how will g change in the spacetime manifold?!”

• If all the partial derivatives of g are identically equal to zero (absolute 0 as opposed to infinitesimal), then any (even non-linear) extrapolation from a local region of any size using only those derivatives must yield precisely that same uniformity throughout the space (or manifold) involved. I.e. there would be no possibility of space (-time) curvature or other deviation from non-uniformity without an asystematic discontinuity of at least some of those partial derivatives. Some new, heretofore super-natural-theoretical entity would need to provide the discontinuous transitions between uniform regions in a piecewise uniform space. No such entity is theorized within relativity, and any attempt to add it would probably sink the ship. So, since relativity posits the existence of uniform gravitational fields, if only in the infinitesimal (locally Euclidean-locally uniform) regions of a manifold, then it cannot correctly model non-uniform gravitational fields or curved space, even if it approximates Newton, even if it improves somewhat on Newton.

• To the extent that a uniform gravitational field is posited to exist in reality — or even in theory — any derivation from that is reasoning from false premises. Essentially anything can be “proven” when reasoning from false premises. Any theory based on false premises and associated false reasoning is gibberish, but if one is careful, one can still model reality to a good approximation. It is just that extrapolations, interpolations and other predictions have lost their rational basis, even though they may still be accurate. If we care to be satirical, we can say that if one is careful to derive — from false premises — only what the traffic will bear, once can be convincingly “scientific”, even “brilliantly scientific”. Many in history have done this; some, like Aristotle, are still highly respected.

And if there were such a thing as a “uniform gravitational field” in our spacetime “quantinuum”, it could only exist in a region that was itself totally devoid of mass — and therefore devoid of matter-energy (in the sense of Einstein), and infinitely far away from any region that did have matter-energy (assuming that the regions are connected by either Newton’s inverse square law or Einstein’s spacetime curvature). Indeed the closest matter-energy would have to be an infinite distance away (still assuming that something like inverse square gravity holds, and that there were “no gotchas” like limits on its range such as cosmological constants, etc.). A theory of gravity with this kind of limitation wouldn’t have the widest range of applicability.

Some will bridle at the idea that — for the theory to actually have a chance — all matter (and therefore energy) must of necessity be an infinite distance away from our “local” region with its “uniform gravitational field”, but it’s a question of how much deviation from the ideal can be tolerated, “locally”, and how rapidly errors will increase with extrapolation-integration back to “global” (not necessarily well-defined) levels of the manifold in question. In general the outlook is poor for such things, as we saw earlier.

Relatedly, and repeating for emphasis: since relativity posits the existence of uniform gravitational fields, if only in the infinitesimal (locally Euclidean-locally uniform) regions of a manifold, then it cannot correctly model non-uniform gravitational fields or curved space, even if it approximates Newton, even if it “improves” somewhat on Newton.

Even a Gedanken Convenience Concept should have more gedanken applicability than (the GCC of) a “uniform gravitational field” has in gedanken fact. And on top of that, it is treated as a theoretical entity, as if “infinitesimal” “uniform gravitational fields” can actually exist in accordance with that theory, and be modeled successfully both locally and globally by “manifolds”. The wrong kind of synergy is happening here.

If it remained a purely Gedanken Convenience Concept, and if it were acknowledged publicly as such, and if its shortcomings were properly addressed, this need not be a fatal... oversight (??). But, given its actual status, and the fact that it synergizes with other flaws (see Part II, below), it deserves a strong rating (????).
RATING AS AN... OVERSIGHT: ?? or more likely ????

THE “EQUIVALENCE PRINCIPLE”
“Uniform Gravitational Fields” PART II

In order to have a “uniform gravitational field”, all partial derivatives of this “uniform gravitational field” (including the “curvature of spacetime”) would have to be identically equal to zero, or at most “infinitesimal” (“small enough to never be a big problem”, and this “never” itself never happens). But this zeroing of all derivatives in fact does not happen no matter how “infinitesimal” one makes the local regions. Some, such as partial derivatives with respect to distance, perforce remain non-zero. (An important example is the rate of change of the gravitational field with distance, relating to the “curvature of spacetime”.)
 

• Mathematics: remember that, by definition, the limit, as the region gets infinitely small, of the ratio of the variation in the field strength, g, to the 1-dimensional measure of the size of the region in a given direction is in fact the first derivative of the field strength in that direction (you can think of it as a partial derivative, independent of any coordinate system) with respect to distance, and it can have any value no matter how small the region and therefore no matter how small the change or variation in field strength over that region is.)

It is no mere coincidence that the first derivative of a 1/r² gravitational field strength (with respect to distance, r) is a 1/r³ term, and like the gravitational field itself (conceived in Newtonian terms) it only goes to “zero” “at infinity”, and that it also appears in tensors for curvature of spacetime. It is this 1/r³ term that gives us a Roche Limit type acceleration apart for 2 (assume spherical) test particles whose density is low enough (i.e. “infinitesimal in the limit”; this is assuming that the particles are interacting gravitationally, and that they are spherical and touching; see comment, just below).
 

• Contrary to “popular” scientific opinion, the Roche Limit effect (for spherical particles that are in contact, i.e. where they start to be torn apart by the gradient of the gravitational field if they are too close to e.g. Saturn) really depends on the densities of the particles-bodies, not on their size or mass per se. (If they are spherical and in contact, which gives their gravitational interactions the best chance to keep them together, the density determines the mass and distance relationships and their relative accelerations, given the larger, non-uniform gravitational field; if they are not in contact, then we are back to mass and distance relationships that are not determined by the density, but since they are further apart, they are more likely to be torn apart yet more quickly by the larger gravitational field difference.) These needn’t be “infinitesimal” here, but should probably fit in the gedanken elevator just to keep things nice. Although the rate of acceleration apart can get very small for small particle sizes, since we are conducting gedanken experiments, we can certainly gedanken notice the “infinitesimal” differences being greater than absolute zero.

• NOTE: an “infinitesimal” difference is greater than an absolute zero difference, even though its only “infinitesimally” greater. We like to think that it doesn’t make much difference, and indeed it often doesn’t; this is why we approximate some e.g. masses as “infinitesimal”, because we have a relatively good idea that we will still get a reasonable approximation to the correct values by thinking of the mass as essentially zero. But when we extend “infinitesimal” differences back to the system as a whole (i.e. to the whole “manifold”, even though “locally Euclidean” is a property that never holds in reality), we get error terms that become indefinitely large. Think of the value of
              ∫_a^b 0 (or ∫_a^b 0dx) = 0
  versus the value of
              ∫_a^b 1dx = b-a
  or better yet 
              ∫_a^b e^xdx = e^b - e^a
  which (here an error term) literally increases exponentially when integrated from the infinitesimal level back to the finite level of a system/manifold.

• Gedanken Experiment that falsifies the “equivalence principle”:
  There is a simple gedanken experiment that can be performed in any gedanken elevator: since in the real world we cannot have a “uniform gravitational field”, we will have ∂g/∂r ¹ 0 (i.e. Δg ¹ 0 between at least 2 points in the elevator) and therefore a Roche (“tidal force”) type effect: i.e. any 2 test particles can be placed at any 2 points in the elevator which have a different g (and we just established that such points exist) in the elevator, with zero relative velocities (for convenience), and the ∂g/∂r ¹ 0 (or Δg ¹ 0) will ensure that they experience an acceleration relative to each other, unless their masses cause an interaction that just happens to precisely counterbalance this. If the masses are “infinitesimal”, then this is not a factor. This acceleration (most probably apart) cannot happen in the elevator that is in a zero gravitational field but experiencing uniform acceleration.
  
  One simple way to do the above is to use the Roche Limit type acceleration result referred to above. Put 2 test particles with “infinitesimal density” (spherical and in contact) and with zero relative velocities (for convenience in measuring their relative-to-each-other/elevator accelerations) in any-and-every part of the elevator (if they are not in contact, then the further apart their centers of mass are, the more pronounced the Roche type acceleration apart is likely to be), and with the line between their centers of mass taking any direction (since some directions may be neutral with regard to a Roche Limit type acceleration). The 1/r³ (vector) term (from the ∂g/∂r of an inverse square gravitational field; this is our old friend who shows up in so many of relativity’s tensors) will be non-zero in at least some directions in some regions within the elevator if-and-only-if there is a non-uniform gravitational field. (Reminder: any non-uniformity in the gravitational field will do, e.g. from multiple overlapping 1/r² fields.)
  
  The “infinitesimal density” (spherical and in contact) test particles will thus accelerate apart if-and-only-if there is a real world-style gravitational field as opposed to “uniform acceleration” due to “mechanical forces”, even if only “infinitesimally” quickly. In an elevator in a “uniform gravitational field”, we will not be able to gedanken notice even an “infinitesimal” acceleration apart (or together if they are not in contact and we take into account the masses of the particles). (There is no need to even measure the acceleration apart, if-when it occurs, since any gedanken noticeable gedanken acceleration apart can “only” be explained by a gedanken non-uniform gravitational field — since, as we have seen, a truly “uniform gravitational field” cannot generate an acceleration either apart or together — supplied by something other than the test particles. We should not totally forgo mentioning that the test particles themselves must introduce a non-uniform gravitational field into the gedanken elevator, even if only “infinitesimally” non-uniform.)
  
  We should actually NOTE: the density of the test particles can be varied so as to get either acceleration apart, “together” (non-zero force if in contact), or a neutral state (zero force if in contact). This makes this alternative gedanken experiment result even clearer.
  
  A “uniform gravitational field” is one level of approximation to a gravitational field in a local region, and as an approximation it is good for some things... but not for others. If we use an approximation to a gravitational field that is just “infinitesimally”  better, Einstein’s famous gedanken experiment fails.
  
  A second failure of the concept of “uniform gravitational field” relates to the masses of objects in e.g. our gedanken elevator (or in an “infinitesimal” region within a manifold). If the objects have “finite” mass, then they must perturb the “uniform gravitational field” correspondingly, and our gedanken experiment fails before it even starts.  And we must ask the question that everyone so far has failed to ask: “in the gedanken elevator that is experiencing ‘uniform acceleration’ as opposed to a “uniform gravitational field’, do the objects have mass and gravitational interactions?”  Even if we consider the objects to have “infinitesimal” mass, then again the “uniform gravitational field” is perturbed into non-uniformity, even if only “infinitesimally”, enough so that we must consider it to be non-uniform in the absolute sense.  And given enough time they will accelerate together, unless we have other non-uniformities that allow a Roche-type effect to accelerate them apart. The gedanken experiment comparing a “uniform gravitational field” to a “uniform acceleration” of our elevator requires that all objects within the elevator (and also within any “infinitesimal” region within a manifold) must have absolute zero mass and corresponding gravitation. So if we use even a better approximation to the gravitational fields supplied by the masses within our gedanken elevator,  even if it only is “infinitesimally”  better, Einstein’s famous gedanken experiment again fails.
  
  This relatively simple gedanken experiment distinguishes between real world-style gravitational fields and uniform acceleration, and thus falsifies the “equivalence principle”, a sine qua non of Einstein’s relativity.

The “uniform gravitational field” of equivalence principle is a level of approximation to a gravitational field that is very, very far from any mass (i.e. from any matter-energy), valid if we are correspondingly very careful about applying it only within its domain of applicability. There may be only “slight” differences between the “uniform gravitational field” and the next better level of approximation to an actual gravitational “field” (using Newtonian terminology), but the differences between Einstein and Newton are similarly slight, and they relate to the differences in these levels of approximations. I.e. relativity has essential parts of its foundations kicked out from under it when we go to even a slightly better level of approximation to real world-style gravitational fields. That is not to say that it fails altogether to approximate reality better than Newton in some gratifying ways, but it effectively reasons from false assumptions (see also next section), and that is... bad.

Einstein’s “equivalence principle” is so fundamental to general relativity that this failure of it should eventually require an immediate general overhaul of relativity.
RATING AS AN... OVERSIGHT: ?????

“RELATIVITY” REQUIRES ABSOLUTE SPACETIME —
Acceleration of Lighter and Heavier Bodies

That relativity requires absolute spacetime is well known by at least a few. The ever-present concept of “inertial reference frame” requires it. An inertial reference frame is one that is moving with a constant velocity “relative” to the absolute reference frame. Relativity assumes that we will never be able to determine the absolute frame itself, but does assume that the concept of a frame of reference that is in uniform motion relative to it does make sense. It can only make sense as a Gedanken Convenience Concept, however, a point that has been overlooked. It remains to be seen if this concept is used to reason from false premises, as we commented on above.

The Earth is in fact not an inertial reference frame, although it is usually considered to approximate one. In fact, there can be no inertial reference frame that is in any way fixed to matter (or energy). Any frame of reference fixed to any matter-energy must — in reality — be experiencing forces that will cause it to accelerate relative to the absolute frame of reference (perhaps in many directions at the same time, as with systems of particles experiencing Roche type “tidal” forces). This is why

• the concept of an inertial reference frame can only be a Gedanken Convenience Concept

unless there exist other parallel planes of existence — e.g. astral planes — from which we can observe our own “gross material plane of existence” (or so the yogis call it) sans any acceleration with respect to its relativistically necessary absolute reference frame, and perhaps also bypassing the Uncertainty Principle.

• Relativity is called relativity, not because it says there is no absolute frame of reference, but because we seem to have no way of determining such an absolute frame and studying the laws of physics from it. The assumption is made that, since we can’t determine the absolute reference frame, creating a physics whose laws must be invariant (the same) in every reference frame in uniform motion relative to the absolute frame is necessary — and, far more subtly, sufficient. Why didn’t Einstein make the far more general leap to creating a physics whose laws must be invariant in any frame of reference?!

Relatedly, Einstein made the fundamental theoretical assumption that in fact (i.e. not just a Gedanken Convenience Concept even though this is a gedanken experiment) lighter and heavier “test particles” will always “accelerate” at the same rate. But Newton’s theory predicts that, if only 2 bodies are involved, the heavier one will have a greater (Newtonian) attraction for e.g. the Earth which will accelerate toward it faster than it would toward the lighter body, and in the relative frame of reference of the Earth, the heavier body will accelerate toward the Earth faster than the lighter body will. (In the relative frame of the lighter/heavier body, it is the Earth that appears to accelerate-fall faster.) I.e. if released in separate trials the lighter and heavier bodies will always “fall-accelerate” at different rates relative to the frame of reference of e.g. the Earth... unless we have an absolute frame of reference in which we can measure their absolute acceleration (which is strictly forbidden in Einstein’s relativity). This is also found to be true in fact, as evidence by their distances from their common center of mass around which they orbit, and their angular acceleration in doing so.

• If we release a 1 kg body and a 2 kg body near the surface of the Earth, in separate trials, the falling rate difference relative to the Earth will be ~1.67 parts in 10²⁵ of the standard acceleration of ~9.8 m/s² (i.e. the ratio of the 1 kg mass difference of the 2 bodies to the mass of the Earth); this is not nearly within the current experimental limit of ~1 part in 10¹¹; 10^-25 would here be totally overshadowed experimentally by... just about anything.

Assume the is a space with a gravitational field that derives from any number of masses placed in various locations. We get a gravitational field at every point that will be the same at that point no matter what the mass of any test particle that we happen to place at that point. If we look at the instantaneous acceleration at the instant of release in the absolute Newtonian frame of reference, the mass of the test particle-body in question will not affect its initial instantaneous acceleration, although it will affect the initial instantaneous acceleration of all the other bodies, and therefore the acceleration by the test particle-body experienced at subsequent instants will be different since in general the mass will have caused all other bodies to shift differently and therefore the landscape of the gravitational field and its potential will be different.

• The Earth is only approximately an inertial frame of reference. This approximation is enough to explain why we would measure a different initial instantaneous acceleration for lighter and heavier “test particles” using Earth as the reference frame. But in general only the acceleration at the initial instant is the same. After that initial instant other masses will approach the heavier test particle more quickly, and the gravitational-potential field(s) will be different — stronger (ignoring arithmetic sign) — for the heavier particle than for the lighter particle, and the heavier particle will in general accelerate faster.

So it turns out that we must resort to the fact that both in Newton’s theory and in reality, lighter and heavier bodies will accelerate at the same rate (and only at the instant of release) only if we use Newton’s absolute spacetime (where time is conceived somewhat separately) — that even Newton had trouble with and Einstein tried to reject completely, thus the name “relativity” — and take only the absolute motion of the bodies into account. We... well it is more than merely embarrassing if relativity requires absolute spacetime to get such a fundamental assumption to work. (And we do not improve the situation if we try to say that we actually have a reference frame that is not tied to a mass or other piece of matter-energy, and is therefore in some kind of uniform motion with respect to the absolute Newtonian reference frame.)

Mass Difference: There is a certain... irony?... satire?!... here: many will object that Einstein was speaking of (note: as a gedanken convenience) “infinitesimal test particles” (i.e. speaking of them as accelerating at the same rate independent of whether one is lighter or heavier than the other), which theoretically will not affect other masses in this embarrassing way. Let us ask an embarrassing question:

• If the 2 test particles are both “infinitesimal”, how can we speak of them as having a true mass difference, i.e. a mass difference that is non-“infinitesimal”?!

• And if there is a true mass difference that is “infinitesimal”, why don’t we look for and recognize the corresponding “infinitesimal” difference in test particle acceleration?! I.e. if  the 2 test particles have only “infinitesimally different” masses, then we can reasonably expect at most “infinitesimally different” accelerations due to that difference. Although we normally fail miserably to gedanken distinguish “infinitesimal differences” from “absolute zero differences”, they are in fact essentially “infinitely different”. 
  (Think of the difference between a definite integral of:
       0dx  (i.e. “zero”dx)
  and a definite integral of:
      f(x)dx.
  There can literally-mathematically be an “infinite difference”.)
  It is one thing to have a mass difference that is “finite” (i.e. “infinitely greater than infinitesimal”) yield only an “infinitesimally different” acceleration, and it is quite another to have an “infinitesimal mass difference” yield an “infinitesimal difference” in acceleration (reminder: a difference that we standardly fail to distinguish from an “absolute zero difference” in acceleration), since the dy/dx properties are completely different. (See next point, immediately below.)
  There is also a variant of circular reasoning potentially taking place here — arguing from the “infinitesimality” of the mass difference to the “infinitesimality” of the acceleration difference, thence to “no difference” — along with the failure to distinguish a Gedanken Convenience Concept from a theoretical concept. Why aren't we looking explicitly for these “infinitesimal differences” in acceleration to gedanken distinguish them from “absolute zero acceleration differences”? (See Why “infinitesimal” is infinitely greater than “absolute zero” in More Detail.)

• The use of “infinitesimals” does not even remotely relieve us of the calculus-derivative problem, that “infinitesimals” are used to define quantities like  ∂g(m_t)/∂m_t (≠ 0), i.e. the (partial) rate of change of the acceleration of the test particle as a function of the mass of the test particle.

They can only have an at most “infinitesimal” mass difference, which in gedanken terms is supposed to mean “small enough to never ‘make a difference’, to never be noticeable or have a noticeable effect”, i.e. “effectively zero”. It can hardly be surprising that lighter and heavier bodies which can only have an “infinitesimal” mass difference have the same (or only “infinitesimally” different?!) acceleration if the difference in their accelerations can only be due to their mass difference. This is the kind of problem one has when one mistakes Gedanken Convenience-Entities — used e.g. in gedankening, evolving, evaluating, and revising a theory, and hopefully in putting it right — for theoretical entities and-or especially for real world entities.

This acceleration rate difference can be contrasted with both Newton’s theory and actual fact in another way: lighter and heavier bodies, if released at the same time, always “fall” at different rates relative to the third body (in whose field they are “falling”, e.g. the Earth) except when they occupy Lagrangian points L4 or L5; in actual fact this falling rate difference is the underlying mechanism of Lagrange’s Trojan asteroids, which are even more readily detectable astronomically than the “infinitesimal” advance in the perihelion of the orbit of Mercury (which does not seem to be accounted for by Newton, but does seem to be accounted for by Einstein). Further, because of the asymmetry of their masses and gravitational interactions, if released simultaneously they will also accelerate at different rates even if we take that acceleration as “relative to the absolute” Newtonian space that Einstein rejected in relativity.

Digressive NOTE: Further, scientists have in recent years attempted to make careful measurements of the Earth and the Moon to see if they can find a difference in the rate at which they accelerate toward the Sun. They currently find no difference (to ~1 part in 10¹¹), in keeping with relativity. Depending on the structure of the experiment, they should be able to find the difference predicted by Newton’s theory. But since they have not been looking for it, in fact are still not even aware of its existence, this is not so strange or disappointing as a negative result.

The non-equal acceleration rates of lighter and heavier “test particles” or the need for an absolute spacetime in which they accelerate equally only for the first (truly “infinitesimal”, or less) instant... In the latter case we almost lose relativity before we even start. This... oversight suggests that even more fatal flaws lurk around the next corner (so at least ???), but we will give Einstein the benefit of the doubt, since Newton’s theory has survived both, so far. (So only...)
RATING AS AN... OVERSIGHT: ???

MANIFOLDS AND SINGULARITIES

If we try to put a non-“infinitesimal” mass in a local “infinitesimal” region — ostensibly a locally Euclidean region in a manifold — we will wind up with a “singularity”. It is difficult to conceive of a “singularity” as locally Euclidean, as required by relativity, and this is also not likely to yield a region within which particles move with the “uniform velocity”, also required by relativity. So, we are stuck with only regions within which there is at most “infinitesimal” mass (matter-energy).

If we put an “infinitesimal” mass in a local “infinitesimal” region, we have one of those ambiguous situations. We can ask “what is the mass density?” to get an idea of whether the gravitational field strength approaches “singularity” proportions. In any case, the problem of extrapolating-integrating such “infinitesimal” regions back up to non-“infinitesimal” region size remains problematic.

This one is hard to call, but its synergy with all the others suggests another nail in the coffin.
RATING AS AN... OVERSIGHT: ????

“STRAIGHT LINE” DEFINITIONS

In our everyday modern world we have an “ampolyguity” of ways to evaluate the “shortest distance between 2 points”: spatial distance (e.g. “as the crow flies” [e.g. what’s a ‘crow’?!] or “shortest driving distance” [e.g. what’s a ‘driving’?!]), or temporal distance (e.g. “quickest way home... uhh, that goes past the store... uhh, make that ‘quickest’ by my significant other’s internal clock” [dittoes]). Einstein, with his concept of “spacetime” should have made this a fundamental observation or question in his relativity, but he didn’t. Is this an... oversight?!

This should have been openly discussed and evaluated. It will probably hit pay-dirt eventually, but for now...
RATING AS AN... OVERSIGHT: ???

WHY DOESN’T E.G. ELECTRICAL CHARGE CURVE SPACE (-TIME)?!

If “matter (-energy)” curves space (-time), and this curvature effects the movements of particles along geodesic “straight lines”, why do we not also say that “electrical charge curves space (-time)” so as to effect (or at least affect) the movements of particles?! Of course, a charged particle has mass that curves spacetime, but its motion does not follow a “straight line” in that mass/gravitationally curved space (-time), it follows a path determined rather more by its charge and the electromagnetic field it finds itself in. Any classical mass (much larger than atomic/molecular) will (usually) have a net electrical charge that is extremely small compared with the maximum potential charge for that mass (i.e. assume the mass is entirely protons, or worse, electrons), but the trajectory of that mass may be determined rather more by that charge than by the mass itself. Maybe both mass and charge act to curve space (-time). Is this an... oversight?!

This should have been openly discussed and evaluated. It will probably hit pay-dirt eventually, but for now...
RATING AS AN... OVERSIGHT: ???

“GRAVITATIONAL LENSING”

It turns out that the bending of light by the Sun due to atmospheric refraction — which we know exists more certainly than we think gravitational lensing exists, Newton, again — is approximately of the same order of magnitude as current estimates for “gravitational lensing”, but no one (as of the late 1990s) has taken it into account, at least not with any publicity. When taken into account (especially along with Einstein’s estimate for the effective Newtonian mass and associated deflection of photons), what is left over deviates unacceptably from the value needed to fully support Einstein’s  gravitational lensing.

Too, the solar eclipse photos that ostensibly demonstrate gravitational lensing show erratic displacements of the stars, as if the Sun’s turbulent atmosphere — and it’s an absolute certainty that the Sun’s atmosphere is turbulent as far as density goes — is “roiling” the apparent positions of the stars so much that some even appear closer to the Sun by quite a bit, rather than the uniformly further away predicted by gravitational lensing. If only gravity were lensing the light, we should expect very very regular star displacements, all in the proper direction (away from the Sun).

Any astronomical mass has an atmosphere, and it is this atmosphere that is probably responsible for all or most of the apparent lensing by galaxies, etc. It is a next order of business to study the atmospheric refraction of the planets (all of them, but especially the massive ones) and the bending of star light in their atmospheres, and compare observations to relativity based and atmospheric refraction based estimates. Perhaps gravity helps bend starlight, but atmospheric refraction is a certainty.

There is worse, but it needs More Detail; see “Gravitational Lensing”.

Atmospheric refraction knocks the stuffings out of the bending of starlight around the Sun as a support for Einstein’s gravitational lensing being the only source of the bending. Plus, the roilingly erratic bending, characteristic of atmospheric refraction but anti-characteristic of gravitational lensing, was totally obvious from the eclipse photos, even the first ones by Eddington, but not from later simplified diagrams, that its lack of public acknowledgement itself deserves the rating.
RATING AS AN... OVERSIGHT: ?????

HYPER-DIMENSIONAL, HYPER-TOPOLOGICAL (SPACE-) TIME

It shows up much more clearly if one approaches it through an analysis of the... oversights in the concept of entropy, but Einstein also overlooked that his relativistic time dilation — with accurate clocks running at different but locally accurate rates as one approached the speed of light, and displaying different times (but still accurate for each clock) when the clocks became local to each other again (and especially when we put this together with quantum mechanics and its implications for the existence of particles doing the same type of odd things as our gedanken clocks, you know, the “there’s a quantum mechanical Mac truck driving through our living room right now, but it’s kind of whispy so it doesn’t smash into anything, at least not always”) — meant that:

• Time must be “hyper-dimensional” and “hyper-topological” (i.e. well beyond our usual concept of multi-dimensional, and it must also have a topology that is extremely far from Euclidean, one that at least in part is event dependent). Both Einstein’s clocks remain correct, and when they meet again they are at different times even though they seem “local” to each other both in time and space.

(It is not enough to have multi-dimensionality. E.g., a Klein bottle is “hyper-dimensional” and “hyper-topological” in a very simple way. Further, we must learn to distinguish carefully between-among the general hyper-dimensional spacetime that the particular event of our general hyper-dimensional spacetime is embedded in, and the particular hyper-dimensional spacetime general event that is embedded in it, and the particular hyper-dimensional events that are embedded in that... uhh... our “hyper-dimensional description languages” are still naively inadequate.)

This “hyper-dimensionality” and “hyper-topologicality” of time (and therefore also spacetime) should have been at least noticed and openly discussed and evaluated. It will definitely hit mother lode pay-dirt eventually, so...
RATING AS AN... OVERSIGHT: ?????

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