A little background is needed because most people assume or have been taught that GR is dependent entirely on the curvature of both space and time, because of the use of geodesics, because Riemannian curvature is taught as a prerequisite, because the whole tensor-covariant formulation of the laws of physics is taught as a prerequisite, and because they have never seen a formulation of GR with a particularly simple set of coordinates.

In a 2013 paper http://arxiv.org/abs/1308.0394 Fromholz, Poisson and Will - all eminent authorities; Cliff Will is NASA's goto guy for all GR experimental verification missions - demonstrate that some coordinates are not only simpler than others, but that commonly used coordinates may have conditions for which there are not even solutions.  In particular, the Landau-Lifshitz formulation is compared to standard textbook derivations of the Schwarzschild metric, and to Schwarzschild's original.  They conclude that the Landau-Lifshitz formulation "is much more complicated, has one aspect for which we have been unable to find a solution, and gives an explicit illustration of the fact that the Schwarzschild geometry can be described in infinitely many coordinate systems."

One can understand the coordinate issues oneself by working in two dimensions plus time with the Schwarzschild metric.  A good place to find a clear analysis from which to start is the class notes of professor Herter at Cornell http://rogachev.dyndns-at-home.com:8080/Copy/Physics/Cornell%20A2290_34%20(Schwarzschild%20Metric).pdf

On page 3 he gives the "spatial part of the metric," which for fixed angular position he reduces to: dσ = dr / (1-2M/r)-1/2  Then on page 4 he gives the local time in terms of ephemeris time (of a distant observer) as dτ=(1-2M/r)-1/2dt .  That the local time is slow by this amount (time dilation) is independent of coordinates.  We can measure the time dilation of clocks in a gravitational field by the signals they send (spectra and so forth, or literal signals from an atomic clock in a lab).  So dt and dτ are physically meaningful.  If dr refers to local coordinates in which the speed of light is measured as 3x108 meters per second, locally, then we know that either the dimensions of clocks (particularly light clocks) relative to ours have increased, or light travels more slowly, or some of both.  These are the only methods of achieving the required time dilation.

If any coordinates can be chosen, then it is possible to choose length measurements to be the same as those of the remote observer, and all of the time difference to be accounted by a slower light relative to the remote observer (though using clocks whose timing depends upon the speed of light, the local observer still gets 3x108 m/s). 

This is the choice most explanations of the behavior of light in a gravitational field adopt.  But often a relativist (a name for a GR expert) will switch to a coordinate system in which the speed of light is constant relative to the remote observer, and radial distances are stretched.  This gives those neat images of depressions in a plane that illustrate spatially curved geometry.  Our intuition is seduced by them because obviously objects would fall down in the depressions, but this is slight of hand.  We have unwittingly imposed a pre-existing notion of gravity on the image.  You never, ever see a "geodesic" drawn on one of these images to illustrate gravity, because it can't be done without a time axis and they don't illustrate time.  For the non-expert (even for physists) the switching (without really any notice often) between these coordinate choices leads to confusion and a sense that this subject matter is just beyond them. 

But it may not be a true impression.  The first choice of coordinates in which light moves slower in a gravitational field has time dilation (curved time, you might informally say, but I wouldn't debate the terminology) but actually flat Euclidean spatial coordinates, unchanged from the remote observer.  These spatial coordinates are physically meaningful.  If you do the math in GR, you see that meter sticks would locally correspond to these coordinates.  Even local radar distances will correspond.  But of course remote radar distances will seem longer as light in the gravity field slows down (in this coordinate choice) with respect to the remote observer.

But can you explain gravity in spatially flat coordinates?  (remember, time still varies in the gravity field)  It turns out that you can by a simple Hamiltonian formulation (energy conservation between kinetic and potential energy) which uses time dilation.  I have uploaded a draft exposition of this (currently in review) to support this discussion thread, here: https://www.researchgate.net/publication/264419808_Hamiltonian_analysis_using_time_dilation?ev=prf_pub

@Mozafar, thanks for your post.  That is exactly what I had assumed.  But you have made me aware of the terminology "coordinate velocity" and "measured velocity". 

The short version of my long post above is that it seems for some widely used GR metrics, using coordinates that correspond to the measured values (locally measured and integrated over distances) would yield flat spatial coordinates plus time dilation.  So it should be possible to understand gravity without Riemannian space.

Robert ~

I expect that when you see this you will have seen my remarks about coordinates on a different RG thread (“Is it possible to derive the constant & uniform velocity of light…”, which you yourself initiated. As is usual with discussion threads, it rapidly wandered away for the topic! – but that’s OK…). My recent remarks there could equally well be regarded as responses to the question you’ve posted here.

A coordinate system says nothing at all about the physics.

You ask “is spatial curvature necessary?”

There are in the literature some obscure “bimetric” theories. Think of an atlas, that represents areas on the surface of the earth (a sphere) on flat pages in a book. There are transformations that cartographers know, that translate the longitude/latitude coordinate system to a flat representation (an interesting topic in its own right...). That kind of thing can be done with the curved spacetime of GR. We would then have a formulation of GR in which Einstein’s gij is just a field in a flat spacetime. The physics is unchanged. But that doesn’t simplify gravitational theory. It makes it more complicated.

Robert ~

The metric inside a small laboratory on the surface of a large planet, deduced from the Schwartzschild solution is

ds2 = fc2dt2 – f -1dz2 – dx2 – dy2

Where f = 1 – k/z, z = z0 + h. h is height above the ground, presumed small. The measure of clock time is T = t√f.

T1/T2 = √[f(h1)/f(h2)] is the well known gravitational time dilation. If you take that into account, you can’t neglect the corresponding (less often considered) gravitational length contraction √[f(h2)/f(h1)]. In other words, vertical length measurements vary with height. The acceleration of falling bodies due to the gravitational field of the planet will not be a constant but will depend on h. The 3-space metric is f -1dz2 + dx2 + dy2.  You can write this 3-space metric as dZ2 + dx2 + dy2, but that describes the geometry as seen by an inertial observer, and that's not what we want - an inertial observer is a freely falling observer, not an observer standing on the ground at fixed h. There may be a way of getting around this to deduce the observed paths of light rays in the laboratory, without going to the trouble of integrating the geodesic equation, but I can’t see it just now…

@Eric Lord, the extra acceleration can be derived exclusively from time dilation, see publication below.  The contraction of height was discussed earlier on another thread and it was shown that if one makes local measurements and integrates, the upper and lower observers get the same measured value.  The apparent difference over longer distances is due to time again, and its effect on radar measurement.

OK. Thanks, Robert. I need to give it more thought.

The question related to the necessity or not of the spatial curvature I believe that it is just a question of the existence of energy at some place. No energy, no nothing. Presence of energy implies curvature, and if enough energy then we have complete curvature: i.e., a black hole.

@Antonio, see the Hamiltonian paper referenced in my first post on the first page of this discussion.  Energy and all the weak and strong field effects appear to be determinable from time dilation only.  I can't find a role for spatial curvature except to produce time dilation, but in that case the spatial curvature is removable by proceeding directly to the final value of time dilation.  See first post.

@Mozafar, I would refer you to the long discussion of light bending, in which almost everyone including me and Charles, evolved our opinions over time.  https://www.researchgate.net/post/Is_it_possible_to_derive_the_constant_uniform_velocity_of_light_the_Lorentz_transform_without_starting_from_the_principle_of_relativity?_tpcectx=qa_overview_following&_trid=53e385a7d5a3f258228b45b7_ pages 35 through about 42.

Robert,

In the Abstract of your post you say

"Frequency and therefore time dilation is a function of potential in a gravitational field, as in for example "gravitational redshift"

My question is: frequency is equal to the ratio of the speed of light c and the wavelength L of a photon: nu = c/L  . Here we have two properties, c and L, that both may be a function of potential in a gravitational field. One may define in physics the usual system of units , mass length and time, or another one like mass length and speed. In that sense frequency is the ratio of two properties, speed and length. And these two properties may be different functions of the potential in a gravitational field. Then I do not see how you can reduce everything to just one property: time.

I believe Charles (in post above) is saying almost the same thing I am asking or speculating about.  This is not surprising since I got the idea from him, but still I didn't know how he would like the way I phrased it.  As for weak vs. strong field, I think for now I'd be happy to have and understand such a formulation for weak fields, as long as they are post-Newtonian.

When Charles speaks of embedding Schwarzschild coordinates into a Euclidean space, he is answering the protest about curved "spaces" being provably not mappable to flat ones.  It turns out a curved "space" has a peculiar mathematical definition.  And the protest may be true in some context.  But we are only talking about curvature of space, not "a curved space" like a sphere, torus or cylinder (incidentally cylinders are considered flat, oddly enough).  Finite sections of any of these are mappable, and if coordinate singularities are allowed the sphere is mappable to a plane (two points, usually the poles, will map to lines rather than points).

Antonio is helpfully asking the obvious questions and setting up a discussion of metrics I was thinking about overnight, as well as Mozafar and Eric who used metric equations in earlier posts.  First regarding frequency, we get no information about the absolute relation of c and L as Antonio points out.  If an observer at a stationary point A measures the frequency of an annihilation photon locally, and then lowers a pair of particles into a gravitational field to a relative stationary position, allows them to annihilate there, and measures the frequency back at the same location as before, then this produces an accurate measure of the difference in potential energy between the two locations.  From this a Hamiltonian may be set up solely on the basis of this frequency measurement at the distant observer's location.  The draft paper analyzing this is now linked in the question description at the top of the page.

That frequency difference gives no information about "how" the frequency change came about, and I imagine Charles would be quick to point out the choice of whether to take it all as time dilation, or instead as a dilation (expansion) of lengths so that clock vibrators emit slower vibrations due to traveling further.  Or instead of dilated lengths of space, the lengths of objects could be contracted.  Or any combination of these, and possibly others I haven't thought of.

I also imagine Eric would be quick to point out that it makes sense to choose physically meaningful coordinates, either such as local observers would choose, or such as would be convenient for expressing the laws of physics.  The covariance mechanism in GR takes care of expressing the laws of physics, but results in a complex expression and things like "coordinate velocity" such that our intuitive notions don't apply.  We have to drag in covariance.  So I would agree with Eric also.  But there are still multiple choices!

The two we have discussed on another thread are radar distance over long distances, and meter stick measurements which are essentially local and must be integrated over distances.  And we reviewed and re-derived the well known fact that radar distances made over short distances and integrated equal the meter stick distances.  But radar distances in a gravitational field from high in the field appear longer because of the light time delay.  Conversely from deep in a field, measurement of the same distance by one step radar gives a shorter measurement.

Back to Antonio's comment, whether we take this to be more meter sticks or a slowing of light due to time dilation is arbitrary.  It might be a proportioned combination of the two.  But it is not fully both because that would give double the correct value.  It would be a duplication.

Now the usual way of speaking about this in much of the literature is to use the time dilation terminology, and hold that light slows in a gravitational field.  Since local clocks tick slowly, light is always locally measured at c.  Still this gives lots of people fits and you see perpetual questions on discussion boards.  How can light or anything falling "slow down" as if falls?  It only does that relative to the distant observer.  Think of it this way, relative to the distant observer there is a fixed distance to the center of a massive object.  Such an observer can measure this by moving around and triangulating in the approximately Euclidean space surrounding and distant from the massive object.  If the in-falling object takes too long to reach the center, or in the case of an event horizon never gets there, then it must be slowing down.

Still you occasionally see the flip side point of view, and authors of popular articles do not carefully distinguish.  This is that the distance to the center has increased, the "funnel" shaped coordinate space.  It is a mathematically consistent view as well in which no slowing down occurs, and is physically meaningful to the local observer.  To some people this is the "least preposterous" way to explain observations, as the slowing of falling things is counterintuitive.

So both viewpoints have physical meaning at least to some observers.  I would think both would satisfy Eric if expressed carefully and logically.  And I'm pretty sure Charles could think of a half a dozen more that make sense.

The distant observer's time dilation explanation, with the gravitational field embedded in Euclidean space as Charles mentioned, and as I described above with the distant observer running around in the space far from the gravitating object, spotting and mapping the positions of everything within to extensions of the surrounding Euclidean coordinates, is something that I spent a lot of time thinking about a few years ago.  I was not aware of the subtleties of coordinate choices at the time, and as Kassner pointed out, I made a particular choice of coordinates and formulated the laws of physics in those coordinates.  This I published in 2011 as "Isotropy, Equivalence and the Laws of Inertia."  Now that I have given this paper a new context, I'll link it again below.  I think Eric at least may be interested in it. 

In this point of view, watching things remotely which are time dilated is like watching a slow motion movie.  If one were to write a set of laws of physics to explain such a movie, it would be the equations in that paper.  Which are pretty simple compared to GR, by the way, but as Charles points out, very complex if multiple large masses are involved.  It, I think, amounts to embedding a weak field metric in Euclidean space. 

The explanation for things slowing down is to assume their mass has increased (though this is only their mass in regard to motions relative to the massive object, not a total system mass increase), and due to conservation of momentum, they must reduce their velocity.  Forces and acceleration and many other things change by a simple scale factor, so that all the laws of physics have their usual beginner simple form, with this extra mass factor, for which I used the symbol Γ (big gamma).  I used this because the symbol has a role similar to that of little gamma γ in SR, and Charles pointed out it conflicts with the symbol for the Christoffel symbols in GR.  However, once I began to understand that the Christoffel symbols take care of preserving ordinary operations like derivatives in various coordinate spaces, then I think by serendipity I have indeed chosen the correct symbol.  My usage is just a scalar version of it for a particularly simple set of coordinates in which only time changes.

Now to the question of using simple metric equations like dS2=... in our present discussion.  These representations make a lot of assumptions about spacetime which this question "questions."  So they can't be used to answer the question.  I'll give a simple example.  Take the time-independent (or you might want to say "stationary," nothing moving) Euclidean metric dS2=dx2+dy2+dz2.  If we say we are going to embed a weak field in Euclidean space, then we mean exactly that that metric holds.  Regardless of what happens with time.  We can't put a dt2 in there, or even a tτ2.  (Locally for SR we can do that of course, but the mutual relative distortion of lengths between local moving observers says nothing about these stationary lengths.)

This is a different way of looking at things.  But equally valid.  It is in fact the meter sticks method.  In developing GR Einstein dropped meter sticks.  He did not say why.  I can think of some reasons, but see nothing added by speculation.  In SR, the [mutual] length contraction is always associated with the de-synchronization of clocks in the other frame.  No such thing happens with gravity.  Once the ticking rates are accounted for, no skew in time appears when one moves from one location to another.  It is difficult for me to figure out how to even write a metric including time for the Euclidean embedding from only the distant observer's point of view (Charles, can you?).

Article Isotropy, equivalence and the laws of inertia

Robert:

I that think you ask a useful question and that the answer may be 'yes.'

Observationally, I think people are finding that the large-scale observed universe is (at least close to) 'flat' (within observational accuracy). People interpret such as correlating with 'just the right amount' of 'density.'

Permit me to provide some thoughts that correlate with a 'yes' answer to your question and that avoid the possible explanation of 'flat' as a 'coincidence with just the right amount of density.'

As you (and some other people who have commented regarding your question) may know, I have a math model that correlates with the list of known elementary particles and that points to possibly yet-to-be-discovered elementary particles. For zero-mass bosons (other than gluons), the overall list includes what I call 2e2 (photons, spin 1), 4e4 (gravitons, spin 2), 6e6 (spin 3), and 8e8 (spin 4). The math also correlates with 'coherences' for these particles.

For any 2 similar-sized, nearby clumps of 'stuff,' the dominant forces (within each clump and between the clumps) evolves during and after the big bang - from 8e2468 (an R^(-8) repulsive force) to 6e246 (an R^(-6) attractive force) to 4e24 (an R^(-4) repulsive force) to '4e4 plus 2e2' (R^(-2) forces). Here, R characterizes linear sizes and separations. Here, for example, 4e24 is a 'coherence' involving a photon and a graviton. For observable large-scale objects (they are larger than galactic superclusters), 8e2468 dominates (from early in the big bang) for a few billion years, 6e246 dominates for the next some billion years, and 4e24 dominates now. (The progression - increasing acceleration, decreasing acceleration, increasing acceleration - correlates with observations.)

For reasons I mention in my book "Physics Math Reset" (see attachment), each of 8e2468, 6e246, 4e24, and 2e2 correlates with 'flat.' So, the large-scale universe may 'always' (after the start of the big bang) have been and perhaps always will be flat (independent of energy density).

In Section 23 of "Physics Math Reset" (not included in the attachment), I address the 'what if people used "flat" coordinates' possibility. People may find the treatment speculative, but it may be OK. Some key points. In a 'co-moving flat' coordinate system, gravity includes a 'magnetic-like field.' (This is better phrased and explained in the book. There is also a way to think about a 'current' that would correlate with such.) Also, photons exhibit a non-zero 'g-mass' (the energy, relative to an observer, of the photon) and zero k-mass (as in a quantum mechanical version of E^2 - (c^2)(P^2) = ±(m c^2)^2; I include the ± for completeness; it appears that the 'minus' applies to quarks; please, don't let this be a distraction here). The photon g-mass would correlate with curved trajectories for photons.

By the way, it might be useful to think about standard uses of general relativity. Curved trajectories of light, perihelion shifts, … are observed. But, GR calculations may treat photons and planets as having no effect or space-time curvature. Perhaps this is OK for 2-body calculations in which 1 body has relatively small mass/energy. But, beyond such, …?

By the way, people might differ with your statement 'speed of light varies by direction.' Unless I misinterpret your remarks, people interpret experiments as 'speed of light does not vary by direction.' And, I think people generally confine their use of math to correlate with these results.

In any event, the answer could be 'yes.'

Thomas, thanks for interjecting the cosmic flatness viewpoint.  I think Charles and others have pointed out a multi-body problem is still very messy even in weak field Euclidean formulation (or in Newtonian gravity for that matter).  My statements about the speed of light and direction dependence was intended only to illustrate that non-isotropic coordinates probably do not have physical sense for local observers.  If the coordinates are isotropic, the speed of light does not vary by direction.  One may say in such cases as it does that it is only a "coordinate" velocity.  Very careful experiments in the 1960s found no anisotropy with respect to the sun or galactic center.

It helps me to understand physics when I can think in pictures. The picture here is helpful to me, so it may be helpful to others. One can’t visualise a curved 4-space or a curved 3-space, but a curved surface is no problem. The familiar funnel-shaped surface represents the θ = 0 surface in the spatial part of the Schwartzschild metric, the surface in which measured lengths are given by dL2 = (1 – k/r)-1dr2 + r2dφ2. The red line in the funnel (very crudely drawn – I sketched it by hand in photoshop) is meant to represent the path of a light ray falling in. The speed along this path of course involves also the temporal part of the metric (1 – k/r)dt2. This light speed can be measured by inertial observers. An inertial observer in GR is a freeling falling observer, not an observer at a fixed point (r, φ); the paths of inertial observers look similar to the light ray path, but at a steeper angle. Each inertial observer can therefore only make ‘local’ measurements and will conclude that the speed of light in a gravitational field is a constant, c.

My picture also includes a flat plane, onto which the funnel is projected by orthogonal projection. (This is analogous to the way in which distances are not - and cannot be - correctly represented in a map of a large region of the earth). r and φ become ordinary polar in the projected image the - the metric of the plane is dr2 + r2dφ2. In this flat picture, the speed of light appears to vary, continually decreasing. That’s because the component of velocity in the direction of the projection axis has been lost in the ‘flat’ description.

This picture helps me to understand that the question of whether the speed of light in GR is a constant or whether it varies is a matter of choice of description. I would also venture to say that whether space is curved or flat is also a matter of choice of description (unless we have global topological properties that forbid flatness, as in the "finite but unbounded" cosmologies). The physics is the same, however we choose to describe it.

Eric, nice picture.  Where did you get it? 

Charles, thanks for the confirmation of what I was beginning to suppose by looking at many metrics (with the k2 and k-2). 

"...nice picture. Where did you get it?" ~ Robert

I got an image of 'concentric circles' from Google Images, squashed to ellipses in Photoshop, and then kept using 'cut, copy and past' in Photoshop (It's a very versatile software for making diagrams). The edges of the funnel and the red trajectories were drawn freehand - that's why they look rough!

Eric, may I use the image if an occasion arises in a paper?

Charles, thanks for further clarification, especially about the lightspeed.

Robert ~ yes, of course you can use it!

Charles ~

Thank you for your comments.

The funnel-shaped surface has to be pictured as if embedded in a 3-dimensional Euclidean space because the only way we can visualize anything is to imagine it in 3-dimensional Euclidean space – an unfortunate limitation of the human mind. (Even the flat plane in the picture is embedded in 3-dimensional Euclidean space). In the context of GR, of course, there is no such embedding! The picture is meant only as a simple (and perhaps naïve) way of illustrating two ways of thinking about the behaviour of light in GR.

The constancy of the speed of light is established by Maxwell’s equations and by experimental measurement, and led to SR. To me it feels right and natural to carry this fundamental fact of nature over into GR. It can therefore be confusing, or even seem paradoxical, when physicists adopt the concept of variable light speed in a gravitational field. Einstein himself was the first to do this, even before he had arrived at GR in its final form, in Einfluss der Schwerkraft auf der Ausbreitung des Lichtes (Annalen der Physik 35, 1911), where the formula c = c0(1 - Φ/c2) appears for the first time. That, of course, is only what a non-inertial observer would see. (With respect to the gravitational field of the earth, we ourselves are non-inertial observers...)

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I said in my previous post “I would also venture to say that whether space is curved or flat is also a matter of choice of description.” I would like to qualify that by saying that the only way to treat it as flat would be to introduce the hideous concept of spacetime-dependent units of measure for lengths and times! Some people don’t like "curved spacetime" and hope for a gravitational theory that won’t need it, to supercede GR . (The apparent inconcilability of GR and QM motivates some of this.) As I see it, the equivalence principle inevitably imposes curvature of spacetime.

@Eric, thank you (re: figure).

@Charles, I want to emphasize two of your points in short bullets:

• "The speed of light is local constant because the physical coordinates [locally] are defined using light."
• " 'The hideous concept of spacetime dependent units...' is precisely what we do when we define coordinate systems for a curved space time."

I would add that on the first, this is true regardless of whether one assumes an objective reality for Minkowski spacetime, or whether one assumes a more limited applicability to quantum systems and electromagnetic systems.  The practical result is the same.  And the second speaks for itself to anyone who has tried to plod through covariance and tensors!

@Eric, re: "'Some people don’t like "curved spacetime" and hope for a gravitational theory that won’t need it.'  It is also true, by my unsystematic observation, that a greater number of people are in love with curved spacetime and have not considered whether it is sound, or proved, or necessary.  I was for 40 years one of those and would still be had I not begun to study inertia without using or knowing some of the finer points of GR (some of which I have absorbed since) and stumbled onto another way. 

We cannot do without curvature of some kind.  There is plenty of experimental evidence.  But as to exactly what curvature is necessary, and as to the proper interpretation of it, that is the subject of this question thread.  Previously (i.e. on other threads) we came up with an alternate explanation of experimental curvature.  If measurements are made by entities constrained to a Lorentz limit, of some "effect" which is unconstrained or constrained to a higher limit, the measurements will show curvature.  There was a long argument with Kassner about whether said entities would be able through simple increases in energy by their measurements to distinguish whether there was "real" curvature or only apparent curvature due to observation of more Lorentz-stiff effects (a term I coined).  The argument is complex but I will repeat it or find the old posts if needed. 

Putting that aside, even within our current framework of thinking, a framework which basically you and Charles have admitted is "hideously complex,"  : D and therefore not easy to think through, coordinates are somewhat arbitrary.  We have both Charles word for that and your clever picture.  The objections from others I think are answered in two parts:

So, I would like to throw out here a proposition.  Given the empirical data that exists (NOT anything predicted by GR, just the data), is there anything that rules out the possibility that we could describe the universe in terms of flat space, time dilation, and appropriate local physically meaningful measures and statements of the laws of physics?  If so what data is it?  If not, what is an experiment that should be proposed?

Charles, Einstein used a rotating disk.  I have thought about this a lot and read numerous papers on the subject.

The orbital situation might at first appear different because orbits have a natural velocity, and there is arguably an SR effect with that.  But as an SR effect, it is not agreed by other observers.  I'll come back to this.

Note that the problem of forming an Einstein synchronized frame on the disk was analyzed by M. Cranor, E. Heider and R. Price, "A circular twin paradox," Am. J. Phys. 68, 1016 (2000); http://dx.doi.org/10.1119/1.1286313 .  One cannot do so.  If one proceeds via 2-way light signals to synchronize adjacent clocks around the disk then when one comes back to the starting point there is a discontinuity.

Consider a stationary disk.  You fly around it in a rocket making measurements with your meter stick.  The circumference appears too small.  You let a string out behind our rocket which traces the edge of the disk and mark off meter marks to prove your point. Note that this reading has to do with your flight path and is true even if you have only imagined the disk.

Now you change places with an observer at a fixed point on the disk and measure the rocket's meter stick as it flies around.  It is shorter than your meter stick.  But the interval between the marks on the string are longer than your meter stick.

The disagreements about the lengths of the meter sticks you can trace to corresponding disagreements about clocks, and also the difficulty that Cranor et. al. point out.  It is an SR type of disagreement.  A thinks B's clock runs slow and vice versa.

But in gravity we have:

• No disagreement of clocks at the same radius, except for exactly that accounted for by SR.  There is no gravity contribution unless the radius changes.  Since the clocks are linked to the lengths, there is no gravity component to the length argument either, it is all SR.
• At different radii, the disagreement about clocks is the opposite of SR.  A thinks B's clock is slow, and B quite agrees (regarding the relative disparity).

By citing an orbital problem, you superimpose the rotating disk problem (the circular motion of the orbit) with the gravity problem (which could be done with stationary observers, i.e. having a constant measured distance between them, a constant radii, and zero angular momentum (or no ephemeral rotations).

Factor the two parts of the problem from each other, and you have an SR puzzle with the rotating disk (which like the Twins puzzle, is no longer puzzling when understood), and simple time dilation and no information about spatial coordinates with the gravity part.

Anyway, what I asked about was not thought experiments, but empirical data. 

Since light travel times can be explained in several ways, and conveniently by the changing coordinate (non-local) velocity of light in a gravitational field, there is no reliable information about distance from non-local radar measurements.  So we don't learn anything about curvature from radar measurements or even light bending measurements.  All we learn is that the equivalence principle is valid.  Light falls like everything else in a gravitational field, and bends due to the coordinate velocity gradient.

As far as I know, every prediction of GR regarding curvature has, when tested, showed space to be flat.  Time is obviously dilated, and that is sufficient to explain any measurement of curvature I'm aware of.

No one has an experimental result of non-flatness that cannot be explained by time dilation alone?  If so, we have a big gap in the experimental verification of GR's most loved prediction. 

Charles ~

I’m sorry that we seem to be misunderstanding each other. In my previous post I was for the most part agreeing with you, not trying to get into an argument.

Your remark “…you could do with revising the distinction between intrinsic and extrinsic curvature” was uncalled for. The embedding of a surface in a three-dimensional space is describable by the “second fundamental form”. A flat plane can be embedded in E3 as a cone or a cylinder, or as some other ruled surface. Its “intrinsic curvature”, a property of the “first fundamental form” (the metric), remains zero. In differential geometry this can be extended to embedding of a subspace of a curved space of n dimensions. I don’t like the terminology “extrinsic curvature” because this isn’t “curvature” in the usual sense of that word.

I have to disagree strongly when you say “The 'hideous concept of spacetime-dependent units of measure for lengths and times' as you call it is precisely what we do when we define coordinate systems for a curved space time. That is the basis of the mathematics of general relativity.” You are mistaken. When we use coordinate systems in GR and appeal to the “principle of general covariance”, the coordinates are dimensionless numbers. Labeling one of them “t” (as is common practice when dealing with the Schwartzschild solution for example) doesn’t make it into a measure of time! The units of length and time come from the metric properties of spacetime – basically a coordinate-independent concept - not from the coordinate system.

What I meant by the "hideous concept of spacetime-dependent units of measure for lengths and times" would be a transformation of the form gij → Mik(x)Mjl(x)gkl, keeping the coordinate system fixed. I think you will agree that this kind of thing would not be advisable! - curved spacetime provides the simplest interpretation of gravitational effects. The only instance I know of in which this kind of transformation was attempted was Weyl’s rescaling gij → σ(x)gij that he hoped might be interpetable as the electromagnetic gauge transformation.

@Charles:

@All, we (those of us following these related threads the last few weeks) have a much greater understanding of equivalence than anyone I've ever talked with.  Previously I had overlooked that full bending was explained by equivalence, and was unduly worried about some 2nd order effects.  This gives me much more confidence to use equivalence as an established premise to develop things mathematically, as Charles suggests.  So maybe it's time to recap how I got here asking about curvature.  Maybe this time someone would understand?

In 2007-8 I began to develop an old idea about inertia I got from a 1968 book by R. Good, something which appeals to electrical engineers.  I had been totally enchanted by GR and curvature for all my life and could not have imagined being without it.  It would have left me unable to explain the creation of the universe and without hope of an eventual future of exotic space travel with wormholes or warp drive.  Yes I was green, old but naive.

Turned out Good misunderstood Sciama and that idea was wrong, but I had fallen into a book by Ghosh which referenced Einstein's 1912 inertia paper, of which Sciama wasn't even aware.  And when I saw the simple equations for inertia in terms of potential energy, I realized immediately they were right and I abandoned all my previous work.

Why was this concept not more widely known in physics?  Einstein favored it.  It should be taught in all physics classes.  Eventually I learned of some of the objections.  It required thinking about the effects of mass at great distances, because with a 1/R fall off in effect and an R2 increase in shell volume, there is more effect from distant mass than nearby.  The horizon problem.  Some people also though there was an isotropy problem.  Thinking electromagnetically (Einstein had called it an "electromagnetic analogy,") people thought inertia would be a vector quantity, but careful experiments had demonstrated it was not, leaving physicists divided.

Being fond of SR analysis, and proficient at it, I decided to investigate using the equivalence principle.  I quickly found that an increase of inertia is manifest lower in the elevator, and that it is isotropic.  If it were not isotropic, clocks would tick at different rates depending on orientation.  They do not.  This was the careful experiment that was done in the 1960s.

When transforming reference frames, it is as I mentioned previously like watching a movie played at the wrong speed.  There are infinite possible ways this can be done.  To pick just one, a point of contact is needed.  In SR, a physical point of contact is possible as objects pass, and conservation of momentum is used to pick just one transformation.  It was natural for me to assume lateral momentum was conserved across transforms between the top and bottom of the elevator because it is essentially an SR transformation.

In the elevator, there is no effect on transverse lengths.  So with length of horizontal oscillators the same, and time slowing, then velocity must slow.  By conservation of momentum, inertia has increased.  If the bottom of the elevator is taken as equivalent to being lower in a gravitational field (i.e. if we accept the EEP) then proximity to gravitating objects must cause inertia.  This is just Mach's principle.  BUT, it is derived mathematically from well established principles, SR and the EEP.  So you see why my confidence in stating inertia theory is related to my confidence in the EEP.  Well, it increased a whole lot during this discussion.

So far all good.  Then the bad news started to come in.  Gravity started emerging from inertia.  First there was light bending, obviously half of it due to time dilation.  For a long time I could not determine whether this was part of GR or something extra.  In any case, no curvature other than time dilation is necessary.  But it turns out there is no irreconcilable conflict with GR after all, as Charles has put all that together nicely.

Then I made a mistake, I suspect a Freudian slip.  There was some suggestion that all of gravity emerged from inertia, but a question about how gravitational acceleration was affected by inertia.  Was it diminished like EM and everything else?  Or was it constant?  If the latter, gravity fully emerges.  No curvature needed.  I was not willing to go there.  I separated the paper into two parts, and intended to submit the innocuous first part to Physical Review (somewhat audacious I now realize).  I sent the wrong part.

The reviewer, intending to be disrespectful, said I would have to make it a field theory.  He didn't say it was ridiculous or impossible.  I started thinking about the field theory and within three days I had a basic idea how inertia accounted for gravity and equivalence, and shortly after had some math and a free parameter.  As you now I recently nailed down the free parameter with the Hamiltonian analysis.

Now I am in possession of a pretty good theory of gravity without the spatial part of curvature.  Something I never intended.  What do I do with it?  Just sit on it and ignore it isn't a credible or intellectually honest option, and makes me feel like I've been given something very valuable and just threw it away.  Can't do that.

In the intervening years, aside from nailing down the free parameter, I've discovered that GR verification authorities admit curved space is only a strong argument, not a proof.  And that no evidence has been found for it.  As you see, I didn't start out as a flat-spacer at all.  But the evidence is pointing away from curvature.

Hi Charles, often you are right about things that require insight and intuition, not to mention mathematical derivation.  I'm guessing your Jungian type is probably INTJ, or something with an "N" (for intuitive)?  I am an N-type, at any rate, and grow impatient with the S (sensor) types who want to plow through the data.  But whether it was assumed curvature was the only viable way to satisfy equivalence ... or not ... is a piece of data to be found in some early document.

Or, I suppose, you could provide your own derivation of gravity, which in a way you are doing, and argue that curvature is necessary from first principles.  But I did not see that.  In any case, this would be an interesting side discussion between us.  My question is in regard to the official position of physics:

• I have read and been told that it was assumed curvature was the only way to implement equivalence, that this is not part of the "proof."
• We in this and preceding threads have demonstrated that a substitution of spatially flat (curved or dilated time if one is not too picky about definitions) coordinates and associated reasonable physical laws explains all known data regarding the shape of space.
• I am in possession of a very good theory of inertia and therefore time dilation which has no need of the spatial part of curvature.  It was derived from equivalence and naturally includes equivalence, and happens to match the old and very famous intuitions of Mach (and Einstein).
• ⇒Spatial (not spacetime, just spatial) flatness becomes the de facto simplest and most plausible expectation and spatial curvature becomes more of an extraordinary claim requiring great proof.

Charles, a reference to something Einstein wrote would be convincing.  There is none among your remarks.  : )  I will begin looking for a specific reference to the "assumption."

Meanwhile, does anyone else want to weigh in on this, or is Charles the only one who will even challenge that curvature was a convincing assumption rather than a proved result?  (That would say something!)

See Einstein 1916 The Foundation of the Generalized Theory of Relativity http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity

In section 2 "About the reasons which explain the extension of the relativity-postulate" Einstein introduces the assumption that gravity will be generated by geometry.  This is a brief philosophical discussion.  There is no mathematical rigor in it.  He begins with an astrophysical version of Newton's "bucket" thought experiment.  The analysis Einstein gives is actually incorrect.  Neither he nor Newton nor Mach actually got it right.  I have a peer reviewed paper coming out shortly in Physics Education giving the correct analysis, and I'll send a preprint to anyone interested.

About half way through the section, saying "Besides this momentous epistemological argument," Einstein admits the bucket argument is not a mathematical proof, or any kind of proof.  Then he launches into a discussion of equivalence without at first calling it by name.  It is a short 3 and a half paragraphs:

• "Let there be a Galiliean co-ordinate system K relative to which (at least in the four-dimensional region considered) a mass at a sufficient distance from other masses move uniformly in a line. Let K' be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative to K' any mass at a sufficiently great distance experiences an accelerated motion such that its acceleration and its direction of acceleration is independent of its material composition and its physical conditions.
• "Can any observer, at rest relative to K', then conclude that he is in an actually accelerated reference-system? This is to be answered in the negative; the above-named behaviour of the freely moving masses relative to K' can be explained in as good a manner in the following way. The reference-system K' has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to K'.
• "This conception is feasible, because to us the experience of the existence of a field of force (namely the gravitation field) has shown that it possesses the remarkable property of imparting the same acceleration to all bodies.[2] The mechanical behaviour of the bodies relative to K' is the same as experience would expect of them with reference to systems which we assume from habit as stationary; thus it explains why from the physical stand-point it can be assumed that the systems K and K' can both with the same legitimacy be taken as at rest, that is, they will be equivalent as systems of reference for a description of physical phenomena.
• "From these discussions we see, that the working out of the general relativity theory must, at the same time, lead to a theory of gravitation; for we can "create" a gravitational field by a simple variation of the co-ordinate system.  [emphasis added] Also we see immediately that the principle of the constancy of light-velocity must be modified, for we recognise easily that the path of a ray of light with reference to K' must be, in general, curved, when light travels with a definite and constant velocity in a straight line with reference to K."

Here again Einstein does not pretend to have offered a proof.  He seems even more tentative.  It is not even an argument.  It is just "discussions."

Yet this is the key step in which it is ASSUMED that geometry must explain gravity.  Previously, in the introduction, he has asserted that the generalization of SR will proceed by assuming the interpretation of Minkowski that SR is geometry of spacetime, that it is a kinematic-geometric theory not physical dynamics, and so GR by definition is an extension of this "geometric" theory.

Now you can see the importance of the previous QA thread on constructive relativity and the derivation of Lorentz and the discussions about the sonic world, etc.  I do not view SR as a geometric theory.  Neither do a lot of other famous physicists (Swann, Bell, de Broglie, et. al.).  The reader may of course differ.  I have several friends on RG whose thinking I admire, even though I know they take Minkowski spacetime as an objective reality, not as a mathematical model of certain types of coupled systems.  That I admire their thinking in other areas does not mean they are right about Minkowski.

Anyway, Einstein goes in committed to geometry without any proof, just "discussion."  If I am allowed such loose discussion, I can show nearly anything by clever argument. 

This is only the first mistake.  Later in the derivation of the specific geometry Einstein proceeds without regard to or use of QM generally, or the Planck relation specifically, or the Heisenberg uncertainty principle.  The Planck relation together with the isotropy of clocks (insensitivity to orientation, which Einstein could have figured out easily) is sufficient to show that spatial geometry has no role in the energy relation that controls orbits.  The uncertainty principle suggests causal mechanisms but is not strictly necessary to reach the correct conclusion.  But Einstein is already beginning to choose to ignore the quantum revolution he boosted so greatly in 1905. 

If any of you who are very facile with the mechanics of GR and not opposed philosophically to exploring this avenue would like to go through with me the mechanics of the derivation, in private discussion, I believe we could find specifically where Planck and/or Heisenberg can be substituted for the spatial component, and the derivation carried to a different conclusion.

(P.S. to Charles, I would love to do this with you but I'm afraid you are philosophically unwilling, am I correct?)

Robert  and the others

There is a question which is close to this thread, Why length contraction in GR is not being proven experimentally yet?

please if any one has the desire just follow the link:

https://www.researchgate.net/post/I_want_to_know_why_length_contraction_in_GR_is_not_proven_experimentally_until_now?_tpcectx=qa_overview_following&_trid=53e7cb78cf57d7af0e8b4608_#53e7d5cdd11b8b9f198b45ee

sorry for interrupting.

regards

Charles, I normally don't bite the "will you accept" line because I can lead anyone on a merry chase with it.  I understand the satellite completely and I don't believe you can tell me anything new about it.  I will summarize briefly for your satisfaction.  Some other readers may be interested.  It provides insight into how one can use either time dilation or distance distortion to explain a given orbit.  Ultimately, the satellite's orbit is a consequence of its natural measurement of its environment.

In an approximately circular orbit, half as much relative inertia increase (and time dilation) comes from SR as from GR (or from gravity).  This is in my 2011 paper, linked again below.  SR and gravitational inertia must be used together to get the precession right.   Here are the two explanations:

So the satellite involves the same arbitrary choices of which coordinate to distort, time, length, or the appropriate combination.  You don't need both.  The one we reliably measure remotely is time.  The simplest course is to take time as the governing factor.

Another motivation for this choice is that a quantum action to cause the spatial distortion is not known, and those proposed cause renormalization problems, but it is easy to come up with a universal measurement-like interaction that causes inertia, and measurement-like interactions are massless (appropriate for something which causes mass) and do not cause any renormalization problems.  The theory is very mature, going back in quantitative detail to Einstein 1912, and qualitatively to Mach.

Perhaps the best reason to choose the time coordinate is that EQUIVALENCE produces exactly the right amount of time dilation.  But equivalence produces zero transverse length distortion, and only 2nd order radial distortion which is not nearly enough.  I had not even thought of this before, as much as I have thought about the problem.  Charles, thank you very much for provoking this insight.  You are the perfect debating partner!

Article Isotropy, equivalence and the laws of inertia

Charles ~

I said that it was not my intention to get into an argument with you, but I seem to have no choice (-:

(1) "this kind of curvature [extrinsic "curvature"] is not interesting in gtr"

Agreed!!

(2) "I think most people would think of a cylinder as a curved surface".

I wouldn't. I think of curvature as an "intrinsic" property. Studying Riemannian geometry trained me to think like that. But that's a matter of terminology, not worth arguing about.

(3) "Mathematically all quantities are pure numbers"

But we are not dealing here with pure mathematics, we are talking about physical quantities, which involve units of measure (mass, length, time, charge, etc). Coordinates are not physical quantities.

(4) "The time coordinate in Schwarzschild is exactly what I would mean by a measure of time".

Yes, that is true, but the Schwarzschild spacetime is a very particular case of the Riemannian geometry of GR. The identification of t as time is possible in this case because a specific kind of coordinate system has been chosen, adapted to this very special geometry. In that coordinate system you can make the identification, if you wish (but then you have to be careful to specify: whose time?). It is conceptually misleading to think like that in the context of more general solutions, and it violates the correct interpretation of the "principle of general covariance". (If I’d said r instead of t, you will agree, I hope, that r can be interpreted as a length measure tangentially, but not radially...).

(5) "It is actually rather awkward to attach units to tensors. Apart from anything else, in general we do not have the same units in different slots."

I don't see a problem. In physical applications all components of a tensor are required to have the same units.

Reverting to the original question that started this thread:

“If any coordinates will do in General Relativity, is spatial curvature necessary?”

The straightforward answer is YES, curvature is necessary to properly describe gravitational phenomena. The curvature properties of spacetime follow from the equivalence principle, as Einstein correctly deduced. Curvature is a metrical property intrinsic to Riemannian geometry and has nothing to do with coordinates. The metrical properties are determined by ds2, which is invariant under change of coordinates. If ds is directed along a timelike curve, we obtain a measure of time; if it’s directed along a spacelike curve we get a measure of length. These measures are independent of the choice of coordinate system. That is the meaning of the “principle of general covariance”.

This point led recently to a disagreement between Charles and myself, which was exacerbated by a discussion of the “embedding” problem, which is a red herring, irrelevant to GR. (There’s a paper somewhere in the literature that shows that any spacetime that satisfies Einstein's equations can be embedded in Euclidean space, but in general a Euclidean space with as many as ten dimensions is needed!: a curious, but not very useful or relevant fact…)

___________________________________________________

My simple figure seems also to have been misunderstood. The mapping between the the two surfaces has nothing to do with “embedding” or “extrinsic curvature”. The figure is simply an illustration - an aid to conceptualization - of how the speed of light can be observed as a constant by an inertial observer and as a varying quantity when observed by a non-inertial observer who remains at a fixed spatial point in a gravitational field. The mapping between the two surfaces shown in the figure is not a coordinate transformation, it is a change in the radial measure of length.

___________________________________________________

The original question also asked “Is GR more complex than necessary, and is curvature verified?”

My answer is that the equivalence principle (which I take to be well verified) inevitably leads to the conclusion that the curvature of space inherent in GR offers the simplest possible interpretation of gravitational effects.

Thanks, Charles ~

You and I have different ways of understanding GR. Your way works for you; my way works for me. No problem. The physical world doesn't care either way (-;

"For the metric tensor, some slots refer to time, some to space, some to both". But time occurs in relativity as a length (ct), so there is consistency in the units (L2) of gij. The EM tensor fij (i, j = 0...3) deserves some thought. When thinking about EM - which is rarely - I like to choose units so that c = 1, ε0 = 1, μ0 = 1, which tends to hide the problem of units. However, I am confident that, with dimensionless coordinates, the overall unit for the components fij would work out OK (it has to!). Not in the mood to think about that just now...

Eric, I didn't ask about curvature generally, only whether a reasonable choice of coordinates can be found in which for objects stationary relative to each other dx2+dy2+dz2 is invariant implying spatial flatness.  Then I gave an example of explaining an orbit with such a coordinate system.  Since you didn't point out any problem, I'll take your yes as a no, spatial curvature is not required.

Charles, since you didn't point out any problem either I assume that you are poised to agree if I answer your question.  But first tell me what inconsistency you are talking about.  And yes, I gave TWO explanations of an orbit, both involving some kind of curvature:

There is an objection to my argument which should have been given by any GR expert.  It's a good question that gets right to the heart of the matter, and exposes another powerful reason to choose inertia theory rather than GR.  I was thinking about my answer all night.  Now I don't get to use it!

I see I didn't state that, but it was intended in my #1 explanation called "time dilation" since I didn't specify otherwise.  In the Hamiltonian paper, which you reviewed long ago, I specified Euclidean spatial coordinates overlayed on the region containing the mass.

Come on, neither of you will ask the big question?  Clue: how did I produce gravity vs. how does GR produce it?  What did I deliberately discard and why?  How then do I obey the equivalence principle?  What can actually be derived from equivalence without any assumptions about the effect of coordinates?

Charles, Euclidean coordinates is what I asked about in my question, and in the last two posts "coordinate" was the only word I used, not "geometry."

Perhaps you are asking the question I was anticipating but using a different word.  Here is a definition:

• Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

Shape, size and relative position are addressed by coordinates which are physically meaningful as Eric has been insisting.  That is always the sort I have been addressing, and in particular I have tried to show that meter sticks would correspond to the coordinates I'm using.  That leaves the overly general phrase "properties of space" which would contain nothing surprising, except for Einstein's new claims in GR:

• Matter tells space how to curve.  Space tells matter how to move. [I think the exact quote is attributable to Wheeler]

So the phrase I highlighted would be a property of space in Einstein's theory.  I assumed everyone would understand that in even asking about "flat" space and talking about time (or inertia) being the cause of this or that, I was discarding any reliance on space to tell matter how to move, and using other mechanisms.  This is the crux of the matter.

In fact, that "space tells matter how to move" is already patently untrue even in GR.  It's role is limited to telling matter how NOT to move, i.e. it only applies to the absence of disturbing forces.

Einstein originally expected his theory of gravity to contain inertia.  He stated this point blank in his 1912 paper on inertia, and defended it over and over, including in his 1922 book and subsequently, but finally giving up late in his life in the 1950s.  I think the references are in my 2011 paper on inertia.

This leaves GR no better off than Newtonian dynamics.  There is still an absolute inertia.  This in spite of Einstein's great love of Mach's ideas and "intent" to include them.  If you look at the 1912 paper closely, and compare it with the opening sections of Foundations of GR, you can see what happened.  He over constrained the possible causal elements in GR, wishing to use only geometry as a motion causing agent, and when he was done he had only the cause of the motion of free falling bodies with no degree of freedom in the geometry to do anything else.  He had expected inertia as formulated in 1912 to just fall into place.  It didn't.

I started with Mach, and with Einstein 1912, and proceeded to develop a theory of inertia FROM EQUIVALENCE.  That is, I derived Machian inertia from equivalence.  Until now I did not realize the significance of that.  I thought equivalence was a flaky basis because I "thought" it did not predict correct bending.  You guys, but especially Kassner and Charles, set me straight about that (even though it was unintentional with Kassner).

Now I believe that the first order predictions of equivalence must ALL be taken to be true in a gravitational field.  Both what it predicts and what it doesn't:

• It DOES predict inertia is a result of gravitational potential, giving exactly the formula of Einstein 1912 and Sciama 1953.  This represents the same answer by three substantially different methods.  I believe this answer very strongly as a result.  Four different methods when you count Mach's epistemological argument.
• Equivalence DOES NOT lead to transverse length distortions.  One might at first think it leads to some radial distortions as a result of SR factors, but a clear accounting reveals that these are just the local SR differences and that a stationary observer does not see any first order length changes.  I covered all this in the 2011 paper.  (The problems we worried with earlier were 2nd order.)

So, the GR geodesic does not do enough to justify itself.  It cannot yield Mach's inertia.  Then came the big surprise.  A thorough formulation of inertia does tell matter how to move under ALL conditions, both free fall and under pressure from non-gravitational forces. 

I greatly appreciate all participants in this question.  I think my understanding of the matter has been significantly improved.  Even though I had most of the pieces, I could not clearly define what was the problem or benefit of the coordinate systems I was asking about when we started.  Now I can and just did.  I think all that's left is to just agree or disagree based on opinion and conjecture.  Inertia theory is more comprehensive with respect to dynamics, but a lot of work on cosmology might need to be re-done and I expect this to influence people's opinions.

By the way, Einstein's 1912 paper is so hard to find that Sciama missed it entirely.  It is still hard to find despite its age and obvious expiration of copyright.  My wife was kind enough to re-key it, including all formulas, and I re-drew the figure to avoid any copyright questions and have made it available via the link below:

http://mc1soft.com/papers/1912-Einstein_Inertia.pdf

Charles, "metre" is not in my vocabulary.  I will not engage in word games.  I think I have stated the case for inertia theory in Euclidean space (inertia implying time dilation) clearly enough for a high school physics student to understand it.  Sorry I cannot help you.  Geodesics are not going to survive.

Sorry, I never saw "metre" before.  I'm in America.  Our language was deliberately modified early in the nation's history to introduce differences from British English.  We do have some French words, but not that one.  I thought you were introducing some new concept.  Let me read your remarks again . . .

OK, you say "inertia means geodesics."  That is the objection that I wrote earlier I was "expecting."  In the flat space I derived, one can no longer use geodesics, so it is not a GR space.  That was a bit of a clarification for me, which came out of this QA thread.  Earlier you got me on to the arbitrariness of coordinates, and I created this thread to explore the limits of that, and I think I found them.  So to get where I'm going, we DO have to depart from GR.  I was not sure.  I kept hoping, like Einstein, to fit inertia into GR (inertia in the broader sense as response of an object's motion to application of a force, not just the path of an unforced object).  But he eliminated it without at first realizing it, and I now see clearly that it was the geodesic principle that removed his 1912 principle of inertia from GR.

So, I think this thread should end about here, of course with any closing comments one wants to make.  The answer is that if one is using geodesics to describe free fall, one must use the full Schwarzschild metric (or whatever appropriate in a given situation).  This cannot be transformed away.  But if one is willing to give them up, there is nothing to prevent going back to Euclidean coordinates.  And they are meaningful as Eric wishes.  We can actually get that much out of the metric, just not the geodesics.

My closing thought is this ... the Schwarzschild metric is inconsistent, because:

• It describes changes to the time part of spacetime that fully account for time dilation as observed remotely, and using the Planck energy relation and a Hamiltonian these fully account for all trajectories.  There is no "room" for a substantive role of anything else.
• It describes further changes that occur (in the same place and time) to the spatial part of the metric that if taken literally would result in additional time dilation which has not been observed.  But these changes are necessary to "point" the geodesic in the right direction and GR cannot do without them.

I should post this on Sadeem's GR thread I suppose.

Charles, what do you mean "you have not described a flat space."  Do you mean you are expecting me to give some mathematical description, or do you mean something I described as flat is you believe other than flat (and if so please exactly what).  Thanks.

Keep in mind that you cannot entirely rely on a GR mindset to describe something outside of GR.  Which I have concluded it is.  In other words, GR people would automatically say any space with a gravitational field is not flat.  But that is only a GR concept, not a universal concept.  It seems to me quite simple.  If static measurements give C=2piR at all locations, the space is flat.  Satellites would be subject to SR effects, of course, and would differ only in that regard to static measurements.

Robert and Charles and the others

Robert 

"the Schwarzschild metric is inconsistent" 

to Robert and the others

I agree with you Robert, the problem is initially in the Schwarzschild metric. Its not enough describing the space time geometry. Nevertheless we can't build a position to deny space time curvature due to that reason. We must search to modify it in a more consistent form. But space time curvature can't be assumed as a theoretical assumption only because its there and you can recognize it from experimental observations. 

Robert 

Thank you for the paper of Einstein nice displayed.

Robert ~

"coordinates ... are physically meaningful as Eric has been insisting".

No, I've been insisting in my conversation with Charles that, on the contrary, "coordinates are not physical quantities".

The result of that conversation was my realization that Charles was attaching the units of length and time to the coordinates, wereas I conceive of them them as attached to the components of gij. Since the key expression is the invariant gijdxidxj, both ways of thinking would seem valid - and, at least for special geometries and the special coordinate systems adapted to them, such as the Schwarzschild or the Robertson-Walker cosmologies, it makes no difference.

My diagram of a surface θ = const. in the 3-space of Schwartzschild represented a light trajectory as observed by a set of inertial observers and as observed by a set of observers located at fixed spatial points. The plane in the lower part of the diagram does hint at the possibility that the meter sticks of this latter set of observers might make it possible for them to treat a surface θ = const. as flat, and in calculating orbits in that surface, to deal only with the effect of the geometry on time. I'm by no means sure of this, it needs further investigation. Could this be what you've been trying to tell us?

Eric, I was referring to an older post.  Maybe I misunderstood.  No problem.

Sadeem, a few points on your post: "the problem is initially in the Schwarzschild metric. . . . Nevertheless we can't build a position to deny space time curvature due to that reason"

• I never denied that curvature was an option (in the spirit of coordinates being arbitrary), but that there exists a flat space with time dilation (I'd call this curved time but surely someone would object) in which objects obey intuitive physical laws (described in my 2011 paper) and gravity emerges from inertia.
• The Schwarzschild metric is exactly the spacetime shape required of geometry, or more specifically geodesics, determine the trajectories of inertial observers in the vicinity of a large mass where the masses of other objects can be neglected.  This is provable I believe.  Ask Charles.  It is the one and only such metric.  It may be expressed in many coordinates.
• If one accepts an inconsistency in the Schwarzschild (here I'm referring to the sufficiency of the time dilation alone based on Hamiltonian analysis, and the surplus contribution that arises when the spatial distortion is analyzed from the point of view of time), then one must accept also that there is an inconsistency in the rigorously related concept of the geodesics directing motion.  In other words, all metric theories of gravity fall due to a flaw in something they all share.

The redundancy of the geodesic is something I was thinking about many years ago.  I had nearly forgotten it, becoming lost in the details of discovering many interesting things.  When Charles accepted that equivalence explains full light bending, and used the phrase "Huygens refraction", I was lulled into a false sense of "ok-ness" with geodesics, i.e. maybe they just another way of looking at effects that could be described from multiple perspectives. 

But then when Charles and I got crosswise about - whatever it was, Charles never actually articulated an objection I could understand, but what I had expected was for him to object that geodesics wouldn't correspond to freefall trajectories in the coordinates I adopted - anyway as a result of that discussion I realized the inertia-theory coordinates absolutely rule out geodesics.

Charles, you are making the mistake of using radar length measurement over non-local distances.  No one does that.  NASA doesn't do it.  They have an elaborate book of corrections to give the distances that would result from integrating local measurements (which could be either radar or measuring rods).

I really think this QA thread has ended.  We appear to be starting over on the long discussion of radar distances, and I do not have time to repeat posts already written, so I'm hitting the exit button.  :D

Why didn't you answer some of the interesting comments I made?  Why did you try to restart the long thread on radar measurements?

This QA thread is closed.  The question has been answered.  The result is:

• It is NOT possible to construct a FLAT space in which geodesics form the trajectories of free falling objects. 
• It IS possible to describe gravitational motion as a Hamiltonian function with time dilation as the potential term, in Euclidean space, putting spatial curvature terms necessary for geodesics in conflict with conservation of energy.

@Charles, I will send you the document on corrections if you like.

I deduced that you were using radar measurements (consciously or not).  I realize you did not use the word.  If measurements are made either by integrating local radar measurements (equivalent to meter sticks) or by correcting long radar measurements using data from the clocks, then it is possible to choose either stretched time or stretched distance as an explanation for the time dilation, but not both (part of both is OK, but not all of both).  If choosing stretched time, then the spatial coordinates will remain Euclidean.

@ALL, following is a simple explanation of radar measurement issues relative to coordinate distortions that I wrote for a friend this morning who is not on RG.  I thought anyone puzzled by the discussion of radar measurements might enjoy it:

If you measure a distance with a radar reflection, and the signal from a clock at the other end is slow, you have several ways you could go in interpreting the results:

• The distance is true (in your coordinates) and the light speed (in your coordinates) is unchanged. In that case, if the local distances at the other end agree with your distances then lightspeed has changed locally at the other end.
• If we have a message from an observer at the other end saying lightspeed has not changed, then we know one of the other two suppositions is false, thus either:
• - the distance is true
• - the light speed in our coordinates is changed

If the distance is true, then time at the other location is dilated because clock pendula swing through greater distances, thus ticking slower.

If lightspeed is less, then clock pendula (or photons if a light clock) move more slowly through the same distances.

The first is spatial coordinate stretching. The second is time coordinate stretching. Either one alone produces the observed time dilation. Both together, slower light through greater distance, would produce twice as much time dilation.

- - - - - [end of explanation] - - - - -

A CONTRADICTON (discovered this argument just now)

Consider three observers about 1 AU apart, A B and C.  They measure their distances as "1 AU" (distance from earth to sun) in empty space using radar.

Now bring the sun very near to B (suppose he has good thermal tech and doesn't burn up).  They observe radar times are longer between AB and BC, and clocks tick slower (both are actual empirical results).  Comparing clocks, they choose to assume that light is slowed by gravity (i.e. the time coordinate is stretched) and they conclude the spatial distance is unchanged.  This is the way NASA makes measurements.  Me too.

Of course it is possible to "assume" the distance is greater.  But it is an assumption.  You don't have to assume any stretching of time coordinates in such case, as the greater dimensions of clocks will require pendula to move farther and explain the slow time.  The fact that objects are stretched as much as lengths, however, means that the distance in terms of multiples of a measuring rod would NOT be changed.  So this assumption leads to a contradiction.

OK, let's try another variant.  Assume the distance is greater but not the distances within quantum coupled objects (material objects).  Then the measuring rod distance will increase.  But since coordinate lightspeed is unchanged, these clocks do not tick slower, which contradicts the data.  So this method does not work either.

So, all the variants I can think of for the stretched distance assumption lead to contradictions.  This is even worse than I thought.  I cannot imagine how GR became an accepted theory?

-

Keep going Charles.  I would never have discovered these simple and effective arguments without your prodding. : )

Charles, please object to a specific point.  I suspect you have not because you cannot.

Construct a simple tensor in one dimension plus time which corresponds to my spaceship thought experiment, with all the measurement assumptions evident, and I'll respond to your statement about tensors.  Again, I believe you cannot construct one without making measurement assumptions that will betray the limitations of tensors.

I have simplified my thought experiment to only two observers.  A and B are traveling on spaceships, parallel, 2 AU apart.  They determine distance by radar, and navigate (maintain course) by ephemeral stars.

The sun passes between them.  They notice the radar delay between them increases.  The ephemeral stars near line-of-sight to the sun change positions but no others do (so they easily remain on course).  The two observers make different assumptions.

• "A" assumes light slows and bends in gravitational fields.  Reasoning from this A concludes also that time slows, and sending a probe with a clock, confirms this.  With a slow clock on the probe, A realizes it still measures a locally constant speed of light.  Correcting for the slowing and bending of light, A retrieves the same 2 AU distance measurement as previously.
• "B" assumes that space is distorted around the sun.  Space is stretched so that light takes longer.  So the distance to B is greater than 2 AU.  So far so good.  But B wonders what happens to time.

B reasons that if space is stretched, then objects in the space may or may not be stretched a similar amount.  B analyzes these options:

B sends a probe and finds that time is dilated and chooses option 2, that objects are stretched as much as space. 

A and B get together and compare notes.  A points out that if B's assumptions are correct, that objects stretch as much as space, which is required by B's empirical results on time dilation, then the same number of measuring rods would fill the space between A and B as before the sun passed between them.  Therefore the distance would be 2 AU, not stretched at all, contradicting B's assumption.

Eric, simple, you are mixing coordinate time at 1, with proper distance integrated through the interval.  Try using both proper time and proper distance, in which case both will be integrals.

@others ... it is so easy to mix up coordinate and proper time or make other mistakes with the equations that it is better to stick with the thought experiment.  Once you understand the thought experiment, then you have a better chance of figuring out your equations.

Charles, I'm shocked, I never said you were incompetent.  The thought experiment I constructed is very simple.  I have studied the math of GR and found it flawed.  It sounds like to me you cannot answer my puzzle and are becoming emotional.  Please edit your response to remove personal attacks.

Charles, there is nothing wooly about my puzzle.  Eric is up to the challenge.  I'm looking at his response now.

Eric, are you talking to me?  I assume so, and that you are giving a GR analysis of the puzzle.

If  α = (1 – k/r) then it will be smaller at smaller r.  The time T which you compute I will label as T1, as it is measured at r1, which is the greater r.  Using α1 it is a smaller interval than an ephemeral observer, reflecting some time dilation at r1.  The t1 and t2 values you give, however, are both measured at r1 and are experimental, not calculated, in your setup. 

Just to keep my thinking organized and us on the same page, I'll note in passing that if an observer at r2 < r1 made similar measurements, this would give a smaller proper interval T2 < T1, due to α2 < α1.

However, the radial integral you give has just "α" unsubscripted.  I assumed you meant the function α(r) through the integral.  This quantity is independent of which observer performs it.  But the distance cannot agree with both T1 and T2.

Eric, thanks much for this example!

Robert ~

I found mistakes in my calculation and have deleted it till I can think it through. It's not as simple as I'd thought. Nevertheless, I stand by my statement that "the idea that there's something "inconsistent" about the Schwartzschild field is quite absurd."

@All, Charles seems to be disturbed this afternoon.  I have successfully passed 4 calculus and a differential equations class and have two degrees in engineering.  I certainly know well what an integral is.  Charles... desist with the derogatory remarks.

Here we begin with the setup as stated earlier by Eric, observer A at radius r1 and B at r2 such that r2 < r1.  But instead of computing times I will only compute distance.

Taking a general form of the Schwarzschild metric for constant angle dφ=dθ=0:

dτ2 = α dt2 - dr2/α                  (1)

Where τ is proper time for a local observer accumulated on his clock, wherever he has moved, and t is ephemeral (distant) time, and r being defined as C/2π. Units are c=G=1. And α=(1-k/r) where k=2M. Proper distance is S:

dS2 = - αdt2 + dr2/ α               (2)

Notice that α < 1. If measurement is made along the path at synchronized ephemeral time (dt=0) then:

dS = dr/ α1/2                            (2b)

S = ∫AB (1 / a(r)1/2)dr                         (3)

If the distance (r1-r2)

Robert ~

Here is my corrected analysis:

An observer at some point (r1, t1) in a Schwartzschild field sends out a radar signal to an object located in a radial direction at some unknown coordinate r2. The arrival of the signal has coordinates (r2, t2). Note that t and r are dimensionless coordinates, not measures of time and distance.

The trajectory of the signal is given by αc2dt2 - α-1dr2 = 0, were α = α(r) = (1 – k/r). That is, dr/cdt = α(r). Hence t(r) is a known function

t(r) = ∫ r1r[1/cα(r)]dr. (It’s probably expressible analytically, but I’ve forgotten all the tricks of integration I knew at school! But at least a numerical solution can be obtained.) The inverse function r(t) is then also known.

The time for the signal to reach r2, measured by the clock of the observer at r1 (ie., half the time observed for the signal to return) is

T2 – T1 = α(r1)1/2(t2 – t1). The actual distance to r2 is then

D = ∫ r1r2 [α(r)] -1/2dr = ∫ t1t2 [α(r(t))]1/2cdt .

This latter expression involves only quantities known to the observer at r1

[t1 = T1 α(r1)-1/2, t2 = T2 α(r1)-1/2 and the function r(t)].

It’s reasonable to assume that algorithms based on this, and on more complicated situations where the observer at r1 and object at r2 have different latitude/longitude coordinates, and when the object is moving, are well known and in daily use by NASA people and others who have the responsibility of tracking satellites.

The mathematics of GR is difficult, but there is no inconsistency in the theory. It correctly predicted perihelion precession, bending of light around the sun, and the behaviour of clocks in the earth’s gravitational field, without any ad hoc adjustments.

I’m prone to making silly mistakes when doing mathematics, but I firmly believe that mathematics is a far more secure guide when thinking about physics, than verbal arguments, which can so easily lead one astray. I believe (and I know that Charles would agree) that they are leading you astray.

Robert ~

Let me put it this way:

Geodesics in a Riemanian geometry are well-defined geometrical objects.

Schwartzschild spacetime is simply a particular Riemannian manifold. It’s a well-defined geometrical object. Obviously, a geometrical object cannot be inconsistent with itself.

The Schwartzschild spacetime arises as a particular solution to a set of differential equations. A solution to a set of differential equations cannot be inconsistent with itself.

Therefore, the only possible kind of inconsistency, if there were one, would have to be inconsistency with the physics that the spacetime and its geodesics are supposed to describe.

What, then, in your view, are the observations that contradict the predictions implied by the Schwartzschild geometry and its geodesics?

I’m not aware of any.

Eric & Charles, yes the inconsistency is in the spacetime itself, I thought that was obvious. Yes the S... metric describes both time and space stretching.  That is the problem.  I doubt there is any error in the derivation of the Schwarzschild metric, so the problem is back of that.

It derives from an error in the derivation of GR, when it was assumed geodesics are responsible for free fall paths.  This didn't come from equivalence, it was a separate assumption. I believe all the derivation was probably correct once this assumption was made, but it was not carried far enough in the midst of all the complexity and new concepts to see a conflict.  Specifically, the Planck relation was not introduced.  I doubt without that I would have poked around long enough to find the time conflict. 

This is a big finding.  I will work to explain it better and promote understanding.  I appreciate the help of you guys and also Sadeem and Kassner, even though 3 out of 4 of you are unwilling you nevertheless helped by prompting me to think about it and clarify the problem.  The idea of redundancy of the geodesic concept has been in the back of my mind ever since gravity emerged from inertia in about 2008 or 9. Had Kassner and Charles not straightened me out on bending and equivalence I'd have never pinned it down.  I will continue to post wherever it is relevant in hopes of getting additional feedback and assistance, which is what RG is for.

It is normal and natural to first deny.  On the face of it the greatest probability is that such a thing is simply a mistake, and of course I will keep looking at it.  Coursework only directs one toward the proven paths in a complex theory, and graduate work is directed at the frontier not the basics.  It just was a happenstance I was working on inertia, not gravity, without realizing (at first) they were joined at the hip. 

P.S. also want to acknowledge role of Asif's theory in getting me motivated to re-examine bending in equivalence, and the long debate between Charles & Crothers in another thread which I didn't participate in but which raised my awareness of the meaning of the coordinates in the Schwarzschild metric.

The geodesic is a geometric concept which is defined as the locally straightest path.  The "assumption" was that this had physical meaning.

All components of motion must have an energy value and be able to be represented at that value in a Hamiltonian or similar energy conservation relation.  Time dilation is easily represented by the Planck relation of frequency (reciprocal of time) and energy.  Spacetial curvature has no intrinsic energy meaning, but Einstein needed it to make the idea of geodesic's work.  Einstein "knew the right answer" because he was also deriving from equivalence, which as we saw recently really does have all first order effects in it.  GR is complex enough that the path to connect back to ambiguity over the energy value of curved space was overlooked (though many people have been concerned about energy conservation in GR).  All I've done is look at the energy value of curved space by examining whether just space is stretched or also the objects in it.  There is a problem either way.  If the objects are stretched (which is what you'd expect from SR), the curvature is meaningless and vanishes.  If they are not stretched, spatial curvature takes on an energy value equal in magnitude to the time dilation and in the same direction, which we don't measure, since from a distance it all gets rolled into frequency.

Newton's first law is a special case of the second law F=ma in which F=0.  This can be proved to be equivalent to the Hamiltonian H=T+V which I have used to argue that each thing which influences motion must have an energy value.  The advantage of the Hamiltonian formulation is well known.  One does not need to identify "forces."  This is what makes it so useful in describing the propagation of quantum waves.  It is also useful with gravity since gravity is not a force (or has not been thought to be one for over 100 years).  For an accounting of energy in a gravitational fields solely in terms of time dilation and kinetic energy, see the paper below.

The first law of motion says objects " move at a constant velocity, unless acted upon by an external force." ( from http://en.wikipedia.org/wiki/Newton's_laws_of_motion )  A geodesic in flat space accomplishes this, and in curved space it does not.  Obviously Einstein thought he might use the notion (roughly the analogous to a "straight line") to explain gravity, which due to equivalence doesn't seem to have exactly the properties of a force.  But doubtless Newton would have rolled over in his grave.  (Maybe he did, has anyone checked?)  ; )

Newton's first law is upheld if an object continues at the same velocity.  There is no fuzz on this.  And he viewed free fall in a gravitational field as acceleration.  Now if the proposition is correctly called "Einstein's law of motion" referring to geodesics, then obviously I am pointing out that conflicts with conservation of energy and you have to pick one.  It is an easy choice for me to pick conservation of energy.

Newton's first law says "When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force."  from: http://en.wikipedia.org/wiki/Newton's_laws_of_motion

This is a special case of Newton's 2nd law F=ma where F=0, from which energy conservation can be derived.  In other words it is closely related to the Hamiltonian H=T+V.  The Hamiltonian is more useful for propagation of quantum waves or in a gravitational field when specific forces may not be identifiable.  Each factor (force or otherwise) which influences motion will have a corresponding energy value which contributes to the Hamiltonian.  This is completely consistent with and derivable from Newton's first and second laws.

A geodesic is the analogy of "continuing in a straight line" in curved spacetime.  Newton actually said "continues to move at a constant velocity."  Geodesics in curved spacetime will cause velocity to change.  Newton viewed gravity as a force and Einstein observed that it was probably naïve to assume gravity was a force (on account of the universality of equivalence), and so Einstein suggested the geodesic  principle that objects in free fall follow a geodesic.  (It's a nice educated guess.  Newton of course viewed such objects as accelerated.)

The trouble with Einstein's guess, or assumption, or perhaps we should call it "Einstein's law of geodesic motion," is that Einstein did not assign energy values to the spatial and time curvature.  We can use the Planck relation to assign energy to time (the reciprocal of frequency) and measure the time contribution easily.  Where does that leave the spatial contribution?

Einstein was deriving also from equivalence.  As we have seen earlier on this thread, equivalence does indeed contain all the right answers as to motion in a gravitational field.  So Einstein got the right numerical answer to motion.  But as I have shown in the paper linked below, the time changes implied by only the time coordinate of a Schwarzschild metric account for all energy changes and all motion in a gravitational field.

The spatial curvature is necessary to "point the geodesic" in the right direction to get what Einstein knew was the right answer (from equivalence).  Unfortunately, the spatial component will have to be associated with energy also, and then there will be too much energy change predicted by the Schwarzschild metric.  This has been overlooked because of the many different ways to calculate sizes and coordinates and other things energy depends upon, but in the argument above I showed that one way or another, either there is a conflict of the geodesic with energy conservation, or the extra spatial curvature goes away (because measuring rods are also transformed, thus it would be in that case a difference that makes no difference).

When given a choice between conservation of energy, and Einstein's guess about the appropriate geometric analogy of straight line motion (as a substitute for Newton's constant velocity), I'll put my money on energy conservation.

I realize this will be hard for GR experts to take.  I did not set out to get involved in anything sure to be so controversial.  Give it time.  I think it will sink in.  Keep remembering that I started from inertia and only accidentally found that gravity emerges.  Then I had to spend 6 years studying gravity (the learning about things Charles mentions) to understand which was right and what the problem was.  I'm pretty sure this is now a consistent explanation, but of course I expect people will go over it with a fine tooth comb and make every possible objection.  : )

Robert ~

What you say is correct, the geodesic hypothesis was, in the beginning, Einstein’s ‘guess’ – an additional hypothesis over and above the field equations. If my memory serves me correctly, he did later produce an argument to indicate that the geodesic motion of “test particles” is a consequence of the field equations. Unfortunately I cannot recall the reference. A google search came up with many publications relating to this topic, which I haven’t the time or patience to go through in any detail - these considerations seem redundant to me because, as I understand it, the geodesic “hypothesis” is a straightforward consequence of the equivalence principle, which obliges us to generalize these special relativistic principles: for an inertial observer Newton’s first law applies to a massive particle and light travels in straight lines with constant speed. These principles of special relativity, expressed in a general coordinate system, take the form UiUj;i = 0 (semicolon denoting covariant differentiation employing the Christoffel symbols); the equivalence principle, once accepted, compels us to take this geodesic equation over into GR.

Here is my own attempt at a simple derivation of the geodesic “hypothesis” for a massive test particle, without appealing to the equivalence principle. (A “test particle” is a particle whose own contribution to the gravitational field can be neglected as insignificant.) The energy-momentum tensor density of “matter” satisfies Tij;i = 0. For “dust” (a distribution of massive “test particles” that exert no forces on each other) we have Tij = ρ UiUj where Ui is four-velocity and ρ is rest-mass density. Then Tij;i = (ρ Ui);iUj + ρ UiUj;i. But conservation of rest-mass is (ρ Ui);i = 0, so we get the geodesic equation UiUj;i = 0 for the worldlines of the individual dust particles.

                      __________________________________________

My problem, Robert, is that my habitual way of thinking about relativity is so very different from yours. Consequently, I have great difficulty understanding your longer posts (especially the “stretching” of space and of time, which is alien to my way of thinking). It is clear to me that you have a quarrel with the "equivalence principle" and that you are convinced you are “onto something”. Maybe you are. But I don’t “get it”.

Eric, Charles, you two are doing admirably well in debunking the weaknesses of Robert's claims.

I was contemplating to point out the irony that manifests itself in, on the one hand, his insistence on all gravitational effects being contained in the equivalence principle, whereas, on the other hand, he refutes geodesic motion, which is precisely that aspect of general relativity that can be "derived" from special relativity using the equivalence principle.

You have essentially said it all in your preceding posts, but let me nonetheless give a short version of what I was going to say.

Robert claims to be an expert in special relativity. As such he must know how to formulate special relativity in general coordinates, for example in the Rindler coordinates of an accelerating frame. What is the equation of motion of a free particle in *special* relativity in its most general form? Well, we discussed this before: d2xi/dτ2+Γikl dxk/dτ dxl/dτ = 0.

(As an aside, when Charles explained the Christoffel symbols to Robert, the fact that Christoffel symbols may be nonzero even in a non-curved spacetime made me think that the explanation referring only to curved geometries should be refined. But then I decided that this explanation was as good as any to a beginner in the field.)

Anyway, the equivalence principle says that it is always possible to transform to a local frame of reference, in which the gravitational field is absent and the laws of special relativity hold. In such a frame, the above equation of motion, which describes motion along a geodesic, must be valid. The only additional assumption that is then needed to have it hold in general relativity is that there are no higher order corrections (e.g., in powers of the curvature), i.e. that this is already the exact law. That it holds on small scales is already a consequence of special relativity. That it is in fact exact is the simplest assumption that one can make in generalizing, and that this assumption is valid has been tested experimentally to a certain (high) degree of accuracy.

Of course, one can also derive the geodesic equation from the field equation. This is a consistency requirement for the theory. It means that general relativity could in principle have been developed by skipping special relativity. But one does not *have* to derive the geodesic equation this way. The field equation is the part of general relativity that cannot be easily guessed via generalizing special relativity. The geodesic equation is a straightforward generalization from special relativity.

Charles ~

Of course I don’t think the geodesic hypothesis was a “guess” – that was my point. I was paraphrasing Robert: “… an error in the derivation of GR, when it was assumed geodesics are responsible for free fall paths.” (Meeting him half way before presenting my argument....) The distinctions among the terms “hypothesis”, “assumption” and “guess” are subtle.

I agree with everything you just said.

Dear Robert, it is good to have a look at this paper attached. There are still problems in physics about who says that photons lose energy in a gravitational field and those who don't. The answer is in this paper, wanted also by J.A Wheeler, and in the GP-A experiment. The Einstein equivalence principle and the tested clock hypothesys if put together seem to have some issues...

Article Gravitation, photons, clocks

Stephano ~

This is indeed a difficult phenomenon to interpret correctly. Some of the puzzling aspect of these experiments come from the fact that the experimenter is not an “inertial” observer. In a “thought experiment” an inertial observer (carrying a clock) would be an observer falling down the tower alongside a beam of light. He would not see any red shift, I think. Compounding the problem is that the energy of a photon E = hν is a quantum-mechanical concept, whereas the only viable gravitational theory we have is a classical theory. This kind of mixed theorising is treacherous. Perhaps something could be done with the classical theory of light (Maxwell) in Einstein’s gravitational theory. The problem then is that it becomes in principle impossible to to formulate energy (and momentum) exchange between electromagnetism and gravitation (that might account for any loss or gain in the energy of photons) in a coordinate-independent way.

A lot of mysteries remain, even in "well-established" physics   (-:

Thanks to everyone for their posts.  Some items like the paper I will have to look at, but I want to address what other people are saying about my opinions on equivalence and geodesics, and special relativity, and correct any mis-impressions I may have given.

On equivalence, I have been fascinated by it for many years and read many arguments about its validity without having a position, but essentially accepted it as fact.  When encountering the ideas of Einstein 1912, Sciama, Wheeler-Ciufolini, etc. on inertia, and the anisotropic objection raised in the 60s on account of which some discarded that inertia theory, I decided to try using equivalence to analyze it.  We we accept equivalence, and inferred inertia from equivalence is isotropic, then gravitationally inferred inertia (sorry for the vague term, but my papers are available for any to read) would have to be isotropic, overcoming this objection.  I didn't intend to get into any part of GR.  I considered it beyond my scope.  My efforts were successful.  Lacking professional guidance, I expressed some concern about 2nd order effects in the 2011 paper and proposed a way to sort of paper over them.  Later on this thread I learned from Kassner, Charles and others that no one worries about 2nd order effects and I breathed a lot easier.  Because equivalence has to be absolutely accepted for my argument about isotropy to work. 

During that analysis I had a difficult time with the bending and chose to ignore it entirely since it didn't seem to bother other people, and basically no one that I was aware of was proposing that bending should come entirely from equivalence.  When discussions on RG got into bending, just for a lark I fished out my analysis and asked about the lack of bending in equivalence.  This annoyed Kassner so much he quit posting (glad to have you back!) but while searching his references I found that the last paper on the subject in AJP accepts the bending, and shortly thereafter Charles endorsed this and wrote it up on his website.  I was very glad to have my opinion updated and once again to have equivalence in full force!

On special relativity, I'm referring in the narrow sense to the original theory, prior to Minkowski's intervention.  My familiarity with Minkowski spacetime and related items is more or less average.  There is a specific reason for this, which none of you will like very much.  It is because I have been influenced by many authors in regard to whether Minkowski space or metrics in general represent an objective reality, or a mathematical model with limits on its applicability.  The evidence presented by Swann in 1960 convinced be beyond doubt of the latter.  Please let us not get into a long discussion of this, but just acknowledge disagreement.  You will need to consider in my responses, however, that they will not be based on the assumption of objective reality behind SR metrics (GR metrics are another question ... as Charles suggests by considering static fields and static or slow moving objects we can avoid the SR component of the metric).

I strongly favor constructive relativity, and some of you joined this discussion on my Lorentz thread, which essentially sought ways of deriving SR constructively.  In the last couple of months, I've begun to think about applying the same constructive ideas to GR metrics, i.e. that the spacetime changes come about because of the behavior of waves in EM or QM coupled systems.  When I give analyses which are influenced by this thinking, they will sound very strange to you guys.

On geodesics, learning more about them has been a major project for me since I found gravity emerging from inertia in 2008 (rather than the other way around as generally supposed by the references I gave).  My first thought was that if gravity emerges in this way, there is no role for geodesics, and that is a problem because no one will be interested in what I have found about inertia.  I took two approaches:

• I separated out everything I could which did not require any revision of gravity and put that in the 2011 paper.
• I started studying GR.  That was in 2008.  But I did not have the motivation or time to take a full university class (which is only offered in Houston once every two years in the summer ... the next one in 2015), and my motivation for trudging through a lot of it was low since I was only interested in certain parts, and suspected the rest was based on incorrect premises.  Have you ever tried to learn a complex theory you were pretty sure was incorrect?  A computer can do this, but humans cannot.  Anything of importance in life will take precedence over learning incorrect theories.  And it's not like at 63 I will be starting a new career as an astrophysicist.  I needed just enough to make my case on inertia.  Eventually, partly as a result of you guys, I got it.

I am a very thorough and careful logician.  This was an important aspect of my formal graduate training and I'd have never gotten a degree from Rice without it.  In my opinion most people, even mathematicians, are not thorough because they skip over the assumptions too quickly and go right to the details, and the error is usually in the assumptions.  Either something is left out, or something extra creeps in.  In the case of economists and the equity premium, something was left out.  In the case of GR, something crept in.  That was geodesics.

I appreciate Eric's comments in this regard, as he acknowledges how the geodesic assumption might have slid in.  He thinks this might have been firmed up later, but to me as a logician any use of the field equation or any other result is circular.

Regarding a metric representation of equivalence, I would prefer to analyze it from first principles within the original theory.  As I've already stated, I don't trust the metrics to apply in all situations.  But I'm confident in my expertise to analyze from the basics.  I would doubt that the Rindler coordinates can be trusted, in other words.  And I'd point to many papers that argue to what extent various GR-like metrics can be derived from equivalence.  But I wouldn't go there myself.  I'd simply analyze from the basics.  Equivalence occurs in flat spacetime, and we had many detractors on this very thread when trying to make statements beyond that.  I'm not sure everyone concurred even with the idea that continuous acceleration mimics spacetime.  And the 100 year debate on whether bending is included shows how difficult it is to analyze the finer points of equivalence.

Going only with the basics, geodesics looks like an assumption.  I perfectly understand that to someone who has accepted Minkowski space as objective reality (and I know others, not you guys, who get downright religious about this) and revised Newton's first law from "constant velocity" to "straight line in Minkowski space" it seems reasonable to assume also straightest possible line in some other space. 

But I would point out, gently, that extending Minkowski straight lines to geodesics in curved space as representing Newton's first law is still an assumption even accepting that both Minkowski and Riemannian spaces are entirely objective realities.  The reason is because of energy.  Straight lines in Minkowski space do not change kinetic energy.  Geodesics in Riemannian space do.  This is the basic assumption of "geometric gravity."  But if one puts energy first, one cannot get there. 

I believe if you step back you will find that if you hadn't spent years in graduate school and professional life based on the geometric assumption (which creates the feeling of needing to protect an investment, which I fully understand, I'm still clinging to Diana Shipping and a leveraged Russia fund even though they seem to go only down) then the assumption that for energy to change there has to be energy in the agent which causes the change would look reasonable.

About answering.  When you answer that it has been shown or Hamiltonians have been used for this or that, it doesn't really read like an answer to me.  I am definitely not seeking the opinion of an "authority."  I am seeking to understand exactly how everything is worked out.  This is absolutely necessary when examining fundamental assumptions.  So I need either a reference to a specific paper (the ones Kassner provided were very helpful) or for you to repeat the essence of the argument (which Eric often does, and sometimes Charles does on his webpages).

Thanks again for your interest.