Depending of the k - the curvature constant, to keep it simple.

-it could be a sphere if k > 0

-a hypersphere if k

The RW metric does not correspond to any known non-Euclidean metric of constant curvature. They confuse the circumference of a hyperbolic circle with length. The hyperbolic length is tanh R and they write sinh R or sin R by the substitution R->iR.

Bernard lavenda is right. I am waiting for a nice answer to his question...

FRW to be exact, allow different configurations for the shape of the Universe depending on a constant curvature of space, noted with k. You can see te discution of k values in post above.

I do not pretend to answer directly Bernard's question, because I do not have access to the mentioned books, or to the subtlety that he evokes. I'm sure Bernard will say that I do not understand what he means and he will probably be right.

However, I would like to mention what I like about this metric: it allows to link the radius of the universe to its curvature, from a simplified cosmological model of Big Bang, developed by A. Friedman and independently G. Lemaitre, wherein the universe is represented by homogeneous and isotropic expanding fluid. In this model, the length is given by the Robertson-Walker metric:

ds2 = c2dt2 – R(t)2 [dr2/(1 + r2/Rc2) + r2(dφ + sin2φdθ2)]

Assuming an isotropic universe (which is hardly the case), the coefficients of length elements are not functions of the angles θ et ¢.

The factor R(t) ) is called the «scale factor.» It is a dimensionless number that gives the distance scale at a given time. It describes the expansion (or contraction) of the space over time. The space is assumed to be homogeneous, since this number multiplies in the same way the three space coordinates.

The presence of the term (1 + r2/Rc2) indicates that the space has a radius of curvature Rc. The curvature affects the distances between points. Two points separated by an interval dr, where dt = dφ= dθ = 0, are remote of: dl =R(t) dr(1 + r2/Rc2)-1/2.

The radial coordinate r is normalized by setting |k| = (Rc)–2 :

ds2 = c2dt2 – R2(t) [dr2/(1 +kr2) + r2(dθ2 + sin2 θdφ2)]

The factor k signs the curvature of space. If k = 0, space is flat. if k = 1, the space is of spherical type; a "closed" universe of finite volume. If k = –1, space is hyperbolic; an "open" universe of infinite volume.

In a complete cosmological model, R(t) is a function that depends only on the cosmic time t and is called metric radius of the Universe. This function reflects the large-scale evolution of the Universe. In fact, the cosmological time permits to define in each point a referential locally at rest but which contributes to the overall expansion of the Universe.

This is consistent with the accepted physics and satisfies me. Bernard says «The RW metric does not correspond to any known non-Euclidean metric of constant curvature. They confuse the circumference of a hyperbolic circle with length. The hyperbolic length is tanh R and they write sinh R or sin R by the substitution R→ iR». If he is right and if this equation has a report with what he denounces, what would it be necessary to change?

@Russell,

First let me say I appreciate your comments. To quote from my own book, "A New Perspective of Relativity", "There is a fine line when talking about the metric, length, distance, and say 'lengths are shorter and time goes more slowly as we proceed further out on the disc.' However, to a person who lives there he sees no difference, nor can he measure one." Only to us Euclideans looking in from the outside is there a difference.

According to John Stachel, the uniformly rotating disc was the 'missing link' in Einstein's formulation of his general theory. The question then boils down to whether the RW captures the features of the rotating disc? It was known to Einstein that the ratio circumference of a uniformly rotating disc to its radius is greater than 2\pi. This corresponds to k=-1 in RW metric. Set r=constant and integrate the square root of angular part of the metric with \theta=\pi/2. The ratio is 2pi if r= R sinh\chi, where \chi is some 'angle'. Then do the same for k=+1 where r=R sin\chi. You come out with the same answer for the ratio, 2\pi. How is this possible? Because the angular part of the metric is independent of k. A different result would be gotten if r=R\tanh\chi for then R sinh\chi=r/(1-kr^2)^{1/2}, where k=1/R^2.

@Werner,

If r is not r and 0=0 is an equation of state, it doesn't seem like general relativity is a viable option for a cosmological theory. Moreover, since the RW metric is does not belong to non-Euclidean geometries then to what does it belong?

-The RW metric doesn't define distance in terms of the cross-ratio.

-It is not isomorphic to the half-plane metric.

-It does not correspond to hyperbolic or elliptic geometry, no matter what value of k you choose.

What replaces the parallel postulate? and what is the angle of parallelism? There is a whole theory of non-Euclidean geometry (for constant curvature) that goes back to Gauss, Bolyai, Lobachevsky, Poincare', and Klein. Before attempting to invent "new" geometries of constant curvature it might have been better to look at what was already there. And it is even more surprising to see that Grossmann was expert in non-Euclidean geometry! What was he trying to do? invent new non-Euclidean geometries of constant curvature?

After almost a century, all we have the 3 classic tests of Einstein's theory. They can be derived as diffraction phenomena in a much more simple and intuitive way in the same way that Maxwell derived his 'fish eye' for elliptic geometry. The metric contains the centripetal potential while the gravitational potential influences the varying index of refraction. Hence, there is no equivalence principle, not even a local one. The origins of the two forms of acceleration are completely different, and one cannot "cancel" the other. And space-time, what is it? It is not space time but the ratio of space to time. The cross-ratio is completely compatible with the velocity addition law that is satisfied by the hyperbolic tangent, not the hyperbolic sine!

On the contrary the 'r' in the Beltrami metric is a radial distance.

In noneuclidean geometies distance is measured by the logarithm of the cross-ratio. To do so you need to define an absolute constant (no need of a Bureau of Standards). This is a distance in the same way that the velocities add according to the relativistic velocity addition law. The cross-ratio goes all the way back to Desargues' projective geometry in 1639, and have become part-and-parcel of the artist's perspective view. Sadly, they have not become known to physicists. Measures of distance also apply to spherical geometry. They are not something that has meaning only in the k->0 limit. Since the solutions to Einstein's equations do not coincide with these metrics, for constant curvature, then there is no other conclusion then to say they are inaccurate (a very mild statement).

If you ask the question how many lines through a point that is parallel to a given line are there? then it becomes apparent that hyperbolic (infinite) and elliptical (zero) geometries are unique and there are no others for constant curvature.

Planck, Einstein, Sommerfeld, Ehrenfest, & Co. were all aware of this. Kaluza ( of the Kaluza-Klein fame) wrote a two page paper on it but could not present it at a conference in 1910 for health reasons. Einstein and Ehrenfest did Varicak in when he expressed similar views. Sommerfeld published a paper in 1909 showing that the addition law for the hyperbolic tangent is the same as the Poincare' addition law, which Einstein later generalized to non-planar velocities. (He later disowned the paper.) Finally, there is the statement of Planck, in his Columbia lectures, that noneuclidean geometry is child's play in comparison to Einsten's relativity. I have tried to document all this in:

http://www.amazon.com/gp/product/9814340480/ref=s9_psimh_gw_p14_d0_i1?pf_rd_m=ATVPDKIKX0DER&pf_rd_s=center-2&pf_rd_r=0JE6BP1K7CZGGFK18NW2&pf_rd_t=101&pf_rd_p=1389517282&pf_rd_i=507846

I don't see how you can justify your claims. If you look at the metric under the section

Reduced-circumference polar coordinates, in the Wikipedia article you cite, and set r=const, theta=pi/2 and take the square root. Then integrating from 0 to 2pi you get 2pi r. Have I missed something?

Why the RW seeming works is because at lowest order, the Einstein equations reduce to the virial and Newton's law. The pressure provides only small corrections: the ratio of the pressure to rest (not internal) energy is always of the order of the Schwarzschild radius to the radius of the star. This is known to give the mass defect, bending of light, red shift, etc. All this is very well stated in Sexl and Sexl, "White Dwarfs, Black Holes", but they don't say why it works. You can't do any more than that with the RW metric, or for that matter, Einstein's equations in general. Inflationary cosmology is wrong because the pressure can't go negative and dominate the energy density. There is no thermal equation of state like the Gruneisen one relating the internal energy density to the pressure.

And what about black holes? If you use the Beltrami metric, it is clear that the singularity occurs on the rim of the disc; it takes an infinite amount of time to be reached because rulers shrink and clocks continually to go slower as you approach the rim. But the inhabitants doesn't notice that because they shrink along with the rulers and their heart beats decrease along with the slowing of the clocks. It is only to us Euclideans that these effects are perceptible. But where is the black hole?

An example of the confusion in general relativity, and why it is wrong, see Moller's book, "The Theory of Relativity" eqn (76a) on page 241 and eqn (18) on page 349. Moller has to admit that "the left-hand side of (18) cannot in general be interpreted as angular momentum, since the notion of a 'radius vector' occurring in the definition of angular momentum has an unambiguous meaning only in a Euclidean space."

This is the situation in GR but not in hyperbolic geometry. There you modify the dot product to an inner product. So too you must modify the cross product in order that sin^2+cos^2=1. That modification for angular momentum of a uniformly rotating disc brings you exactly to eqn (76a) in Moller's book. The radius vector does not need Euclidean space for it to be defined!

Werner, Then how do you rationalize the calculation of say the advance of the perihelion of Mercury? In the derivation to lowest order the orbit is an ellipse, and angular momentum conserved and is given by its Euclidean expression. The whole calculation rests with a cubic term (non-Euclidean) in the d.e. for the orbit of the particle. But, to the same order you have that angular momentum is no longer conserved. Moller would agree with you when he says that "the notion of a 'radius vector' occurring in the definition of angular momentum has an unambiguous meaning only in a Euclidean space." Are we to assume that the change in the angle variable over two successive perihelions is not something measurable? Then what are we supposed to do with the prediction that the advance of the perihelion of an angle 42.9" per century? Supposedly, this is in satisfactory agreement with what is measured even though we are not in "a Euclidean space." Similar remarks pertain to the gravitational deflection of light, and the gravitational shift of spectral lines.

How can you neglect all the physics and still call general relativity (gr) a physical theory? According to Einstein himself "It is true that the theory of relativity, particularly the general theory, has played a rather modest role in the correlation of empirical facts so far...[but] it has special appeal because of its inner consistency and the logical simplicity of its axioms." That was written 71 years ago, and the first part is still true to this day. The second part I would disagree with, but you are trying to eliminate even that! He attributes to gr, the general covariance of the laws of nature, and their nonlinearity. Who needs covariance when you know (or should know) the geometry you are dealing with? The nonlinearity he is referring to is the propagation of waves, and nonlinear waves have nothing in common with linear waves except they both propagate.

Very interesting discussion! Unfortunately I'm neither a physicist nor a mathematician, so I con only comprehend superficially. However, you've raised a couple of points that I find most interesting. I'm not really even interested in 'doing physics', but as an information systems analyst it seems to me that GR is not a complete, physical theory since, as I understand, abstract geometries can only cause physical effects to the extent that they describe some physical structure. I think many physicists would consider me simply naive, but while the effects of gravitation certainly correspond to the dimensional coordinates used to describe them, there seems to be no described aspect of spacetime that could produce the observed physical effects. I can only guess that some malleable physical property, perhaps vacuum energy density, for example, is actually represented by the described geometry of spacetime and physically produces the described effects.

One important aspect of spacetime that I think has not been appropriately considered is that the accretion of massive objects is a two-way street - it also necessarily extracts (generally disperse) mass-energy from spacetime, in effect producing the vacuum. This process may also help to explain the physical distortion of some physical property of spacetime...

The other most interesting item regard the inverse-cubed term used to adjust proximal gravitational effects. Apparently I'm not alone in questioning the process Einstein used to reach the final form of the field equations. Please see: John D. Norton, (2007), "History of Science and the Material Theory of Induction: Einstein’s Quanta, Mercury’s Perihelion," http://philsci-archive.pitt.edu/3562/1/HS%26MTI.pdf

- section 5, "Conclusion: Virtues of the Material Approach"

"... For we still do not know whether an Einstein inferring from the anomalous perihelion motion of Mercury should be portrayed as secretly computing Bayes’ theorem or secretly sifting an hypothesis space for the best explanation."

While it was the bending of light from background stars during Solar eclipse or accurately describing Mercury's perihelion precession that provided the most compelling proof of GR, I suspect that Einstein selected a term that worked well. While it's described as a proximal 'perturbance', its precise physical nature does not seem to have been established.

Very briefly, in classical terms, I suspect that Newton's Principia Proposition 75 Theorem 35, concluding that a spherically symmetrical massive object can be considered a point mass for external bodies, is not valid for sufficiently proximal masses. For example, for a test body at a distance of 1 from a spherical mass with a diameter of 10, a component point mass at a distance of 1 from the test body will produce a much greater gravitational effect that an identical point mass at a distance of 11, and that the average of the two's inverse-squared effects will not be the same as if they were at a distance of 6 (the center of the spherical mass). I think Einstein also considered the Sun to be valid point mass in his evaluation of his compensatory factor.

Sorry I can't do the math - I hope you can understand my tedious descriptions.

I wasn't aware there was a theorem to the effect that the Schwarzschild metric describes a spherical mass distribution. Einstein's equations contradict themselves since the Schwarzschild metric is solved by the condition that the Ricci tensor is zero. Hence, space is empty by Einstein's own definition. You can't put in mass where there is none. And even if you could by arguing a weak field at great distances, it wouldn't be Newton's potential that you would use because that is absolute space, not relative space. Then there is the error pointed out by Levi-Civita that the trace of the energy-momentum tensor of gravitational radiation cannot be a differential invariant of the first order because such an invariant doesn't exist. Einstein was led to the incorrect result that spontaneous waves should radiate, and said that "quantum theory should intervene by modifying not only Maxwell's electrodynamics, but also the new theory of gravitation.' Maxwell's equations don't radiate and don't discriminate between advanced and retarded solutions (clearly the advanced solutions are unphysical unless you have a crystal ball).

It is not the numerical factor which is important but the logic in the explanation of a phenomenon. Soldner predicted the bending of light by a gravitational field back in 1804. Ritz derived the perihelion advance in 1908 from a force equation mimicking Amperes' law. [cf. A New Perspective on Relativity].

Even the fact that gravity should propagate at the speed of light is still not a solid fact because there is no gravitational aberration which should exist if it does. You can explain the bending of light by the relative motion of the earth, like a projectile that would pass through a moving train would be seen by the passengers as a curved trajectory.

Then there is the question of what is space-time? I have no idea at all what the Gaussian curvature of what space time is. At the heart of relativity are the Doppler effect and aberration. These deal with velocities--the ratio of space to time. That nothing can propagate faster than the speed of light (nothing Euclidean that is) is welded into the Poincare' addition law of the velocities. One-half its logarithm is the definition of hyperbolic distance. That IS distance defined through the cross-ratio.

Finally, how many times have there been proofs of wrong results? A case in point is the Strominger-Vafa derivation of the Bekenstein entropy of a black-hole. Clearly, the result is wrong, so how can the proof be correct?

I should have said that GR's prediction of Mercury's orbit and that light was curved by the Sun's gravitation was the evidence that distinguished GR from Newtonian gravitation and compelled the public to believe that GR was a groundbreaking new theory. As I understand, the inverse cubic proximal gravitation term was the major change made before the final publication of the field equations - that corrected those predictions.

I agree that http://en.wikipedia.org/wiki/Birkhoff_theorem_(relativity) is the analog to Newton's Proposition 75, Theorem 35.

Please see http://en.wikipedia.org/wiki/Mathematical_Principles_of_Natural_Philosophy#Book_1.2C_De_motu_corporum

"Section XII (Propositions 70–84) deals with the attractive forces of spherical bodies. This section contains Newton's proof that a massive spherically symmetrical body attracts other bodies outside itself as if all its mass were concentrated at its centre. This fundamental result enables the inverse square law of gravitation to be applied to the real solar system to a very close degree of approximation."

Please also see http://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA270 - scroll down to Proposition 75. Theorem 35. pg 270

In the Newtonian case, http://en.wikipedia.org/wiki/Universal_gravitation#Bodies_with_spatial_extent explains:

"If the bodies in question have spatial extent (rather than being theoretical point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which constitute the bodies. In the limit, as the component point masses become "infinitely small", this entails integrating the force (in vector form, see below) over the extents of the two bodies.

"In this way it can be shown that an object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its centre.[3] (This is not generally true for non-spherically-symmetrical bodies.)"

I contend that this suggested detailed evaluation only prescribes that the component point masses be evaluated throughout the two-dimensional spatial extent of the object presented - that if the detailed evaluation included the _depth_ of three dimensional component point masses then their contributing gravitational effects would differ significantly from that obtained by evaluation of a single point mass.

Likewise, I suggest that the same effect should be considered for proximal spherically symmetrical objects with apparent 'spatial extent' in GR - that both Prop. 75 Theorem 35 and Birkhoff's theorem are very convenient and useful methods of approximation that yield adequate results only at sufficient distance.

Moreover, unless there is some other physical rationale for GR's inverse-cubic (proximal) incremental gravitational effect, I suggest that the significant object depth/distance relation of mass distribution causes the discrepancy in proximal gravitational effects.

As mentioned, the more compelling evidence of GR's physical correctness is the prediction of time and length dilation. However, as I understand, the dimensional coordinates of spacetime are 'merely' used to describe those effects - not to identify a physical cause. So it can be shown that time dilation is produced by gravitation, and the effect can be precisely predicted, but what causes it to occur? Perhaps more easily considered, what causes length dilation - physically causing the length of a ruler to contract? I may be naive, but I find it very difficult to accept that a geometric abstraction of spacetime dimensions can cause such physical effects...

As for QM, it is only presumed that gravitation is actually a material force interaction, mediated among quantum particles through the exchange of force carrying bosons. However, I doubt that such a simple exchange interaction could explain the effects of dimensional dilation (contraction). GR describes not an interaction between objects, but the fundamentally distinct interactions between condensed mass-energy and spacetime - the vacuum. I think in this way gravitation is not a quantum property of particles of matter but rather the interaction between potential mass-energy and, I suspect, some kinetic energy contained within the vacuum of spacetime.

I think you dismiss too easily the importance of the process of accretion. If you consider back to the primordial quark-gluon plasma, the subsequent condensation of matter not only localized mass-energy but extracted mass-energy - _producing_ the vacuum of spacetime. Likewise, the continued accretion of massive objects evacuates spacetime, extracting potential mass-energy. I think this is crucial to understanding the physical-energy relationship between objects of mass and the vacuum of spacetime.

The inverse cubic term you refer to comes from the Beltrami metric for the angular component when you introduce M=\rho r^3 (exchange constant density \rho for constant mass M) Eqn (7.5.13). In fact, GR gets a quartic term and makes the weak field approximation to get rid of it Eqns (7.5.12). Also GR finds that angular momentum is not conserved while the Beltrami metric does (i.e., it does not distinguish between the Euclidean and hyperbolic definitions of angular momentum). You are right to say that all relativistic corrections are to be found in the inverse cubic term for the equation of the trajectory. But this has nothing to do with space-time. All the tests of GR are static, except for the red shift which is Doppler

Using a combination of Weber and Riemann forces Ritz calculated the advance of the perihelion of Mercury [Sec. 3.8.2] in 1908. Had he not died prematurely, I'm sure he would have gone on to use his gravitation theory to determine the mass defect, deflection of light, spectral line shifts, etc. They all follow from the inverse cubic term in the potential (3.8.24). I have shown how he would have gone about it in Sec. 3.8.2.1 and 3.8.2.3. [All equations, sections, and pages refer to my book].

GR does not predict time dilation and length contraction. They come from the Lorentz transformation for an inertial system. Length contraction was introduced by FitzGerald, and a little later by Lorentz, (1894) to explain the Michelson-Morley null effect to second order in the ratio of the velocity, v, of light to the speed of light,c. However, the assumption of an outward velocity c+v is a ballistic assumption. If you assume the relativistic law for the addition of velocities, you get the null result without the necessity of introducing length contraction [p. 115].

It is not that GR offered compelling physical evidence. A study of the history of relativity shows just the opposite. It is much like Weber versus Maxwell theories of electromagnetism. Weber's theory is not inferior to Maxwell's, but who hears of his theory today?

There is no mass if I choose to put the integration constant in Schwarzschild's solution =0. According to de Sitter KNAW Proc 191, 1917 pp. 527-532], "the need for the introduction of distant masses arises from the wish to have the gravitational field zero at infinity in ANY system of reference. This wish, however well founded in a theory based on absolute space, is contrary to the principle of relativity."

"The condition that the gravitational field shall be zero at infinity forms part of the conception of an absolute space, and in the theory of relativity it has no foundation." There is a footnote saying that Einstein found a set of values of the g_ij at infinity. It is given by a 4 by 4 matrix with infinities in the last line and last column. This does away with setting the constant of integration equal to the mass for large distances and weak fields. Actually it is an illogical procedure. You start with a T matrix which has zero components. There is nothing in the Ricci tensor which relates to energy, so you can't put it in by hand.

The crux of the matter lies with Einstein's equivalence principle. Accelerations due to centrifugal forces is not the same as acceleration due to gravitational attraction. For this would allow special relativity to be used in free-falling systems as well as in inertial systems. The uniformly rotating disc is proof of this, where the angular velocity is the inverse of the free-fall time.

I appreciate your responses, to the extent that I can comprehend...

My discussion of the inverse-cubed term is based solely on http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity#Effective_radial_potential_energy

"The third term is attractive and dominates at small r values, giving a critical inner radius r[inner] at which a particle is drawn inexorably inwards to r=0; this inner radius is a function of the particle's angular momentum per unit mass or, equivalently, the a length-scale defined above."

I also find other sources refer to this term as a 'perturbation', but that's as close as I could come to a physical explanation of this proximal 'perturbational' influence. At these magnitudes, curving background light obliquely traversing the Sun's gravitational field, for example, it seems that the spherically symmetrical inverse-squared gravitational effects must be augmented by this third inverse-cubed influence. Dismissing my 'black swan' suggestions is quite reasonable, but I'm merely trying to determine whether there is a physical explanation for this seemingly arbitrary inverse-cubed third term, or it is merely a compensatory factor inserted to produce results that agree with observation. So, is there some explanation for the proximally effective supplemental gravitational effect?

BTW, I can only guess that this supplemental proximal gravitation also enters into the predicted decay of extremely massive binary orbits - usually attributed to 'energy loss' through gravitational wave propagation...

As far as the seemingly necessary physical properties of dimensional spacetime, is there a physical explanation for the length dilation of a ruler in a strong gravitational field?

I appreciate any guidance you can offer!

James,

The clearest and simplest exposition of this term can be found in Sexl & Sexl, White Dwarfs and Black Holes (Academic Press,1979) page 103. They attribute it to GR; it is attractive and a product of the centrifugal potential and the ratio of the Schwarzschild radius to the radius of the star. It appears in the d.e. for the orbit in eq. (28) on page 350 in Moller, The Theory of Relativity.

In 1864 Seegers proposed to treat the advance of the planets as a gravitational force in the same way that Weber's force holds for charges in motion. Scheibner and Tisserand found secular variations this way in 1872. Ritz added a term to gravitational acceleration that was the inverse quartic [Sec. 3.8.2, A New Perspective to Relativity]. I have shown that it can be accounted for by a varying index of refraction [p. 381] in the exact same way that Maxwell derived his 'fish eye' which is a simple example of an absolute optical instrument in a medium with a varying index of refraction. This he did in 1854--the same year that Riemann proposed a very general modification of planar distances which happens to be the same expression.

You can get this term from the Beltrami metric by assuming that the absolute constant of the hyperbolic geometry is c/omega where omega is the angular velocity. If you set this equal to the inverse of the free-fall time and go from constant density to constant mass it comes out also. But, this is not consistent because you go from a metric of constant curvature to one in which it is not constant.

Bernard,

Thank you so much for your very kind consideration and helpful explanation and guidance! To the extent that I am beginning to comprehend how the Schwarzschild metric relates to the field equations, I think I see now that what appeared to me to be an explicit 'patch' fix to the equations of classical dynamics to correct for the anomalous curvature of light and Mercury's orbital precession is more correctly the term that describes the deviation of the field equations' results from the linear gravitational effects described by Newton. In simplistic terms then, as I currently understand, the inverse-cubed term essentially describes the nonlinear effects of gravitation as a function of radial distance, representing the 'curvature of spacetime' (I think actually describing a topological map of gravitational accelerations) produced by a spherically symmetrical discrete mass.

As I find in http://en.wikipedia.org/wiki/Schwarzschild_metric#Orbital_motion

"Noncircular orbits, such as Mercury's, dwell longer at small radii than would be expected classically. This can be seen as a less extreme version of the more dramatic case in which a particle passes through the event horizon and dwells inside it forever. Intermediate between the case of Mercury and the case of an object falling past the event horizon, there are exotic possibilities such as "knife-edge" orbits, in which the satellite can be made to execute an arbitrarily large number of nearly circular orbits, after which it flies back outward."

Of course, how Einstein actually arrived at the precise terms specified in the final version of his field equations might arguably still be considered as somewhat of a 'patch' to prior versions, but I realize now that the inverse-cubed term in solutions based on the Schwarzschild metric do not represent the entirety of a 'patch' to Newtonian dynamics - thanks again! If I'm conceptually too far off here - please let me know.

Again, as I understand, there is no physical cause described that produces the effects of gravitation - IMO those dimensional effects are only described by the topology of derived gravitational acceleration. That the predicted dimensional effects have been confirmed by observation certainly indicates a precise correspondence between gravitation and dimensional dilation, but I still don't see physical causation. In my admittedly uninformed opinion, I still think there must be an unspecified, unidentified source of kinetic energy required to physically cause the effects of gravitation... Of course, I may well be wrong, but I'm a pretty obstinate hacker!

James,

There is a very simple way to determine which geometry you are talking about. There are only 3 uniform (homogeneous and isotropic) geometries. In the space metric set r=constant, work in the azimuthal plane by setting theta=pi/2 and integrate ds. If you do that with the Schwarzschild metric you would be forced to conclude that space is Euclidean which it isn't. Moreover, there is no way of reducing it to a metric of known form. To modify the singular r=2m, they introduce 'isotropic' coordinates and if we consider the spatial part alone there is no longer a singularity (it appears in the time part only). Now you do the same thing as before and you don't come out with the ratio of the circumference to the radius is 2pi. So just by a change of coordinates you can change the geometry and get rid of a singularity! This is indeed magic. Just how many different universes you can discover merely by changing the coordinates.

In the calculation of the advance of the perihelion, there is a problem however. Angular momentum is not conserved, and according to Moller, the new expression "cannot in general be interpreted as angular momentum, since the notion of a 'radius vector' occurring in the definition of angular momentum has an unambiguous meaning only in Euclidean space." If you accept this, then it is impossible to determine the advance of the perihelion, since the correction term (the inverse cubic term) lies outside of Euclidean geometry. Staying in Euclidean geometry you would just get the equation of the ellipse. To say the least it is inconsistent: you claim that 'r' has only an unambiguous meaning only in Euclidean space and then go on to determine the advance of the perihelion! And what's worse you compare the angle to observation. If the radius is no longer a radius, then an angle is no longer an angle.

Bernard,

Unfortunately, it's not at all simple for me, but I'll try.

Briefly, is there a requirement that angular momentum be conserved, when Mercury's orbit could be drawing from the Sun's rotational momentum?