I don't think it is appropriate to think of the (square root of the) timelike component of the Schwarzschild metric as a "potential".

The Scharzschild solution is the spherically symmetric, static vacuum solution. Its most general form may be given by ds2 = (1 + C / r)dt2 − (1 + C / r)−1dr2 − r2dΩ2. The value of the integration constant C is not obtained from the solution. It is determined by demanding agreement with the Newtonian prediction in the weak field limit. Apart from the value of C, and apart from trivial (physically irrelevant) transformation of the coordinates, the form of the metric is fully determined by the conditions (spherical symmetry, no time dependence, vacuum solution). Any other solution would necessarily violate at least one of these conditions.

So the time dilation factor will always have the exact form, 1 / √(1 + C / r), whatever the value of C might be. But if you choose a value other than −2GM / c2, the theory is no longer compatible with Newton's in the weak field limit.

That said, I note that in the weak field limit, the value that you propose is really no different from the Schwarzschild solution: So long as GM / c2r

Hi V.T., thanks for the discussion.  I'm aware of the approximations, but was not aware that so much of the solution is an integration constant.  Everything but "r" and the general form, apparently.

So, I was thinking we need a different field equation.  That is what I was asking about.  In other words, it is probably a very complicated question. 

There are two things I was fishing for. 

• Exactly what determines the negative sign and that factor of 2.  You have answered that at the first level.  It is the imposed form which is squared, requiring a square root, and the fact that it appears in the numerator of the metric terms which flips it to the denominator.  Converging these two things to the Newtonian limit produces the final form and no other.
• I have a derivation from the equivalence principle that says that whatever the potential function is, (meaning just a function for difference in energy levels) that it should be positive and in the numerator in its final form.  So a very similar situation.  But again using the Newtonian limit, I get the 1+φ term.

The reason I took the time dilation formula for potential difference is because of the Planck relation.  It represents the energy difference for a photon, for example. 

Anyway, you have given me enough clues and orientation possibly to look at the derivation of the Schwarzschild metric (there is a short one on Wikipedia) and work backward.  But if you have any more ideas I'd appreciate them.

Glad you found my reply useful (note that I made a small correction, a "1 /" was missing fro the next to last paragraph.)

The only other thing I'd add is that the linear approximation (of general relativity) can be written, for arbitrary (weak) gravitational fields as ds2 ~= (1 + 2c−2ϕ)c2dt2 − (1 − 2c−2ϕ)dr2, where ϕ is the Newtonian potential. For an arbitrary distribution of matter, ϕ can be obtained from the Newton-Poisson equation, ∇2ϕ = 4πGρ. For a spherically symmetric source of mass M, outside the source, we of course have ϕ = −GM / r.

Let me put it this way ... what is the field equation which would have instead this solution:

ds2 = (1+C/r)-2dt2 - (1+C/r)2dr2 - r2dΩ2

Hmm, I have been suspecting for years that the conservation of energy-momentum may not be correctly accounted in GR.  If someone will show a tether analysis within a GR context, I can determine whether it is accounted correctly or not.

Hi Charles.  I looked at those links.  However, I cannot think in 4D tensors (and I don't know anyone who can).  I need a time & radius only example, like with the tether I mentioned. 

In a nutshell, not nearly as much energy is extracted on a tether as one might first think.  If one computes F*dr locally, this produces a larger value locally than is felt on a tether for two distinct reasons, and I fear only one has been accounted.  The value of F felt remotely on a tether can be deduced from time dilation dφ and the proper distance ds which the tether actually moves.  This is less by a factor of Γ due to time dilation, and an additional factor of Γ due to the spatial coordinate change.  While only the time dilation produces energy changes, the spatial dilation (for lack of a better word) must produce force changes across the transformation to preserve momentum on the whole.  dφ is always the time dilation factor and always the last word in energy, but dφ is not known beforehand when deriving from a field equation, so any assumptions about coordinate transformations of  force, momentum or energy cannot be checked against it until it is derived, at which point the derivation is somewhat circular.

Would you mind Charles saying a little more? Is mass a four-vector or a second order tensor? We need to make a decision on that, because when we are computing the gravitational field we are supposed to use a four-tensor, when we are computing a path we are supposed to use a vector. In QM we speak of the four-momentum. So what kind of physical magnitude is mass: a four-vector or a second order four-tensor? Einstein considered it a four vector in his derivation of E=m c^2 as we do in covariant classical electrodynamics.

Unless you can show work = force x distance and a few other basics, you will lose track of the relation between measurements in different coordinates, and at different points in the field, and run afoul of energy and momentum.

The tether helps do this.  I'll give an example which will probably surprise you.

Let's get gravity (i.e. the weight of things) out of it by turning the tether 90 degrees at the bottom via a pulley.  The bottom is still in a lower gravitational potential.  And we'll just let you pull on it with a certain force.  What force do you think appears at the top of the tether?

Now you pull the tether through one meter while exerting this force.  How much work did you do locally?  How much work did you do at the top of the tether?

I think you can assume that if you move the tether 1 meter at the bottom, it also moves 1 meter at the top.  If meter sticks shrink or expand, so does the tether.

Einstein based its equations on the conservation principles. There was a big discussion with Hilbert about this and with Emily Noether for the conservation principles. With the help of Hilbert they wrote the Lagrangian version of the equations which is conservative. So far nothing has been accepted in physics which could violate such principles.

Hello,

if I understand your question correctly, you ask on the field equations, which has the solution with the components g_tt = -1/g_rr = 1 + R_g/r of metric tensor, where sign plus figures in front of R_g/r (instead of common minus). The general outer Schwarzschild solution was found in the form g_tt = -1/g_rr = 1+C/r, where "C" is an integration constant, regardless its value is positive or negative, therefore your question is still about the solution found by Schwarzschild (despite the plus sign).

Anyway, this metrics is produced by the Einstein field equations simplified for the case of spherical symmetry. The (most) explicit form of these equations was derived by Tolman (right-hand sides of equations) and, immediately after, improved (the left-hand sides were explicitly given) by Oppenheimer and Volkoff in their famous paper on stable neutron cores. So, you can find the equations in their paper (Oppenheimer J.R., Volkoff G.M.: 1939, "On massive neutron cores",  Physical Review, Vol. 55, pp. 374-381). Or, you can find them also in my recent open-access paper (Neslusan, L.: 2014, "Another way of the continuous linkup of neutron-star-body and surrounding empty-space metrics", International Journal of Astronomy and Astrophysics, Vol. 4, pp. 399-413), in which I just deal also with an application of the "plus" metrics - within the Ni's solution of the above mentioned field equations.

Lubos

Hi again,

I additionally noticed that the factor two is missing in the relation in your question. The appearance or absence of this factor depends on the way of gauging of constant "C". (I saw also the way of gauging, in which the factor is missing, but I do not remember the reference, at the moment, I am sorry.)

Lubos, thanks for this interesting information, especially about gauging C, I was looking for something like that.  I will begin looking at your references.  I may have some questions.  Also, my factor was flipped into the numerator (when expressed as time dilation factor).  That's what made me believe I might need a new field equation.  Obviously it would be preferable to show it is a member of the Schwarzschild family, a variation based on some other assumption.

Dear Robert, if you have a new idea concerning the gauging of constant "C" and you will publish a paper about, I will welcome the reference on it. I was also interested in the gauging in situation, when one assumes the grand unification (of interactions) in the radical form that there is, in nature, no difference between the gravity and electric interaction, therefore the relativity should work in the same way also in the case of latter.

Component g_tt = 1 + C/r, the square root of which figures in time dilation, etc., can be also written, in purpose of the gauging, in form g_tt = 1 + 2U/c^2, where U is the potential and "c" is the speed of light. I suggested to replace the potential with the potential energy, "W_p" and quadrate of speed of light with the rest energy of test body/particle, "W_o". Nothing changes in the case of gravity such the changing, in fact. We obtain g_tt = 1 + 2W_p/W_o.

If one considers the central proton and "test" electron (hydrogen atom) and the electrostatic interaction between them and replaces gravitational "W_p" with the potential energy of electron in the electrostatic field of proton (and the rest energy of electron remains), then one can obtain a relativistic wave vector for an evanescent wave in the form (its size) k = W_o/(hbar*c)*sqrt[-W_p/W_o/(1 + W_p/W_o)]. It enables to construct the hydrogen atom as a crystal-type structure and calculate one corresponding set of energy terms in hydrogen atom (with the same precision as with the help of the Dirac theory).

However, all this is possible only if the factor of 2 is omitted in g_tt. therefore it equals g_tt = 1 + W_p/W_o. So, this is the reason of my interest in the gauging, especially that without the factor of 2.

L. Neslušan,  it would still be very interesting to get (1+2U/c2)1/2 instead of (1-2U/c2)-1/2, as this would eliminate the event horizon, and the only infinity would be at R=0, with no compelling driver to force it.

However, the real puzzle I have is that I have a separate derivation of the gravitational field from a point mass which follows simply (1+U/c2).  While all these are compatible for weak fields, the differences for strong fields are very interesting.  Possibly I made some mistake?  The math of my method is quite simple and easy to follow.  Would you care to look at it?

BTW, I just noticed that (1+2U/c2)1/2 approaches infinity even slower than (1+U/c2).  I wonder if this is simply a difference of coordinate choice?  Perhaps (1+2U/c2)1/2 is correct using Schwarzschild coordinates.  In that case, R is still not the proper distance to the mass point (which is infinite), but it might well increase as the square of the extended Euclidean coordinates I'm using.  I will look into this over the next few days.  I am slow to pick up on coordinate issues, and you may have prompted me to find the key.

Robert Shuler, at the moment, I do not know how to eliminate the factor of 2. Concerning the sign "plus" in front of 2U/c^2 or U/c^2, this sign appears when the internal structure of compact object, e.g. a neutron star, is modelled. One can use the traditional (relatively easy) modelling that was made by Oppenheimer and Volkoff . They established an auxiliary quantity "u" , considered in the object's interior, defined as "u = 0.5*r*(1-1/g_rr)". In the physical surface of the object, form  1-2*u/r should equal to its outer Schwarzschild solution counterpart, which is "1-R_g/r".

In every numerical integration of the equations determining the internal structure of the object, the value of "u", which is positive in the object-centric distances near the outer radius, "R_out", turns to become negative in the region near the inner radius, "R_in" . It appears that this turn and, consequently, the inner physical surface occur always, when the mass of the object is larger than the Oppenheimer-Volkoff limit. [it is valid also for low-mass objects, but I do not want to write here a very long text, which would be needed to provide an explanation, I am sorry].

The turn of sign of "u" is then reflected in the continuation of metrics in the internal cavity of object, i.e. in the region "r < R_in". There is no matter in the central singularity in r=0, but the spacetime below "R_in" is obviously shaped by the matter of circumambient hollow sphere. Since each particle is attracted (by this matter) away from the central singularity, the latter is the non-problematic Big-Bang type singularity.

The turn of the orientation of gravitational attraction, from inward to outward inside the compact object appears to be a new, natural feature in the general relativity (GR). The basic problem is the question of what is the acceleration of a test particle inside the spherically symmetric material shell? In Newtonian physics, one can prove that the net gravity exactly equals zero. In GR, the experts suggested to put the integration constant "C" in g_tt = -1/g_rr = 1 + C/r" zero just because of avoiding the singularity in r=0. If this was not assumed, then the metrics would be outer Schwarzschild with sign plus and particle would be accelerated toward the nearest point of the shell (i.e.outward).

In the equations of the internal structure of neutron star, there is however no constant assumed to be zero, therefore a particle in distance "r" is attracted by the concentric spherical layers with radius larger than "r" upward and, of course, by the layers with lower radius inward. Because the models of these stars are constructed starting the numerical integration from r=0, it is, in fact, impiicitly assumed that the spacetime is shaped in that way that the attraction of lower layers is dominant everywhere (and gravity is oriented only inward). If the integration is started in a distance larger than zero, then the energy density and pressure always (at east in the Oppenheimer-Volkoff problem) decrease to zero in a finite object-centric distance, R_in, i.e. the inner physical surface occurs.

I realize, this is a new knowledge, not yet described in the textbooks, therefore it can be a hard thing to believe. I am however sure that everyone, who will verify my clams, will find that my statements are, simply, the facts. The facts related (also) to the outer Schwarzschild type metrics with sigh plus.

@Neslušan ... I am familiar with deviations from the Newton shell theorem for all sorts of reasons, mainly non-uniformity.  I have often wondered if this might have any effect on a cosmic scale, since there is some degree of large scale non-uniformity, and if accelerated expansion might be due to a phase change in which that non-uniformity is increasing and having a larger effect, but have no ability to make GR calculations nor the necessary mass distribution data.  That GR itself should have such a property I find un-surprising, since it is possible to explain cosmic expansion with GR ... therefore it seems soft of necessary.  But people do not seem to use this terminology in talking about expansion. 

For my problem, I'm not really concerned with the internal structure.  As for how to get rid of the 2, it is not as big a problem as the minus sign.  There seems to be no way to get rid of it when using the Schwarzschild solution.  Nothing else works.  I tried several things just now kind of ad hoc, which isn't a proof of course but I suspect if an alternative was available it would have come to my attention long before now.

@Shuler

To better explain the aim of my last explanation, I moreover note that I described the behaviour of metrics inside the compact object (its structure) in purpose to justify the occurrence of the internal cavity, without matter (only vacuum) and change the orientation of gravity, i.e. the occurrence of sign plus, instead of minus, in this cavity. In other words, I wanted to explain that the occurrence of the cavity is not any ad hoc assumption, but the real objects seem to actually behave in this way. No non-uniformity is needed, it works also for the perfectly spherically symmetric shell. One could, perhaps, explain the cosmic expansion assuming that the known universe is situated inside an extremely large material bubble (which is not seen because its large extent). The gravity of such a bubble would set the field in which all objects in the universe would expand.

The problem with this concept is the "convention" (postulate?), added to the original GR, that constant "C" in the outer Schwarschild (OSCH) metrics describing the spacetime inside the shell is demanded to equal zero (and, in consequence, the metrics is then reduced to flat, Minkowski metrics). If this "postulate" was abolished, i.e. we let the original GR to work, then we would have the OSCH metrics with sign "plus" inside the cavity.

@Charles Francis

If you define the Schwarzschild solution as only that in which the integration constant "C" in g_tt = -1/g_rr = 1 + C/r" is negative, usually "C = -R_g = -2*G*M/c^2", then yes, I agree that the solution does not imply any expansion.

Nevertheless, I insist on the claim, that the gauging with the sign plus, which describes an acceleration due to gravity having the expanding character, is also the possibility (which, btw., obeys the Birkhoff theorem). And, I admit that a note on such the possibility is missing in the textbooks (but this is not a scientific argument against the validity of the claim).

The replacement of the singularity in r=2M by that in r=0 (and concept of full sphere with the concept of hollow sphere) is not my arbitrary idea, but it is the result of an effort to solve a new serious problem of the physics of stable compact objects (with a consequence on the theory of black holes). Namely, I noticed that the metrics describing the space in the object's interior is torn away from the (Schwarzschild) metrics of the adjacent empty space. I think, such a discontinuity is not acceptable in realistic model of, e.g., neutron stars. So, we have to answer, primarily, the question of how to remove the discontinuity? (And, the only possible way seems to be accepting the concept of hollow sphere with all its consequences.)

@Charles Francis

Once more view on the problem: The problem with the discontinuity is most likely caused by the fact that the equations describing the structure of compact objects contain the speed of light, "c", and people constructing the models of these objects supply into the equations the value of "c" which is valid in a free space (vacuum without matter).

In the GR, you likely agree that a  photon moves along the zero geodesic and, hence, "c" inside the curved spacetime of compact object must be lower that in free space. Actually, if one replaces the vacuum-c with the zero-geodesic-c, then the model with the continuous metrics can easily be found, but one can construct the full-sphere model (with gravity oriented inward in the entire object's body) only in the limit of object's zero mass. The Oppenheimer-Volkoff (OV) mass limit thus appears to be zero. (The difference between the common OV limit and zero-mass limit likely corresponds to the difference between vacuum value of "c" and zero-geodesic value of "c".)

The problem can be solved if we accept the concept of hollow sphere, which enables to construct a model for the object of whatever mass (so, the OV disappears) and the outer physical surface occurs to be always situated above the corresponding event horizon.

@Charles Francis

Our contributions met on the way. I response (in part, I am able) to your latter one. Yes, there is the empirical evidence leading to the unique gauging of "C" in the vacuum outside the material objects (planets, stars, etc.). However, this evidence can scarcely be applied in the internal cavity of hollow spheres (if we admit their existence).

I do not think that the singularity in r=0 is problematic. Because nothing can enter it, it does not produce any paradoxes (as e.g. information paradox) or causes another conceptual problems in physics. (The opposite case is the attractive naked singularity in r=0, I agree.) I wonder that the concept of a matter collapsing to a point of infinitely large density is more viable than the "prosaic" hollow sphere.

Maybe, I better explain my point of view saying that the field equations describing the structure of neutron stars (also the classical, full-sphere stars) imply that a particle in the distance R is attracted outward by the star-centric, spherical material layers of stars with radii larger than R. Please do a thinking experiment, in which you will try to answer the question of why this outward action of upper layers is, generally, accepted in the modelling of neutron stars. (It is accepted always when one uses the field equation having this property. I do not know any neutron-star model, in which this feature was eliminated.)  And, if it is accepted when is weaker than the (inward) attraction of the layers with radii smaller than R, but not longer accepted as soon as it becomes dominant? Why this quantitative discrimination?

Concerning the discontinuity, I spoke about the discontinuity of metrics at the outer surface of stable compact objects, as e.g. neutron stars are. The radius of this surface is always larger than the gravitational radius. As I already wrote, I would like to hear, primarily, an answer on how to remove this discontinuity, clearly above the gravitational radius, in a strongly curved, but otherwise normal space (without the concept of hollow sphere).

Hi L. Neslušan, yes I understood it was inside the cavity and that with GR no non-uniformity was required.  I have seen papers supposing that perhaps we are in a cavity-like region.  I was just saying my familiarity with the idea of gravity inside a shell came from non-uniform Newtonian shells. 

I have figured out, I think, how to flip between + and -.  It is a matter of coordinate choice.

@Charles Francis

(1) You wrote "there is no discontinuity in the metric at the outer surface of a stable object. There is not even a discontinuity in the first derivative of the metric." I am confused with this claim, since I have not seen any model (I would welcome a reference if you can provide me with) with a demonstration of the continuity. If one creates a model, he or she has always to calculate the behaviour of g_rr and g_tt (in the limit of non-rotating and, then, spherically symmetric object). Thus, he obtains the values of these components in the outer surface. Having the model, one knows the mass of the relevant object. So, he knows also the metrics in the adjacent empty space, which is (according to the Birkhoff theorem) the OSCH metrics. And, values of this metrics calculated for for the outer radius never equal those obtained from the model of integral structure. As well, their derivatives with respect to "r" are different.

(2) I am sorry, I cannot understand the second part of your message. So, I am not sure if my response below is well relevant. Yes, according to the traditional concept, if there is assumed an existence of objects with the event horizon, there is nothing to stop matter falling through it. I agree that that the field equations say us this. This is not, however, any contradiction to the concept of stable hollow sphere. Namely, the existence alone of a stable model does not mean that all objects must be stable. There can still be collapsing or expanding objects and some can, maybe, collapse below their event horizon. I attempt to explain that the Einstein field equations can describe, at the same time, a real stable object of whatever mass with the outer surface above the corresponding event horizon (when the condition of continuity is satisfied).

I realize, it can be hard to believe. One often needs to perform the appropriate calculations himself to see that things actually behave in the above-described way. If you wish to do such calculations (verification), you can use my code that I emplaced on:

http://www.astro.sk/~ne/PUBLICATIONS/

in the subdir COMP_CODE, when I discussed these problems with the referee of my first article on this topic (Neslusan L.: 2014, Int. J. Astron. Astrophys., Vol. 4, pp. 1 -10.). The code numerically integrates (by the Runge-Kutta method) the equations describing the traditional Tolman-Oppenheimer-Volkoff model of neutron star (except other, the equations are also introduced in my article, which is in the directory PUBLICATIONS; see those for "dtau/dr", "du/dr", and "dnu/dr"). To be able to use this code, please read the instruction file "code_description.txt". [In this code, there is no additional modification to achieve the continuity, because that time I proceeded using the idea, which is today obsolete, that the continuity appears after the speed of light is enlarged in the OSCH solution.] Everytime when you will start the integration in a finite distance, you will find that the inward going part of the integration will finish with zero pressure and energy density not in the object's center, but in a finite distance (inner physical surface). As well, the outward going part of the integration will everytime finish in the outer surface with g_rr, which you will find different from its OSCH counterpart.

@Robert Shuler

Yes, I agree, the position of matter with respect to the coordinate frame is essential. Above the hollow sphere, a point in which we are interested in the metrics is farer from the coordinate origin than the matter and we have the metrics with "+". And, the point is between the coordinate origin and matter in the internal cavity with "-" metrics. (The orientation of the vector [which can be "-" or "+" with respect to the origin] from the point to the mean accumulation of matter determines the sign in the metrics.)

@Robert Shuler

Hi again. I must moreover add that I always in the above discussion assumed that we are speaking about "gravity". Because I also tried, hypothetically, to do a "generalization" of the metrics on electric interaction under the assumption that there is only a single interaction in the nature (in a radical form of this assumption, the gravity and electric force cannot be, in any situation, regarded in a different way; all this is highly hypothetical, of course).

The generalization can be done multiplying both nominator and denominator in fraction 2U/c^2 (or U/c^2) with the rest mass (m) of a test particle (the fraction does not change after this operation, in fact). Then, g_tt = -1/g_rr = 1 + 2mU/(mc^2) = 1 + 2W_p/W_o, where W_p is the potential and W_o is the rest energy of the particle. Now, I suggest to use the metric regardless W_p is the potential energy in the gravitational or electric field (rest energy remains the same). In the case of two particles having the charges of the same polarity, W_p > 0 and, hence, sign "+" also occurs in the metrics. (All this is, however, far from the intended discussion in this thread, I am afraid, so I stop my remark here.)

@Charles Francis

I agree that the continuity should be obvious from the Einstein's equations, but these require a perfectly-consistent approach. It requires, except of other, that the speed limit must be derived from the motion of a photon along the zero geodesic. But this principle has been violated in the construction of neutron-star models. And if the modeling is performed is the full consistency with the GTR (i.e. when the mistake is corrected), then the equations say that it is impossible to construct any model in the form of full sphere (with the maximum pressure and energy density in r=0), but the equations imply the hollow sphere model.

@Charles Francis

Sorry, I forgot to add that there exist some analytical solutions which give the finite and maximum pressure and energy density in the true centre. Combining these solutions with a numerical one, one can create a "composition" model in the form of full sphere. However, the Einstein's equations, when numerically integrated, lead typically to the hollow sphere solution. Thus, this must also be an alternative.

I could use a very short explanation, if someone has one, of how the GR field equation propagates a static field.  By that I mean, what is the equivalent of the 1/r law?  I'm aware that when solving for the Schwarzschild metric, the differential equation:

r dA/dr = A (A - 1)

arises, which has solution 1 / (1 + 1/Sr) where 1/S is chosen for weak field limit.  This term has 1/r in it of course.  But what assumptions or constraints that went into making the field equation resulted in the above differential equation ... in brief.

Interesting.  I see at least what to look for, thanks.

I have updated the question to give more specific rationale on why I think a new field equation is needed.  Hope this will inspire someone.  Thanks all for your answers so far.

Good question that deserves a careful answer.

As far as conservation of energy and momentum, using the method of GR any potential function (or time dilation function) fulfills the need, because the infalling matter stops before more than the original energy of the particle could hypothetically be extracted (or it passes a magical barrier which leaves its energy-momentum trace behind forever).  The functions I mentioned, in the numerator, do not require any special treatment as they do not reach infinite dilation until R=0.  I'm assuming of course that all laws of motion derive from the changes in potential, using either Lagrangian or Hamiltonian forms.

As far as the laws of physics leading to realism, a single real outcome for all observers though they may measure it differently, I do not see how that is violated by any particular choice of potential function either.  If different observers measure time and distance differently, then all physical variables and constants which have time or length components in their "units" will require transformation from one coordinate system to another along with the measurement units.  This is just common sense.  I have given a transformation of the gravitational constant G along these lines in 2013, but it is not in final correct form because I was unable to derive curved spacetime from equivalence at that time.  I can now, and will get around to correcting that when I get all related matters worked out, which might be some time.

Actually, my friend Allen Allen has written a paper showing that an event horizon leads to different realities (the in-faller who is incinerated at the event horizon, vs the in-falling observer who passes unharmed).  I can forward that if you like, though I do not especially like the paper.

On another of my question threads https://www.researchgate.net/post/Is_there_more_than_one_theory_of_gravity_that_can_demonstrate_compatibility_with_the_Einstein_equivalence_principle/1 several respondents point out that many serious theories of gravity have significantly different characteristics. In fact, any kind of field may be supposed and it is still a metric theory if the field couples to the metric and not directly to matter (V. Toth on page 1 of that thread). Presumably the potential function is different for each of these. It is well known that some have event horizons and some do not. Most of these theories conserve energy and momentum and state the laws of physics in co-variant form, leaving us free to preserve those items while considering alternate field equations.

As far as conservation of energy and momentum, using the method of GR any potential function (or time dilation function) fulfills the need, because the infalling matter stops before more than the original energy of the particle could hypothetically be extracted (or it passes a magical barrier which leaves its energy-momentum trace behind forever). The functions I mentioned, in the numerator, do not require any special treatment as they do not reach infinite dilation until R=0. I'm assuming of course that all laws of motion derive from the changes in potential, using either Lagrangian or Hamiltonian forms.

As far as the laws of physics leading to realism, a single real outcome for all observers though they may measure it differently, I do not see how that is violated by any particular choice of potential function either. If different observers measure time and distance differently, then all physical variables and constants which have time or length components in their "units" will require transformation from one coordinate system to another along with the measurement units. This is just common sense. I have given a transformation of the gravitational constant G along these lines in 2013, but it is not in final correct form because I was unable to derive curved spacetime from equivalence at that time. I can now, and will get around to correcting that when I get all related matters worked out, which might be some time.

Actually, my friend Allen Allen has written a paper showing that an event horizon leads to different realities (the in-faller who is incinerated at the event horizon, vs the in-falling observer who passes unharmed). I can forward that if you like, though I do not especially like the paper.

On another of my question threads https://www.researchgate.net/post/Is_there_more_than_one_theory_of_gravity_that_can_demonstrate_compatibility_with_the_Einstein_equivalence_principle/1 several respondents point out that many serious theories of gravity have significantly different characteristics. In fact, any kind of field may be supposed and it is still a metric theory if the field couples to the metric and not directly to matter (V. Toth on page 1 of that thread). Presumably the potential function is different for each of these. It is well known that some have event horizons and some do not. Most of these theories conserve energy and momentum and state the laws of physics in co-variant form, leaving us free to preserve those items while considering alternate field equations.

I have updated the question to reflect a better rationale of exactly what I am trying to investigate.  Thanks to Klaus Kassner for an analysis that helped pin this down.

After some months it appears the question is too hard.  So I have devised a much simpler one and tagged it for mathematicians.  See link.

https://www.researchgate.net/post/What_differential_equation_has_a_solution_of_the_form_Fx1_1_Kx-2

Dear Rob:

Don't allow youself to be pulled away from simple geometric thinking. Very few are as able as you are in the method that alone counts. Such children's math questions start way too late.

Take care,

Otto

Very little of the field equation is actually used, only that the curvature tensor = zero (vacuum field equation) ... something Crothers is always complaining about - there is no mass in it - the relation to the central mass comes instead from the weak field approximation.

More specifically ... in the conventional process of using the weak field approximation, "Newtonian gravity" is defined as an inverse square acceleration law.  See box at right 2/3 of the way down in linked derivation below.  This page only gives the process schematically.  I also am attaching the figure.  The upper left corner indicates the acceleration law.

A potential law (1-φ) or an inertia law 1/(1+φ) could be used instead, giving either of my alternate results.  In fact we cannot know from Newton or from the field equation which to select!

However, only gravity defined as an inverse square acceleration is consistent with a vanishing Ricci tensor. [correction from earlier post]

https://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

Yes.

But you may know that my return to the equivalence principle produced a proof of c being a global feature there (in accordance with the axiomatic assumption entered at the outset).

So all solutions which violate c-global need to be re-scaled. And so presumably do the field equations themselves.

Otto, on a separate QA thread, Klaus pointed out that the Schwarzschild solution depends only on a limited subset of the GR axioms, the equivalence principle and the requirement for the Ricci tensor to vanish in a vacuum.

Equivalence does not seem to be at issue in my proposed alternate (potential based) solutions.  It is the vanishing of the Ricci tensor that gets disturbed.  I am still investigating what this means, but am currently guessing that it implies non-locality.  That would certainly explain the difficulty between GR and QM, and Einstein's resistance.

I never got an answer that I understood from you or Sadeem about what c-global means.  Sadeem's answer would seem to eliminate time dilation, which is an observable and cannot be transformed away.  However, the current discussion raises a new issue about compatibility of nearly anything one can think of with GR.  Namely, the Schwarzschild solution depends on very little of GR.  It is already coordinate independent.  As far as the Ricci tensor is concerned in a vacuum, there is no way to rescale zero!  So I think we are both out of luck as to compatibility with GR.

I have managed to come up with a considerably simpler version of this problem.  The stress-energy tensor can be ignored for the moment.  See link.

https://www.researchgate.net/post/What_is_the_non-zero_Ricci_tensor_equation_for_a_metric_field_slightly_non-Schwarzschild

Dear Rob:

It is heroic to enter GR so deeply.

But it is counterproductive before the equivalence principle is completely elucidated - right?

Take care, Otto

Otto, do you refer to derivation of the equivalence principle, or validation (or not) of it?  There are people on here who debate its validity.  I used to have my doubts but am currently confident in it.

With respect to derivation of the equivalence principle, presumably a quantum theory of gravity would be expected to do this.  It is also possible with classical theories.  In fact Thirring used a field theoretic approach (gauge theory and so forth) to derive static gravity and to derive the EP in 1961.  I am philosophically sympathetic to his work, but not fond of the details of his approach.  See Thirring, W.E., “An alternative approach to the theory of gravitation,” Annals of Physics, 16 (1) pp 96-117, Oct. (1961).

I gave a quantum formulation which would explain universal effect, and therefore equivalence, in a 2013 paper which you can find on my profile (the quasi-measurement paper, but be advised I did not know how to obtain spatial curvature when I wrote that).  The purpose with this gravity exercise is to open up the window on permissible theories a little bit, to make room for that formulation, which is a 1/r formulation of potential.  So I'm assuming equivalence for now.  But if I can crack open the window I'll assert my derivation of it.  I'll link both these papers below in case you want to look at them.

Article On Dynamics in a Quasi-Measurement Field

Dear Rob:

I apologize: This level of erudition is important and enlightening.

But at the present moment in time a different, much simpler question is at stake (perhaps):

To re-check the oldest naivest version of Einstein 1907/1911, pure SRT:

What is it really that the SRT predicts rigorously here, over a short "linear" vertical stretch?

Hereby, on this most basic level, a new phenomenon occurs.

If this is true, the foundations of GRT are altered.

Right?

Otto

Otto, all I need to derive gravity is universal free fall, the Planck relation, and the coordinate potential postulate.  Do not really use SR.  I think more will be learned by studying gravity.  Read the latest version of the "Two-step" paper to understand why.  Also you might read my paper on Lorentz as a geometric average.  Pick your battles, and fight one you can win.

Did you see the so far overlooked "relative slantedness" of light hugging the bottom of the Einstein rocketship, when viewed from the tip?

Otto, I have a very surprising paper on a new computational analysis of light bending with new geometric insights I'm working on.  I'll send it to you privately when finished for comment.

I don't know of anyone who's overlooked slantedness of light in SR.  It is a matter of reference frame.  You can always find a frame in which light moves as c, but in longer paths explaining the measurements of other frames.