Dear Demetris

For 2-body GTR in the solar system see V.A. Brumberg, Essential Relativistic Celestial Mechanics, Adam Hilger Editor, Bristol, Philadelphia and New York (1991). Also see S.A. Klioner, P.K. Seidelman and M.H. Soffel (editors); Relativity in Fundamental Astronomy: Dynamics, Reference Frames and Data Analysis,

Proceedings of the IAU Symposium 261 (Cambridge University Press,

Cambridge, 2010).

Way back in 1938 the papers by A. Einstein, L. Infeld and B. Hoffmann, Ann. Math. 39 , 65-100 (1938) and A. Eddington and G.L. Clark, Proc. Roy. Soc. (Lond.) A166 , 465-475, (1938) described the first correct 1PN equations of motion, much work was done afterwards see for instance V.A. Fock, The Theory of Space, Time and Gravitation (Russian edition, Moscow, State Technical Publications, 1955).

as well as L. Infeld and J. Plebanski, Motion and Relativity (Pergamon Press, Ox-

ford, 1960.

The Japanese did a lot of work towards the N-body metric and 2PN level equations of motion, seeT. Ohta, H. Okamura, T. Kimura and K. Hiida, “Physically acceptable solution of Einstein’s equation for many-body system,” Prog. Theor. Phys.50 , 492 (1973), and H. Okamura, T. Ohta, T. Kimura and K. Hiida, “Perturbation calculation of gravitational potentials,” Prog. Theor. Phys. 50 , 2066 (1973), as well as T. Ohta, H. Okamura, T. Kimura and K. Hiida, “Coordinate Condition and Higher Order Gravitational Potential in Canonical Formalism,” Prog. Theor. Phys. 51 , 1598 (1974).

2.5PN level beyond the Newtonian equations of motion was reached in the 1980s by

(1) T. Damour and N. Deruelle, “Radiation Reaction and Angular Momentum Loss in Small Angle Gravitational Scattering,” Phys. Lett. A 87 , 81(1981)

(2) T. Damour; Probl`eme des deux corps et freinage de rayonnement en rel-

ativit´e g´en´erale. 1982. C.R. Acad. Sc. Paris, S´erie II, 294 , pp 1355-1357.

(3) T. Damour; Gravitational radiation and the motion of compact bodies.

1983. in Gravitational Radiation, edited by N. Deruelle and T. Piran,

North-Holland, Amsterdam, pp 59-144.

(4) S.M. Kopeikin, Astron. Zh. 62 , 889 (1985).

(5) G. Schaefer, “The Gravitational Quadrupole Radiation Reaction Force

and the Canonical Formalism of ADM,” Annals Phys. 161 , 81 (1985).

3PN level equations were derived by

(1) Jaranowski, P. and Schaefer, G., “3rd post-Newtonian higher order Hamil-

ton dynamics for two-body point-mass systems ([Erratum-ibid. D 63 ,

029902 (2001)])”, Phys. Rev. D, 57 , 72–74, (1998). [http://arxiv.org/abs/gr-

qc/9712075gr-qc/9712075].

(2) Blanchet, L. and Faye, G., “General relativistic dynamics of compact bi-

naries at the third post-Newtonian order”, Phys. Rev. D, 63 , 062005,

(2001). [http://arxiv.org/abs/gr-qc/0007051gr-qc/0007051].

(3) Damour, T., Jaranowski, P. and Sch¨afer, G., “Dimensional regularization

of the gravitational interaction of point masses”, Phys. Lett. B, 513 , 147–

155, (2001). [http://arxiv.org/abs/gr-qc/0105038gr-qc/0105038].

(4) Itoh, Y. and Futamase, T., “New derivation of a third post-Newtonian

equation of motion for relativistic compact binaries without ambiguity”,

Phys. Rev. D, 68 , 121501, (2003). [http://arxiv.org/abs/gr-qc/0310028gr-

qc/0310028].

(5) Blanchet, L., Damour, T., Esposito-Far`ese, G. and Iyer, B.R., “Grav-

itational radiation from inspiralling compact binaries completed at the

third post-Newtonian order”, Phys. Rev. Lett., 93 , 091101, (2004).

[http://arxiv.org/abs/gr-qc/0406012gr-qc/0406012].

Work is being done at the 4PN level e.g. D. Bini and T. Damour, “Analytical determination of the two-body gravitational interaction potential at the 4th post-Newtonian approximation,” Phys. Rev. D, 87 , 121501(R), (2013) [arXiv:1305.4884 [gr-qc]].

The fact that many scientists use the PN GTR equations daily in very accurate determinations of satellite orbits, planetary orbital integrators etc., which are reliably and accurately calibrated and evaluated by real measurements using planetary radar data, satellite laser ranging and Lunar laser ranging, should be an indication to you that the answer to your question has been solved a long time ago. Yes, it is complex, yes, it is not easy to derive, but this has been done by very competent scientists and we can utilise these solutions fruitfully.

No opinion! Out of my specialties.

No, @Demetris! Please, have a look!

http://glencoe.com/sec/science/physics/ppp_09/animation/Chapter%207/Cavendish%20Experiment.swf

No motion will be produced if R=0 and then " the 'dent' of each mass is absolutely the same as the 'dent' of the other mass". So, the two masses must not only be equal, they are actually the same (which is a bit more than equal) mass. This is of course not physically possible, so you have to follow Robert's answer.

Otherwise, R is very large, and m very small, and spacetime is asymptotically flat, and then no motion is measureable, and can only be calculated....or follow Robert's answer...

Thanks Ljubomir for the link. (Cavedish experiment is not exactly what I wanted. I'd like an experiment to measure the force between m1, m2 and not the constant G). So, where are the detailed calculations of the answer? Thanks.

Dear Robert, since "the two masses both develop identical motion along the line joining them, in an entirely symmetric fashion" and since, as being accepted by the majority of Physicists, the GR >> NEWTONIAN, then the supreme theory should be able to produce that simple fact...

Charles, I do not want to measure G & since "he dents will produce a gravitational attraction almost exactly equal to that of Newtonian gravity" I just want a detailed proof of it. That's all.

Thanks every participant.

Charles, I am not interested for Schwarzschild solution since it holds actually as a limit case when we have M>>m, see also here:

http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html (relation 7.26)

My demand for an absolute equality m1=m2=m cannot be overcome.

Dear Robert, I agree with the first half of your claim only. :)

I totally agree with this statement: "You cannot measure G without measuring the force. "

If gravity is force then it is not the curvature of space-time. Without Newton's law we cannot even define G, so how can we define Einstein's equation?

The Newtonian component of gravity is included via time-dilation in Einstein’s field equations (EFEs). In fact, G appears in both his equations and almost every other formulation of gravity. The best way to think about it is by having each particle as a localized field or wave, e.g. the electromagnetic field. Therefore, when two particle’s interact gravity is essentially due to the time-dilation induced from one particle’s field to another. The trick is that EFEs deform a global metric rather than the field of each particle. This is the equivalence principle, because the coordinates and time-dependence of each particle when compared to a global space-time metric will produce the Euclidean state; i.e. all physics will be equivalent for local frames of reference. To find the time-dependence of a particle you then need to apply the geodesic equations to the space-time metric. One could also say that gravity is due to each particle's field trying to preserve it's Euclidean state with respect to local frames. Although EFEs appear to be flawed for other reasons, the mechanics behind general relativity are straightforward when viewed in this perspective.

So to answer the question, the "dent" due to mass or energy is both time-dilation and metric deformation. The first causes two stationary masses to accelerate towards each other, while the later deviates the path of trajectories similar to a wave traveling through a variable medium.

If the gravitational constant G cannot be defined by GR, then GR is not a self consistent theory like Newton's gravitational law. The so-called good book could be good for some people's dogma.

Who can define the G without using Newton's gravitational law?

GR is only one of so many revised versions of Newton's gravitational theory, so there is no reason to take it as a dogma.

Someone's idiocy is beyond anyone's imagination. :-)

Charles, I don't need to study your RQG, but I can conclude that it is nothing more than a day dreaming version, if it violates the conservation law of energy and momentum like any other theory involved curved space-time.

I think there is a fault in saying that the space time is curved. the curvature indeed is in space only this will keep the conservation of energy. but at the same time this means that the GR itself must be reformulated from the first square.

In order to be as clear I can be, let's say that m1=m2=m=1 Kg and r12=r21=r=1 m.

I am not interested for exotic objects like black holes and other similar.

Please, solve the problem or state that it cannot be solved.

Thank you all.

Dear Charles, I have been watching you at many RG discussions and I'd friendly like to advice you that you have to do something with your mentality and aggressive behaviour in public discussions. I don't want to enter in more details and I will not do it, but you have to think a little my friend... Just for a change. Thanks.

Let's see the things with a different way, in order to accept partially that the 'dents' view is not so much quantitative, why not? Can we accept the next "primary causality scheme" for GR?

Gmu,nu=(8piG/c^4)*Tmu,nu

So, Tmn, stress energy tensor ---> Gmn, ie the gij=metrics?

[ http://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor ]

Or without symbols:

"The existence of mass, energy, momentum is the cause of the metric tensor g?"

[ http://en.wikipedia.org/wiki/Metric_tensor_(general_relativity) ]

Do we agree on that?

Demetris, if one takes your set-up values m1=m2=m=1 Kg and r12=r21=r=1 m, then the GTR solution is the same as the Newtonian solution. Even using much larger masses such as the Earth or Moon, one would typically use the Newtonian approximation and then add linearized GTR components to get to a more exact solution. For instance, if one determines satellite orbits at the cm level, one would calculate the Newtonian potentials (Earth, Moon, Sun, planets) and then add the GTR effects, the acceleration in the Schwarzschild field, then frame dragging (Lense-Thirring precession) and then de Sitter (geodesic precession). Without these GTR contributions, the orbits are not as well determined and the observed - computed residuals increase.

Who can define the G without using Newton's gravitational law?

Any theory giving up the absolute space has to give its own definition first, because the G is defined by Newton's absolute space.

Without a universal flat space-time reference frame (Newton's absolute space), talking about conservation of energy and momentum is nothing more than daydreamer talking nonsense.

Let's be more technical now:

1)Solve for each mass m1,m2 the next sub-problem:

-->Solve Gmn=[(8 pi G)/(c^2)]*Tmn where T^{00}=rho/(c^2)=1 kg/m^3 / (c^2) and other T^{ij}=0, with the mass distribution arbitrarily chosen and not having any kind of known symmetry (: no spherical or other common geometrical symmetry, just arbitrary - the only assumption is that the two bodies are identical - 'twins' ).

2)Find the geodesic that object#2 has to follow due to the space time curvature produced by object#1.

3)Find the geodesic that object#1 has to follow due to the space time curvature produced by object#2.

4)Give the results and tell us what will finally be done.

5)Evaluate the theory, the effort and the results.

You can ignore any kind of 'dent' visualization.

If the two bodies are initially at rest, then the geodesic equations will be reduced from 4 dimensions to 2. My "Theory of Everything" covers how to solve for a single particle falling into a static potential using only special relativity on page 8. You could use these equations to numerically solve for the two body solution by time-stepping at small intervals. I also show on page 27 that the results are equivalent to applying the geodesic equations. However, if we consider some basic postulates of GR, then there will be some modifications to each object's field depending upon mass. For example, say that there are two electrons at finite separation distance. The field of each (electromagnetic field, gravitational field, ect.) will follow the space-time metric that arises from the presence of the other electron. The same thing works for large composite objects, but an exact calculation is very complicated and requires methods discussed in sections 2.5-2.7. In essence you would have each particle within each object affected by the presence of all other particles under consideration. I use a scalar field that makes the space-time metric locally isotropic and much simpler to deal with; i.e. the line element in all directions at a single point has constant magnitude, but it's derivatives do not.

Answering such a low level problem question is not misguided or disingenuous, since every theory and especially a 'supreme' theory like GR, has to be able to solve elementary problems at least in principle and not after making a big set of assumptions. Otherwise we are entering into the normative view, just like in Normative Economics:

http://en.wikipedia.org/wiki/Normative_economics

The later view tells us not to search for what exactly is being done, but what we SHOULD do. Such a view is far away from what I accept for Physics as a natural science and believe me I have read too much normative economics, so I can identify the view from miles away.

Let's return to GR now. As I was writting before the parenthesis, if a theory is not able to find what a sub-ordinate theory can compute relative easy, then it is not 'supreme' by no means. So, if GR will be proved that is not be able to solve this simple problem, then:

*Let's put it in a frame of paintings and let's hang it on the wall, in order to watch its beauty

*Sorry, but the problem of gravity is still open

*The fact that after 100 years the gravity waves have not been found, increases the previously written arguments

*Let the science breathe from such a strong tight, even after a century...

PS I forgot to mention that, although GR cannot solve a two body problem, we have no problem with it and we are trying to quantize it...

@Michael, have you already done the calculations? If so, I am interested for the results. Thank you.

Yes, currently, the conservative dynamics of a two-body system is fully known up to the 3PN level using the usual PN formalism. Many textbooks show how to arrive at the Newtonian formulation from GR. Typically one could decide whether to use the Newtonian approximation or not, depending on requirements by evaluating the relativistic correction factor ε ~GM/(c ^2 r)~v^2/c^2, e.g. surface of the Sun ~ 10^ -5, and surface of a neutron star ~0.1

So depending on your particular application, you would use either GTR or Newtonian approach, they both have a place in science.

,

Demetris,

I provide two ways to approximate the combined curvature for composite objects. The first is an approximation through use of a Dirac delta and the later requires integration for each particle under consideration. Essentially it starts off with 1/r gravitational potentials for each particle in a flat Euclidean space. As each iteration of the continuous fractions are evaluated, the potentials begin to follow the space-time metric induced by every other particle under consideration until convergence. Therefore, you could simulate any type of configuration relatively easily with several steps (here vacuum energy density is proportional to the gravitational potential, so everything can be derived classically).

Static dust solution:

1. Use the initial vacuum energy density of each particle (E/r) to setup the Euclidean space.

2. Find the space-time metric of the current configuration relative to each particle and save the deformation required for each particle's field (I prefer to attach a separate manifold to each particle and use this to conceptualize metric deformation via the equivalence principle).

3. Use the new deformed fields of each particle to find the updated space-time metric from the last iteration. Keep looping here until satisfied.

4. Use the fundamental forms to derive the effective space-time metric and therefore summed curvature.

5. Calculate Christoffel symbols and begin plugging into the geodesic equation to get dv/dt, i.e. the acceleration.

So far I have only done exact solutions using the Dirac delta method for 2 particles and ensured that the use of the second and third fundamental forms produces the correct space-time metric from the vacuum scalar field. I also did some ray tracing through a gravitational potential with these methods but nothing in terms of composite objects as of yet.

So Michael, If I have understood well, you start from the flat space metric eta_mn (ημν) and iteratively you make corrections leading to the correct final metric?

Yes, the first iteration is basically the summed curvature without considering the effect that each particle has on all of the others. But since one particle's field(s) or metric will follow the space-time metric induced by the other particle(s) and vice versa, the other iterations provide corrections to this. If you want to include temperature, pressure or bulk flow, the same concepts apply where you begin with each particles field(s) relative to ημν. Alternatively, I view this as the space-time metric relative to a single particle in vacuum with no other sources (ημν), where r' = r or the Newtonian value.

So, Michael, the answer to the initial problem according to you is that we cannot solve the general problem by only using GR equations, but we could apply a kind of perturbation theory, an iterative approach. But, my initial questions did not want this, it wanted a straight solution of Gμν=(...)Τμν where Τ^{00}=ρ/c^2. Thus, the question is still open...

I read with interest your work about your so called 'theory of everything" and I have some remarks:

1)A concept of "theory of everything" is not correctly defined, since such a thing will never be found. Why, just because knowledge is like the perimeter of a circle: As we know more (the radius of knowledge increases), then we wonder about more (the perimeter of knowledge-circle also increases and as standing on that periemeter we see then many new problems). So, working with such a concept will be proven vain.

2)The problem is not if particles are point-like or vacuum field like, as is your main argument. The problems are many and have to do:

i)with the domination of linear approximation in Physics (quantum and relativistic theories are just linear algebra with applications in Physics)

ii)with the rigid body approximations, obviously all Lagrangian field theories are such a class of theories

3)Dirac's equation is not the 'ultimate equation'.

I could add more, but then I will depart from my main argument, see 1) above.

Anyway, if it gives you satisfaction, you can keep doing it!

I started this question since, by doing a simple general relativity google search for images, you will see next page:

https://www.google.com/search?q=general+relativity&num=20&es_sm=93&source=lnms&tbm=isch&sa=X&ei=ENV-U5reB4Kq0QW93IAI&ved=0CAgQ_AUoAQ&biw=1600&bih=810

So, dents and again dents is what everybody uses in order to explain GR to other people.

If, now, the dent approach is a naive approach and we should use Gμν=(...)Τμν equations for answering the initial problem, then the situation is getting worse.

I think that geometry can always give an answer, although qualitative, to our problems.

And, finally, I insist that according to GR, the two bodies will stay each other inside its dent and will just observe the other, without doing anything like motion and other actions...

I am waiting for a formal solution of the equations...

Otherwise I will close the topic by giving a NEGATIVE answer.

Dear friends, you can downvote as long as you like, but, sorry, you didn't give an explicit answer!

Thank you any way for participating in this question and also for downvoting!

(It means more than you imagine, especially about the mentality of scientists in 'hard' sciences...Of course it is an estimator for some explanations for the sticking of Physics last years and why this science tends more and more to the Inquisition of the Middle Ages)

Only daydreamer can mix up the concept of mathematics and physics.

Curved space-time in GR is incompatible with - Homogeneity of space - Isotropy of space - Uniformity of time, and will violate the conservation laws associated with those symmetries.

Who can define the G without using Newton's gravitational law?

This is a simple question which cannot be answer by mathematicians.

Any theory giving up the absolute space has to give its own definition first, because the G is defined by Newton's absolute space.

Without a universal flat space-time reference frame (Newton's absolute space), talking about conservation of energy and momentum is nothing more than daydreamer talking nonsense.

The Higgs field which was confirmed one year ago is the absolute space.

You are just one year behind the science. :-)

More evidences to prove the existence of absolute space:

http://www.sciencedirect.com/science/article/pii/S0370269311003947

http://arxiv.org/abs/astro-ph/0703694

http://www.newscientist.com/article/dn23301-planck-shows-almost-perfect-cosmos--plus-axis-of-evil.html#.UunXEtGYZok

A preference for spiral galaxies in one sector of the sky to be left-handed or right-handed spirals has indicated a parity violating asymmetry in the overall universe and a preferred axis.

When there exists too much passion, it is not always a good sign...

Anyway, a small recap for scientists that will read this thread after time:

1)GR is a beautiful theory, because it makes the causality scheme mass-energy distribution --> space time curvature, but it cannot solve the simplest two body problem

2)The fact that a theory is mathematical beautiful does not mean that it is also an accurate one. We are living in a local universe and the specific realization of all available mathematical theories(the beauty of Mathematics: They can describe all possible universes!) has to be chosen by experiment and only experiment. What is nice or what we like is irrelevant.

3)Since 1914 Physics has been trapped in a false explanation of gravity, that of GR. this is the main reason for sticking and not progressing. I remind all that last serious model, the standard model, was written in ~1967!

4)What is the theory of gravity? I don't know, the only thing I am sure about is that it is not the combination of (linear) Differential Geometry and (linear) Tensor Analysis that Einstein used.

5)Attempts to build a quantum gravity theory will not have success, since they build on a false environment. That's another reason for not succeeding, even after 30 years, for all (then too) promised new theories, like superstrings.

Anyway, relax everybody, it is just a forum here...

Dear Demetris

For 2-body GTR in the solar system see V.A. Brumberg, Essential Relativistic Celestial Mechanics, Adam Hilger Editor, Bristol, Philadelphia and New York (1991). Also see S.A. Klioner, P.K. Seidelman and M.H. Soffel (editors); Relativity in Fundamental Astronomy: Dynamics, Reference Frames and Data Analysis,

Proceedings of the IAU Symposium 261 (Cambridge University Press,

Cambridge, 2010).

Way back in 1938 the papers by A. Einstein, L. Infeld and B. Hoffmann, Ann. Math. 39 , 65-100 (1938) and A. Eddington and G.L. Clark, Proc. Roy. Soc. (Lond.) A166 , 465-475, (1938) described the first correct 1PN equations of motion, much work was done afterwards see for instance V.A. Fock, The Theory of Space, Time and Gravitation (Russian edition, Moscow, State Technical Publications, 1955).

as well as L. Infeld and J. Plebanski, Motion and Relativity (Pergamon Press, Ox-

ford, 1960.

The Japanese did a lot of work towards the N-body metric and 2PN level equations of motion, seeT. Ohta, H. Okamura, T. Kimura and K. Hiida, “Physically acceptable solution of Einstein’s equation for many-body system,” Prog. Theor. Phys.50 , 492 (1973), and H. Okamura, T. Ohta, T. Kimura and K. Hiida, “Perturbation calculation of gravitational potentials,” Prog. Theor. Phys. 50 , 2066 (1973), as well as T. Ohta, H. Okamura, T. Kimura and K. Hiida, “Coordinate Condition and Higher Order Gravitational Potential in Canonical Formalism,” Prog. Theor. Phys. 51 , 1598 (1974).

2.5PN level beyond the Newtonian equations of motion was reached in the 1980s by

(1) T. Damour and N. Deruelle, “Radiation Reaction and Angular Momentum Loss in Small Angle Gravitational Scattering,” Phys. Lett. A 87 , 81(1981)

(2) T. Damour; Probl`eme des deux corps et freinage de rayonnement en rel-

ativit´e g´en´erale. 1982. C.R. Acad. Sc. Paris, S´erie II, 294 , pp 1355-1357.

(3) T. Damour; Gravitational radiation and the motion of compact bodies.

1983. in Gravitational Radiation, edited by N. Deruelle and T. Piran,

North-Holland, Amsterdam, pp 59-144.

(4) S.M. Kopeikin, Astron. Zh. 62 , 889 (1985).

(5) G. Schaefer, “The Gravitational Quadrupole Radiation Reaction Force

and the Canonical Formalism of ADM,” Annals Phys. 161 , 81 (1985).

3PN level equations were derived by

(1) Jaranowski, P. and Schaefer, G., “3rd post-Newtonian higher order Hamil-

ton dynamics for two-body point-mass systems ([Erratum-ibid. D 63 ,

029902 (2001)])”, Phys. Rev. D, 57 , 72–74, (1998). [http://arxiv.org/abs/gr-

qc/9712075gr-qc/9712075].

(2) Blanchet, L. and Faye, G., “General relativistic dynamics of compact bi-

naries at the third post-Newtonian order”, Phys. Rev. D, 63 , 062005,

(2001). [http://arxiv.org/abs/gr-qc/0007051gr-qc/0007051].

(3) Damour, T., Jaranowski, P. and Sch¨afer, G., “Dimensional regularization

of the gravitational interaction of point masses”, Phys. Lett. B, 513 , 147–

155, (2001). [http://arxiv.org/abs/gr-qc/0105038gr-qc/0105038].

(4) Itoh, Y. and Futamase, T., “New derivation of a third post-Newtonian

equation of motion for relativistic compact binaries without ambiguity”,

Phys. Rev. D, 68 , 121501, (2003). [http://arxiv.org/abs/gr-qc/0310028gr-

qc/0310028].

(5) Blanchet, L., Damour, T., Esposito-Far`ese, G. and Iyer, B.R., “Grav-

itational radiation from inspiralling compact binaries completed at the

third post-Newtonian order”, Phys. Rev. Lett., 93 , 091101, (2004).

[http://arxiv.org/abs/gr-qc/0406012gr-qc/0406012].

Work is being done at the 4PN level e.g. D. Bini and T. Damour, “Analytical determination of the two-body gravitational interaction potential at the 4th post-Newtonian approximation,” Phys. Rev. D, 87 , 121501(R), (2013) [arXiv:1305.4884 [gr-qc]].

The fact that many scientists use the PN GTR equations daily in very accurate determinations of satellite orbits, planetary orbital integrators etc., which are reliably and accurately calibrated and evaluated by real measurements using planetary radar data, satellite laser ranging and Lunar laser ranging, should be an indication to you that the answer to your question has been solved a long time ago. Yes, it is complex, yes, it is not easy to derive, but this has been done by very competent scientists and we can utilise these solutions fruitfully.

Dear Ludwig, thank you for your references, as for the core you already wrote it: Post Newtonian.

Demetris,

I consider EFEs an extension of general relativity, which is based around the geodesic equations and equivalence principle. Gravitational waves still have not been directly detected, which is the defining characteristic between his and my theory (because vacuum energy density is essentially Planck scale fluctuations of space analogous to a 3d spring-mass system). When a charged particle loses momentum in my equations, all changes in relative motion produce EM radiation rather than EM+gravitational waves. Therefore, the question depends upon which theory of GR is preferred (although it appears that you are looking for an EFEs solution).

1. In terms of a “theory of everything”, I strongly believe that there is a logical and mechanical foundation at the Planck scale for everything we see. The only fabric available to give physical essence to fields and particles is Planck scale fluctuations of space itself. Therefore, we do not get stuck in this infinite loop of adding another layer to the theory (such as virtual particles or ad-hoc fields), which lack solid foundations and physicality. The scientific definition for a “theory of everything” however is more along the lines of explaining how all forces originate and link together.

2. With respect to the stated issues within EFEs arising from using point-like sources in a field equation relates to the existence of event horizon and gravitational waves. Although the point-like sources are averaged over space to provide T^(uv), the concept is exactly the same when using a charge density in electrodynamics. When doing per particle solutions with localized field(s), it is observed that black holes can no longer form without infinite energy or mass density. The same thing can be said about gravitational waves, as classically any change in a particle’s momentum is conserved through electromagnetic radiation. Applying these on a per particle basis strongly challenges what EFEs are saying, i.e. they are over simplified and produce erroneous results at high densities.

3. I simply use the Dirac equation to show how general relativity can be incorporated into quantum mechanics via per particle solutions. Although I have some grasp of the standard model, the Dirac equation is something that I fully understand at a fundamental level. I do not think any model of QFT however is the “final theory”, as this is left to the Planck scale fluctuations of space previously discussed. Nonetheless, the foundations I’ve provided for Planck scale fluctuations are compatible with the standard model and weak field limit of EFEs.

The prefix post- "something" is a 'sexy way' to say in other words the same thing, the "something" itself. We say post-modern art and we mean modern art, in a more elegant way. There exist many examples for the use of 'post-...'. So, returning here post-Newtonian theory is just a Newtonian theory. You can see the two cases at the attachment, where in the first case the starting point is GR, while in the 2nd is not. The concept of a gravitational potential is 100% classical Newtonian. We can 'rename' it as we want, but the Poisson kernel is there and is waiting for us to compute the 'all time classic' Poisson integral representation of the solution. That' s the reason for its success.

At the second case we do not make any kind of GR assumptions, we just accept our ignorance and just add more terms in the Taylor series expansion of the true, but hidden to us, theory. We also succeeded to explain perihelion shift of Mercury (Hermes). So, my summary is this:

1)GR can not solve the "two identical body problem"

2)NEWTONIAN theory in its second order approximation can solve it

3)All computations that 'prove' the superiority of GR are (what an irony) made by Newton's extension.

Finally, I wonder if so many institutes like NASA, Max Planck and other famous, after 100 years have not attempt to directly solve GR equations by a super computer. I doubt a lot that they have not done it. Probably what they have found i) they did not like it & ii)thus they kept it secret.

Keep praying to Einstein, but remember: He is not God!

Just to add my last contribution to this question, one cannot just add post-Newtonian corrections or extensions and then assume Newtonian conceptions of space and time. This would result in a complete misinterpretation of GTR solutions and erroneous interpretation of observations. Therefore, as is commonly done in relativistic solar mechanics, astrometry, space geodesy etc., a complete GTR framework approach is required which include timescales, reference frames, interpretation and reduction of observations, modelling etc.

Curved space-time in GR is incompatible with Newtonian absolute space and time, and violates universal conservation laws.

Either Newton or Einstein, one of them must be fundamentally wrong.

@Guoliang, there is a possibility of a compromise, following the Lorentz ether. It is not really a compromise in the metaphysics, but it accepts the physical predictions of GR: The "proper time" of GR is what the clocks are showing. This can be accepted without rejecting absolute space and time - all one has to accept is that the showings of the clocks are distorted by the ether. High density of the ether or large velocity of the ether means higher time dilation.

So, it is possible to have absolute space and time, filled with an ether, where the most important properties of the ether, namely density g^{00}sqrt{-g}, velocity g^{0i}/g^{00}, and stress tensor are described by the gravitational field g_{mn}(x,t), and the equations proposed by Einstein survive as a natural limit of this ether theory.

All this is not some "ether crank nonsense" but a theory published in a serious peer-reviewed journal http://arxiv.org/abs/gr-qc/0205035

@IIja, I agree with you.

The ancient ether has a modern new name now i.e. Higgs field.

@Ilja

can I ask you about your ether properties is it static as supposed before or dynamic?

Is it the same as Lorentz ether or differ?

No, the ether is dynamic and compressible. The ether density rho(x,t)=g^{00}(x,t)sqrt{-g(x,t)} varies in space and time as well as its velocity v^i(x,t)=g^{0i}(x,t)/g^{00}(x,t). The ether stress tensor, which remains on the remaining g^{ij}(x,t) varies too.

But this compressible and dynamic ether lives in a classical, fixed Euclidean space.

If there is no gravitational field, we obtain as in GR g_{mn}(x,t) = eta_{mn} = const., as in SR, and the new ether becomes, in this limit, the classical Lorentz ether.

Thanks Ilja

another question

what about speed of light is it affected by gravity or remains constant?

can the ether move at speed of light in the absence of gravity? or in certain conditions

thanks

The speed of light, in terms of the absolute background coordinates, varies.

But we have no way to measure them. What we measure with rulers and clocks is distorted by the ether. This element is the similar to the Lorentz ether - there clocks and rulers are distorted too, and also in such a way that this prevents a measurement of the undistorted lengths and undistorted absolute time, except by accident if one is in rest relative to the ether. Here we only have some more formulas, about how clocks are distorted by a moving ether and how this depends on ether density and stress.

The absence of gravity is the limit of special relativity, thus, the Lorentz ether, and the Lorentz ether has zero speed and is homogeneous.

The speed of light is the speed of sound of the ether. Thus, usually the speed of the ether itself will be much smaller than the corresponding speed of sound. But at least in principle, these two things have no relation at all. Imagine air inside a supersonic jet. This air has, in the absolute coordinates of the Earth's surface, a velocity higher than the speed of sound. In principle, one can imagine that such things may be possible even without such a jet around, say, some extremal hurrican.

But if one is also part of the ether - and all matter fields are simply different ether properties - one will not even observe this superlight speed, similar to the pilot of the supersonic jet, who does not feel any wind, except if he uses an air conditioner.

Three fundamental constants including the speed of light in vacuum, the Planck constant and the gravitational constant, along with the dimensionless electroweak coupling constant are functions of the gravitational potential.

@Guoling, the situation looks a little bit different.

The speed of light as usually defined (as the result of measurement of this speed using clocks and short rulers) does not depend on the gravitational field. Only if you have in mind the speed of light in terms of some (for some reason preferred) coordinates, then there is a variable speed of light in dependence of the gravitational field.

The Planck constant as well as the electroweak coupling constant are part of theories which have not yet been unified with gravity, thus, there simply is no information at all in the established theories how they depend on gravity.

But, as far as the Einstein equivalence principle can be supposed to hold in such a unified theory, it would not be possible to measure a difference in dependence of the strength of the gravitational field.

Last but not least, in modern theories of gravity, there is no longer a single gravitational potential. Instead, there are ten functions g_mn(x,t) which describe the gravitational field, or, if you like, ten gravitational potentials.

@IIja, my cosmological model only has one gravitational potential defined by Newton's gravitational laws.

Some examples of 'computational general relativity'.

(1)Here we can find a very nice and detailed derivation of Schwarzschild's metric:

http://www.thephysicsforum.com/special-general-relativity/17-solving-einstein-field-equations-example.html

Look the details:

***"The spherical symmetry and the condition that mass and resulting field are static leads to the following simple ansatz : " [equation (5)]

This assumption is a big one!

***"In a vacuum (Tmn=0 ) the Einstein Field Equations (1) reduce to Rmn=0"

So, the existence of a mass, which according to GR leads to a spacetime curvature, has a vanishing effect!

Now we have a clear violation of the causal scheme:

Mass-Energy distribution --> Tmn # 0 --> space time curvature --> geodesic formation --> motion of a test particle.

So, the above derivation is fully incompatible with what the GR argues for...

I will continue another time...

Demetris, first, the Schwarzschild solution is of course only a very special solution. Based on very big assumptions, of course.

What is named "Schwarzschild solution" is the outer part of the full solution of some ideal spherically symmetric and static star. In this outer part, there is vacuum, thus, one has to use for this outer part the Einstein equation for the vacuum. For the complete solution, one needs also a solution for the inner parts of the star, and there also has to be a nice fit on the surface of the star.

Then, no space time curvature means no matter, but not reverse. Solutions for the vacuum do often have nonzero curvature. This holds for the Schwarzschild solution as well as for solutions with gravitational waves and so on.

Your "causal scheme" looks nonsensical.

Ilja, it is not my scheme, but that of GR: Gmn=k*Tmn, where Tmn represents all combined mass-energy et al distribution. If this is not true, then the main equations are nonsensical and not my explanation.

Demetris, I don't understand what you mean with "combined" here.

G_mn(x,t) = k T_mn(x,t). This holds separately for each point (x,t) of spacetime. So, if there are several stars flying around, and between them, at some event (x0,t0) there is vacuum, then at this event the equation is G_mn(x0,t0) = 0. Which is, locally, equivalent to R_mn(x0,t0)=0.

Now, of course, the global solution is a combination of local solutions in the different domains. The local solutions in the different parts have to fit each other for this, else they cannot be combined into a single global solution. But these are solutions which have to be combined.

The energy-momentum tensor describes energy and momentum of matter, as it is locally. If there is, only locally, vacuum, then, only locally, T_mn will be zero there.

Ilja and friends, when in Physics we write an equation y(x1,x2,..,xn)=f(x1,x2,...,xn) we have all agreed to mean that: An n-tuple cause of values (x1,x2,...,xn) leads to the determination of a new (dependent to those variables) quantity 'y'. So, in full deterministic theories like GR (do you remember? Einstein hated playing dice), when the cause is zero, then the result is also zero. Finally, I repeat it, for the next causal scheme:

Mass-Energy distribution --> Tmn # 0 --> space time curvature --> geodesic formation --> motion of a test particle, it is necessary to holds:

Tmn = 0 => Gmn = 0.

If this is not the case, then, sorry, but we have done so many exceptions from the rules in order to accept GR that I do not concern it as a typical physical deterministic theory. We can say that it is something like a 'fuzzy' theory (OK, Tmn=0 but we don't know what will happen next..), but this is a contradiction to the 100% Einsteinian determinism.

Sorry, Demetris, but this does not make sense. A full solution of the Einstein equations may be one with T_mn =/= 0 everywhere. Yes, you can then compute curvature, it will be nonzero everywhere too. And you can write down, for every solution of GR, an equation for geodesic motion of a test particle which does not interact with anything else, including all the matter at that place. But this equation for a mystical hypothetical test particle does not change the solution itself, it is an equation of motion for a test particle for a given metric, and not an equation for the metric. So, there is no relation between the geodesic equation for a test particle and T_mn=0.

Ilja, have you understood my objection?

I cannot accept as valid a theory which it is supposed to explain gravity better than any other theory, but:

1)when the cause of gravity is zero (Tmn=0) then the result (Gmn) it depends...

2)it cannot solve by itself the simplest available gravity problem.

I can accept it as a beautiful hypothesis, which tries to connect the mass-energy distribution with the curvature of space time around it.

But nothing else, just a beautiful hypothesis...

Demetris, I haven't understood your objection and don't understand it now.

1.) The situation in Maxwell theory is similar. If there are no charges, so the cause of the EM field is zero, the results nonetheless depends - there may be light and other EM waves or static fields causes by charges outside the given domain. The same for gravity. In a region without sources of gravity may be gravitational waves or nontrivial fields caused by sources outside the domain.

2.) People with sufficient computer power and a good numerical program for solution of the Einstein equations can solve the Einstein equations as accurate as their computer allows it.

Interesting parallelization Ilja, but we have to remind that Maxwell's equations describe the propagating EM wave, so we do not care about the origin: If it has been produced, then its propagation within a medium where {rho=0, J=0} will be guided by vacuum equations, see proper section here for example:

http://en.wikipedia.org/wiki/Maxwell's_equations

But, in GR, we have a different case: There, the 'spacetime' can do different things when the source of gravity is absent. This is something different than EM theory.

What's your opinion?

Demetris, in metaphysical descriptions this may sound somehow different, and from a mathematical point of view there are some differences too, but all this is quite unimportant.

For gravitational waves we can do the same thing as for EM waves: Not care about the origin where they have been produced (say, some double pulsar system far away) but consider their propagation through the vacuum, which will be guided by the vacuum equations.

The EM field in vacuum can also do "different things".