Dear Peter,

thanks for your kindness.

It is a big problem defining how this 'intrinsic' curvature is manifested in nature at the quantum level. I realized such a quantization only within the linearized (i.e. at first order) theory in my papers Eur. Phys. J. C 65 1-2, 257 (2010), Astropart. Phys. 30, 209 (2008), Mod. Phys. Lett. A 22, 35, 2647 (2007), Gen. Rel. Grav. 42, 1323 (2010), AIP Conf. Proc. 966, 257 (2008), and Mod. Phys. Lett. A 22, 15, 1097 (2007). I reported the main results in page 9 of my Essay. A better definition implies the quantization of the extended field equations (2) of my Essay, which, based on the strong non linear character of these equations, is a goal very very difficult to realize .

I have read your Essay, it is interesting.

The relevance of the extended field equations is important to the physical process described from the empirical evidence which happen on scales larger than the Solar System scales. Examples are Dark Matter and Dark Energy. In fact, General Relativity is very very well tested within the Solar System. Thus, variations from General Relativity have to be very weak in order to be consistent with the Solar System tests. In my geometric approach this means that the spacial hypersurfaces of the intrinsic curve space-time should have a curvature which manifests only on scales larger than the Solar System scales.

Cheers,

Ch.

Dear Jacek,

thanks for your kind words.

I read your Essay and I agree with you that there are strong differences with respect to mine. In particular, your claiming that gravitational waves do not exist in the way they are conventionally defined is very strong. Actually, gravitational waves are solutions of both the linearized and full Einstein Field Equations in both of Standard General Relativity and Extended Theories of Gravity.

On the other hand, you also claim, verbatim, that "the spacetime has elastic properties". But in that case I think that heavy masses with acceleration should radiate gravitational waves based on such elastic properties. What do you think on this issue?

In any case, I wish you a lot of luck in the contest.

Cheers,

Ch.

    Dear Dan,

    be free to call me merely Christian without any suffix.

    Take your time for getting your papers up to professional standards, we are not in a hurry. What other different contest are you attending?

    Cheers,

    Ch.

    Dear Dr. Corda,

    Welcome to the essay contest. Since Dr. Elliot McGucken and you are my neighbors in the essays list, therefore I must visit these pages first.

    All attempts to quantize gravity have utterly failed, and your attempt to quantize gravity will have the same fate; There are a lot of different Extended Theories of Gravity, including yours, which tries to enlarge the Einstein scheme through an addition of corrective terms, but all these theories are not able to explain inertia, mass and the curvature of spacetime. The complete theory of gravitational interaction must be able to explain inertia, mass, and the curvature of spacetime in the same model. Also the true theory of gravitation must be able to unify gravity with the other forces. Since your variant of Extended Theory of Gravity is not able to do it, then your approach also is wrong and even the quantization of the extended Einstein field equations can not help you.

    I agree with you that the intrinsic space-time curvature may be important for Gravity, but not ''by adding an intrinsic space-time curvature to the model''. Instead you must examine how an intrinsic space-time curvature appears and how it works. In other words, you must look for physical solutions rather than for mathematical.

    Also, your essay do not show, if you are in the digital or analog party. In general, I don't found any proofs in your essay that the reality is digital, analog, or digital-analog.

    Sincerely

    Constantin

    Dear Mr. Leshan,

    actually, I do not attempt to quantize gravity. I limit myself to discuss intermediate steps like the extension of General Relativity and/or the quantization of Extended Theories in the weak field approximation.

    On the other hand, I agree with you that the complete theory of gravitational interaction must be able to explain inertia, mass, and the curvature of space-time in the same model. I also agree with you that the true theory of gravitation must be able to unify gravity with the other forces. But, surely, I am not so arrogant and presumptuous to claim that I can find, alone, the complete theory of gravitational interaction. If Albert Einstein and others world's greatest scientists did not find this theory in about 100 years of scientific research, I am absolutely sure that I will not find it being alone!!

    I limit myself to try to give small contributions towards such a goal by following a way that could be the correct one.

    On the other hand, I am not so arrogant and presumptuous to claim that the reality is digital, analog, or digital-analog. Even in this case, I recall you that Albert Einstein and others world's greatest scientists did not find the correct answer to this issue in about 100 years of scientific research. Thus, I prefer limit myself to suggest a potential detectable signal, i.e. the relic gravitational waves, which could clarify the digital rather than analog feature of the gravitational interaction.

    This is absolutely sufficient for me.

    I wish you a lot of luck in the contest.

    Best regards,

    Ch.

    I clarify my reply to Mr. Leshan on the question "Is Reality Digital or Analog?".

    In my opinion, at the present time and at the present status of our scientific knowledge, nobody can claim, with an absolute certainty, that Reality is surely Digital, or surely Analog or surely Digital-Analog. Of course, people have various opinions on this fundamental issue and I respect all the various opinions. I have an opinion too, but I think this is NOT really important. What I think to be really important, at the present status of our scientific knowledge, is the way in which one attempts to arrive to a potential answer to the question, not the question itself.

    I interpreted this beautiful Essay Contest in this spirit and with this spirit I wrote my Essay.

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    • Post #24

    Dear Christian,

    I read that a non-academic received an honorable mention in the 2009 Gravity Research Foundation Essay Contest and that has encouraged me to submit an entry on black holes. The entry deadline is April 1, which is quickly approaching. I believe that I have seen your name listed there, at least once, so undoubtably you're familiar with them.

    Sincerely,

    Dan

    Dear Cristian,

    You write

    ---"As distinct from other field theories, like the Electromagnetic Theory, General Relativity is very difficult to quantize. This fact rules out the possibility of treating gravitation like other quantum theories and precludes the unification of gravity with other interactions. At the present time, it is not possible to realize a consistent Quantum Gravity Theory which leads to the unification of gravitation with the other forces."---

    If a universe is to create itself without any outside intervention, then its particles must create themselves, each other. If (the properties of) particles then are as much the product as the source of their interactions, their energy exchange, then so is the force between them. This means that a force, in principle, cannot be either attractive or repulsive. That is, particles can only exist, have properties (attract/repulse) if they have some kind of backbone so they can, within limits, absorb energy in an increase of their kinetic energy rather than in a change of identity. That said, if the rest energy of particles ultimately is as much the effect as the cause of their energy exchange, their interactions, of a continuing evolution, then interaction energies never can become infinite at infinitesimal distances, so there's no need for string theory. A universe which finds a way to create itself without any outside intervention can hardly stop doing so: gravity, the contraction of masses and the related expansion of spacetime between the mass concentrations they form, is the expression of this continuing creation process, and hence differs fundamentally from the 'other' forces.

    Since according to our present, simplistic ideas particles only are the source of their fields, they either attract or repulse, so the strength of the force between them solely depends on their distance. This belief led to the question how protons can fit in atomic nuclei despite their huge electric repulsion, which is said to be 10^38 times stronger than gravity between them. However, a force never can exceed the counter force it is able to evoke, that is, than the opposition the particles offer to that force: than their inertia. So if we may interpret Einstein's equivalence principle to say that every influence which brings the inertia of a particle to expression as a counter force can be called 'gravity', then there's only gravity. As attractive as it is repulsive, it is much stronger than the weak gravity pulling at Newton's apple, as gravity which is powered by the continuous creation process inherent to a self-creating universe. For details, see my essay the UPDATE 1 post about the strong nuclear force.

    Regards, Anton

    Dear Anton,

    thanks for your comments.

    There are two points of view concerning gravitation: the Einstein's geometric point of view and the Feynman's particle point of view. The Quantum Gravity Theory, if it will be ultimately find, should be the definitive synthesis of them.

    In any case, Einstein's Equivalence Principle is purely geometric as it implies that test masses (particles) have to follow the space-time curvature during their motion. In this geometric vision only mass-energy and curvature are strictly needed. In my attempt to extend General Relativity mass-energy should be produced by variations of an intrinsic space-time curvature. Can this be conciliated with your vision that gravity has to be powered by the continuous creation process inherent to a self-creating universe?

    Best regards,

    Ch.

      Dear Christian,

      You say: "gravitational waves are solutions of both the linearized and full Einstein Field Equations".

      "Looking at the Einstein equations as a set of second-order partial differential equations it is not easy to predict that there exist solutions behaving as waves. Indeed, and as it will become more apparent in this Section, the concept of gravitational waves as solutions of Einstein equations is valid only under some rather idealized assumptions such as: a vacuum and asymptotically flat spacetime and a linearized regime for the gravitational fields. If these assumptions are removed, the definition of gravitational waves becomes much more difficult. In these cases, in fact, the full nonlinearity of the Einstein equations complicates the treatment considerably and solutions can be found only numerically. It should be noted, however, that in this respect gravitational waves are not peculiar. Any wave-like phenomenon, in fact, can be described in terms of exact ``wave equations'' only under very simplified assumptions such as those requiring an uniform ``background'' for the fields propagating as waves." [link:people.sissa.it/~rezzolla/lnotes/virgo/node9.html]

      You also say: "heavy masses with acceleration should radiate gravitational waves based on such elastic properties"

      I assume that "the matter" is a wave itself. In my view e.g. the electron is a real wave (a local spacetime contraction region moving in a wave form). I know it was also the initial Schrödinger's point of view that the electron is a wave but dismissed after the double slit experiment. That time he could not defend his thesis. According to my concept the matter is only a deformed spacetime so there is nothing to explain about the double slit experiment as the electron is a wave and not only has wave properties. And every "massive" object e.g. the Earth is a gravitational wave itself. And the wave is not traveling outward from the source. There is no source - the Earth is a gravitational wave orbiting the Sun along the geodesic. I know my concept is highly speculative but maybe worth to develop because it delivers a chance to create a GUT.

      Best regards,

      Jacek

      I am sorry but I did not read the help on linking sites so I give the the [link: http://people.sissa.it/~rezzolla/lnotes/virgo/node9.html] once more. Sorry, the rest of the text is linked to the site.

      Dear Dan,

      yes, I confirm you that, exactly like the FQXi, the Gravity Research Foundation has a remarkable merit system. What is really important for them is the quality of the research. When I received an Honorable Mention in the 2009 Competition I had no academic affiliation too.

      I well know that the entry deadline is April 1, I am going to attend this year too. Together with some collegues, we are finishing an Essay on black holes exactly like you.

      Best wishes for this Competition too.

      Cheers,

      Ch.

      Dear Jacek,

      the sentence that you quoted by the paper of Rezzolla reads:

      "... it is not easy to predict that there exist solutions behaving as waves."

      which is different from:

      "... it is impossible to predict that there exist solutions behaving as waves."

      In fact, such solutions of exact equations have been found, see for example

      H. Bondi, F.A.E. Pirani, and I. Robinson, Proc. R. Soc. London A 251, 519 (1959);

      H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact solutions of Einstein's field equations, 2nd ed. (Cambridge Univ. Press, Cambridge, 2003) Secs. 24 and 31;

      J. Ehlers and W. Kundt, Exact solutions of the gravitational field equations, in: "Gravitation: an introduction to current research", ed. by L. Witten (Wiley, New York, 1962);

      Y. N. Obukhov, J.G. Pereira and G. F. Rubilar, Class. Quant. Grav. 26, 215014 (2009).

      On the other hand, it is true that

      "the concept of gravitational waves as solutions of (linearized) Einstein equations is valid only under some rather idealized assumptions such as: a vacuum and asymptotically flat spacetime and a linearized regime for the gravitational fields."

      But, even if idealized, such assumptions can be realized with an excellent approximation. Take two neighbouring free-falling test masses within a vacuum cylinder in the Earth's gravitational field. For Einstein's equivalence principle they realize a vacuum and locally flat spacetime. This is EXACTLY the principle which is used for detect gravitational waves with interferometers. In fact, test masses suspended to pendulums represent an excellent approximation of free-falling test masses.

      More, the linearized regime is an excellent approximation even for the gravitational field of the Solar System (historically Einstein used the linearized regime to test astrophysical predictions like the light deflection, the gravitational redshift and the precession of the Mercury's perihelion). The gravitational field of a gravitational wave is various order of magnitude lower than the gravitational field of the Solar System, thus, the linearized regime is an absolutely excellent approximation. In fact, high order terms are lower than 10^-42...

      Best regards,

      Ch.

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      • Post #33

      Hello dear Dr Corda, Happy to see you again,good luck ,it's a beautiful essay,congratulations

      Steve

      Hi Steve,

      nice to see you again too.

      Thanks for congratulations, I see that you do not attend to the Essay Contest this year.

      Cheers,

      Ch.

      Dear Christian,

      I have not said it is impossible to predict that there exist solutions behaving as waves. If we try very hard... And it depends on assumptions. I have just assumed another scenario I will try to find another solutions.

      I have only read the paper by Y. N. Obukhov, J.G. Pereira and G. F. Rubilar and found it interesting. They claim:

      1. any exact gravitational wave solution is highly idealized, and is thus probably more of academic interest

      2. As we have seen, the exact plane-wave solution of Einstein's equation transports neither energy nor momentum. This property appears to be quite unusual, since one could expect that a gravitational wave, like any other gravitational field configuration, should be characterized by nontrivial distribution of energy and momentum. In this sense the exact plane wave solution, although being mathematically well defined, appears to be a physical puzzle.

      As you have maybe noticed in my concept e.g. the Earth is a gravitational wave orbiting the Sun along the geodesics. So it transports the energy and momentum and is not a physical puzzle. That is why I think it is worth further research. Maybe the idea is wrong but very simple and beautiful.

      As far it has not been successful to detect gravitational waves with interferometers. I cannot be sure it is impossible. That is only a consequence of my concept.

      Best regards,

      Jacek

      • [deleted]

      • Post #36

      Hello,Christian

      What is your opinion about alternative to intrinsic space-time curvature -intrinsic space curvature?

      http://www.fqxi.org/community/forum/topic/946

      Hi Yuri,

      I see in your Essay that the first cited example of physical evidences supporting the Ratio 3:1 is the ratio 3 space-like dimensions - one time-like dimension. It is an interesting Essay, I wish you good luck.

      Concerning your question, I think that the space-time should be globally and intrinsically curve.

      Cheers,

      Ch.