Essay Abstract

Towards the goal to quantize gravity, we discuss an intermediate step which consists in extending the picture of standard General Relativity by considering the potential presence of an intrinsic space-time curvature in our Universe, i.e. a curvature which is not due to the mass-energy. The potential existence of an intrinsic space-time curvature is admitted in the framework of Extended Theories of Gravity. In this tapestry, the equations to quantize are not the standard Einstein field equations of General Relativity, but the extended Einstein field equations which take into account the presence of the intrinsic space-time curvature. The traditional relation between mass-energy and space-time curvature, which founds standard General Relativity, results modified in this new picture and, at least at the linearized approximation, variations of this intrinsic space-time curvature generate the mass-energy. Various problems of the Dark Universe, like Dark Energy, Dark Matter and Pioneer anomaly, can be, in principle, solved through this approach, while a definitive endorsement for the effective existence of an intrinsic space-time curvature in our Universe could arrive from the realization of a consistent gravitational wave astronomy. We also discuss the quantization of both mass-energy and space-time curvature in the early Universe by using the process of amplification of vacuum fluctuations which is connected with the primordial production of relic gravitational waves. A future detection of such relic gravitational waves will be an ultimate endorsement for the Digital (Quantum) rather than Classical (Analog) feature of the gravitational interaction.

Author Bio

Christian Corda is a mathematical physicist who works in the research fields of gravitation, astrophysics and cosmology. He received a Ph.D. degree in physics from the Pisa University, Pisa, Italy. He is Editor in Chief and Editorial Board Member of various peer-reviewed international journals in the fields of physics and mathematics.

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Dear Dr. Corda.

Congratulations on the birth of baby David. Soon he will, like you, defy gravity and start walking..enjoy every day of parenthood!

I am unqualified to comment on your paper, but I have a request. Can you please give your impression of my earlier 2005 Beautiful Universe paper on which my present fqxi paper is based?

In my model GR would be greatly simplified in a digital universe, reduced to the 'optics' of refraction in a medium of variable density. A geodesic in such a density field will have the curvature due to acceleration with the velocity of the signal varying from universal lattice node to the next node according to its density or potential. Yes my model proposes a varying speed of light with a maximum of c with Lorentz transformations in an ether of nodes spinning in angular momentum of units (h). Apologies for this intrusion but as an expert on GR you can give valuable feedback. Good luck with your research. Avanti!

Thanking you and with kind regards

Vladimir

    It is good to you submitted a paper here again. I have not read yours yet, so I can't ocmment much now.

    I remember we were considering doing some gravity wave calculations with thermal sources. Yet that got a bit lost in the shuffle.

    Cheers LC

      Dear Dr. Christian Corda,

      as explained in my Essay the derivation of General relativity from the more fundamental canonical theory involves the approximation of the total energy-momentum-stress by the energy-momentum-stress for matter alone. If we relax this approximation and maintain the total energy-momentum-stress, then we do not derive the usual Hilbert & Einstein equations but a generalized equation as the equation 94 in the reference 18 in my Essay.

      That generalized equation is your equation 2, except by the notation (Tab(EXTRA) corresponds to Tab(c) in your equation 2) and because the physical interpretation of the new energy-momentum-stress tensor term is somehow different.

      In your "Conclusion Remarks", you speculate about how the new equation (2) could, in principle, solve the Dark Energy, Dark Matter and Pioneer anomalies.

      We already addressed the issue of Dark Energy in page 53 of the same reference 18. Since we have the explicit form for the new energy-momentum-stress tensor, we compute its value and obtained a cosmological constant of order 10-54 (in Standard Units), which is very close to the last observed value of 10-52. This was then interpreted as an important advance, specially since other approaches to the cosmological constant fail by huge errors. The so called biggest mistake ever in physics!

      We did a more accurate computation recently and obtained a value of order 10-52 in complete agreement with observation. This has still to be verified by other experts, but it seems that a natural solution to the Dark Energy problem is already at your hand.

      Regarding Dark Matter, we are working in this as well. As said in the same reference 18, first research seems to support the idea that the new theory of gravity also solves the Dark Matter problem. We can already explain observational data that cannot be explained neither by GR+DM, nor by other models as MOND, TeVeS, PCG... Moreover, we can combine both the cosmological and the astrophysical models and then solve some other mysteries, such as why the astrophysical constant a0 is numerically close to the Hubble aH. Our new theory predicts a0 = 1/8 aH.

      I know that some authors consider that the Pioneer anomaly must be related to Dark Matter, because the scale of the anomalous acceleration are very close. I have not studied the Pioneer anomaly enough and cannot say. However, there many other foundational issues that you cite are solved in our approach.

      Regards.

      Note:

      General readers that cannot access to the reference 18 can see the pages 9 and 10 of the associated Executive summary available online for free download.

      • [deleted]

      • Post #6

      Dear Dr. Corda,

      You have written a beautiful essay. After the first reading, I was certain that you had gotten your ideas from my essay, but I see by your references that you have a long history of similar innovative concepts. In particular, our essays are complementary on the concepts of intrinsic curvature of global spacetime, the importance of this to DM and DE, how mass-energy is generated by variations of the intrinsic curvature, and that "presence of an intrinsic space-time curvature can generate a model of oscillating Universe". My essay lacks the eloquence and sophistication of yours, as I lack your overall knowledge and experience, so I used a rudimentary conceptual/graphical approach. I have presented a specific General Relativistic mechanism as an alternative to the DM hypothesis and have offered a hypothesis for the formation of structure in which you might be interested. I would be honored if you would read my essay and leave a comment. Congratulations, for in my eyes, you have written a definite winner.

      Sincerely,

      Dan T. Benedict

        Dear Vladimir,

        thanks for your kind congratulations on the birth of baby David.

        I read the abstract of your paper on Beautiful Universe. I think that it could be interesting but you should send it to a peer-reviewed international journal in order to have a concrete scientific feedback.

        Cheers,

        Ch.

        Hi Lawrence,

        nice to meet you again.

        Regarding thermal gravitational waves, give a look to this paper by C. Sivaram and Kenath Arun: http://arxiv.org/abs/1005.3431.

        I recently accepted it for publication in TOAAJ.

        Cheers,

        Ch.

        Dear Christian

        An excellent essay. If we are to find an acceptable route for reality to permeate physics then this will certainly be a strong candidate.

        My own work has taken a more physical view but seems otherwise almost entirely equivalent. It is really a case of defining how this 'intrinsic' curvature is manifested in nature at the quantum level. Perhaps I have gone too far too soon, but have been seduced by the trail of 'magic bullets' followed.

        I'd be very appreciative of your views on my own essay, http://fqxi.org/community/forum/topic/803 and in particular of your opinion on the relevance of the extended field equations to the physical process described from the empirical evidence.

        Best wishes

        Peter

          Dear Dr. Juan R. González-Álvarez,

          thanks for your comments.

          My best wishes for further developing your approach in extending General Relativity.

          Best regards,

          Ch.

          Dear Dan,

          thanks for your kind words.

          I do not know if I have written a definite winner, this will be judged by the expert panel of judges which will be instructed and I have a total respect in FQXi's merit system.

          I have read your Essay and I find it interesting. In particular, I appreciate your relating the cosmological application of GR to the Mach's Principle through FPC and your intuitive-geometric vision of the Universe. Some suggestions: you should try to derive the model in a more rigorous way and send it to a peer-reviewed international journal in order to have a concrete scientific feedback. More, you should improve the references. For example, there are more rigorous reference than Wikipedia on the Friedmann-Lemaître-Robertson-Walker Model, examples are L. Landau and E. Lifsits, Classical Theory of Fields (3rd ed.), London: Pergamon (1971) and C. W. Misner , K. S. Thorne, J. A. Wheeler, "Gravitation", Feeman and Company (1973).

          You cold be interested also in two recent works of mine, where, together with my friend H. J. Mosquera Cuesta, we find ways to remove both of the BH's and Universe's Sigularities. These are Mod. Phys. Lett.A25, 2423-2429 (2010) and Astropart. Phys. 34, 7, 587 (2011).

          I wish you a lot of luck in the contest.

          Cheers,

          Ch.

          Dear Dr. Juan R. González-Álvarez,

          I would like to invite you to submit your next papers on your approach to extend General Relativity to The Open Astronomy Journal, see http://bentham.org/open/toaaj/index.htm, and/or to The Hadronic Journal, see http://www.hadronicpress.com/edit_board.htm.

          These are two international peer-reviewed journal in which I am Editor in Chief.

          Best regards,

          Ch.

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          • Post #14

          Dear Dr. Corda,

          Thanks, for your reply and I greatly appreciate your feedback. I'll be the first to admit that my model needs a more rigorous treatment and that some of my references were somewhat weak. I am presently working on just such a revision, among other things. My rough draft started with over 60000 characters, so it went through a major overhaul and the Wikapedia references were last minute additions, requested by FQXI, that I should have taken more seriously. This is my first attempt at writing a scientific paper. Having a support group would have been helpful, as I do want to be considered seriously. Do peer review journals accept papers from anyone? It was my assumption, that you need to be affiliated with a academic institution to obtain consideration.

          I look forward to reading your latest works, they sound intriguing. My greatest concern with General Relativity has always been the uncomfortable acceptance of singularities, assuming that QG will someday resolve the issue.

          Thanks again and best wishes,

          Dan

          Dear Dan,

          actually, it is not needed to be affiliated with ab academic institution to obtain consideration.

          Even if such an affiliation is, of course, a vantage, what is really important is to write good research papers.

          I would like to invite you to submit your technical papers on your approach to Cosmology to The Open Astronomy Journal, see http://bentham.org/open/toaaj/index.htm, and/or to The Hadronic Journal, see http://www.hadronicpress.com/edit_board.htm.

          These are two international peer-reviewed journal of which I am Editor in Chief.

          Cheers,

          Ch.

          Dear Christian,

          Your essay is very interesting and really innovative. In general I agree with the outcome: from analog to digital. In your essay I have found more concepts that seem to support my own (the details in my essay ). For example you claim: "Dark Energy and Dark Matter have to be considered like pure effects of the presence of an intrinsic space-time curvature in the Universe. Considering this point of view, one can think that gravity is different at various scales because of the existence of the intrinsic space-time curvature, which changes at different scales, and there is room for alternative theories." I am happy - for the first time I have read such innovative view in a public paper. And it is fully compatible with my own view. There are also strong differences. For example I try to prove that gravitational waves do not exist. Moreover in my concept the gravity is an emergent and not fundamental interaction. But I have started out from different assumptions albeit based on geometry as well.

          I think I shall read your essey once more to fully understand it.

          I look forward to reading your publications and finding out a development of your concepts.

          Good luck Christian!

          Jacek

          http://fqxi.org/community/forum/topic/883

          • [deleted]

          • Post #17

          Dear Dr. Corda,

          Thanks for you invitation and advice. It may take some time to get my paper up to professional standards, but I look forward to the challenge. I'm currently working on another essay for a different contest, but the two ideas are complementary so I may have enough material for two papers. I'll work on getting the first one acceptable before I proceed.

          You have been most helpful,

          Dan

          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.