Spacetime as a bit more exotic manifold: Introduction of a particular semi-regular semi-Riemannian manifold Emmanuel Kanambaye To cite this version: Emmanuel Kanambaye. Spacetime as a bit more exotic manifold: Introduction of a particular semi- regular semi-Riemannian manifold. 2021. hal-03240148 HAL Id: hal-03240148 https://hal.archives-ouvertes.fr/hal-03240148 Preprint submitted on 8 Aug 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Spacetime as a bit more exotic manifold: Introduction of a particular semi-regular semi-Riemannian manifold Emmanuel Kanambaye Independent Researcher Email: [email protected] Abstract: Spacetime is a fundamental concept in physics. Einstein‘s general relativity teaches us that spacetime is as an elastic fabric (manifold) what can be infinitely curved or stretched. In this paper, I show that if one makes the hypothesis that spacetime is a particular kind of conformally non- degenerate semi-regular semi-Riemannian manifold; more precisely, if one makes the hypotheses that spacetime is a kind of semi-regular semi-Riemannian manifold: - that can only be curved (deformed) by an energy superior or equal to some minimum energy; - equipped with a metric that degenerates (becomes zero) at any of its point where curvature (deformation) becomes greater than some threshold; then, not only, one gets a mathematically consistent and well-defined spacetime but especially, most of the currents fundamental problems of gravitation (like black hole and initial gravitational singularities problems, cosmological constant problem; black hole entropy interpretation problem etc…) could be elegantly resolved. Keywords: spacetime as exotic manifold; spacetime capable of tearing; spacetime without gravitational singularities; spacetime capable of resolving cosmological constant problem. PACS: 04.50.Kd; 02.40.Ky. 1. Introduction semi-Riemannian manifold Einstein‘s theory of general relativity [1] despite some fundamental problems among which black equipped with the metric g [13] holes gravitational singularity [2] and initial big satisfying the condition: bang singularity [3] is our current best theory of gravitation. gE for 0 Indeed, almost all of its predictions (gravitational time dilation [4], gravitational lensing [5], gE for 0 gravitational time delay [6], black hole [7], gravitational wave [8] etc…) are verified with very high precisions [9, 10, 11, 12]. where denotes the Minkowski General relativity [1] teaches us that spacetime is metric tensor [14] and E the energy a kind of elastic fabric (manifold) capable to occupying the manifold . undergo curvature (deformation). The theory works so well that one forgets B- The spacetime fabric (the manifold) has sometimes that it makes implicitly some non- no yield strength [15] i.e. the curvature at fundamentally necessary hypotheses on the any given point of the spacetime fabric spacetime fabric. can be as big as wanted without that there be neither tearing nor plastic Indeed, for example Einstein‘s general relativity deformation [16]. makes implicitly these following two hypotheses: More precisely, the manifold is assumed to not breakdown at any finite A- There is no minimum required energy to curve (deform) spacetime fabric i.e. any Kretschmann scalar K1 [17]. energy as small as it is, curves (deforms) spacetime, even though the smaller the As we will see; these implicit Hypotheses A and energy, the smaller will be the curvature. B although naturally admitted, are not More precisely, Einstein‘s general fundamentally necessary; indeed one can show relativity assumes that spacetime is a that amending or relaxing them not destroys the consistency of spacetime, better this can lead to spacetime (manifold) capable of overcoming Let‘s notice for example that one could pose: most of the currents fundamental problems of Einstein‘s general relativity. (3) More precisely we will see that making the gf hypothesis that our physical spacetime is a kind of conformally non-degenerate semi-regular where: semi-Riemannian manifold [18, 19] could permit to elegantly resolve most of the currents E g fundamental problems of gravitation like black 2 4 holes and initial gravitational singularities ln x R x e E E problems [2, 3]; the cosmological constant problem [20]; the black hole entropy f (4) interpretation problem [21] etc... g 2 4 Throughout the paper; I will consider the ln x R x e E E signature convention (+ - - -) for the metric. In the same way, Einstein‘s summation convention will be assumed. In which ln , denotes the natural logarithm; 2. First example of exotic manifold , e the natural exponential; the metric As already mentioned; spacetime is according to tensor of Minkowski manifold [14]; the Einstein‘s general relativity a semi-Riemannian g manifold equipped with the non-degenerate expected non-Minkowskian metric tensor; Rx the usual ramp function [22] i.e.: metric tensor g satisfying the condition E gE for 0 xxEE if 0 E R x , with x 1 (5) (1) EE 0 if xE 0 Emin gE for 0 and: where denotes the Minkowski metric tensor and the energy occupying the manifold 1 for Rx 0 xEE R x E E (6) x Yet by slightly relaxing this condition (1); one can E 2 for R xEE x construct some interesting ―exotics‖ manifold (spacetime). To be convinced; one can proceed as follows: Indeed, in this case; one can verify that: Let be a semi-Riemannian four-dimensional f for E Emin manifold equipped with the non-degenerated (7) E metric tensor g . f for E Emin . From there, if is a manifold that can only be curved (made non-flat) by energies superiors or whence a manifold equipped with the metric equals to some minimum amount of energy tensor gf is a manifold that can be E ; then g will not satisfy the condition (1); min only curved (made non-flat) by energies instead, it will satisfy: superiors or equals to some minimum amount of energy E . min g for E Emin (2) it is interesting to note that in the same way; one g for E Emin can assume a manifold equipped with the non-degenerated metric tensor g satisfying for where denotes the metric tensor of the flat example: Minkowski manifold and E the energy occupying the manifold . dS then one could conceive a spacetime capable of g for E Emin overcoming most of the currents fundamental (8) g dS for E E problems [2, 3, 20, 21 etc…] of gravitation. min Let‘s note that with regard to g ; one could for dS where denotes the metric tensor of de Sitter example pose: manifold and the energy occupying . g f f g f (10) Says otherwise, one can assume a de Sitter manifold [23] that can be curved (deformed) only where f is the function given by (4); g the by an energy EE min . metric tensor given by (3) and f a function As on can verify; the interest of such manifolds writing: and is that they permit to conceive ―exotics‖ spacetime capable of naturally K g forbidding for example black holes of energies R x 2 K 4 inferiors to E . ln R x e min K Says otherwise, if our physical spacetime was a f (11) manifold of the kind or ; then black holes g R x 2 K 4 of energies inferiors to E could not exist in the ln R x e min K nature. In the section that follows; I am going to propose an even more interesting manifold. with: xx if 0 3. Second example of exotic manifold KK K1 R xKK , with x 1 (12) 0 if x 0 The ―exotic‖ manifold that I am going to propose K KX here permits to conceive a bit more interesting spacetime. and: Indeed: Let be a conformally non-degenerate semi- 1 for R x x 2xK KK regular semi-Riemannian manifold equipped with K (13) x R x KK 2 for Rx K 0 the metric tensor g [18, 19]. From there, if is such that: Indeed, in this case; one can verify that: 1- It cannot be curved (deformed, made non-flat) by any energy inferior to some f f for E Emin minimal amount Emin ; 2- Its metric degenerates (becomes zero) f f for E Emin (14) at any of its point where the curvature given by Kretschmann scalar [17] f f 0 for K1X K becomes superior to some constant KX ; Let‘s note that as for the previous example; one can get here also a viable manifold by replacing i.e. if the metric tensor g associated to with dS . satisfies: This precision made; one can now discuss of the K1 g for E Emin advantages offers by the manifold . g for E Emin (9) To achieve this; let‘s begin by writing the square g= 0 for KE K 1X of line-element associated to : 2 This little precision made; let‘s now discuss of the ds g dx dx g f dx dx (15) advantages offered by such ―non-usual‖ spacetime. From (9) one infers that: 4. Some important predictions of 2 manifold (spacetime) of metric (15) ds dx dx for E Emin 2 In this section, I am going to show that if our ds dx dx for E Emin (16) physical spacetime is a kind of conformally non- ds2 0 for K K degenerate semi-regular semi-Riemannian 1X manifold equipped with a metric of the kind (15); then most of the fundamental problems of As we are going to see; if our physical spacetime gravitation will be elegantly resolved.