Spacetime as a bit more exotic manifold: Introduction of a particular semi-regular semi-Riemannian manifold Emmanuel Kanambaye
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Emmanuel Kanambaye. Spacetime as a bit more exotic manifold: Introduction of a particular semi- regular semi-Riemannian manifold. 2021. hal-03240148
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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 E 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) 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 E 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 K [17] 1 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 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 K 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. is a conformally non-degenerate semi-regular semi-Riemannian manifold of metric (15); then To succeed in this; I assume the Einstein‘s field most of the currents fundamental problems [2, 3, equation [1]; the sole change that I assume is 20, 21 etc…] of gravitation could be elegantly that spacetime is a manifold of metric (15). resolved. a) Resolution of black holes Indeed, such a manifold characterizes a gravitational singularities problem spacetime of which the fabric is in some sense capable of tearing. A conformally non-degenerate singular semi- This comes from to the fact that (16) says that at Riemannian manifold equipped with a metric of any points or regions (of the manifold) where the kind (15) gives a spacetime capable of overcoming the famous black holes gravitational Kretschmann scalar K1 becomes superior to singularities problem [2]. KX ; the metric degenerates and gives: To see how; let‘s begin by considering for 2 2 2 simplicity, the standard Schwarzschild metric [24] 2 00c dt dx i.e. the metric describing non-charged and non- ds 0 (17) 00 dy22 dz rotating spherical mass M in standard general relativity:
Now as we know that this kind of null-metric (17) 1 not permits defining the usual concepts of length, rr 11SSc2 dt 2 dr 2 duration etc…; one can conclude that if our 2 physical spacetime is a conformally non- ds rr (18) degenerate semi-regular semi-Riemannian r2 d 2sin 2 d 2 manifold equipped with a metric of the kind (15); then it will cease to be defined (i.e. it will tear, breakdown etc…) at any of its points/regions 2 where rS 2 GM c denotes the where Kretschmann scalar becomes superior Schwarzschild radius and ct,,, r the to K . X spacetime coordinates. Says otherwise, a spacetime characterized by a manifold equipped with a metric of the kind (15), By basing on this metric (18); one can calculate is a spacetime that in some sense cannot support that if spacetime is a manifold of metric (15); then (undergo) a Kretschmann curvature superior the gravitational field of a non-charged and non- rotating spherical mass will be given by (see to the threshold i.e. a spacetime that KX Appendix1): breakdown when one applies on it a 1 Kretschmann curvature superior to KX . trrrSS 2 2 2 ftr11 c dt f dr ds 2 rr Indeed, as the Kretschmann scalar is a 2 2 2 2 2 quadric invariant scalar independent on r f d r fsin d coordinate system; it follows that a breakdown or tearing due to it cannot be absorbed (eliminated) (19) by a change of coordinate system. where (see Appendix1):
rS because there is below rX no consistent t r 1 for KK1X S (20) spacetime towards which the mass of the ft 1 r r black hole could collapse further or fall lower. 0 for KK1X Metaphorically speaking; all happens as if the 1 black hole is a deep well of which the bottom is 1 r r 1S for KK localized at such that one cannot fall lower in r S 1X (21) fr 1r r this well since having reached its bottom. 0 for KK1X Says otherwise, in the framework of (19); both the mass of the black hole as well as any matter- 2 energy that will fall into the black hole, will occupy 2 r for K1X K rf (22) the Region2 since there will below this Region2 0 for KK1X be no spacetime (in the usual sense of the term) towards which collapse further or fall lower. 22 22 rsin for K1X K Let‘s emphasize that as strange as this may be rf sin (23) 0 for KK as prediction; everything here is mathematically 1X consistent and well-defined. This precision made; one can also notice that the fact that matter within the black hole occupies the From there, remembering that the Kretschmann non-zero Region2 and thus has finite density scalar of a Schwarzschild black hole of mass M could help understanding the origin of black hole is worth [17]: entropy [21] since in this case the black hole entropy could be naturally interpreted as the 48GM22 measure of the number of microscopic-states of K1 6 (24) the matter-energy inside the black hole i.e. r occupying the Region2. one calculates that, contrary to the usual Likewise, as both density and curvature is finite Schwarzschild metric (18) that separates within this Region2; then the behavior of matter spacetime into two distinct regions: occupying this region should be perfectly - A Region1 for which rr and described by currently known physical laws S (Einstein‘s field equation [1], and quantum - A Region2 for which rr S containing a mechanics [25]); whence one can conclude that the metric equation (19) describes a black hole gravitational singularity at r 0; free from infinite-density gravitational singularity the metric (19) describes three regions: [2].
- A Region1 for which rr S ; b) Exclusion of black holes of energy - A Region2 for which 16 inferior to some minimal value 48GM22 rrS as well as Another interesting consequence of (19) and thus KX (15) is that it excludes Schwarzschild black hole - an exotic Region3 characterized by of mass inferior to a minimum 16 14 48GM22 c3 3 r and where the M min . KX GK4 X metric degenerates. Indeed, as for a Schwarzschild black hole of Now as this Region3 is described by the null mass M; the Kretschmann scalar at the horizon metric (17); then it is a region where spacetime is 2 not defined i.e. a region where the fabric of rS 2 GM c is worth: spacetime is in some sense destroyed/torn.
3 c12 More precisely, the metric (19) predicts that Kr (25) inside the black hole; spacetime (in the usual 1 S 4 GM44 sense of the term) cannot be prolonged below 16 48GM22 it follows that, the soles masses M that (19) the characteristic radius rX allows of collapsing within their Schwarzschild KX 2 22 radius rS 2 GM c are those satisfying the d a43 G p c t (31) 22ft condition: adt33 c 14 c3 3 where (see Appendix2): M (26) GK4 X 1, for KK t 1X ft (32) whence Schwarzschild black hole of mass inferior 0, for K1 KX to this last (26) is excluded by (19).
Now as one knows that the Kretschmann scalar Let‘s note here; that it is suitable to pose: K1 of a homogeneous and isotropic expanding 14 5 (or contracting) universe increases with the 2 c 3 density of that universe; then it results that there Emin c M min (27) GK4 X will be some finite-density X beyond which the equations (30) and (31) will reduce to: where Emin is the previously mentioned minimal required energy to curve (deform) spacetime da 0 (33) fabric. adt
c) Resolution of the initial big bang 2 gravitational singularity problem da 0 (34) adt 2 As for black holes case; one can show that the metric equation (19) permits also overcoming the These lasts describing: initial gravitational singularity [3].
- either infinite, eternal, flat, empty, To be convinced; let‘s begin by writing the two homogeneous and isotropic universe; standard Friedmann equations [26]:
- or the final state of a flat (usual matter 2 da kc228 G c containing) infinite universe expanding 2 (28) from eternity etc…; adt a 3 one infers that, if our spacetime is a manifold of d22 a43 G p c metric (15); then the current observed expanding- universe should be seen as having emerged from 22 (29) adt33 c an infinite-eternal-empty universe (perhaps trough quantum fluctuations [27]); in all cases, In which, a denotes the scale factor; t the time the initial big bang singularity [3] will be erased. coordinate; k the curvature parameter; G the d) Existence of minimal length-scale in Newton‘s gravitational constant; the spacetime cosmological constant; c the speed of light in vacuum; the mass density and p the Another interesting result that can be obtained in pressure. the framework of a spacetime characterized by the metric equation (15), is that if quantum By basing on these lasts (28) and (29); one can mechanics [25] is taken into account; then there calculate that if spacetime is a manifold of metric will be some fundamental length-scale below (15); then the equations describing which spacetime will no longer be defined. homogeneous and isotropic universe in Says otherwise there occurs some minimal expansion (or contraction) will be (see absolute length-scale. Appendix2): Indeed, due to Heisenberg uncertainty principle [28, 29]: 2 22 datt kc8 G c (30) fftt2 adt a 3 (35) xp 2 quantum mechanics ensures that any region of EEX min (39) radius L will contain at least an energy
E c2. L since in this case, the quantum vacuum energy Basing on this; one can calculate by using the [27] will not gravitate. equation (24) that at any region of radius the energy E will give a Kretschmann scalar It is interesting to notice that this last condition (39) can be naturally achieved. worthing: KL1 Indeed by using the fundamental constants , c and G; one can naturally assume: 2 2 12G KL1 28 (36) 1 2 cL c5 EEmin P (40) Now as (16) predicts that the spacetime fabric G ―breakdown/tears‖ at scale where KLK ; 1X which through (27) will imply: then one infers that in the conjoint frameworks of quantum mechanics and spacetime 2 4 characterized by (15); the very concept of 3c5 3 1 spacetime could not be defined below a KX (41) 44Gt characteristic length-scale L given by: P X which in its turn will through (37) give: 14 18 2G 3 L (37) 12 X 2 G c KX LlX 3 2 P (42) c This comes from to the fact that the Kretschmann scalar K L due to the quantum vacuum With this last; the equation (38) will reduce to: 1 12 energy will at any length-scale L LX be 5 1 c EP always superior to K such that the spacetime EX (43) X 2 2G 22 fabric will be always ―torn‖ at length-scales
making impossible to define the L LX , As one can verify; (43) and (40) perfectly satisfy the condition (39) eliminating the cosmological concept of spacetime below the length-scale LX . constant problem [20].
e) Resolution of the cosmological Finally at this stage, one can argue that there is a constant problem simple and reasonable way of elegantly resolving
most of the fundamental problems of gravitation In addition of the other seen interesting results; a [2, 3, 20, 21 etc...]. spacetime characterized by the metric equation The sole price to pay is of accepting that our (15) has all the keys of naturally resolving the physical spacetime is a kind of conformally non- cosmological constant problem [20]. degenerate semi-regular semi-Riemannian
Indeed, as on the one hand, the length-scale manifold of metric (15). ensures that the maximal quantum vacuum 5. Conclusion: energy EX that one can get is worth: As the results obtained in this paper show; if the 3 5 14 18 spacetime within which we live is a conformally ccKX non-degenerate semi-regular semi-Riemannian EX (38) 23L 32G manifold equipped with a metric of the kind (15); X then most of the currents fundamental problems in gravitation could be elegantly resolved. and that on the other hand (15) ensures that any Indeed, as we have seen, such a manifold permits to conceive a spacetime that naturally: energy inferior to Emin not gravitates; then it follows that there will be no cosmological - Permits to avoid black holes and initial constant problem if simply: gravitational singularities,
- Permits to avoid black hole of arbitrarily r small energy, S gtt 1 - Permits to explain the origin of the r entropy of black hole, 1 - Permits to resolve the cosmological rS constant problem, etc… grr 1 (1e) r The simplicity of the hypothesis (spacetime is a gr 2 manifold of metric (15)) as well as the elegancy with which it resolves the most fundamental gr 22sin problems of gravitation make that this hypothesis merits to be considered. In a next paper; I will discuss of how whence thanks to (1a) and (1d) one gets the experimentally test the hypothesis. equations (19), (20), (21), (22) and (23) of the paper. Appendix1: Appendix2: Let‘s recall the metric equation (15) of the paper: Remembering the metric equation (15) of the 2 paper; one rewrites: ds g dx dx g f dx dx (1a) 2 ds g dx dx g f dx dx Remembering that the function f writes: g dx dx (2a) K g where R x 2 K 4 ln R xK e dx dx f dx dx f (1b) g R x This means that if is the metric tensor of 2 K ln R x e4 K FLRW geometry [30]; then the Friedmann equations corresponding to (2a) will be:
2 22 on infers thanks to the equations (12) and (13) of da kc8 G c the paper that: 2 (2b) adt a 3
f1 for K K 22 1X d a43 G p c (1c) 22 (2c) f0 g for K K adt33 c 1X
where: which in its turn gives:
22t dt dt ft (2d) g f g for K1X K (1d) From there; replacing in (2b) and (2c) the term g f 0 for K1X K 2 2 t dt by its value dt ft will lead to:
From there; if g is the metric tensor of 2 22 Schwarzschild geometry [24]; then one has: datt kc8 G c (2e) fftt2 adt a 3
d22 a43 G p c t (2f) 22ft adt33 c
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