Emergence of a quasi-newtonian Gravitation Law: a Geometrical Impact Study.

Transcription

1 Emergence of a quasi-newtonian Gravitation Law: a Geometrical Impact Study. Réjean Plamondon and Claudéric Ouellet-Plamondon Département de Génie Électrique École Polytechnique de Montréal

2 The authors

3 Basic questions Could there be another starting point to redirect our long-term quest for bridging the gap between General Relativity and Quantum Mechanics? Can a modification to Newton s law of gravitation be derived from such a model? What would be the main resulting predictions of this modified Newton s law?

4 Topic A statistical pattern recognition approach Putting general relativity into a probabilistic context (The interdependence principle) The emergence of a quasi-newton s law A symmetric geometry An axisymmetric geometry Take home messages

5 Statistical Pattern Recognition Patterns are generated by a probabilistic system

6 Statistical Pattern Recognition REPRESENTATION scaled features N-dimensional space object random vector INTERPRETATION class density function MAPPING class delimitation discriminating function

7 Topic A statistical pattern recognition approach Putting general relativity into a probabilistic context (The interdependence principle) The emergence of a quasi-newton s law A symmetric geometry An axisymmetric geometry Discussion

8 Putting general relativity into a probabilistic context Two information spaces must be analyzed and compared.

9 REPRESENTATION coordinates +metrics 4-dimensional space metric quantified coordinate arbitrary specific feature of the manifold MAPPING Einstein tensor: G INTERPRETATION curvature space First information space: the structure of a manifold

10 REPRESENTATION Second information space: the content of a manifold coordinates +metrics 4-dimensional space metric quantified coordinates localization of events MAPPING Momentum-Energy tensor: T INTERPRETATION Mass-energy density, energy flux, momentum density and stress components

11 Einstein s equation G = KT The Einstein s equation can be seen as making a link between the two interpretation spaces. BUT, according to the statistical pattern recognition paradigm, these interpretation spaces could be given a probabilistic meaning How?

12 Topic A statistical pattern recognition approach Putting general relativity into a probabilistic context (The interdependence principle) The emergence of a quasi-newton s law A symmetric geometry An axisymmetric geometry Discussion

13 Interdependence principle Spacetime curvature (S) and matter-energy density (E) are two inextricable information spaces defining the physically observable universe (U); they must be mutually exploited to describe any subset U i of this universe. In terms of expectations, the probability of observing a subset (U i ) is: PU ( ) = PS (, E) < 1 i i i

14 Corrolary The probability of observing and describing a given subset of the universe P(U i ), that is the joint probability of P(S i,e i ), can be studied from two equivalent modi operandi: either by analyzing the structure of the spacetime as an interpretation space associated with an a priori given matter-energy density or by analyzing the matter-energy density as an interpretation space associated with an a priori given spacetime structure.

15 In terms of Bayes law (conditionnal probabilities) P( Ui) = P( Si, Ei)= P( Si/ Ei)P( Ei)=P(E i/s i)p( Si) f( Si/ Ei) f( Ei)= f(e i/s i) f( Si) f( S / E )= f(e /S ) i i i i f( Si ) f( E ) i

16 A link with Einstein s law? f( Si ) f ( Si/ Ei) f (E i/s i) G KT f( E ) i f ( Si/ Ei) k1trg f( Si ) ktrt f( E ) i f (E /S )? i i

17 Introducing quantum mechanical concepts? f(e /S ) kψ ψ k f ( S ) i i 3 E E 3 ψ i i i kk ( ) 3 G fψ Si T k1 f ψ ( S )? i

18 Topic A statistical pattern recognition approach Putting general relativity into a probabilistic context (The interdependence principle) The emergence of a quasi-newton s law A symmetric geometry An axisymmetric geometry Discussion

19 A statically spherically symmetric system low speed and weak field condition : 1 1 R00 Φ KT 00 fψ ( Si ) c f ψ ( S )? i T 00?

20 Estimating the probability of presence Building a star from scratch by adding numerous identical particles (N ), each one with its own wave function, density function and associated space-time, as seen from a locally flat tangent space. Making the convolution of their corresponding density functions. The central limit theorem predicts that the ideal form of the global probability density f(x) will be a Gaussian multivariate function.

21 Emergence of Newton s law of gravitation: the density factor: 1 x f( x) exp 4 4πσ σ x σ σ r 1 σ f( Ei / Si) k3fψ ( Si) k3f r exp 4πσ r r

22 Estimating the momentum- energy tensor component p p cδτ σ σ σ π π 4π T r 3 Mc σ Mσc 4πσ r 4π r 00 3

23 Emergence of Newton s law of gravitation: the Sun s Laplacian 1 1 R00 Φ KT 00 fψ ( S i ) c KMc 4 Φ exp 3 5 4πσ σ r σ r

24 Emergence of Newton s law of gravitation: the field 4 KMc σ r 3 gr () Φ exp 4πσ r r gr () 4 KMc GM = 3 r r ( 4πσ )

25 Emergence of Newton s law of gravitation: the potential 4 KMc π σ Φ erfc () r erfc Φ () 3 erfc r 4πσ σ r 4 KMc GM Φ erf ( r) πσ r 6r 40r r

26 Brief pause According to the present paradigm, the Newton s law is not empirical. It is an approximation of a more general law. It can be seen as an emerging phenomenon when the proper representation and interpretation spaces are used to describe a physical manifold. The resulting erfc potential can be incorporated in a metric to study statically symmetric system.

27 Topic A statistical pattern recognition approach Putting general relativity into a probabilistic context (The interdependence principle) The emergence of a quasi-newton s law A symmetric geometry An axisymmetric geometry Discussion

28 The symmetric metric and the field equation π σ ds 1 GM erfc c dt c σ r 1 π σ 1 GM erfc dr r dθ r sin θdφ c σ r No coordinate singularity No intrinsic singularity Temporal offset at infinity Radial delays

29 Exact analytical solutions: Ricci components R αc βu 16π r 00 3 R βu 16παr 11 3 R ( α 1) rβ 33 R ( 1) r sin α β θ Φ GM 4π π u α 1 1 erfc c c u 4π r β GM c r u exp 4 π r

30 Exact analytical solution: Einstein components β ( α 1) G00 αc ΚT 00 fψ ( Si ) r r 1 β ( α 1) G11 Κ T 11 fψ ( Si ) α r r u β G Κ T fψ ( Si ) 16π r u β G33 sin θ ΚT 33 fψ ( Si ) 16π r

31 Exact analytical solutions: curvature scalars K r β u 3π r 16β 4 α π r r r R s βu 4β ( α 1) 3 8π r r r

32 The second order geodesics t βtr α 0 β r αβct r αr θ sin θφ 0 α r θ θ θ θφ r (sin cos ) 0 r φ φ ( cotθ ) θφ 0 r

33 The first order equatorial geodesics αt k h α r Φ c ( k 1) r r φ h

34 The geodesics: a modified Kepler s law φ r 4 u GM exp 3 π r 1 1 K GM exp u K u erf 3 r c cr π r cr 4 π r

35 Massive particle effective potential 3cδτ h h 3cδτ Veff Kerfc Kerfc 16π πr r cr 16π πr Converges toward Einstein s predictions at large r and when the constant term of the erfc potential is neglected.

36 Most striking properties New set of exact analytical solutions Converge towards Schwarzchild s predictions at large distance Differs from Einstein s predictions at small distance No singularities No unstable inner circular orbits No gravitational collapse Gauge dependent?

37 erfc vs erf functions erfc z 1 erf z

38 Topic A statistical pattern recognition approach Putting general relativity into a probabilistic context (The interdependence principle) The emergence of a quasi-newton s law A symmetric geometry An axisymmetric geometry Discussion

39 A rotation term c π σ GM erfc c dt σ r 1? π σ GM 1 GM π erf c dt dφdt c σ r ωstc σ

40 An expansion term ? st GM erfc dr c r GM erf dr c r v GM GM erf c c r G c π σ σ π σ σ π π σ σ σ 1 M erfc drdt r π σ σ

41 Two new components A rotation term: ω st dφ K = + d φ dt dt ω st An expansion term: v st dr Kvst K u = + 1 erf dt c c 4π r 1

42 In other words The axisymmetric metric can be seen as explaining why any the massive body in the universe is rotating and its associated space-time looks like expanding.

43 An axisymmetric solution ds g dt g dφdt g drdt g dr g dθ g dφ π σ π ds 1 GM erf c dt GM d dt c φ σ r ω st σ 1 v st π π σ 1 GM GM erf c σ c σ r 1 π σ 1 GM erfc c σ drdt r 1 π σ 1 GM erf dr r dθ r sin θdφ c σ r

44 Very complex exact analytical solutions 6 different covariant metric components 7 non-null contravariant metric components 10 non-zero analytical expressions for the Christoffel symbols of the first kind 1 coefficients of the second kind 16 Riemann tensor covariant components 9 different Ricci tensor components 9 different Einstein tensor components.

45 The second order geodesics K 4 Kβv ( ) st αc K r Kvstr ( αc K) t βc rt φ 0 ωst α ( αc K) αα ( c K) 4 Kvt st βcr cr βct r θ ( rsin θφ ) 0 αα ( c K) ( αc K) ( αc K) r θ θ θ θφ r (sin cos ) 0 Kt ω st r θ r φ r inθ cos θ) θφ ( sin θ) φ 0 ( sin ) ( s r Φ 8π πgm u α 1 1 erfc c uc 4π r β GM u exp cr 3 π r

46 Very Brief Pause The axisymmetric metric can be seen as explaining why any the massive body in the universe is rotating and its associated space-time looks like expanding. There will be no gravitational collapse in systems described by such a metric. Black holes without any intrinsic singularity Dark matter might not be necessary to explain the orbital rotation velocity at and beyond the visible outer edge of distant galaxies

47 Dark Matter: rotational velocity of galaxies v B Aexp r exp Predicted and measured values of the orbital rotation velocity of the NGC 801 galaxy as a function of its radius. The experimental data are from Table 4, chapter 7 of Broeils (199)

48 Dark Matter: rotational velocity of galaxies Predicted vs measured values of the orbital rotation velocity of some distant galaxies. from table 1 and table 4 of Blok and McGaugh (1997)

49 Take home messages The present paradigm provides partial answers to the initial questions. Two new metrics that will require further investigations. But the whole approach raises more numerous interrogations and point out many challenges. In the long run, it might provide a new pathway to bridge the gap between General Relativity and Quantum Mechanics More investigations, theoretical and experimental, will be required to survive the Occam s razor. This is the tip of an iceberg.

50 To investigate further

51 Final take home message The main message conveyed throughout the chapters of this book is that the four basic interactive forces of physics, which are considered to be empirical facts, can be seen as emergent phenomena described by specific mathematical patterns, when seen through the appropriate representation and interpretation schemes.

52 Final take home message Similarly, in such a model, once a coherent set of physical units is defined, the values of the fundamental constants can be seen as numerical parametric patterns that can be predicted after taking into account the various projections that are required to perform these measurements as well as the physical environment and the specific context in which these estimates are made.

53 Thank you for your attention!