From local to global relativity

Transcription

1 Physical Interpretations of Relativity Theory XI Imperial College, LONDON, 1-15 SEPTEMBER, 8 From local to global relativity Tuomo Suntola, Finland

2 T. Suntola, PIRT XI, London, 1-15 September, 8 On board of Galileo ship Newtonian space defines no limits to space or physical quantities; - the velocity of an object increases linearly under the influence of a constant force - the state of rest is inseparable from the state of rectilinear motion dp F= = m a dt time c velocity kinetic energy ( v) dx v v v d m Ekin = dx= m d = m v dv= ½mv dt v dt c velocity

3 T. Suntola, PIRT XI, London, 1-15 September, 8 3 From Galileo ship to space craft Relativistic space: - increase of velocities are limited to the velocity of light which breaks the linear linkage between force to acceleration time v' dp F= = m a dt F' Rectilinear dp = = dt m ( 1 β ) 3 a c velocity Ekin = ½mv 1 Ekin = mc 1 = c Δm 1 v c kinetic energy c velocity

4 T. Suntola, PIRT XI, London, 1-15 September, 8 4 From Galileo ship to space craft Relativistic space: - increase of velocities are limited to the velocity of light which breaks the linear linkage between force to acceleration Mathematical description (SR): redefinition of coordinate quantities: dt ' = dt 1 β dr ' dr 1 β β = = vc time v' dp F= = m a dt F' Rectilinear dp = = dt m ( 1 β ) 3 a c velocity Ekin = ½mv 1 Ekin = mc 1 c m = Δ v 1 c kinetic energy c velocity

5 T. Suntola, PIRT XI, London, 1-15 September, 8 5 Gravitation in modified coordinates via equivalence principle Schwarzschild solution of GR field equations: dr ' = dr 1 β dr ' = dr GM 1 rc GM β ff ( Newton) = rc ct ds ϕ dϕ r' ds r' β ff= v ff /c dt ' = dt 1 β GM dt ' = dt 1 β rc M β ff ( Newton ) 1 β The (coordinate) velocity of free fall in Schwarzschild space goes to zero at the critical radius r ( ) c Schwd GM = c.5 β ff, Schwarzschild r GM c

6 T. Suntola, PIRT XI, London, 1-15 September, 8 6 In search of fundamental physical limits In his lectures on gravitation in early 196 s Richard Feynman [1] stated: If now we compare the total gravitational energy GM Σ /R to the total rest energy of the universe, M Σ c, lo and behold, we get the amazing result that GM Σ /R = M Σ c, so that the total energy of the universe is zero Why this should be so is one of the great mysteries and therefore one of the important questions of physics. and further []...One intriguing suggestion is that the universe has a structure analogous to that of a spherical surface... It might be that our three-dimensional space is such a thing, a tridimensional surface of a four sphere

7 T. Suntola, PIRT XI, London, 1-15 September, 8 7 Zero-energy balance in spherically closed homogeneous space R 4 m n F n Fn 3-dimensional space

8 T. Suntola, PIRT XI, London, 1-15 September, 8 8 Zero-energy balance in spherically closed homogeneous space E = c p = c mc m m E g V ( ) dv r = ρmg r m E g GmM " = R 4 GmM " = R 4 M" The balance of motion and gravitation. mc GmM " = R 4

9 T. Suntola, PIRT XI, London, 1-15 September, 8 9 Zero-energy balance in spherically closed space Energy of motion Contraction Expansion Time Energy of gravitation

10 T. Suntola, PIRT XI, London, 1-15 September, 8 1 Zero-energy balance in spherically closed space E = c p = c mc m dc c Δ t = year time E g GmM " = R"

11 T. Suntola, PIRT XI, London, 1-15 September, 8 11 The energy of motion in local space m E = c p = c mc m E g GmM " = R 4 M"

12 T. Suntola, PIRT XI, London, 1-15 September, 8 1 The energy of motion in local space Im m E = c p = c mc m Em = c mc ( ) E g GmM " = R 4 E g GmM " = R 4 Re M"

13 T. Suntola, PIRT XI, London, 1-15 September, 8 13 Buildup of kinetic energy maintaining the zero-energy balance Em Im ( ) Re Im = c mc mc φ p φ ( Re) Ekin Re = c cδm Ekin = c mδc mc φ p ff mc local ( ) Global relativity (DU) : E = c Δ p = c cδ m+ mδc kin ΔE g ( global ) M E c m Local relativity (SR): kin = Δ

14 T. Suntola, PIRT XI, London, 1-15 September, 8 14 Rest energy in nested frames Im Im E rest( ) E rest ( ) m = m 1 β ( ) rest n n i= i n n i= ( 1 ) c = c δ i E = c p = c m c 1 δ 1 β rest n ( ) ( i) i n i=

15 T. Suntola, PIRT XI, London, 1-15 September, 8 15 The system of nested energy frames Hypothetical homogeneous space local galaxy group frame Milky Way frame Solar frame Earth frame n ( ) rest = 1 i 1 i i= E m c δ β Relativity of rest energy

16 T. Suntola, PIRT XI, London, 1-15 September, 8 16 The system of nested energy frames Hypothetical homogeneous space local galaxy group frame Milky Way frame Solar frame Earth frame n f = f 1 δ 1 β (,) ( ) local i i i= Relativity of characteristic frequencies

17 T. Suntola, PIRT XI, London, 1-15 September, 8 17 Satellite in Earth gravitational frame 1 f δ,β /f, GR DU β = δ f( ) = f, ( 1 δ) 1 β f, 1 δ β β δβ DU f( ) = f, 1 δ β f, 1 δ β β δβ GR 8

18 T. Suntola, PIRT XI, London, 1-15 September, 8 18 Free fall and orbital velocity in Schwarzschild space and in DU space β ff ( Newton ) Schwarzschild space: - the velocity of free fall goes to zero at the critical radius r ( ) c Schwd GM = c 1 β.5 Limit for stable orbits - velocity at circular orbit exceeds the velocity of free fall at r = 3r c(schwd) β β orb ( ) ( Schw ) ff Schw r GM c DU space: - the velocity of free fall goes to zero at the critical radius r ( ) cdu GM = c β 1.5 β ff ( Newton ) β ff DU ( ) - velocity at circular orbit goes to zero at r = r c Slow orbits maintain the mass of the black hole! β orb ( DU ) r GM c

19 T. Suntola, PIRT XI, London, 1-15 September, 8 19 Orbital period near black hole DU DU P = c π r ( 1 δ ) 3 π r Schwarzschild P = ( = Newton ) c δ δ P min( DU ) 16π r = c c r /r c Schwd Sgr A*: M 3.7 million solar masses r c(du) 5.3 million kilometers DU 6 min. 4 Schwd r /r c Observed 17 min rotation period at Milky Way Center, Sgr A* [R. Genzel, et al., Nature 45, 934 (3) ]

20 T. Suntola, PIRT XI, London, 1-15 September, 8 Distinctive characteristics of local and global relativity Local relativity (SR): Global relativity (Dynamic Universe): What is finite? Velocity in space Total energy Description of finiteness dt ' = dt 1 β dr ' = dr 1 β E total GM " = c p = R 4 Rest energy Erest = mc E rest = c mc ( ) = mc 1 δ 1 β i i i Kinetic energy Ekin = c Δm kin ( ) E = c Δ p = c cδ m+ mδc by local momentum via insert of mass equivalence by gravitation via tilting of space

21 T. Suntola, PIRT XI, London, 1-15 September, 8 1 Jump to Galileo ship characterized as a velocity frame Velocity of mass object on board v = v vship = v 1 v ship v Velocity of light on board? c c v ship h pλ = c= mλc λ characterized as a momentum frame Momentum of mass object on board p= mv= p 1 h v ship vship Momentum of light on board p= mλc= 1 c= p 1 λ c c (as reduced by the Doppler-effect) v ship v p λ h c = = h f λ The observed phase velocity is conserved c λ λ ( 1 β ) ( 1 β ) β β = = = = Tβ T T λ c

22 T. Suntola, PIRT XI, London, 1-15 September, 8 Jump to Galileo ship characterized as a velocity frame Velocity of mass object on board v = v vship = v 1 v ship v Velocity of light on board? c c v ship h pλ = c= mλc λ characterized as a momentum frame Momentum of mass object on board p= mv= p 1 h v ship vship Momentum of light on board p= mλc= 1 c= p 1 λ c c (as reduced by the Doppler-effect) v ship v p λ h c = = h f λ Michelson - Morley interferometer: no change in momentum in different arms zero result guaranteed

23 T. Suntola, PIRT XI, London, 1-15 September, 8 3 Distinctive characteristics of local and global relativity QM, SR : Dynamic Universe: Planck constant h 3 h= π e μ c= hc [ ] h kg m Quantum of radiation E h = hν Eλ = cc = c k c= cmc λ λ Energy momentum four-vector ( ) ( ) E = mc + pc energy momentum mass ( ) Im( ) ( ) ( ) ( ) ( ) m = ptot = + E c c mc c p ( ) ( Re( ) ) ( Im ) ( Re( )) kc = k c + k c φ φ φ k = k + k φ φ φ wave number kφ = kim( ) + k φ Re( φ)

24 T. Suntola, PIRT XI, London, 1-15 September, 8 4 Magnitude versus redshift: Supernova observations 5 μ ,1,1,1 1 z 1 Data: A. G. Riess, et al., Astrophys. J., 67, 665 (4)

25 T. Suntola, PIRT XI, London, 1-15 September, 8 5 Supernova observations: Predictions by FLRW and DU 5 μ 45 + SN Ia observations Dynamic Universe ( z) μ = μ +.5 log z 1+ 4 Standard model (FLRW) Ω m =.3, Ω Λ =.7 Standard model (FLRW) Ω m = 1, Ω Λ = 35 μ = μ + 5log 1+ ( z) z ( 1 ) ( 1 ) ( ) + z +Ωmz z + z Ωλ 1 dz 3,1,1,1 1 1 Data: A. G. Riess, et al., Astrophys. J., 67, 665 (4)

26 T. Suntola, PIRT XI, London, 1-15 September, 8 6 Angular size of galaxies and quasars, observations log (LAS) (d) z Largest angular size (LAS), open circles: galaxies, filled circles: quasars Collection of data: K. Nilsson et al., Astrophys. J., 413, 453 (1993)

27 T. Suntola, PIRT XI, London, 1-15 September, 8 7 Angular size of galaxies and quasars, observations / predictions log (LAS) Standard model Ω m = 1, Ω Λ = Standard model + dark energy Ω m =.3, Ω Λ =.7 (d) Dynamic Universe: Euclidean θ = r 1 R z z Largest angular size (LAS), open circles: galaxies, filled circles: quasars Collection of data: K. Nilsson et al., Astrophys. J., 413, 453 (1993)

28 Where is the anti-energy of a localized object? 8 cmc

29 It is everywhere it is the gravitational energy of matter in the rest of space! 9 GM " m R" cmc GM " m Em + Eg = cmc = R"

30 Thank you for your attention. 3