with classical methods Visualisations Introduction The first - PowerPoint PPT Presentation

with classical methods

 Visualisations  Introduction ▪ The first problem  Problem ▪ The second problem  Given Problems  Analysis  Simplification  Classification  Discussion  Complex Handling  Lagrangian  Conclusion  Constants of motion  Classic Approximations  dr/dt  Bizarre Characteristics  Solution  Series Solution ▪ t(r) ▪ phi(r)

 Quamvis movet! (E pur si muove!)  Galilaeo Galilaei  Quamvis trahitur! (However, attracted)  Issaco Newtono  Quamvis procedit! (However, proceed)  Alberto Einsteino

   2 2 r r              2 2 2 2 dtL mc dt f ( r ) sin 2 2 c f ( r ) c r   1  c f ( r ) r 2    r     c f ( r ) 1   2 r

     2 2  2 2 2 r r r r               2 2 2 2 2 L mc f ( r ) sin mc f ( r ) 2 2 2 2 c f ( r ) c c f ( r ) c Assumption     2 2 2 r r      L ' f ( r ) 2 2 c f ( r ) c 2

 What this action means?

 Schwarzschild Solution  Non-rotating non-charged solution  Reissner-Nordstrom Solution  Non-rotating charged solution

   d L     0      dt    2 L mr     p    L '  2 r  f ( r ) 2 c f ( r )   L ' 2 p   1 2 2 2 m c r 2 mc       H p q L f ( r ) i i L ' i

2 p   1 2 2 2 m c r  2 H ( r , p ) mc f r  2 r  f 2 c f    2 6 3 p m c f       2 2 2 r c f 1   2 2 2 2   H m c r

dt dr        t ( r ) dt dr   dr r   2       2 2 2 4 2 4 p p 1 m c 3 m c               1 1 1 f f       2 2 2 2 2 2 2 2   2 H m c r 8 H m c r             dr   cf        Expanding, r     2   2 2 4 4 8 4 7 2 4 c p r m c 3 m c r 3 r m c log r 3 m c                      s c t 1 ... log r r 1     s 2 4 4 2 2       c 2 H 8 H c 8 H 2 H r 2 H   r 0   r   4 7 r r 3 r m c log r c       2 2 2 2 s c   log r r m c r p  s 4 2   c c 8 H 2 H r r 0

        1 m H r 1 , p 0 m H r p   c c 30 40 20 30 10 20 10 1 2 3 4 5 6 10 0 1 2 3 4 5 6

        1 m H r 1 , p 0 m H r p   c c 10 10 5 5 1 2 3 4 5 6 1 2 3 4 5 6 5 5 10 10

dt      d L ' dt L ' ( r ) dr dr

        1 m H r 1 , p 0 m H r p   c c 2 4 6 8 10 2 4 6 8 10 2 2 4 4 6 6 8 8 10 10 12 12 14 14

        1 m H r 1 , p 0 m H r p   c c 10 10 5 5 2 4 6 8 10 2 4 6 8 10 5 5 10 10

  p L ' ( r )       dr dr  2  r mr r

 See Octave

    H K V K V , r r eff     r     d L dr L L    r  r            K p dr dt r r dt        r            dt r dt r r r 0 r 0   r 0 2 2 p GMp GMm       V eff , r 2 2 3 r 3 mr mc r

    cos x yi cos x cosh y i sin x sinh y  See the comparison of the inside orbit with two different complex realisations.

 Apparently, the right choice of the Lagrangian gives the right result, and vice versa.  At least, from the inside result, the implication of new concept is required.