with classical methods Visualisations Introduction The first - - PowerPoint PPT Presentation

with classical methods

 Introduction  Problem

• Given Problems
• Simplification
• Discussion

 Lagrangian

• Constants of motion
• dr/dt

 Solution

• Series Solution

▪ t(r) ▪ phi(r)

• Visualisations

▪ The first problem ▪ The second problem

 Analysis

• Classification
• Complex Handling

 Conclusion

• Classic Approximations
• Bizarre Characteristics

 Quamvis movet! (E pur si muove!)

• Galilaeo Galilaei

 Quamvis trahitur! (However, attracted)

• Issaco Newtono

 Quamvis procedit! (However, proceed)

• Alberto Einsteino



• r

r r f

c

 1 ) (

2

2 1 ) (         r r r f

c

 

 

    

2 2 2 2 2 2 2 2

sin ) ( ) (       c r r f c r r f dt mc dtL

Assumption

 

2 2 2 2 2 2 2 2 2 2 2 2 2 2

) ( ) ( sin ) ( ) ( c r r f c r r f mc c r r f c r r f mc L                  

2 2 2 2 2

) ( ) ( ' c r r f c r r f L       2   

 What this action means?

 Schwarzschild Solution

• Non-rotating non-charged solution

 Reissner-Nordstrom Solution

• Non-rotating charged solution



• 

'

2

L mr L p  



                 L dt d

2 2 2 2 2 2

1 ) ( ) ( ' r c m p r f c r r f L



   

) ( '

2

r f L mc L q p H

i i i

   

          

2 2 2 2 3 6 2 2 2 2

1 r c m p H f c m f c r





f c r f r c m p f mc p r H

r 2 2 2 2 2 2 2

1 ) , (    



  Expanding,

dr cf r c m p f H c m r c m p f H c m r dr dr dr dt dt r t

   

                                                   

2 2 2 2 2 2 4 2 2 2 2 2 2 4 2

1 8 3 1 2 1 1 ) (

 

 

r r c s s r r c s s

p r c m r H c H r c m r r r c r c r H c m r H p c H r c m r r r c r H c m H c m c r t

2 2 2 2 2 4 7 4 2 4 2 2 2 2 4 7 4 4 8 4 2 4 2

2 8 log 3 log 2 3 1 2 8 log 3 log ... 8 3 2 1                                                

 

 

1 2 3 4 5 6 10 20 30 40 1 2 3 4 5 6 10 10 20 30

1    



p r H m

c

, 1    



p r H m

c

1 2 3 4 5 6 10 5 5 10 1 2 3 4 5 6 10 5 5 10

1    



p r H m

c

, 1    



p r H m

c

dr dr dt r L dt L d

 

  ) ( ' ' 

1    



p r H m

c

, 1    



p r H m

c

2 4 6 8 10 14 12 10 8 6 4 2 2 4 6 8 10 14 12 10 8 6 4 2

1    



p r H m

c

, 1    



p r H m

c

2 4 6 8 10 10 5 5 10 2 4 6 8 10 10 5 5 10

dr r mr r L p dr r

 

    

2

) ( '



 

 See Octave

eff r r

V K V K H

,

   

       

     

                      

  

  

r r r r r r r

dt r L r r L r dt dt dr r L dt d dr p K

3 2 2 2 2 ,

3 r mc GMp mr p r GMm V eff

r  

   

  See the comparison of the inside orbit with

two different complex realisations.

y x i y x yi x sinh sin cosh cos cos   

 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.