A Theory of Black Holes


 
If we assume there is a singularity at the centre of a black hole:
 
For an observer at the centre of a black hole (of mass M) the force on a small mass (m0) approaching from infinity when at distance r from the centre

F  =   - GMm
r2


Where        m  =  mo        by special relativity, which is towards the centre.
(1 - v2/c2)½





d  r  =  GMm
dtr  m




and    dr  =  GM
dtr  dt



or    dr  =  GM  =  GM  =  GM
dtr  drrro



and    dr  =  GM    -  as ro  o
dtro



The kinetic energy  =   
 
and the added mass  =     /c    +  mo 
 
so that the black hole has infinite mass, which is clearly not true by observation
 
 
If on the other hand the mass of the black hole is contained in a neutron star at absolute zero, with radius ro;
 
If a particle of rest mass mo arrives from infinity

Force (F)  =   - GMm
r  


Where        m  =  mo        invoking special relativity, and
(1 - v2/c2)½



d   r/dt    =  -GMm/r   m/r   m


So        dr  =          GM
dtr  dt



and    dr  =  GM  =  GM  =  GM
dtr  drrro



Particle arrives at r0 with kinetic energy moGM/2ro and the added mass  is  moGM/2ro c    +  mo
 
Note that dr/dt can exceed the velocity of light if M is large enough
And -dr/dt  = vo say.
 
For example, for G =  6.67X10-11 , M = 4×1033  kg and ro= 1.343×106m the particle arrives at the neutron surface at c
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The value for r0 requires some justification.
If a neutron has a cross section of one barn its radius is 5.64×10-5m assuming it is incompressible.
A spherical neutron star of mass M with hexagonal close packing has a radius of (5.656 M / mo)1/3×5.64×10-5 = 1.343×106m.


An alternative way of calculating the escape velocity from ro allowing for r inside the neutron star is as follows:
 
F =  -GMm/r    .r   /ro    when r <  ro
 
and d   r/dt    =  -GM/r   r   /ro  
 

dr  =    GMr
dt  ro  dt



 dr  =  GMr  =  GMr    =  GM
dtro  dr2ro  2ro





Therefore        dr  =  GM  =  - vo        giving the same value for vo as before.
dt2ro


 

J F Fleming  24 February 2011