By Kalashnikov, V L

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Additional info for Introduction to relativistic astrophysics and cosmology through Maple

Example text

5 Schwarzschild black hole Now we return to Schwarzschild metric. > get_compts(sch); 2M  −1 + r           0 0 0 0 1 2M 1− r 0 0 0 0 0 0 r2 0 0 r 2 sin(θ)2            One can see two singularities: r =2M and r =0. What is a sense of first singularity? e. g0, 0 and 47 g1, 1 ) change signs. The space and time exchange the roles! The fall gets inevitable as the time flowing. As consequence, when particle or signal cross the gravitational radius, they cannot escape the falling on r =0.

If Vmax >E > 1 the motion is infinite (it is analogue of the hyperbolical motion in Newtonian case). Vmax >E =1 corresponds to parabolic motion. When E lies in the potential hole or E = V, we have the finite motion. The motion with energy, which is equal to extremal values of potential, corresponds to circle orbits (the stable motion for potential minimum, unstable one for the maximum). The existence of the extremes is defined by expressions: > > > numer( simplify( diff( subs(r(tau)=r,V), r) ) ) =\ 0:# zeros of potential’s derivative solve(%,r); 1 (L + 2 √ L2 − 12 M 2 ) L 1 (L − , M 2 √ L2 − 12 M 2 ) L M √ As consequence, the circle motion is possible if L ≥ 2M 3 , when r ≥ 3(2M ).

4 Msun (so-called Chandrasekhar limit for white dwarfs). The smaller masses produce the white dwarf with non-relativistic electrons but larger masses causes collapse of star, which can not be prevented by pressure of degeneracy electrons. 4 Msun < M < 3 Msun . Collapse for larger masses has to result in the black hole formation. 5 Schwarzschild black hole Now we return to Schwarzschild metric. > get_compts(sch); 2M  −1 + r           0 0 0 0 1 2M 1− r 0 0 0 0 0 0 r2 0 0 r 2 sin(θ)2            One can see two singularities: r =2M and r =0.