By Kalashnikov, V L

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Angular part of metric is equal to 0) collapse of dust sphere with mass M. 8 3 ∂ r (this The collapse is the motion from the right point of zero velocity ∂t is the velocity for remote observer) to the left point of zero velocity. These points are > solve(subs(r(t)=r,pot_1)=0,r); 2 M, 2 M, −2 M −1 + E 2 that are the apoastr (see above) and the gravitational radius. Hence (from the value for apoastr and pot 1 ) ∂ r)2 = ( ∂t (1− 2 rM )2 ( 2 rM −1+E 2 ) E2 = M ) (1− 2 rM )2 ( 2 rM − 2R M 1− 2R = (1 − Rg 2 r ) 1− Rg r R 1− Rg 1− .

We will investigate the 2-surface z=z (r ). As dz= dz dr dr, one can obtain Euclidian 2 2 dz 2 2 line element: ds = [1+ ( dr ) ] dr + r dphi 2 > > > > > > subs(theta=Pi/2,get_compts(sch));#Schwarzschild metric\ dr^2*%[2,2] + dphi^2*%[3,3] = \ (1+diff(z(r),r)^2)*dr^2 + r^2*dphi^2;#equality of\ intervals of flat and embedded spaces diff(z(r),r) = solve(%,diff(z(r),r))[1]; dsolve(%,z(r));# embedding 2M −1 + r 0 0 0 0 0 0 1 2M 1− r 0 r2 0 0 0 0 1 r 2 sin( π)2 2 0 34 dr 2 ∂ + r 2 dphi 2 = (1 + ( z(r))2 ) dr 2 + r 2 dphi 2 2M ∂r 1− r ∂ z(r) = ∂r z(r) = 2 √ 2 (r − 2 M ) M r − 2M 2 r M − 4 M 2 + C1 Now let’s take the penultimate equation and to express M through ρ0 = const: > > > > > > rho_sol :=\ solve(M = 4/3*Pi*rho0*R^3,rho0);#density from full mass fun1 :=\ Int(1/sqrt(r/((4/3)*Pi*rho0*r^3)-1),r);# inside space fun2 :=\ Int(1/sqrt(r/((4/3)*Pi*rho0)-1),r);# outside space rho sol := fun1 := fun2 := > > 3 M 4 π R3 1 2 1 −4 3 2 r π ρ0 2 dr 1 dr r −4 3 π ρ0 fun3 := value(subs(rho0=rho_sol,fun1));# inside fun4 := value(subs(rho0=rho_sol,fun2));# outside −R3 + r 2 M fun3 := − −R3 + r 2 M rM r2 M 35 4 fun4 := r R3 − 4M M R3 The resulting embedding for equatorial and vertical sections is presented below (Newtonian case corresponds to horizontal surface, that is the asymptote for r–> ∞ ).

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. This fact (Γα ) can be illustrated by infinite value of acceleration on r=2M, which is - g0, 0,0 0 ( Γ are the Christoffel symbols): > > > D1sch := d1metric( sch, coord ): Cf1 := Christoffel1 ( D1sch ): displayGR(Christoffel1,%); The Christoffel Symbols of the First Kind non − zero components : [11 , 2 ] = M r2 [12 , 1 ] = − [22 , 2 ] = − M r2 M (r − 2 M )2 [23 , 3 ] = r [24 , 4 ] = r sin(θ)2 [33 , 2 ] = −r [34 , 4 ] = r 2 sin(θ) cos(θ) [44 , 2 ] = −r sin(θ)2 48 [44 , 3 ] = −r 2 sin(θ) cos(θ) > > -get_compts(Cf1)[1,1,2]/get_compts(sch)[1,1];#radial\ component of acceleration − M r 2 (−1 + 2M ) r Such particles and signals will be expelled from the cause-effect chain of universe.

### Introduction to relativistic astrophysics and cosmology through Maple by Kalashnikov, V L

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