[ home ] Motion in the Schwarzschild metric General Relativity. Note: « 10 »
Unnamed Web page

Motion in the Schwarzschild metric

In the Schwarzschild metric the geodesic equations, d(gabub)/ds = 1/2 gbc,aubuc, for a=t,θ,φ are:

d/ds [(1− 2M/r ) dt/ds ]=0 ;  d/ds [r2 dθ/ds ]= r2sinθcosθ( dφ/ds )2 ;  d/ds [r2sin2θ dφ/ds ]=0 .
Instead of the a=r geodesic we shall divide the expression for the Schwarzschild metric by ds2:
1=(1− 2M/r )( dt/ds )2 −(1− 2M/r )−1( dr/ds )2 −r2[( dθ/ds )2+sin2θ( dφ/ds )2]

The first three equations can be integrated as θ=π/2 , r2 dφ/ds =J , (1− 2M/r ) dt/ds =E , where J and E are constants. The fourth equation then bocomes

1=(1− 2M/r )−1E2 −(1− 2M/r )−1J2r'2− J2/r2  ,
where r'= dr/dφ . Traditionally one makes a variable substitution r=1/u
(1− 2M/r )u=E2−J2u'2−J2u2(1−2Mu)
which is the sought equation of motion.

Differentiating it once more and assuming u'!=0 gives

u''+u= M/J2 +3Mu2

Exercises

  1. Calculate Rθθ and Rrr from note8.

  2. Show that in a synchronous reference system (ds2=dτ2+gαβdxαdxβ, where α,β=1,2,3) the time lines are geodesics.

  3. Show that a light ray can travel around a massive star in a circular orbit much like a planet. Calculate the radius (in Schwarzschild coordinates) of this orbit. (Answer: r= 3/2 rg)

  4. Find explicitely the Schwarzschild coordinates t, r as function of the Lemaitre coordinates τ, ρ. Answer: ρ−τ= 2/3 r3/2/rg1/2 .

Copyleft © 2005 D.V.Fedorov (fedorov @ phys au dk)