Schwarzschild metric: a static spherically symmetric solution of vacuum Einstein equations

A general spherically symmetric static metric can be written as
ds2=Adt2Bdr2r2(dθ2 +sin2θdφ2),
(1)
where A and B are functions of only the radius r. Calculating the Christoffel symbols1 and the Ricci tensor2 and then solving the vacuum Einstein equations Rab=0 gives3 the famous Schwarzschild metric
ds2 =
1− R

r

dt2 dr2

1− R

r
r2(dθ2+sin2θdφ2).
(2)
The integration constant R is determined from the Newtonian limit, R=2M where M is the mass of the central body4. It is called gravitaional or Schwarzschild radius.

Motion in the Schwarzschild metric

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

ds


1− 2M

r

dt

ds

=0 ; 
(3)

d

ds

r2 dθ

ds

= r2sinθcosθ
dφ

ds

2

 
 ; 
(4)

d

ds

r2sin2θ dφ

ds

=0 .
(5)
Instead of the r-geodesic we shall divide the expression for the Schwarzschild metric (2) by ds2:
1=
1− 2M

r


dt

ds

2

 

1− 2M

r

−1

 

dr

ds

2

 
r2

dθ

ds

2

 
+sin2θ
dφ

ds

2

 

(6)
The first three equations can be integrated as
θ = π

2
 ,  r2 dφ

ds
=J ,  
1− 2M

r

dt

ds
=E ,
(7)
where J and E are constants. The fourth equation then becomes
1= E2

1− 2M

r
J2

r4
r2

1− 2M

r
J2

r2
 ,
(8)
where r′ ≡ [(dr)/(dφ)]. Traditionally one makes a variable substitution r=1/u,
(1−2Mu)=E2J2u2J2u2(1−2Mu),
(9)
and then differentiates the equation once. Assuming u′ ≠ 0 this gives
u"+u= M

J2
+3Mu2  .
(10)
In this form the last term is a relativistic correction to the othewise non-relativistic equation.
The light rays travel along the null-geodesics where ds2=0. Consequently instead of ds one needs to use some parameter dλ in the geodesic equations [(Dka)/(dλ)]=0, where ka=[(dxa)/(dλ)] and also the unity in the left-hand side of equation (6) has to be substituted with zero. This apparently leads to the equation
u"+u=3Mu2  ,
(11)
which describes the trajectory of a ray of light in the Schwarzschild metric.

Exercises

  1. Consider a non-relativistic equatorial (θ = π/2) motion of a planet with mass m around a star with mass M discribed by a Lagrangian5
    L= 1

    2
    m(

    r
     
    2
     
    +r2

    φ
     
    2
     
    )+ mM

    r
    .
    Write down the Euler-Lagrange equations

    t
    L


    q
     
    =L

    q
     ,  q=r,φ .
    Using the first integral r2[(φ)\dot]=J rewrite the r-equation as an equation for the function u(φ), where u=1/r, and compare with (10).
  2. 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](2M).

Footnotes:

1  Γrrr=[1/2] [(B′)/(B)], Γttr=[1/2] [(A′)/(A)], Γrtt=[1/2] [(A′)/(B)], Γθθr=[1/(r)], Γrθθ=−[(r)/(B)], Γφφr=[1/(r)], Γrφφ=−[(r sin2θ)/(B)], Γφφθ=cotθ, Γθφφ=−sinθcosθ.
2  Rtt = [(A")/(2B)]+[(A′)/(B)]([1/(r)]−[(B′)/(4B)]−[(A′)/(4A)]), Rθθ = 1−([(r)/(B)])′−[1/2]([(A′)/(A)]+[(B′)/(B)])[(r)/(B)], Rrr = −[(A")/(2A)]+[(AB′)/(4AB)]+[(A2)/(4A2)]+[(B′)/(rB)].
3  Making a linear combination BRtt+ARrr=0 gives AB+AB′=0  ⇒    [(A′)/(A)]+[(B′)/(B)]=0   ⇒   AB=1. Then Rθθ=0 gives B=[1/(1−[(R)/(r)])], A=1−[(R)/(r)], where R is an integration constant.
4 in the units where G=1
5 [(A)\dot] ≡ [(∂A)/(∂t)]


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On 22 Oct 2007, 18:45.