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The Newtonian limit: slow motion in weak fields

In this limit all fields are weak and all velocities are small. Only the 00 component of the energy−momentum tensor is non−vanishing, T00=μ where μ is the mass density of the matter. Therefore we shall only consider the 00 component of the Einstein's equation, R00=κ(T00− 1/2 g00T).

In the Newtonian limit g00=1 + 2φ, Γ α00=−φ,α, R00=−φ ,α,α=Δφ, where α =1,2,3. The Einstein equation thus turns into the the Poisson's equation Δφ= 1/2 κμ which is equivalent to the Newtonian theory if we put κ= 8πG/c4 , where G=6.67⋅10−11m3kg−1s−2 is the Newton's gravitational constant.

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

A general spherically symmetric static metric can be written as
ds2 = Adt2 − Bdr2 − r2(dθ 2 +sin2θdφ2),
where A and B are functions of only the "radius" r. The Christoffel symbols Γ abc= 1/2 gad(gdb,c−gbc,d+gcd,b) are given as
Γ rrr= 1/2 B'/B ,  Γ ttr= 1/2 A'/A ,  Γ rtt= 1/2 A'/B ,  Γ θθr= 1/r ,  Γ rθθ=− r/B , 
Γ φφr= 1/r ,  Γ rφφ=− r sin2θ/B ,  Γ φφθ=cotθ ,  Γ θφφ=−sinθcosθ .
The Ricci tensor Rab=Rcacb, where Rabcd= Γ abd,c−Γ abc,d +Γ ebdΓ aec −Γ ebcΓ aed is then equal
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 + A'B'/4AB + A'2/4A2 + B'/rB , 
The vacuum Einstein equations are Rab=0. Making a linear combination BRtt+ARrr=0 we find A'B+AB'=0 ⇒ A'/A + B'/B =0 ⇒ AB=1. From Rθθ =0 we then find r/B =r−R, where R is an integration constant, and B= 1/1− R/r , A=1− R/r . Finally the famous Schwarzschild metric is
ds2=(1− R/r )dt2−( 1/1− R/r )dr2− r2(dθ2+sin2θdφ2).
The integration constant R is determined from the Newtonian limit, R=2GM, where M is the mass of the central body. It is called gravitaional radius, or Schwarzschild radius.

Exercises

  1. Calculate the energy−momentum tensor Tab for a particle with mass m. The action is S=−m∫ds.

  2. For the metric ds2 = (1− R/r )dt2− 1/1− R/r dr2 − r2(dθ2+sin2θdφ2) write down the geodesic equations for a massive body and for a ray of light.

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