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Geodesic from the variational principle (d'Inverno, 7.6; t'Hooft, page 29)
Geodesic is the straightest curve in a curved space. Along this curve the integral
S = ∫AB ds,
between the two points A and B in space-time has an extremum:
δ S = 0,
Maxwell equations in the gravitation field (d'Inverno, 12.5; t'Hooft, 10)
The first pair does not change its form
∂aFbc+∂bFca+∂cFab = 0.
In the second pair we use the covariant derivative D/dxk
DFab/dxb = 4π/c ja .

Equation of motion of a charged particle in the presence of both gravitational and electro-magnetic field
mDua/ds = e Fabub .
Exercises
  1. Prove, that the no-acceleration trajectory,
    Dua/ds ≡ dua/ds - Γbac ubuc = 0,
    is equivalent to the shortest trajectory,
    dua/ds - 1/2  ( dgbc/dxa )  ubuc = 0 .
  2. In the Rindler space ds2 = (1 + gξ/c2)2(cdη)2 - dξ2 make the non-relativistic limit and prove, that the geodesic equation reduces to the Newtonian non-relativistic equation d2ξ/dη2=g.
  3. Prove that the Kronecker delta δab={1 if a=b, 0 otherwise} is a tensor.
  4. (non-obligatory) Explain the twins paradox from the point of view of the accelerating observer.
Copyleft © 2005 D.V.Fedorov (fedorov @ phys au dk)