For relativistic massive particle, the action is S = −m0∫ds = −m0∫dλ √gμν˙xμ˙xν = ∫dλ L,
Answer
I) The equation of motion for a scalar massless relativistic point particle on a Lorentzian manifold (M,g) is
˙x2 := gμν(x) ˙xμ˙xν ≈ 0,
where dot denotes differentiation wrt. the world-line parameter τ (which is not proper time). [Here the ≈ symbol means equality modulo eom.] Thus a possible action is
S[x,λ] = ∫dτ L,L = λ ˙x2,
where λ(τ) is a Lagrange multiplier. This answer (B) may seem like just a cheap trick. Note however that it is possible by similar methods to give a general action principle that works for both massless and massive point particles in a unified manner, cf. e.g. Ref. 1 and eq. (3) in my Phys.SE here.
II) The corresponding Euler-Lagrange (EL) equations for the action (B) reads
0 ≈ δSδλ = ˙x2,
0 ≈ −12gσμδSδxμ = d(λ˙xσ)dτ+λΓσμν˙xμ˙xν.
III) The action (B) is invariant under world-line reparametrization τ′ = f(τ),dτ′ = dτdfdτ,˙xμ = ˙x′μdfdτ,λ′ = λdfdτ.
References:
- J. Polchinski, String Theory Vol. 1, 1998; eq. (1.2.5).
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