3.3 World-Line Fields
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3.3 World-Line Fields
In this section we consider the dynamics of a free point-particle of mass m and charge
q from a field-theoretical viewpoint. The particle position now appears as four scalar
fields on a 1-dimensional manifold L, with values in the space-time manifold M,
and we consider the associated action. We also discuss the geometry and symmetries
associated with L, especially in the m, q = 0 case, a feature which will be useful in
the construction of the observer space. Finally, we derive the particle equations of
motion in the tetrad formalism.
3.3.1 Geometrical Considerations
Geometrically speaking, a path of a point-particle in M is a map from a onedimensional manifold L to the space-time manifold M
γ : L → M ,
(3.3.1)
and its image γ(L) ⊂ M is the particle’s world-line. Given a coordinatization
λ : L → R and the space-time coordinates x
μ
: M → R
4 , we can describe the
γ map through four functions γ
μ
(λ). The latter is a set of four scalars under
L-diffeomorphisms (LD). In the passive version (PLD) that are the
λ-reparametrizations, we have
˜
λ = ˜
λ(λ) ,
⇒ ˜
γ
μ
( ˜
λ) = γ
μ
(λ) ,
(3.3.2)
while in the active version (ALD) we have the action of the Lie derivative with
respect to a vector field κ on L
δ κ γ
μ
= −L κ γ
μ
+ O(κ
2
) ≡ −κ∂ λ γ
μ
+ O(κ
2
) .
(3.3.3)
In the case of PMDs, the γ
μ transform as coordinates
˜
x
μ
= ˜
x
μ
(x) ,
⇒ ˜
γ
μ
(λ) = ˜
x
μ
(γ(λ)) .
(3.3.4)
In the case of AMDs, however, there is a subtlety. If we interpret γ as simply some
continuous collection of points in M with no relation to the fields whatsoever, then
our pullback definition of AMDs would suggest
δ ξ γ
μ
= 0 ,
(3.3.5)
since we move the fields while keeping the points fixed. However, here the γ
μ we are
interested in is not any path, but one that must ultimately obey the geodesic equation,