2.1 Test particles: geodesics and a_ne parameterization

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Let C be a timelike curve with endpoints A and B. The action for a particle

of mass m moving on C is

I = 􀀀mc2

Z B

A

d_ (2.1)

where _ is proper time on C. Since

d_ =

p

􀀀ds2 =

p

􀀀dx_dx_g__ =

p

􀀀x__x_ _g__d_ (2.2)

where _ is an arbitrary parameter on C and x__ = dx_

d_ , we have

I [x] = 􀀀m

Z _B

_A

d_

p

􀀀x__x_ _g__ (c = 1) (2.3)

The particle worldline, C, will be such that _I=_x(_) = 0. By de_nition,

this is a geodesic. For the purpose of _nding geodesics, an equivalent action

is

I [x; e] =

1

2

Z _B

_A

d_

_

e􀀀1(_)x_ _x_ _g__ 􀀀m2e(_)

_

(2.4)

where e(_) (the `einbein') is a new independent function.

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Proof of equivalence (for m 6= 0)

_I

_e

= 0 ) e =

1

m

p

􀀀x__x_ _g__ =

1

m

d_

d_

(2.5)

and (exercise)

_I

_x_ = 0 ) D(_)x__ = (e􀀀1e_)x_ _ (2.6)

where

D(_)V _(_) _

d

d_

V _ + x_ _

_

_

_ _

_

V _ (2.7)

If (2.5) is substituted into (2.6) we get the EL equation _I=_x_ = 0 of the

original action I[x] (exercise), hence equivalence.

The freedom in the choice of parameter _ is equivalent to the freedom in

the choice of function e. Thus any curve x_(_) for which t_ = x__(_) satis_es

D(_)t_V _ = f(x)t_ (arbitrary f) (2.8)

is a geodesic. Note that for any vector _eld on C, V _(x(_)),

t_D_V _ _ t_@_V _ + t_

_

_

_ _

_

V _ (2.9)

=

d

d_

V _ + x_ _

_

_

_ _

_

V _ (2.10)

= D(_)V _ (2.11)

Since t is tangent to the curve C, a vector _eld V on C for which

D(_) = f(_)V _ (arbitrary f) (2.12)

is said to be parallely transported along the curve. A geodesic is therefore a

curve whose tangent is parallely transported along it (w.r.t. the a_ne connection).

A natural choice of parameterization is one for which

D(_)t_ = 0 (t_ = x__) (2.13)

This is called a_ne parameterization. For a timelike geodesic it corresponds

to e(_) = constant, or

_ / _ + constant (2.14)

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The einbein form of the particle action has the advantage that we can

take the m ! 0 limit to get the action for a massless particle. In this case

_I

_e

= 0 ) ds2 = 0 (m = 0) (2.15)

while (2.6) is unchanged. We still have the freedom to choose e(_) and the

choice e = constant is again called a_ne parameterization.

Summary

Let t_ =

dx_(_)

d_

and _ =

_

1 m 6= 0

0 m = 0

_

.

Then

t _ Dt_ _ D(_)t_ = 0

ds2 = 􀀀_d_2 (2.16)

are the equations of a_nely-parameterized timelike or null geodesics.

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