2.2 Symmetries and Killing Vectors

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Consider the transformation

x_ ! x_ 􀀀 _k_(x); (e ! e) (2.17)

Then (Exercise)

I [x; e]! I [x; e] 􀀀

_

2

Z _B

_A

d_ e􀀀1x__x_ _ ($kg)__ + O

􀀀

_2_

(2.18)

where

($kg)__ = k_g__;_ + k_

;_g__ + k_

;_g__ (2.19)

= 2D(_k_) (Exercise) (2.20)

Thus the action is invariant to _rst order if

$kg = 0 (2.21)

A vector _eld k_(x) with this property is a Killing vector _eld. k is associated

with a symmetry of the particle action and hence with a conserved

charge. This charge is (Exercise)

Q = k_p_ (2.22)

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where p_ is the particle's 4-momentum.

p_ =

@L

@x__ = e􀀀1x_ _g__ (2.23)

= m

dx_

d_

g__ when m 6= 0 (2.24)

Exercise Check that the Euler-Lagrange equations imply

dQ

d_

= 0

Quantize, p_ ! 􀀀i@=@x_ _ 􀀀i@_. Then

Q ! 􀀀ik_@_ (2.25)

Thus the components of k can be viewed as the components of a di_erential

operator in the basis f@_g.

k _ k_@_ (2.26)

It is convenient to identify this operator with the vector _eld. Similarly for

all other vector _elds, e.g. the tangent vector to a curve x_(_) with a_ne

parameter _.

t = t_@_ =

dx_

d_

@_ =

d

d_

(2.27)

For any vector _eld, k, local coordinates can be found such that

k = @=@_ (2.28)

where _ is one of the coordinates. In such a coordinate system

$kg__ =

@

@_

g__ (2.29)

So k is Killing if g__ is independent of _.

e.g. for Schwarzschild @tg__ = 0, so @=@t is a Killing vector _eld. The

conserved quantity is

mk_ dx_

d_

g__ = mg00

dt

d_

= 􀀀m" (" = energy/unit mass) (2.30)

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