Models and symmetries 1253.5.1.1 Killing vectors
A space or spacetimesymmetry,orisometry, is a transformation that drags the
metric along a certain congruence of curves into itself. The generating vector field
ξiof such curves is called aKilling vector (field)(or ‘KV’), and obeys Killing’s
equations,
(Lξg)ij= 0 ⇔∇(iξj)= 0 ⇔∇iξj=−∇jξi, (3.67)
whereLXis theLie derivative. By the Ricci identities for a KV, this implies the
curvature equation:
∇i∇jξk=Rmijkξm, (3.68)
and hence the infinite series of further equations that follows by taking covariant
derivatives of this one, e.g.
∇l∇i∇jξk=(∇lRmijk)ξm+Rmijk∇lξm. (3.69)The set of all KVs forms a Lie algebra with a basis{ξa},a= 1 , 2 ,...,r,of
dimensionr≤^12 n(n− 1 ). ξaidenotes the components with respect to a local
coordinate basis,a,bandclabel the KV basis andi, jandkthe coordinate
components. Any KV can be written in terms of this basis, withconstant
coefficients. Hence, if we take the commutator[ξa,ξb]of two of the basis KVs,
this is also a KV, and so can be written in terms of its components relative to the
KV basis, which will be constants. We can write the constants asCcab, obtaining
[ξa,ξb]=Ccabξc, Cabc=Ca[bc]. (3.70)By the Jacobi identities for the basis vectors, these structure constants must satisfy
Cae[bCecd]= 0 (3.71)(which is just equation (3.60) specialized to a set of vectors with constant
commutation functions). These are the integrability conditions that must
be satisfied in order that the Lie algebra exist in a consistent way. The
transformations generated by the Lie algebra form a Lie group of the same
dimension (see Eisenhart [24] or Cohn [11]).
Arbitrariness of the basis: We can change the basis of KVs in the usual way;
ξa′=λa′aξa⇔ξai′=λa′aξai, (3.72)where theλa′aare constants with det(λa′a)=0, so unique inverse matricesλa
′
a
exist. Then the structure constants transform as tensors:
Cc′
a′b′=λ
c′
cλa′
aλ
b′
bCc
ab. (3.73)Thus the possible equivalence of two Lie algebras is not obvious, as they may be
given in different bases.