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The group structure is not commutative. It defined by η 00 = 1, η =−1 when i> 0 and
ii
is compatible with the homotopy equivalence η ij = 0 when i = j. An endomorphism α :
m
m
relation so that it induces the group structure of R → R is an element of the Lorentz group if
the homotopy group π (X, x ) of X (based at x ). and only if
0 0 0
Let us now consider the homotopy groups
a b j
π (X, x ) and π (X, x ) at two different points A η ab A = η ij α(E ) = A E .
i
i
j
j
i
0 0 0 1
of X and a path γ : I → X connecting the two
points, i.e., γ(0) = x and γ(1) = x . We can Lorentzian manifold A pair (M, g) formed
0 1
define a group isomorphism i between π (X, x ) by a manifold M and a Lorentzian metric g on
0
0
and π (X, x ) given by M, i.e., a pseudo-Riemannian metric of signature
1
0
either (1,m − 1) or (m − 1, 1).
−1
i(λ) = γ ∗λ∗γ ∈ π (X, x ) λ ∈ π (X, x )
0
0
0
1
where γ −1 is the path γ −1 (t) = γ(1 − t). luminescence Spontaneous emission of
radiation from an electronically or vibrationally
Lorentz group The special orthogonal excited species not in thermal equilibrium with
m
group SO(1,m − 1) of isometries of R with its environment.
the standard indefinite metric η of signature
(1,m−1). LetuschooseanorthonormalbasisE i lyate ion The anion produced by hydron
m
i
in R , the metric is expressed by η = η E ⊗E j removal from a solvent molecule. For example,
ij
i
(where E is the dual basis). The matrix η is the hydoxide ion is the lyate ion of water.
ij

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