Metric Signature
The
signature
of a
metric tensor
(or more generally a nondegenerate symmetric
bilinear form
, thought of as
quadratic form
) is the number of positive and negative
eigenvalues
of the metric. That is, the corresponding real
symmetric matrix
is
diagonalised
, and the diagonal entries of each sign counted. If the matrix is
n
×
n
, the possible number of positive signs may take any value
p
from 0 to
n
. The signature may be denoted either by a pair of integers such as (
p
,
q
), or as an explicit list such as
-+++
. The signature is said to be
indefinite
if both
p
and
q
are non-zero. A
Riemannian metric
is a metric with a
(positive) definite
signature. A
Lorentzian metric
is one with signature (
p
,1) (or sometimes (1,
q
)).
See also
:
pseudo-Riemannian manifold
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