The term metric signature refers to the signature of a metric tensor 
on a smooth manifold 
, a tool which quantifies the numbers of positive, zero, and
negative infinitesimal distances of tangent vectors
in the tangent bundle of 
and which is most easily defined in terms of the signatures
of a number of related structures.
Most commonly, one identifies the signature of a metric tensor 
with the signature
of the quadratic form 
induced by 
on any of the tangent spaces 
for points 
. Indeed, given an orthogonal vector basis 
for any tangent space 
, the action of 
on arbitrary vectors 
and 
in 
is given by
 |
(1)
|
whereby the signature of 
is defined to be the signature of any of the forms 
, i.e., the ordered triple 
of positive, negatives, and zero values for the inner
products 
.
This value is well-defined due to the fact that the signature of 
remains the same for all points 
in 
. For non-degenerate quadratic forms, the value 
will always satisfy 
, whereby the signature of 
will be the ordered pair 
.
Alternatively, one can view the signature of a metric tensor in terms of matrix signatures. For an 
-dimensional differentiable manifold 
whose tangent space 
has basis 
, the tensor 
induces an 
matrix 
whose 
-entry 
is given by
 |
(2)
|
Because the signatures of the matrices 
are the same for all 
, one may define the signature of the metric tensor 
to be the matrix signature of 
for any 
. Moreover, by rewriting 
on any pointwise
tangent space 
,
it follows that this definition is equivalent to the quadratic signature definition
mentioned above.
In many contexts, one finds it beneficial to express the metric tensor 
itself as a diagonal matrix,
usually denoted 
and whose components are sometimes called the "metric coefficients" associated
to 
.
In such circumstances, the signature of 
is precisely the matrix signature of 
. For example, in Minkowski
space,
![eta_(ij)=[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 -1],](/images/equations/MetricSignature/NumberedEquation3.svg) |
(3)
|
which corresponds to the fact that the metric tensor in 4-dimensional Lorentzian space has signature 
(Misner et al. 1973). This viewpoint requires one
to define a local basis for the action of 
, but by Sylvester's inertia
law, this definition is well-defined independent of the choice of basis vectors.
In an 
-dimensional
pseudo-Euclidean space, the metric tensor
is often denoted 
and its signature is defined to be the ordered pair 
where 
, respectively 
, denotes the number positive, respectively negative, terms
in the expansion of 
:
 |
(4)
|
The transition between the 
notation and the 
notation is summarized by the identity
 |
(5)
|
where 
are suitably-chosen basis vectors (Snygg 2012).
For 
-dimensional
Euclidean spaces, the metric signature is 
. For 
-dimensional Lorentzian space 
, the metric signature is 
, e.g., 
(as above) for the Minkowski space of special relativity.
Note that in (1) above, the order of the positive- and negative-squared
terms is sometimes swapped, under which convention the signature would be given by
, e.g., 
for 
-dimensional Euclidean spaces and 
for 
-dimensional Lorentzian spaces. This convention may also carry
over to the case where 
is a matrix 
, e.g., in equation (2) above where
may be replaced
by 
.
General tensors of signature 
come about in the study of Clifford
algebras.
