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Metric Signature


The term metric signature refers to the signature of a metric tensor g=g_(ij) on a smooth manifold M, a tool which quantifies the numbers of positive, zero, and negative infinitesimal distances of tangent vectors in the tangent bundle of M 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 g with the signature of the quadratic form Q_p=<·,·>_p induced by g on any of the tangent spaces T_pM for points p in M. Indeed, given an orthogonal vector basis e_p^1,...,e_p^n for any tangent space T_pM, the action of Q_p on arbitrary vectors v=sum_(i=1)^(n)v^ie_p^i and w=sum_(i=1)^(n)w^ie_p^i in T_pM is given by

 Q_p(v,w)=sum_(i=1)^nv^iw^i<e_p^i,e_p^i>_p,
(1)

whereby the signature of g is defined to be the signature of any of the forms Q_p, i.e., the ordered triple (p,q,z) of positive, negatives, and zero values for the inner products <e_p^i,e_p^i>_p. This value is well-defined due to the fact that the signature of Q_p remains the same for all points p in M. For non-degenerate quadratic forms, the value z will always satisfy z=0, whereby the signature of g will be the ordered pair (p,q).

Alternatively, one can view the signature of a metric tensor in terms of matrix signatures. For an n-dimensional differentiable manifold M whose tangent space T_pM has basis e_p^1,...,e_p^n, the tensor g_(ij) induces an n×n matrix A_p whose (i,j)-entry a_(ij)^p is given by

 a_(ij)^p=g_(ij)^p(e_p^i,e_p^j).
(2)

Because the signatures of the matrices A_p are the same for all p in M, one may define the signature of the metric tensor g to be the matrix signature of A_p for any p. Moreover, by rewriting g_(ij)^p(·,·)=Q_p(·,·) on any pointwise tangent space T_p(M), 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 g_(ij) itself as a diagonal matrix, usually denoted eta_(ij)=g(e_i,e_j) and whose components are sometimes called the "metric coefficients" associated to g. In such circumstances, the signature of g_(ij) is precisely the matrix signature of eta_(ij). For example, in Minkowski space,

 eta_(ij)=[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 -1],
(3)

which corresponds to the fact that the metric tensor in 4-dimensional Lorentzian space has signature (3,1) (Misner et al. 1973). This viewpoint requires one to define a local basis for the action of g, but by Sylvester's inertia law, this definition is well-defined independent of the choice of basis vectors.

In an n-dimensional pseudo-Euclidean space, the metric tensor is often denoted (ds)^2 and its signature is defined to be the ordered pair (p,q) where p, respectively q, denotes the number positive, respectively negative, terms in the expansion of (ds)^2:

 (ds)^2=sum_(j=1)^p(dx^j)^2-sum_(j=p+1)^n(dx^j)^2.
(4)

The transition between the g_(ij) notation and the ds notation is summarized by the identity

 (ds)^2=sum_(i=1)^nsum_(j=1)^ng_(ij)(u^1,...,u^n)du^idu^j
(5)

where du^1,...,du^n are suitably-chosen basis vectors (Snygg 2012).

For n-dimensional Euclidean spaces, the metric signature is (n,0). For n-dimensional Lorentzian space R^(n-1,1), the metric signature is (n-1,1), e.g., (3,1) (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 (q,p), e.g., (0,n) for n-dimensional Euclidean spaces and (1,n-1) for n-dimensional Lorentzian spaces. This convention may also carry over to the case where g_(ij) is a matrix eta_(ij), e.g., in equation (2) above where eta_(ij)=diag(1,1,1,-1) may be replaced by eta_(ij)=diag(-1,1,1,1).

General tensors of signature (p,q) come about in the study of Clifford algebras.


See also

Clifford Algebra, Diagonal Matrix, Lorentzian Manifold, Lorentzian Space, Matrix Signature, Metric Tensor, Minkowski Space, Orthogonal Basis, p-Signature, Pseudo-Euclidean Space, Quadratic, Quadratic Form, Quadratic Form Rank, Quadratic Form Signature, Smooth Manifold, Sylvester's Inertia Law, Sylvester's Signature, Tangent Bundle, Tangent Space, Tangent Vector, Vector Basis

This entry contributed by Christopher Stover

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1105, 2000.Marsden, J. E.; Ratiu, T.; and Abraham, R. Manifolds, Tensor Analysis, and Applications, 3rd ed. Springer-Verlag Publishing Company, 2002.Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. New York: Springer, 2006.Snygg, J. A New Approach to Differential Geometry using Clifford's Geometric Algebra. New York: Springer Science+Business Media, 2012.

Cite this as:

Stover, Christopher. "Metric Signature." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/MetricSignature.html

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