nLab
pseudo-Riemannian metric

Idea

The notion of pseudo-Riemannian metric is a slight variant of that of Riemannian metric.

Where a Riemannian metric is governed by a positive-definite bilinear form, a Pseudo-Riemannian metric is governed by an indefinite bilinear form.

Definition

A pseudo-Riemannian “metric” is a nondegenerate quadratic form on a real vector space ${ℝ}^n$. A Riemannian metric is a positive-definite quadratic form on a real vector space. The data of such a quadratic form may be equivalently given by a nondegenerate symmetric bilinear pairing $⟨,⟩$ on ${ℝ}^n$.

A pseudo-Riemannian metric

$$Q:{ℝ}^n\to ℝ$$Q: \mathbb{R}^n \to \mathbb{R}

can always be diagonalized: there exists a basis $e_1,\dots ,e_n$ such that

$$Q(\sum _{1\le i\le n}{x}_i{e}_i)=x_1^2+\dots +x_p^2-x_{p+1}^2-\dots -x_n^2$$Q(\sum_{1 \leq i \leq n} x_i e_i) = x_1^2 + \ldots + x_p^2 - x_{p+1}^2 - \ldots - x_n^2

where the pair $(p,n-p)$ is called the signature of the form $Q$. Pseudo-Riemannian metrics on ${ℝ}^n$ are classified by their signatures; thus we have a standard metric of signature $(p,n-p)$ where $\{e_1,\dots ,e_n\}$ is the standard basis of ${ℝ}^n$.

More generally, there is a notion of pseudo-Riemannian manifold (of type $(p,n-p)$, which is an $n$-dimensional manifold $M$ equipped with a global section

$$\sigma :M\to S^2(T^{*}M)$$\sigma: M \to S^2(T^* M)

of the bundle of symmetric bilinear forms over $M$, such that each $\sigma (x)$ is a nondegenerate form on the tangent space $T_x(M)$.

Certain theorems of Riemannian geometry carry over to the more general pseudo-Riemannian setting; for example, pseudo-Riemannian manifolds admit Levi-Civita connections, or in other words a unique notion of covariant differentiation of vector fields

$$\nabla :(X,Y)\mapsto {\nabla }_X(Y)$$\nabla: (X, Y) \mapsto \nabla_X(Y)

such that

$$X\cdot ⟨Y,Z⟩=⟨{\nabla }_{X}Y,Z⟩+⟨Y,{\nabla }_{X}Z⟩$$X \cdot \langle Y, Z\rangle = \langle \nabla_X Y, Z\rangle + \langle Y, \nabla_X Z\rangle

$$[X,Y]={\nabla }_{X}Y-{\nabla }_{Y}X$$[X, Y] = \nabla_X Y - \nabla_Y X

In that case, one may define a notion of geodesic in pseudo-Riemannian manifolds $M$, and we have a notion of “distance squared” between the endpoints along any geodesic path $\alpha :[0,1]\to M$ (which might be a negative number of course). The term “pseudo-Riemannian metric” may refer to such distances in general pseudo-Riemannian manifolds. (I guess.)

Is there an accepted notion of “distance squared” between two points in a pseudo-Riemannian manifold, in cases where there are multiple geodesics between them? How does one choose? (Edit: might one choose the value which is least in absolute value?)

A typical example of pseudo-Riemannian manifold is a Lorentzian manifold, where the metric is of type $(1,n-1)$. This is particularly so in the case $n=4$, where such manifolds are the mathematical backdrop for studying general relativity and cosmological models.

• The terminology “metric” is not optimal of course: the values of the quadratic form would need to be nonnegative to avoid terminological conflict with metric as it is more commonly understood (and even in that case, the values of “metric” refer to the square of the metric rather than the metric itself). Caveat lector.