3
$\begingroup$

I wonder what are some statements that although they can be formulated for pseudoriemannian manifolds of arbitrary signature they turn out to be true only in the lorentzian case.

I admit things like: "no analog to cauchy hypersuface can be defined in arbitrary pseudoriemanniann manifolds, just in lorentzian ones". I don't know if this last statement is true, it's just to exemplify.

Another one could be: the equivalent topological condition for a given compact manifold to admit a pseudoriemannian metric of a given signature is to have 0 euler characteristic. I think this is true only for lorentzian manifolds.

Thanks in advance!

$\endgroup$

2 Answers 2

8
$\begingroup$

A pseudo-Riemannian manifold of signature $(p,q)$ with $p, q\geq 2$ certainly does not admit Cauchy hypersurfaces, since spacelike submanifolds have dimension at most $p$. However, you could define a notion of "Cauchy $p$-surface" as a spacelike submanifold of dimension $p$ intersecting any inextensible timelike $q$-disc at one point. Though it is not explicitly stated, recent progresses in the study of discrete group actions on pseudo-hyperbolic spaces tend to construct higher signature analogs of globally hyperbolic Cauchy compact anti-de Sitter spacetimes.

Concerning the Euler characteristic: A manifold $M$ admits a pseudo-Riemannian metric of signature $(p,q)$ if and only if its tangent bundle $TM$ admits a splitting as $E^{(p)} \oplus F^{(q)}$. This implies the vanishing of the (rational) Euler characteristic when $p$ or $q$ is odd.

More generally, the specificities of Lorentzian geometry should really come from the fact that the "timelike dimension" is $1$. For instance, there cannot be a distinction between "past" and "future", hence no notion of causality in higher signature.

$\endgroup$
4
  • $\begingroup$ Thank you. Can you explain why is true your last sentence please? $\endgroup$ Jan 13 at 10:05
  • 2
    $\begingroup$ In lorentzian signature, the set of non-zero tangent vectors on which the metric is non-positive has two connected components. In higher signature, it is connected. $\endgroup$ Jan 13 at 13:06
  • $\begingroup$ Also if they have closed timelike curves? $\endgroup$ Jan 13 at 21:31
  • $\begingroup$ Yes, now you can add physical conditions. My point is that these conditions don't really make sense in higher signature. To put it differently, in signature $(p,q)$, $q\geq 2$, there are closed "timelike" curves in any neighbourhood of a point. $\endgroup$ Jan 14 at 12:29
5
$\begingroup$

The reverse Cauchy-Schwarz inequality: for any two causal vectors (orientation in this `squared' formulation is not relevant) $v, w$ we have $$ \vert g(v,w)\vert^2\ge g(v,v) g(w,w) $$ is false in pseudo-Riemannian spaces of signature $(p,q)$, $p,q\ge 2$. The proof uses a certain convexity of the unit sphere (not its compactness), and this property holds only for the Lorentzian signature.

$\endgroup$
1
  • $\begingroup$ I like this one. It can be used to prove that if $v$ is timelike and $w$ is orthogonal to it then $w$ is lightlike or spacelike. Thanks! $\endgroup$ Jan 24 at 19:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.