'local' closed timelike curves?

[principalbundles@yahoo.it]

Hi all,
is it true that in any spacetime M (aka real smooth 4-dim connected
Hausdorff paracompact manifold without boundary with smooth
lorentzian
metric) every point in M admits a neighborhood which contains no
closed timelike curve? If not, any counterexample?
Thank you.

[Marc Nardmann]

principalbundles@yahoo.it wrote:

> is it true that in any spacetime M (aka real smooth 4-dim connected
> Hausdorff paracompact manifold without boundary with smooth
> lorentzian
> metric) every point in M admits a neighborhood which contains no
> closed timelike curve? If not, any counterexample?

For every n, every point p in an n-dimensional (not necessarily
Hausdorff, not necessarily paracompact, not necessarily connected) C^1
manifold equipped with a C^0 Lorentzian metric g admits a neighbourhood
without closed C^1 timelike curves. The case n<1 is trivial, so let us
assume n>=1.

Choose coordinates (x_1,...,x_n) on a neighbourhood of p such that the
corresponding coordinate vector fields \partial_2,...,\partial_n are
spacelike in the point p. Since g is continuous, there exists a smaller
neighbourhood U of p such that \partial_2,...,\partial_n are spacelike
on U. Assume that there exists a C^1 timelike path w: [0,1] --> U with
w(0) = w(1).

Then its x_1-component w_1 is a C^1 map from [0,1] to R such that
w_1(0)=w_1(1). There exists a number t with 0<t<1 where w_1 is maximal
or minimal and therefore satisfies w_1'(t)=0. This means that the vector
w'(t) is spanned by \partial_2,...,\partial_n, and is thus spacelike or
zero. This is a contradiction to the assumption that w is timelike. So U
contains no closed timelike curve.


-- Marc Nardmann