  Hawking's stable causality implies Sorkin and Woolgar's K-causality. The work
investigates the possible equivalence between the two causality requirements,
an issue which was first considered by H. Seifert and then raised again by R.
Low after the introduction of K-causality. First, a new proof is given that a
spacetime is stably causal iff the Seifert causal relation is a partial order.
It is then shown that given a K-causal spacetime and chosen an event, the light
cones can be widened in a neighborhood of the event without spoiling
K-causality. The idea is that this widening of the light cones can be continued
leading to a global one. Unfortunately, due to some difficulties in the
inductive process the author was not able to complete the program for a proof
as originally conceived by H. Seifert. Nevertheless, it is proved that if
K-causality coincides with stable causality then in any K-causal spacetime the
K future coincides with the Seifert future. Explicit examples are provided
which show that the K^+ future may differ from the Seifert relation in causal
spacetimes.
