Stanchian

The Stanchian of a system is the condition by which all causal paths in the metric of the system tend to zero.

For a system S with metric $g^{\mu \nu }$, the condition of being Stanchian is:

$\int _{\omega }f_{\mu }\partial _{\nu }(g^{\mu \nu }f_{\nu })\rightarrow 0$

as $\omega \rightarrow o(k\cdot f)$.

where k are the "heights" of the weighted path, according to its time-like affinity.

Note that all heights in a Stanchian system tend to zero. So, Stanchian systems are self-Stanchian when the paths are themselves the heights of the system.

The condition of being Stanchian is frame-independent. Thus, in all Stanchian systems,

$J\;o(k\cdot \epsilon _{s})\rightarrow 0,$

where $J$ is the Jacobian of the transformations $\epsilon _{s}$.

The Stanchian was first defined by Xela M. Alakas by considering tachyonic signals in F-theory, but was given the modern form in terms of differential geometry by Agellio Stanch (both Cretans).