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 ${\displaystyle g^{\mu \nu }}$, the condition of being Stanchian is:

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

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

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,

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

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

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).