Start with an inertial reference system S considered to be at rest. Somewhere out there, a particle moves along an arbitrary trajectory. The space coordinates of the particle parametrized by the time measured in the rest system S may be for example X(t),Y(t),Z(t), suppose a clock moves with the particle; during a time interval dt measured by the rest frame clock, The time measured on the clock traveling with the particle will change by
This is the special relativity expression for the proper time of a particle moving at constant velocity. It is a postulate (supported by experiment) that this same expression carries over to a particle moving in an any manner ("clock postulate" see the Relativity FAQ: Does a clock's acceleration affect its timing rate?). What this means is that we don't add acceleration terms and higher deriviatives to the proper time formula. The total elapsed time then is:
Four Dimensional Space-Time Frenet Formulae
Represent the space time location of the particle with the four vector (or quaternion)
From this the tangent vector T(t)
Which is the same as the four velocity. Note that the vector T(t) is dimensionless and time-like. It determines the velocity direction at some point. ds/dt determines the magnitude.
Next the acceleration of the partcle is represented by the Normal vector N
where
Again N is dimensionless,k has units of inverse length. c2 k has units of acceleration.
N determines the direction of the acceleration c2 k determines the magnitude of the acceleration.
The Binormal B is defined as
where tau is
B is dimensionless; the torsion tau has units of inverse length.
The three 4 vectors T(t),N(t) and B(t) are all hemitian. T(t) is time like: T*T=-1; N and B are space like N*N=B*B=1. All three are mutually orthogonal: T*N=T*B=N*B=0. In four dimensional space time, we can define a four vector C(t) that is mutually orthogonal to T,N,B:
Generally in four dimensions, a quaternion that is mutually perpendicular to three quaternions q1,q2,q3 is
(Hamilton quaternions here. The magnitude of Vol3 is volume of the parrallelpiped with edges q1,q2,q3; see Sudbery )
Define the trinormal C(t) then as:
It can be shown that C(t) can also be differentially defined as:
where
We can also show that
These can be summarized as a matrix differential form equation
where
Rev 8/20/0 Copyright ©2000 George Raetz <bestwork.1@pcisys.net>