# More physics behind the concepts of length and rest length

1. Dec 12, 2008

### bernhard.rothenstein

Consider a particle that moves with speed u relative to the inertial reference frame I and with speed u’ relative to the inertial reference frame I’. Let V be the speed of I’ relative to I. All speeds show in the positive direction of the permanently overlapped OX(O’X’) axes. The speeds mentioned above are related by the transformation equation for parallel speeds
u=(u’+V)/(1+u’V/cc) (1)
The particle starts at the origin of time in the two frames (t=t’=0) from the origins O and O’ of the two frames located at that very moment at the same point in space. After a given time of motion the particle travels a distance
L=u(t-0) (2)
when detected from I and a distance
L’=u’(t’-0) (3)
when detected from I’. Combining (1) and (2) as
(L/t)=(L’/t’)(u/u’)=(L’/t’)(1+V/u’)/(1+Vu’/cc) (4)
we could consider that
L=L’Г(V)(1+V/u’) (5)
t=t’Г(V)(1+Vu’/cc) (6)
where Г(V) represents a function of the relative speed but not of the physical quantities involved in the transformation process (linearity). Imposing the conditions that (6) accounts for the time dilation we obtain that Г(V)=g(V) represents the Lorentz factor and so the two lengths defined above are related by L=g(L’+Vt’) (7)
an equation that relates the two lengths measured from I and I’ respectively. If the particle is at rest in I’ then L’=0 (7) leading to
L(u’=0)=gVt’ (8)
Defining Vt’=L(0) as rest length [(t’-0) proper time interval] we obtain that rest length and length are related by
L(u’=0)=gL(0) (9)
Please let me know if you detect violations of special relativity theory.