- #1
SiennaTheGr8
- 497
- 195
I thought this was nice. In what follows, ##c=1##.
For a particle undergoing constant proper acceleration ##\alpha## in the positive ##x##-direction, an inertial observer can use:
##x (\tau) = x_0 + \dfrac{\gamma(\tau) - \gamma_0}{\alpha}##,
where the Lorentz factor is:
##\gamma (\tau) = \cosh \phi (\tau)##
and the rapidity is:
##\phi (\tau) = \phi_0 + \alpha \tau##
(##\tau## is the particle's proper time, naught subscripts indicate values at ##\tau = 0##).
When ##\phi(\tau) \ll 1##, the Taylor series for the ##\cosh## function gives ##\cosh \phi(\tau) \approx 1 + \frac{1}{2}(\phi_0 + \alpha \tau)^2##:
##x (\tau) \approx x_0 + \dfrac{ \left[ 1 + \frac{1}{2} (\phi_0 + \alpha \tau)^2 \right] - \gamma_0}{\alpha} \\[20pt] \qquad = x_0 + \dfrac{ 1 + \frac{1}{2} \phi^2_0 - \gamma_0}{\alpha} + \phi_0 \tau + \dfrac{1}{2} \alpha \tau^2 .##
If also ##\phi_0 \ll 1##, then we can use ##\tau \approx t## (assume ##t_0=0##), ##\phi_0 \approx v_0##, ##\phi^2_0 \approx 0##, ##\gamma_0 \approx 1##, and ##\alpha \approx a##:
##x (\tau) \approx x_0 + v_0 t + \dfrac{1}{2} a t^2##.
Usually this reduction is done from a simplified version of the ##x(\tau)## function (##\phi_0 = 0##), so that the ##v_0 t## term isn't included.
For a particle undergoing constant proper acceleration ##\alpha## in the positive ##x##-direction, an inertial observer can use:
##x (\tau) = x_0 + \dfrac{\gamma(\tau) - \gamma_0}{\alpha}##,
where the Lorentz factor is:
##\gamma (\tau) = \cosh \phi (\tau)##
and the rapidity is:
##\phi (\tau) = \phi_0 + \alpha \tau##
(##\tau## is the particle's proper time, naught subscripts indicate values at ##\tau = 0##).
When ##\phi(\tau) \ll 1##, the Taylor series for the ##\cosh## function gives ##\cosh \phi(\tau) \approx 1 + \frac{1}{2}(\phi_0 + \alpha \tau)^2##:
##x (\tau) \approx x_0 + \dfrac{ \left[ 1 + \frac{1}{2} (\phi_0 + \alpha \tau)^2 \right] - \gamma_0}{\alpha} \\[20pt] \qquad = x_0 + \dfrac{ 1 + \frac{1}{2} \phi^2_0 - \gamma_0}{\alpha} + \phi_0 \tau + \dfrac{1}{2} \alpha \tau^2 .##
If also ##\phi_0 \ll 1##, then we can use ##\tau \approx t## (assume ##t_0=0##), ##\phi_0 \approx v_0##, ##\phi^2_0 \approx 0##, ##\gamma_0 \approx 1##, and ##\alpha \approx a##:
##x (\tau) \approx x_0 + v_0 t + \dfrac{1}{2} a t^2##.
Usually this reduction is done from a simplified version of the ##x(\tau)## function (##\phi_0 = 0##), so that the ##v_0 t## term isn't included.
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