- #1

Kaguro

- 221

- 57

- Homework Statement
- Derive time dilation and length contraction using invariance of spacetime interval.

- Relevant Equations
- ##ds^2 = dx^2+dy^2+dz^2 - c^2 dt^2##

Deriving time dilation was easy:

Imagine two events in frame O' at the same location.

##ds^2 = -c^2 dt'^2##

The same viewed in O frame is:

##ds^2 = dx^2+dy^2 + dz^2 - c^2 dt^2##

##\Rightarrow dx^2+dy^2 + dz^2 - c^2 dt^2 = -c^2 dt'^2##

##\Rightarrow (\frac{dx}{dt})^2+(\frac{dy}{dt})^2+ (\frac{dz}{dt})^2 - c^2 = -c^2(\frac{dt'}{dt})^2##

But since these events are at the same location in O', the dx, dy, dz is hence the amount by which O' moves in O in time dt.

Therefore,

##v^2 -c^2 = -c^2 (\frac{dt'}{dt})^2##

##\Rightarrow (\frac{dt'}{dt})^2 = 1-\frac{v^2}{c^2}##

##\Rightarrow (\frac{dt'}{dt}) = \sqrt{1-\frac{v^2}{c^2}}##

##\Rightarrow dt = \gamma dt'##

But for length contraction

I choose two events in O which are simultaneous. So dt=0

##ds^2 = dx^2+dy^2 + dz^2 = dx'^2 + dy'^2 + dz'^2 - c^2 dt'^2##

Here I can not equate any of these with the amount by which O' moves in O.

Please help.

P.S.: Why has the website become so weird ? The preview button is shifted and works differently, and I am asked to "submit homework statement" and "relevant equations" twice...

Imagine two events in frame O' at the same location.

##ds^2 = -c^2 dt'^2##

The same viewed in O frame is:

##ds^2 = dx^2+dy^2 + dz^2 - c^2 dt^2##

##\Rightarrow dx^2+dy^2 + dz^2 - c^2 dt^2 = -c^2 dt'^2##

##\Rightarrow (\frac{dx}{dt})^2+(\frac{dy}{dt})^2+ (\frac{dz}{dt})^2 - c^2 = -c^2(\frac{dt'}{dt})^2##

But since these events are at the same location in O', the dx, dy, dz is hence the amount by which O' moves in O in time dt.

Therefore,

##v^2 -c^2 = -c^2 (\frac{dt'}{dt})^2##

##\Rightarrow (\frac{dt'}{dt})^2 = 1-\frac{v^2}{c^2}##

##\Rightarrow (\frac{dt'}{dt}) = \sqrt{1-\frac{v^2}{c^2}}##

##\Rightarrow dt = \gamma dt'##

But for length contraction

I choose two events in O which are simultaneous. So dt=0

##ds^2 = dx^2+dy^2 + dz^2 = dx'^2 + dy'^2 + dz'^2 - c^2 dt'^2##

Here I can not equate any of these with the amount by which O' moves in O.

Please help.

P.S.: Why has the website become so weird ? The preview button is shifted and works differently, and I am asked to "submit homework statement" and "relevant equations" twice...