- #1
Cancer
- 14
- 0
Hi everybody,
I know that there are a lot of threads in this forum about Rindler coordinates but none of them have helped me
I'll explain you my problem. First of all, my coordinates [itex](x^0,x)[/itex] (Cartesian coord., where [itex]x^0=ct[/itex]) are related to the Rindler coordinates [itex] (\omega ^0,\omega)[/itex] (where [itex] \omega ^0[/itex] is the proper time) as follows:
[tex] x = \frac{c^2}{g}\left[ \cosh \left( \frac{g\omega^0}{c^2}\right)-1\right]+\omega \cosh \left( \frac{g\omega^0}{c^2}\right)[/tex]
[tex] x^0 = \sinh \left( \frac{g\omega^0}{c^2}\right) \left( \frac{c^2}{g}+\omega \right)[/tex]
Which are defined in an exercise I'm trying to solve, and c is the speed of light and g is the constant acceleration. It says that I have to proof that some signals sent from an inertial frame I won't never reach an observer which is at rest in the frame R i.e., which is at constant [itex]\omega[/itex].
So the first thing I do is isolate [itex]\omega^0[/itex] from the second equation, getting:
[tex]\frac{g\omega^0}{c^2}=\sinh^{-1} \left( \frac{x^0}{\frac{c^2}{g}+\omega}\right)[/tex]
We put this into the first equation, using this mathematical relation:
[tex]\cosh (\sinh^{-1}(x))=\sqrt{1+x^2}[/tex]
Finally we arrive to
[tex]x= \left(\left(\frac{c^2}{g}+\omega\right)^2 +(x^0)^2\right)^{1/2} -\frac{c^2}{g}[/tex]
If I have properly done the math, that's the path which will follow an observer at rest in the R frame in the [itex](x,x^0)[/itex] coordinates. BUT I don't see why light signals won't reach that observer! If I take x as ct (i.e. the diagonal of the light cone, a photon) I will always be able to find a solution for [itex] x^0[/itex], i.e., the light will eventually reach the observer.
I don't know what's happening here, I've read all I've found in Google about Rindler coordinates but nothing helps me, because nobody uses this definition for the Rindler coordinates...
I hope some of you could help me with this!
Thanks,
Victor
I know that there are a lot of threads in this forum about Rindler coordinates but none of them have helped me
I'll explain you my problem. First of all, my coordinates [itex](x^0,x)[/itex] (Cartesian coord., where [itex]x^0=ct[/itex]) are related to the Rindler coordinates [itex] (\omega ^0,\omega)[/itex] (where [itex] \omega ^0[/itex] is the proper time) as follows:
[tex] x = \frac{c^2}{g}\left[ \cosh \left( \frac{g\omega^0}{c^2}\right)-1\right]+\omega \cosh \left( \frac{g\omega^0}{c^2}\right)[/tex]
[tex] x^0 = \sinh \left( \frac{g\omega^0}{c^2}\right) \left( \frac{c^2}{g}+\omega \right)[/tex]
Which are defined in an exercise I'm trying to solve, and c is the speed of light and g is the constant acceleration. It says that I have to proof that some signals sent from an inertial frame I won't never reach an observer which is at rest in the frame R i.e., which is at constant [itex]\omega[/itex].
So the first thing I do is isolate [itex]\omega^0[/itex] from the second equation, getting:
[tex]\frac{g\omega^0}{c^2}=\sinh^{-1} \left( \frac{x^0}{\frac{c^2}{g}+\omega}\right)[/tex]
We put this into the first equation, using this mathematical relation:
[tex]\cosh (\sinh^{-1}(x))=\sqrt{1+x^2}[/tex]
Finally we arrive to
[tex]x= \left(\left(\frac{c^2}{g}+\omega\right)^2 +(x^0)^2\right)^{1/2} -\frac{c^2}{g}[/tex]
If I have properly done the math, that's the path which will follow an observer at rest in the R frame in the [itex](x,x^0)[/itex] coordinates. BUT I don't see why light signals won't reach that observer! If I take x as ct (i.e. the diagonal of the light cone, a photon) I will always be able to find a solution for [itex] x^0[/itex], i.e., the light will eventually reach the observer.
I don't know what's happening here, I've read all I've found in Google about Rindler coordinates but nothing helps me, because nobody uses this definition for the Rindler coordinates...
I hope some of you could help me with this!
Thanks,
Victor