- #1
Tomer
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Thanks for reading!
I have been given a proof for the lorentz transformations (which I only partly understand) that relied on the two relativity postulates (equivalence of inertial systems and the speed of light being constant) for the case of two standard inertial systems: the system S' moved with velocity v along the x axis, and the systems are parallel.
Now I've been asked to generalize the transformation for the case the the systems are parallel, but the velocity isn't necessarily in the x axis. (a general v vector).
I need to show, that in this case, this equation holds:
1. [itex]\vec{r'}[/itex] = [itex]\vec{r}[/itex] + ([itex]\frac{\vec{v}\cdot\vec{r}}{v^{2}}[/itex]([itex]\beta[/itex] - 1) - [itex]\beta[/itex]t)[itex]\vec{v}[/itex]
2. t' = [itex]\beta[/itex](t - ([itex]\frac{\vec{v}\cdot\vec{r}}{v^{2}}[/itex]))
The development of the transformation for the simpler case (where v = (v,0,0) ) arrives to the point where it shows that the size x2 + y2 + z2 - c2t2 is invariant. It doesn't to that point use the assumption that the systems are parallel.
It is then sugested that from the fact that the space is isotropic y = y' and z = z'.
This is however no longer the case. So that's where, I guess, I need to work with new tools.
By using [itex]\vec{r}[/itex]2 = x2 + y2 + z2 I've tried to make an analogical development, treating |[itex]\vec{r}[/itex]| as x was. I just don't seem to derive the equation, and I don't know how to use the fact that the systems are parallel.
I'm pretty rusty with algebra, I must say, haven't touched it for almost 2 years.
I'd appreciate any help!
Tomer.
Homework Statement
I have been given a proof for the lorentz transformations (which I only partly understand) that relied on the two relativity postulates (equivalence of inertial systems and the speed of light being constant) for the case of two standard inertial systems: the system S' moved with velocity v along the x axis, and the systems are parallel.
Now I've been asked to generalize the transformation for the case the the systems are parallel, but the velocity isn't necessarily in the x axis. (a general v vector).
I need to show, that in this case, this equation holds:
1. [itex]\vec{r'}[/itex] = [itex]\vec{r}[/itex] + ([itex]\frac{\vec{v}\cdot\vec{r}}{v^{2}}[/itex]([itex]\beta[/itex] - 1) - [itex]\beta[/itex]t)[itex]\vec{v}[/itex]
2. t' = [itex]\beta[/itex](t - ([itex]\frac{\vec{v}\cdot\vec{r}}{v^{2}}[/itex]))
Homework Equations
The development of the transformation for the simpler case (where v = (v,0,0) ) arrives to the point where it shows that the size x2 + y2 + z2 - c2t2 is invariant. It doesn't to that point use the assumption that the systems are parallel.
It is then sugested that from the fact that the space is isotropic y = y' and z = z'.
This is however no longer the case. So that's where, I guess, I need to work with new tools.
The Attempt at a Solution
By using [itex]\vec{r}[/itex]2 = x2 + y2 + z2 I've tried to make an analogical development, treating |[itex]\vec{r}[/itex]| as x was. I just don't seem to derive the equation, and I don't know how to use the fact that the systems are parallel.
I'm pretty rusty with algebra, I must say, haven't touched it for almost 2 years.
I'd appreciate any help!
Tomer.