Textbooks tend to only give you the Lorentz transformation along a single coordinate axis, but it is not always convenient to keep redefining the coordinate system for problems with several different velocities. To derive a more general formula using vector notation, use the idea that the part of a position vector $\vec{r}$ that is parallel to the velocity is the part that is changed by the transformation, while the part that is perpendicular to the velocity is unchanged.
- Assume that you wish to transform from your inertial frame (the ($\vec{r}, ct$) frame] to the “primed” inertial frame ($\vec{r}’, ct’$) moving with velocity $\vec{v} = c\vec{\beta}$ that points in some arbitrary direction (e.g., it has an $x$, $y$ and $z$ component). You should find the following: $c t’ = \gamma (c t − \vec{r} \cdot \vec{\beta})$ and $\vec{r}’=\vec{r}+(\gamma −1)(\vec{r}\cdot \hat{\beta})\hat{\beta}−\gamma ct \vec{\beta}$.
- Show that in the case that the velocity is in the x-direction, you get back the usual transformation.
- How would you write this using 4-vector notation? What is $\Lambda_{\nu}^{\mu}$?
We derived the transformation of velocity in 1D (i.e., when there is one frame moving at a speed $v$ in the $x$ direction, $S’$ relative to the other, $S$) using the Lorentz transformations. We found that
\[u = \dfrac{u' + v}{1+\frac{u'v}{c^2}}\]
- Derive the relationship between the velocity components in each frame (for both the $y$ and $z$ directions) for the same scenario. Recall that length measurements will be the same in both frames!
- Derive the relationship between the acceleration measured in the $S$ frame and the $S’$ frame in just the x-direction.
- Show check the limits of your results in part 2 when $v$ approaches 0. Does you result make sense? What about when $v$ approaches $c$?
3. Rapidity
It is common in nuclear physics to talk about “rapidity” of a particle, defined as an angle $\phi = \cosh^{-1} \gamma$ (here $\gamma$ is the usual relativistic gamma factor, and that’s an inverse hyperbolic $\cosh$).
- Prove that the usual relativistic $\beta = v/c$ is given by $\beta = \tanh \phi$, and then show $\beta \gamma = \sinh \phi$. With these, rewrite the Lorentz transformations in matrix form entirely in terms of the rapidity angle. The result you get might remind you of a rather different kind of transformation, please comment!
- Suppose that observer B has rapidity $\phi_1$ as measured by observer A, and C has rapidity $\phi_2$ as observed by B (both velocities are on the x-axis). Show that the rapidity of C as measured by A is just $\phi_1 + \phi_2$, i.e. rapidities “add” (unlike velocities, which do not “properly” add in relativity!)
Here is a hyperbolic identity you might find useful:
\(\tanh(a+b) = \dfrac{\tanh a + \tanh b}{1 + \tanh a \tanh b}\).
4. Invariance of the space-time interval
Prove that the interval between two events is Lorentz Invariant:
\[I = \Delta {x}'_{\mu} \Delta {x}'^{\mu} = \Delta {x}_{\mu} \Delta {x}^{\mu}\]
Recall that the Lorentz transformation is $\Delta {x}’^{\mu} = \Lambda_{\nu}^{\mu} \Delta x^{\nu}$.