Velocity is a defined quantity:
$$\mathbf{u}=\dfrac{\Delta\mathbf{r}}{\Delta t} = \langle \dfrac{\Delta x}{\Delta t},\dfrac{\Delta y}{\Delta t},\dfrac{\Delta z}{\Delta t}\rangle$$
In another inertial frame, seen to be moving to the right, parallel to x, observers see:
$$\mathbf{u'}=\dfrac{\Delta\mathbf{r'}}{\Delta t'} = \langle \dfrac{\Delta x'}{\Delta t'},\dfrac{\Delta y'}{\Delta t'},\dfrac{\Delta z'}{\Delta t'}\rangle$$
Is velocity a 4-vector?
1. Yes
2. No
Note:
* Correct answer: B
Which of the following equations is the correct way to write out the Lorentz scalar product?
1. $a \cdot b = -a^0b^0 + a^1b^1 + a^2b^2 + a^3b^3$
2. $a \cdot b = a_0b^0 + a_1b^1 + a_2b^2 + a_3b^3$
3. $a \cdot b = a_{\nu}b^{\nu}$
4. None of these
5. All three are correct
Note:
* Correct Answer: E
Imagine this quantity:
$$u^{\mu} \equiv \begin{pmatrix}c\\\ \frac{\Delta x}{\Delta t}\\\ \frac{\Delta y}{\Delta t}\\\ \frac{\Delta z}{\Delta t}\end{pmatrix}
$$
Is this quantity a 4-vector?
1. Yes, and I can say why.
2. No, and I can say why.
3. None of the above.
Note:
* Correct answer: B
Imagine this quantity:
$$\eta^{\mu} \equiv \frac{1}{\Delta \tau}\begin{pmatrix}ct\\\ \Delta x\\\ \Delta y\\\ \Delta z\end{pmatrix}
$$
Is this quantity a 4-vector?
1. Yes, and I can say why.
2. No, and I can say why.
3. None of the above.
Note:
* Correct answer: A
In my frame ($S$) I measure two events which occur at the same place, but different times $t_1$ and $t_2$ (they are NOT simultaneous)
Might you (in frame $S'$) measure those SAME two events to occur simultaneously in your frame?
1. Possibly, if I'm in the right frame!
2. Not a chance
3. Definitely need more info!
4. ???
Note:
* Correct Answer: B