The displacement between two events $\Delta x^{\mu}$ is a contravariant 4-vector. Is $\Delta x^{\mu}/\Delta t$ also a 4-vector (where $\Delta t$ is the time between in events in some frame)? 1. Yes 2. No Note: * Correct Answer: B

The displacement between two events $\Delta x^{\mu}$ is a contravariant 4-vector. Is $\Delta x^{\mu}/\Delta \tau$ also a 4-vector (where $\Delta \tau$ is the proper time)? 1. Yes 2. No Note: * Correct Answer: A

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