The geometry of spacetime: special and general relativity

A probe travels from Earth to a star $12$ light-years away at speed $v = \frac{12}{13}$ (units with $c = 1$). How many years pass on the probe's clock during the trip?

Which separation vectors $(\Delta t, \Delta x, \Delta y, \Delta z)$ between two events are timelike (so that a massive particle could travel from one event to the other)? Use $c = 1$. Select all that apply.

A spacecraft moves at $\frac45 c$ relative to Earth and fires a probe forward at $\frac45 c$ relative to itself. What is the probe's speed relative to Earth?

The reverse triangle inequality $\tau(v) + \tau(w) \le \tau(v + w)$ is stated for future-pointing causal vectors. Where does the proof use that both point to the future?

Let $q - p$ be future timelike in Minkowski spacetime. Among future-directed causal curves from $p$ to $q$, which statement about proper time is correct?

Compute the Schwarzschild radius $r_s = 2GM/c^2$ of the Sun, in kilometres, rounded to the nearest whole number. Use $GM_\odot = 1.327 \times 10^{20}\ \mathrm{m^3\,s^{-2}}$ and $c = 2.998 \times 10^8\ \mathrm{m\,s^{-1}}$.

Starlight grazing the Sun is deflected by $1.75''$ according to general relativity. Treating light as a Newtonian particle moving at speed $c$ (Soldner, 1801) gives a different value. Which?

The lesson's Lean theorem twin_paradox has hypotheses h₁ : ‖x‖ ≤ t₁ and h₂ : ‖y‖ ≤ t₂ and conclusion √(t₁^2 - ‖x‖^2) + √(t₂^2 - ‖y‖^2) ≤ √((t₁ + t₂)^2 - ‖x + y‖^2). What would happen if both hypotheses were deleted?

For light in the Schwarzschild metric ($G = c = 1$), $\big(\frac{dr}{d\lambda}\big)^2 = E^2 - L^2 \frac{1 - 2M/r}{r^2}$, and the impact parameter is $b = L/E$. A ray from infinity is captured exactly when $1/b^2$ exceeds the maximum of $(1 - 2M/r)/r^2$. Find the critical value of $b^2/M^2$.

A polygonal worldline is given by a list of events (t, x, y, z) (with $c = 1$). Write proper_time(events) returning the total proper time $\sum_k \sqrt{\Delta t_k^2 - \Delta x_k^2 - \Delta y_k^2 - \Delta z_k^2}$, and raising ValueError if some step is not future-pointing causal ($\Delta t_k \le 0$ or $\Delta s_k^2 > 0$).