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The Lumenary · First-year graduate

The geometry of spacetime: special and general relativity

Name: ______________________

Date: ______________________

You will learn to

Use the Minkowski metric to compute proper time, explain time dilation and the twin paradox, and describe the Lorentz group
Prove the reverse triangle inequality for timelike vectors and explain why timelike geodesics maximise proper time
Formulate the equivalence principle and Einstein's field equations in the language of Lorentzian geometry, and recover Newtonian gravity in the weak-field limit
Derive the motion of light and planets from geodesics of the Schwarzschild metric — light bending, perihelion precession and the innermost stable circular orbit — to leading order
Distinguish coordinate singularities from curvature singularities using curvature invariants

1.
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?
2.
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. $(5, 3, 0, 0)$
- B. $(3, 5, 0, 0)$
- C. $(13, 3, 4, 12)$
- D. $(-7, 1, 2, 2)$
3.
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?
- A. $\frac{40}{41}c$
- B. $\frac85 c$
- C. $c$
- D. $\frac45 c$
4.
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?
- A.To know that $t_1 t_2 - |\mathbf x||\mathbf y| \ge 0$ , so that squaring the reversed Cauchy–Schwarz inequality is legitimate.
- B.In the Cauchy–Schwarz inequality $\langle \mathbf x, \mathbf y \rangle \le |\mathbf x||\mathbf y|$ .
- C.In the identity $(t_1t_2 - |\mathbf x||\mathbf y|)^2 - (t_1^2 - |\mathbf x|^2)(t_2^2 - |\mathbf y|^2) = (t_1|\mathbf y| - t_2|\mathbf x|)^2$ .
- D.Nowhere: the inequality holds for all timelike vectors.
5.
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?
- A.The straight segment has the largest proper time, and there is no smallest: the infimum is $0$ .
- B.The straight segment has the smallest proper time, as in Euclidean geometry.
- C.All causal curves from $p$ to $q$ have the same proper time, because proper time is invariant.
- D.The minimum is attained by a null curve and equals $\tau(q - p)/2$ .
6.
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 $c = 2.998 \times 10^8\ \mathrm{m\,s^{-1}}$ .
7.
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?
- A. $0.875''$ , half the relativistic value.
- B. $1.75''$ , the same value: light bending is a Newtonian effect.
- C. $3.5''$ , twice the relativistic value.
- D. $0''$ : Newtonian gravity does not act on light.
8.
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?
- A.The statement would become false, so it could not be proved: for example t₁ = 2, t₂ = -1, x = y = 0 violates it.
- B.It would still be true, because Real.sqrt returns 0 for negative arguments.
- C.It would still be true for spacelike vectors, only false for null ones.
- D.Lean would reject the statement as ill-typed.
9.
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 $b^2/M^2$ .
10.
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 $\Delta s_k^2 > 0$ ).
Computational — Practice

Answer key — The geometry of spacetime: special and general relativity

1.
$\Delta t = 12 \cdot \frac{13}{12} = 13$ years and $\Delta x = 12$ , so the proper time is $\sqrt{13^2 - 12^2} = 5$ years. Equivalently $\gamma = 13/5$ and $\Delta\tau = 13/\gamma = 5$ .
2.
(a) $\Delta s^2 = -16$ : timelike. (b) $+16$ : spacelike. (c) $0$ : null. (d) $-40$ : timelike (past-pointing). The causal character is Lorentz invariant, and so (for orthochronous transformations) is the distinction between future and past for causal vectors.
3.
$u \oplus v = \frac{4/5 + 4/5}{1 + 16/25} = \frac{8/5}{41/25} = \frac{40}{41}$ . Check with Lorentz factors: $\gamma(\frac45) = \frac53$ and $\gamma(\frac{40}{41}) = \frac{41}{9} = \frac53 \cdot \frac53 \cdot (1 + \frac{16}{25})$ , as $\gamma(u \oplus v) = \gamma(u)\gamma(v)(1 + uv)$ requires.
4.
(a). From the identity, $(t_1t_2 - |\mathbf x||\mathbf y|)^2 \ge \tau(v)^2\tau(w)^2$ , and to conclude $t_1t_2 - |\mathbf x||\mathbf y| \ge \tau(v)\tau(w)$ one needs the left side non-negative, which follows from $t_1 \ge |\mathbf x|$ , $t_2 \ge |\mathbf y|$ . In Lean these are exactly the hypotheses ‖x‖ ≤ t₁ and ‖y‖ ≤ t₂.
5.
(a). For a broken line, repeated use of $\tau(v) + \tau(w) \le \tau(v + w)$ gives total proper time $\le \tau(q - p)$ , with equality only for the straight line; smooth timelike curves are limits of broken lines. Going from $p$ to $q$ along two null segments (out to the light cone and back) gives proper time $0$ , and timelike approximations of it have arbitrarily small positive proper time. In curved spacetime, timelike geodesics are locally longest in the same sense.
6.
$GM/c^2 = 1.327 \times 10^{20} / 8.988 \times 10^{16} \approx 1476$ m, so $r_s \approx 2953$ m $\approx 3$ km. The Sun's radius is about $696{,}000$ km, so its metric is very nearly flat: $2GM/(c^2 R_\odot) \approx 4 \times 10^{-6}$ , and twice this, $4GM/(c^2R_\odot) \approx 8.5 \times 10^{-6}$ rad $= 1.75''$ , is the deflection of starlight at the limb.
7.
(a). A Newtonian particle passing a mass $M$ at speed $c$ and impact parameter $b$ is deflected by $2GM/(c^2b)$ for $b \gg GM/c^2$ ; the geodesic computation gives $4GM/(c^2b)$ . For the Sun, $0.875''$ versus $1.75''$ . In the weak-field metric $-(1 + 2\Phi)dt^2 + (1 - 2\Phi)d\mathbf x^2$ , the $g_{00}$ part alone gives the Newtonian half and the spatial part $g_{ij}$ the other half.
8.
(a). The hypotheses ‖x‖ ≤ t₁, ‖y‖ ≤ t₂ say exactly that (t₁, x) and (t₂, y) are future-pointing causal vectors. Without them the claim fails (for a future and a past vector), so it is unprovable. A good formal statement includes precisely the hypotheses the mathematics needs — no vacuous strengthening, no missing conditions.
9.
$f'(r) = (6M - 2r)/r^4$ vanishes at $r = 3M$ , where $f(3M) = \frac{1/3}{9M^2} = \frac{1}{27M^2}$ . Rays with $1/b^2 > 1/(27M^2)$ , i.e. $b \lt 3\sqrt3\,M \approx 5.2M$ , have no turning point and fall in. So $b_c^2/M^2 = 27$ : the black hole's shadow has radius $3\sqrt3\,M$ , about $2.6$ times the horizon radius $2M$ .

10.

import math

def proper_time(events):
    total = 0.0
    for (t0, x0, y0, z0), (t1, x1, y1, z1) in zip(events, events[1:]):
        dt = t1 - t0
        ds2 = -dt ** 2 + (x1 - x0) ** 2 + (y1 - y0) ** 2 + (z1 - z0) ** 2
        if dt <= 0 or ds2 > 0:
            raise ValueError('each step must be future-pointing and causal')
        total += math.sqrt(-ds2)
    return total

Each step contributes $\sqrt{-\Delta s^2}$ , the time shown by a clock moving uniformly along it. The twin test is the lesson's example: legs of proper time $4$ each, against $10$ for the straight worldline — an instance of the reverse triangle inequality.