World · Temporal mechanics

How time travel works on paper

Real physics and mathematics that time-travel stories lean on, kept honest. Every panel says what kind of thing it is: tap a label to see what it means.

Established scienceMathematical modelFictional mechanismUnresolved story clue

Light cones and worldlines

Established science

In relativity every event has a light cone. Anything you can reach lies inside your future cone; anything that could have reached you lies inside your past cone. A traveller's path through spacetime, the worldline, must stay inside the cones at every step. The time a traveller's own clock measures along it is the proper time, τ.

Is this a time loop?

Established scienceMathematical modelFictional mechanismUnresolved story clue

Eight events, one unit of time apart. Every step stays inside its light cone, so the path is allowed: the traveller's clock reads τ = 6.2221 against 7 outside. Whether a drawn path counts as a loop depends on how time is joined up as a whole, which the light cones alone cannot show.

Closed timelike curves

Established scienceMathematical model

General relativity has exact solutions in which a worldline can return to its own past: Gödel's rotating universe (1949), the inner region of a spinning Kerr black hole (Carter 1968) and Tipler's rotating cylinder (1974). Misner's spacetime (1967) is the simplest picture: as one coordinate passes zero, the light cones tip over until a circle becomes a path a traveller could follow. Whether nature allows any of this is open; Hawking's chronology protection conjecture (1992) suggests it may not.

Can a timeline contradict itself?

Mathematical modelUnresolved story clue

Physicists asking whether time travel can be self-consistent (Friedman et al. 1990; Echeverria, Klinkhammer and Thorne 1991; Deutsch 1991) first need a list of events and arrows that does not contradict itself. Kahn's algorithm (1962) checks that: it removes, again and again, an event with no unmet causes. If events are left over, the arrows contain a loop. Try it on a toy graph in the explorer above, or test an interpretation of the story on the World map.

Why time has a direction

Established science

The laws of motion run equally well backwards. What gives time its arrow is entropy: there are vastly more mixed-up arrangements than tidy ones, so a tidy start almost always gets messier (Boltzmann; Albert, Time and Chance, 2000).

Thirty-two steps to a different timeline

Mathematical model

The logistic map x → 4x(1 − x) is fully deterministic, yet two starts that differ by 2⁻³² (about one part in four billion) part ways: the gap roughly doubles every step, because the map's Lyapunov exponent is ln 2 ≈ 0.6931 (May 1976; Lorenz 1963). Starting from 0.2, the gap first exceeds 0.1 at step 30.

Thirty-two episodes, five switches

Mathematical modelUnresolved story clue

32 = 2⁵. Write episode e as the five-bit Gray code of e − 1: g(n) = n XOR (n shifted right by one). Each episode then differs from the next in exactly one switch, and #32 differs from #1 in one switch too: the whole series is a closed walk around the corners of a five-dimensional cube.

Episode	e − 1	Gray code
#14	01101	01011
#17	10000	11000
#18	10001	11001
#19	10010	11011
#20	10011	11010
#23	10110	11101
#31	11110	10001
#32	11111	10000

Lights that hold their place

Established scienceFictional mechanism

Two heavy bodies orbiting each other leave two quiet spots, L4 and L5, at the tips of equilateral triangles. A small third body there stays put if the mass ratio μ is small enough: 27μ(1 − μ) < 1, so μ < ½(1 − √(23/27)) = 0.03852 (Routh 1875). Sun and Jupiter, μ ≈ 0.000954, pass easily; the Trojan asteroids sit there.

Sources

E. F. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed., W. H. Freeman, 1992.
C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W. H. Freeman, 1973.
K. Gödel, “An example of a new type of cosmological solutions of Einstein's field equations of gravitation”, Rev. Mod. Phys. 21, 447 (1949).
B. Carter, “Global structure of the Kerr family of gravitational fields”, Phys. Rev. 174, 1559 (1968).
F. J. Tipler, “Rotating cylinders and the possibility of global causality violation”, Phys. Rev. D 9, 2203 (1974).
C. W. Misner, “Taub-NUT space as a counterexample to almost anything”, in J. Ehlers (ed.), Relativity Theory and Astrophysics I: Relativity and Cosmology, Lectures in Applied Mathematics vol. 8, American Mathematical Society, 1967, pp. 160–169.
S. W. Hawking, “Chronology protection conjecture”, Phys. Rev. D 46, 603 (1992).
J. Friedman et al., “Cauchy problem in spacetimes with closed timelike curves”, Phys. Rev. D 42, 1915 (1990).
F. Echeverria, G. Klinkhammer and K. S. Thorne, “Billiard balls in wormhole spacetimes with closed timelike curves: classical theory”, Phys. Rev. D 44, 1077 (1991).
D. Deutsch, “Quantum mechanics near closed timelike lines”, Phys. Rev. D 44, 3197 (1991).
A. B. Kahn, “Topological sorting of large networks”, Commun. ACM 5(11), 558 (1962).
T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, 4th ed., MIT Press, 2022, §20.4 “Topological sort” (§22.4 in the 3rd ed., 2009).
D. Z. Albert, Time and Chance, Harvard University Press, 2000.
E. N. Lorenz, “Deterministic nonperiodic flow”, J. Atmos. Sci. 20, 130 (1963).
R. M. May, “Simple mathematical models with very complicated dynamics”, Nature 261, 459 (1976).
E. J. Routh, “On Laplace's three particles, with a supplement on the stability of steady motion”, Proc. London Math. Soc. s1-6, 86–97 (1875; issue dated November 1874), doi:10.1112/plms/s1-6.1.86.
C. D. Murray and S. F. Dermott, Solar System Dynamics, Cambridge University Press, 1999.