Time travel fascinates because it merges geometry, physics, and fate. Recent work by Germain Tobar and Fabio Costa shows that closed time-like curves (CTCs) can coexist with free choice. Yet the underlying mathematics is deep. To understand how paradox-free loops might emerge—and why nature resists our attempts to escape the arrow of time—we need to examine the geometry of general relativity, the constraints imposed by information theory, and the emergent behaviour of repeated time loops. We then propose a model for the global dynamics of a civilisation perpetually looping to avoid an asteroid.
Throughout this discussion we tell it like it is. Mathematics is unforgiving; we cannot hide behind catchy analogies. Our approach is forward-thinking—imagining new structures that could generalise known theories—but rooted in traditional physics. We value the past precisely because equations uncovered decades ago still guide our thinking. We encourage creative speculation, yet we acknowledge fundamental constraints.
A time-like curve on a Lorentzian manifold \((M,g)\) is a smooth curve whose tangent vector \(u^\mu\) satisfies \(g_{\mu\nu}u^\mu u^\nu>0\) everywhere along the curve (using the \(+,-,-,-\) signature). A closed time-like curve (CTC) is a time-like curve that loops back to its starting point. In such a spacetime, an observer following the curve returns to their own past without ever moving faster than light.
Kurt Gödel’s 1949 solution of Einstein’s field equations is the prototype of a spacetime with CTCs. In rectangular coordinates \((x^0,x^1,x^2,x^3)\) the metric takes the form \[\mathrm {d}s^2=a^2\!\left[(\mathrm{d}x^0)^2-(\mathrm{d}x^1)^2+e^{2x^1}(\mathrm{d}x^2)^2-(\mathrm{d}x^3)^2+2e^{x^1}\,\mathrm{d}x^0\,\mathrm{d}x^2\right],\] where \(a>0\) is a constant. Under a change to cylindrical coordinates \((t,r,\varphi,y)\), one finds an equivalent form \[\mathrm{d}s^2=4a^2\!\left[\mathrm{d}t^2-\mathrm{d}r^2-\mathrm{d}y^2+(\sinh^4 r-\sinh^2 r)\,\mathrm{d}\varphi^2+2\sqrt{2}\,\sinh^2 r\,\mathrm{d}\varphi\,\mathrm{d}t\right].\] The cross-term \(\mathrm{d}\varphi\,\mathrm{d}t\) tilts the light cones such that for \(r>0\) there exist closed loops at constant \(r\) and \(y\) that are time-like. Gödel’s universe is homogeneous and solves the field equations with dust and a negative cosmological constant. However, it contains closed time-like curves through every point, meaning global causality is violated.
Another class of solutions arises from a rotating cylinder of matter. Frank Tipler showed that an infinitely long cylinder spinning sufficiently fast could tilt light cones so much that world-lines near the cylinder wrap around in time. In a spacetime diagram, the light cones tilt backwards; a geodesic that winds around the cylinder can trace a closed time-like curve. Tipler’s original solution required an infinite cylinder. Hawking later argued that a finite time machine would require negative energy densities, aligning with the chronology protection conjecture—nature forbids macroscopic regions with CTCs.
Closed time-like curves invite paradoxes. Novikov’s self-consistency principle states that events on a CTC must be globally consistent: the probability of events that cause paradoxes is zero. Tobar and Costa’s work formalises this in quantum information theory. They model a CTC as a sequence of \(N\) regions \(R_1,R_2,\dots,R_N\) through which a quantum system cycles. Each region performs a local unitary operation \(U_i\) on the system. Let \(\rho\) be the density matrix at \(R_1\). The system evolves through the loop as \[\rho \mapsto U_N\circ\cdots\circ U_2\circ U_1(\rho)=:\mathcal{U}(\rho).\] A fixed-point condition \(\mathcal{U}(\rho)=\rho\) ensures the output state equals the input state. When at least two of the \(U_i\) commute (maintain causal order), there exist fixed points for all choices of the remaining operations. This demonstrates that complex dynamics can occur on a CTC while preserving deterministic consistency.
Time travel is not free from thermodynamics. In any physical process, information storage and erasure incur entropic costs.
Rolf Landauer showed that erasing one bit of information at temperature \(T\) dissipates at least \[E_{\mathrm{erase}}\ge k_B T\ln 2 \quad \text{(per bit)}.\] If time travellers repeatedly loop back with knowledge of the future, they bring information into an earlier epoch. Either this information must be erased—expending energy per [eq:landauer]—or stored, increasing entropy.
For a region of radius \(R\) containing energy \(E\), Jacob Bekenstein derived an upper limit on the entropy \(S\) contained in that region: \[S\le \frac{2\pi k_B R E}{\hbar c}.\] Because entropy is proportional to the number of bits \(N\) via \(S=N k_B\ln 2\), [eq:bekenstein] implies a bound on information capacity: \[N\le \frac{2\pi R E}{\hbar c\,\ln 2}.\] No finite region of spacetime can indefinitely accumulate information without increasing its energy or volume. If time travellers keep adding memory, they either saturate the bound or must dissipate heat to increase \(E\).
Arthur Eddington introduced the arrow of time: time flows in the direction of increasing entropy. At microscopic scales, the fundamental laws are nearly time-reversal invariant, but macroscopic systems overwhelmingly evolve toward higher entropy. Time travel that decreases entropy locally would have to be balanced by increases elsewhere, or else the second law of thermodynamics would be violated.
Cosmic strings are hypothetical one-dimensional topological defects predicted by some grand-unified theories and brane-inflation models. Benjamin Shlaer and Henry Tye examined the possibility that two rapidly moving cosmic strings could create a “Gott time machine.” They argued that the formation of closed time-like curves in this scenario is unstable: a single photon or graviton is sufficient to bend the strings and eliminate the CTC. Since our universe is awash with cosmic microwave background photons, any attempt to engineer such a spacetime would fail. Their work supports Hawking’s chronology protection.
String theory goes further: it suggests the Big Bang may not have been the ultimate beginning. Gabriele Veneziano writes that string theory implies the Big Bang was “not the origin of the universe but simply the outcome of a preexisting state.” If the universe underwent cycles of contraction and expansion, the concept of time becomes global, and local CTCs may appear as ripples on a larger background. However, there is no evidence that such cycles allow macroscopic time travel.
Consider a civilisation facing an inevitable asteroid impact at global time \(t=0\). Assume they possess a mechanism to travel back in time by a fixed interval \(\Delta t>0\) via a CTC. Let the \(n\)-th generation of travellers (those who have jumped \(n\) times) arrive at time \(t=-n\Delta t\). We aim to model the information and energy flow through repeated loops.
Let \(P_n\) be the number of travellers in the \(n\)-th cohort (arriving at time \(-n\Delta t\)), and \(I_n\) the total additional information they carry relative to the native population at that epoch. Suppose each cohort is of size \(P\) and each traveller carries \(i\) bits of future knowledge. Then \[P_n=P,\qquad I_n=iP_n+I_{n-1},\] with \(I_0=0\) at \(t=0\) (no extra information before the first jump). Iterating, we obtain \[I_n=iP\sum_{k=1}^{n}1=niP.\] Thus, after \(n\) loops, the total extra information scales linearly with \(n\). Storage constraint: by the Bekenstein bound [eq:capacity] applied to the Earth’s radius \(R_\oplus\) and mass–energy \(E_\oplus\), the maximum number of bits Earth can hold is \[N_{\max}=\frac{2\pi R_\oplus E_\oplus}{\hbar c\,\ln 2}.\] Energy-dissipation constraint: by Landauer’s principle [eq:landauer], erasing \(I_n\) bits of information requires at least \[E_n\ge k_B T\ln 2 \times I_n = k_B T\ln 2 \times n i P.\] At room temperature (\(T\approx300\,\mathrm{K}\)) each bit costs about \(3\times10^{-21}\,\mathrm{J}\). So erasing \(10^{30}\) bits—small by astrophysical standards—releases \(\sim 3\times10^{9}\,\mathrm{J}\). For large \(n\), the dissipation becomes astronomical. Without an unbounded energy reservoir, the planet overheats.
A more refined model treats the entropic flux as a function \(S(t)\) defined on continuous time. Let \(s(t)\) denote the rate of entropy injection per traveller at arrival time \(t\) (entropy per unit time). If travellers arrive at intervals \(\Delta t\) with constant population \(P\), the entropy injected into the epoch \((t,t+\mathrm{d}t)\) is \[\mathrm{d}S(t)=s(t)\,P\,\delta_{\Delta t}(t)\,\mathrm{d}t,\] where \(\delta_{\Delta t}\) is a Dirac comb with period \(\Delta t\). The total entropy in the interval \([-n\Delta t,0]\) is then \[S_n=P\sum_{k=1}^{n}s(-k\Delta t).\] Assuming \(s\) is constant yields the linear growth [eq:info-growth]. If \(s\) decreases over loops due to memory loss or cultural assimilation, the series may converge. This suggests: perpetual time-jumping only remains feasible if the information carried decays sufficiently quickly.
Einstein’s field equations relate geometry to energy–momentum: \[G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu},\qquad \kappa=\frac{8\pi G}{c^4}.\] Each additional cohort contributes to the stress–energy tensor. If \(\rho\) is the average energy density of travellers, then after \(n\) loops the total energy density at time \(t=-n\Delta t\) is \(\rho_n=n\rho\), assuming travellers do not die or disperse. Substituting into [eq:einstein], the Ricci curvature \(R_{\mu\nu}\) and hence the light-cone structure depend on \(n\). For large \(n\), back reaction could tilt light cones such that the CTC ceases to exist or collapses into a singularity, enforcing chronology protection. One can model this by considering the effective metric \(g_{\mu\nu}(n)=g_{\mu\nu}^{(0)}+\delta g_{\mu\nu}(n)\), where \(g_{\mu\nu}^{(0)}\) is the original CTC metric and \(\delta g_{\mu\nu}(n)\) is the perturbation due to the accumulated stress–energy. A simplistic linearisation yields \[\delta g_{\mu\nu}(n)\approx \kappa \int_0^{n} T_{\mu\nu}^{(\mathrm{trav})}(k)\,\mathrm{d}k.\] If \(T_{\mu\nu}^{(\mathrm{trav})}\) has non-zero off-diagonal components (e.g., due to rotation or motion), the perturbation can eliminate the cross-term required for the CTC.
Combining the information and geometric constraints, a stability condition for repeated time loops emerges. Let \(n\) be the number of loops, \(i\) bits carried per traveller, and \(P\) travellers per loop. Define \[\eta=\frac{iP}{N_{\max}},\qquad \xi=\frac{k_B T\ln 2\, iP}{E_{\mathrm{avail}}}.\] Here \(E_{\mathrm{avail}}\) is the energy available to absorb erasure heat and \(N_{\max}\) the Bekenstein information capacity [eq:nmax]. The system remains below the storage and energy thresholds if \[\eta n<1,\qquad \xi n<1.\] Thus the maximum number of loops is \[n_{\mathrm{max}}=\min\!\left(\eta^{-1},\,\xi^{-1}\right).\] Beyond \(n_{\mathrm{max}}\), either information saturates the Bekenstein bound or the energy cost of erasure exceeds \(E_{\mathrm{avail}}\). In both cases, the time machine becomes physically untenable. Should travellers attempt to exceed this, the back reaction from [eq:metric-perturb] would likely destroy the CTC, aligning again with chronology protection.
We can also treat \(t\) as continuous and consider travellers who attempt to stay within a finite time window \([t_0,t_0+T]\), continuously looping inside. The number density of travellers at time \(t\) is \[\rho(t)=\sum_{n=-\infty}^{\infty} P\,f\!\left(t-n\Delta t\right),\] where \(f\) is a normalised arrival profile (e.g., a Gaussian). The stress–energy tensor is then a convolution of \(\rho(t)\) with the energy per traveller. The periodicity in \(\rho(t)\) implies that the spacetime metric inherits an approximate periodicity. If \(\Delta t\) is small, the average energy density may grow unbounded, again leading to a breakdown of the CTC. To avoid this, the arrival distribution must have a decaying amplitude so that \(\rho(t)\) remains finite.
The model suggests that mass migration into the past faces stringent limits. Even if a closed time-like curve exists, the information and energy budgets bound the number of times we can loop before saturating physical resources. Each loop injects extra information and stress–energy into the earlier epoch. Unless travellers willingly forget most of what they know (reducing \(i\)) and disperse to dilute their energy, the loops quickly hit the thresholds [eq:thresholds]. By the time the \(n\)-th cohort arrives, the Earth of \(-n\Delta t\) may be unrecognisable—overheated, information-saturated, and gravitationally distorted.
General relativity provides the geometric possibility of closed time-like curves, but the chronology protection conjecture suggests that quantum and thermodynamic effects conspire to prevent large-scale causal violations. The energy-condition violations needed for Tipler cylinders, the instability of cosmic-string time machines, and the finite information capacity of physical systems all point to natural limits. The stability condition [eq:thresholds]–[eq:nmaxloops] encapsulates how these constraints manifest in an engineered CTC: repeated use of a time machine drains the same region’s energy and erodes its causal structure.
This model is speculative and not derived from a full theory of quantum gravity. Nevertheless, it respects traditional principles: General relativity—Einstein’s equations [eq:einstein] and known exact solutions [eq:godel-rect]–[eq:godel-cyl]—grounds the reasoning. Thermodynamics—Landauer’s principle [eq:landauer] and the Bekenstein bound [eq:bekenstein]—imposes unavoidable limits on information and energy. Causality—Novikov’s self-consistency—ensures paradoxes cannot occur.
Mathematics offers a clear lens through which to examine time travel. Exact solutions like Gödel’s rotating universe and Tipler’s cylinder show that general relativity allows closed time-like curves. Quantum self-consistency ensures that such curves can, in principle, be paradox-free. Yet information theory and thermodynamics impose powerful constraints: erasing information dissipates heat, and finite regions have bounded entropy. Combining these with Einstein’s equations, we obtain a stability condition for repeated loops. This condition reveals that perpetual time travel is self-limiting: after a finite number of loops, a system’s energy or information capacity is exhausted, and the CTC collapses.