Time travel is a staple of science fiction, but it has also been studied in theoretical physics for decades. Einstein's general relativity describes gravity as curvature of spacetime, and some exact solutions of its field equations mathematically allow paths that return to the past. These paths are called "closed timelike curves" (CTCs). The question is how far physical law can rule out the possibility that an observer who keeps moving toward the future ends up back at the same point in spacetime where they started. It goes to the heart of causality in general relativity.
Hongsheng Zhang, a physicist at the University of Jinan in China, has now used a remarkably concise mathematical proof to show that closed timelike curves cannot exist in spacetimes that satisfy certain conditions. In a preprint posted to arXiv on October 24, 2026 (arXiv:2610.33706), he uses the "null eikonal equation," which describes the propagation of light, to prove that timelike paths cannot form closed loops. All that is needed is a basic geometric property relating null and timelike directions. There is no need to invoke the specific equations of the gravitational field or energy conditions on matter.
This does not mean that time travel has been proved impossible in every spacetime, including our own universe. For the theorem to hold, a strict premise is required: that a single-valued null coordinate exists across the entire spacetime. For spacetimes that do not satisfy this condition, the existence of closed timelike curves cannot be ruled out. The significance of the work lies in clarifying the geometric conditions under which time travel is forbidden, and in showing concretely how those conditions can fail.
Closed timelike curves and the conditions for a global null foliation
When general relativity is used to consider time travel, the objects of study are not imaginary time machines but mathematically defined paths in spacetime. The path followed by a massive object or an observer is called a worldline, and its tangent vector is "timelike," corresponding to motion slower than light. A closed timelike curve is a worldline of this kind that forms a loop and eventually returns to the same point in spacetime where it began.
Zhang's paper defines a CTC as "a future-directed, piecewise smooth closed curve whose tangent vector is timelike wherever it exists." Mathematically, the curve is required to be piecewise \(C^1\). Simply returning to the same place in ordinary space does not make a closed timelike curve. It must return to exactly the same point in spacetime, including the time coordinate as well as the spatial position.
The theorem concerns spacetimes described as smooth, connected four-dimensional Lorentzian manifolds. The premise of the proof is that a "global null foliation" exists across the whole spacetime. Specifically, it assumes a \(C^1\) scalar function \(u\) defined on the entire spacetime whose gradient never vanishes and is everywhere "null." Here, null means that the squared norm of the vector, computed with the metric, is zero.
The surfaces on which the scalar function \(u\) is constant are called null hypersurfaces. These are not the trajectories of individual light rays but three-dimensional hypersurfaces that contain null directions. A structure that divides the whole spacetime into such hypersurfaces without gaps is a null foliation. Slicing by ordinary spacelike hypersurfaces involves choices about time and observers, whereas null hypersurfaces can be defined from the light-cone structure of spacetime. That said, the metric does not always determine the null foliation uniquely; the theorem requires only that at least one foliation satisfying the conditions exists.
Particularly important is the requirement that the function \(u\) be "globally single-valued." Even if null hypersurfaces can be constructed locally, there is no guarantee that they can be joined consistently across the entire spacetime. In spacetimes with complicated topology, a null coordinate that is unproblematic locally may fail to return to its original value after going around a closed path. Zhang's theorem applies to regions where this problem does not arise and the null coordinate can be defined as a single function over the whole spacetime.
Traditionally, one widely used way to investigate the causal structure of spacetime is to use the existence of a global time function. This defines temporal order using a function that increases along every future-directed causal path. In particular, the existence of a smooth time function whose gradient is timelike everywhere is closely tied to "stable causality." Zhang's theorem, by contrast, uses a function with a null gradient rather than a timelike one. The two conditions are not identical, and the study also shows that the existence of a global null coordinate is logically independent of stable causality.
A three-step proof built on the non-orthogonality of null and timelike directions
The hallmark of Zhang's proof is that it requires neither complicated tensor calculations nor Einstein's field equations. Its starting point is a basic property of Lorentzian geometry: a null direction cannot be orthogonal to a timelike direction. Combining this property with the null eikonal equation rules out the existence of closed timelike curves.
Let the metric tensor be \(g_{ab}\) and its inverse \(g^{ab}\). The condition that the gradient \(\partial_a u\) of the scalar function \(u\) be null is expressed by the following null eikonal equation.
\[ g^{ab}\partial_a u\partial_b u=0 \]
The theorem also requires that \(\partial_a u\neq 0\) hold throughout spacetime. Without this condition, a constant function would also satisfy the equation, and no information about causal structure could be obtained.
In the first step of the proof, let \(T^a\) be a future-directed timelike vector. The proof uses the fact that it cannot be orthogonal to the null gradient \(\partial_a u\). In Lorentzian geometry, the space of vectors orthogonal to a timelike vector is a spacelike subspace with a positive-definite metric. There is no room in it for a nonzero null vector. The contraction of a timelike vector with a null gradient therefore always takes a nonzero value.
\[ T^a\partial_a u\neq 0 \]
The second step shows that the sign of this value remains constant across future-directed timelike directions. The set of future-directed timelike vectors forms a connected convex cone, and \(\partial_a u\) is continuous. Because the contraction never vanishes, its sign cannot switch partway from positive to negative or from negative to positive. In other words, the direction in which the function \(u\) changes is consistent for any future-directed timelike motion.
The third step applies this property to an observer's worldline. If the worldline's parameter is \(\lambda\), the rate of change of \(u\) can be written as follows.
\[ \frac{du}{d\lambda}=T^a\partial_a u\neq 0 \]
Because this value keeps a constant sign along a timelike path, \(u\) is either strictly increasing or strictly decreasing. As long as the observer keeps moving toward the future, \(u\) therefore cannot return to a previous value.
Now suppose the observer's worldline forms a closed timelike curve. If the curve is closed, its start and end are the same point in spacetime, and the value of \(u\) there must be the same. But since \(u\) changes strictly monotonically along the path, it cannot take the same value at the start and the end. This contradiction proves that closed timelike curves cannot exist.
Importantly, neither the energy density of matter nor any specific property of the gravitational field appears anywhere in this argument. No condition requiring positive energy is needed, and neither are assumptions about geodesic completeness or compactness of spacetime. All that is required is the existence of a global function satisfying the null eikonal equation, together with the basic directional relations of the Lorentzian metric.
In this respect, Zhang's criterion differs from the time-function-based methods of judging causality developed by Stephen Hawking and others. In the conventional theorems that use time functions, a timelike gradient tied to stable causality is key; in this method, a null gradient suffices. For example, a spacetime in which a null direction is identified circularly may contain closed null curves, yet if another global null coordinate exists, closed timelike curves are still excluded. Such a spacetime does not satisfy stable causality, which shows that the new criterion covers a range different from that of conventional time-function tests.
However, what the theorem forbids is closed timelike curves, not closed null curves. For a null path, if its tangent vector coincides with a direction along the null foliation, the rate of change of \(u\) can be zero. A null curve could therefore still close within a single leaf. This distinction matters so that the absence of closed timelike curves is not confused with stronger causality conditions.
Where the theorem applies: from Kundt to Robinson–Trautman spacetimes
The value of the theorem also lies in its applicability to concrete spacetime models known from general relativity. In relativistic spacetimes, the properties of a null geodesic congruence are characterized by three optical quantities: expansion, shear, and twist. These describe geometrically how a bundle of light rays spreads, changes shape, and rotates, respectively. Note, however, that these quantities are defined not for spacetime itself but for the chosen congruence.
The main examples of application that Zhang cites are "Kundt spacetimes" and "Robinson–Trautman spacetimes." Both are classes of spacetimes characterized by twist-free null geodesic congruences. If the corresponding null coordinate \(u\) is globally single-valued over the whole region of interest, the theorem rules out closed timelike curves. Instead of tracing timelike curves in each individual solution, one can decide the question simply by confirming that the null coordinate exists.
Kundt spacetimes are characterized by a null geodesic congruence whose expansion, shear, and twist are all zero. Some of these spacetimes include a "rotation one-form \(W\)," which represents a component related to the rotation of gravitational waves. But the presence of a rotation one-form does not mean the null geodesic congruence itself has twist. Zhang shows that the null foliation is maintained regardless of the value of the rotation one-form, and concludes that CTCs do not exist in regions where a global null coordinate holds.
This result also bears on gyraton-type solutions, which describe waves carrying angular momentum. Even if a wave carries angular momentum, that alone does not necessarily give the null geodesic congruence a twist. If the twist of the congruence is zero and the null coordinate can be defined globally, the presence of rotation-related components does not prevent the theorem from holding. One must distinguish rotation in the general sense from the geometric twist of a null geodesic congruence.
Robinson–Trautman spacetimes, on the other hand, are a class in which the null geodesic congruence expands while shear and twist are zero. Unlike Kundt spacetimes, the bundle of rays spreads out, but if the null coordinate is globally single-valued, CTCs are again excluded. Zhang also gives examples of other metrics, with nonzero expansion and shear, for which the same conclusion holds. In other words, the magnitude of expansion or shear is not in itself a factor that produces time loops in spacetimes with a global null coordinate.
This conclusion does not depend on the value of the cosmological constant \(\Lambda\), either. Whether it is positive, zero, or negative, the theorem's conclusion is unchanged as long as the metric in question satisfies the null eikonal condition. This does not mean that the cosmological constant has no effect on the shape of spacetime; it means that if the existence of a global null coordinate is preserved when its value changes, CTCs are not formed.
A familiar theoretical example is Schwarzschild spacetime, which describes a non-rotating black hole. Outside the event horizon, one can use the retarded null coordinate \(u=t-r^*\) of Eddington–Finkelstein coordinates, where \(r^*\) is the tortoise coordinate, a transformation of the radial coordinate. Because this null coordinate can be defined globally in the exterior region, the theorem is consistent with the known result that no CTCs exist outside a Schwarzschild black hole.
These conclusions, however, concern only the spacetime regions under consideration. For example, even if the theorem holds in a local coordinate patch of a Robinson–Trautman spacetime, the same conclusion does not automatically hold for the entire extended spacetime. Moreover, these are results about mathematical spacetime models in general relativity, not phenomena newly confirmed by astronomical observation or experiment. The study clarifies which geometric conditions prevent the formation of time loops in families of solutions describing gravitational waves and radiation.
| Spacetime model / class | Main geometric features | Conclusion from the theorem | Caveats |
|---|---|---|---|
| Kundt spacetimes | Expansion, shear, and twist of the null geodesic congruence are zero. May include a rotation one-form \(W\) | CTCs can be excluded in regions where a globally single-valued null coordinate exists | The rotation one-form is distinct from the twist of the congruence |
| Robinson–Trautman spacetimes | Nonzero expansion; shear and twist are zero | CTCs can be excluded in regions where the null coordinate is globally single-valued | The conclusion is limited to the region considered and does not automatically apply to the extended spacetime |
| Schwarzschild exterior | Static and spherically symmetric. The retarded null coordinate \(u=t-r^*\) can be defined | No CTCs exist outside the event horizon | This coordinate does not directly settle the interior of the horizon or the vicinity of the singularity |
| Kerr spacetime | The principal null congruence has nonzero twist | The null foliation required by the theorem cannot be built from the principal null congruence | Twist of the principal congruence alone does not rule out another null coordinate. A globally single-valued null coordinate cannot exist over a spacetime that contains CTCs |
| Gödel spacetime | A rotating-universe model in which the flow of matter has nonzero vorticity | Because CTCs are known to exist, no single-valued null coordinate can cover the whole spacetime | Vorticity of matter must be distinguished from twist of a null congruence. It is not necessarily a model of the real universe |
Two escape routes for time loops, and the limits of the theorem
To understand the theorem, it is important to grasp not only under what conditions CTCs are forbidden but also what happens when those conditions fail. Zhang classifies the main mechanisms that prevent the theorem from applying, within the range of spacetimes covered in the paper, into two: the "twist" of the null geodesic congruence, and the breakdown of the "global single-valuedness" of the null coordinate. These, however, are reasons the theorem's conditions may not be met; they do not by themselves mean that time travel becomes possible.
The first mechanism is the case where the null geodesic congruence has nonzero twist. According to Frobenius' theorem, whether hypersurfaces orthogonal to a vector field can be constructed locally is determined by the integrability condition that the vector field satisfies. For a null geodesic congruence, if the twist is zero, null hypersurfaces orthogonal to that congruence can be constructed locally. If the twist is nonzero, a null foliation orthogonal to that congruence cannot be built.
Importantly, though, twist is a problem only for a particular null geodesic congruence. Even if some congruence has twist, the possibility that a null foliation can be built from another null direction in the spacetime is not ruled out. To apply the theorem, it is enough that at least one global null coordinate exist. Having twist in a particular congruence and the premises of the theorem failing for the whole spacetime must therefore be strictly distinguished.
The second mechanism is the case where the topology of spacetime destroys the single-valuedness of the null coordinate. Even if a coordinate satisfying the null eikonal equation exists locally, if part of the spacetime is identified circularly, that coordinate may not be definable as a single real-valued function over the whole spacetime. For example, going once around along a null direction may geometrically bring you back to the original point while the coordinate value shifts by an amount corresponding to the period. In such a structure, the global, single-valued \(u\) required by the proof does not hold.
A representative example is Misner space. In Misner space, even where the local geometry is regular, compactifying a null direction changes the causal structure. In certain regions, the periodically identified direction becomes timelike, and closed timelike curves form. However, periodically identifying a null direction does not necessarily produce CTCs. If another, non-compact global null coordinate remains, the theorem can still exclude CTCs.
Zhang also applies this topology-driven mechanism to Ori's time machine and to a series of spacetime solutions by Ahmed. In these models, the periodically identified direction becoming timelike is tied to the formation of CTCs. Even when quantities such as the cosmological constant, expansion, and shear change, what directly governs whether time loops arise is the causal character of the compactified direction, he says. This is an analysis of the specific solutions examined in the study, however, and it cannot be generalized to every spacetime model.
The two categories of twist and global single-valuedness also have limits of applicability. The paper mentions a third logical possibility: that no nowhere-vanishing null direction field exists on the spacetime at all. It states, though, that this problem does not arise in the spacetime regions the study treats concretely. The two escape routes are therefore a classification within the range of spacetimes the paper addresses, not an exhaustive account of how the premises can fail for every Lorentzian manifold.
An important example of these differences is Kerr spacetime, which describes a rotating black hole. The principal null congruence of Kerr spacetime has nonzero twist, so a null foliation orthogonal to it cannot be constructed. In addition, in mathematically extended Kerr solutions, CTCs are known to exist near the ring singularity. Combining this known fact with the present theorem shows that no globally single-valued null coordinate can exist that covers the whole Kerr spacetime including the CTCs.
Still, the mathematical fact that CTCs exist in Kerr spacetime does not mean one could safely travel to the past using a real black hole. The region in question lies inside the black hole, and it is not clear how the geometry near the singularity should be interpreted as a physical process. CTCs are also known to exist in Gödel spacetime, in which the entire universe rotates, but that is no evidence that the real universe has such a structure. What this study treats is only causal structure in mathematical solutions of general relativity.
As a subject for future research, Zhang points to Hauser's twisting type-N solution. It is known as a vacuum solution with a null geodesic congruence that has both expansion and twist, and it is one of the spacetimes to which the present theorem cannot be applied directly. To the author's knowledge, whether closed timelike curves exist in this solution is an open problem. Whether causality really breaks down when twist is present, or whether some other geometric condition excludes CTCs, requires further investigation.
The study was released as a preprint on October 24, 2026, and has not been peer reviewed by an academic journal. What it presents is a theorem showing that CTCs do not exist under specific geometric conditions; it does not directly determine whether time travel is possible in nature. Whether spacetime structures that permit CTCs can form in the real universe, or whether quantum gravity effects and the like would prevent their formation, remains unresolved.
What the study shows is that one condition forbidding time travel follows from the basic geometry of spacetime rather than from the specific properties of the gravitational field or the energy of matter. As long as a globally single-valued null coordinate exists, a timelike path cannot return to its original point in spacetime. Whether CTCs actually form when that condition fails must be examined separately. With a proof of only a few steps, the boundary between the region where time travel is mathematically forbidden and the region that still leaves room for investigation has become clearer.