What if a wormhole could contain two rotating black holes? What if these black holes were connected to another region of spacetime by enormous cosmic strings? And what if the same wormhole also contained a naked ring singularity and paths that could theoretically allow time to loop back on itself?
These are not ideas from a science-fiction movie. They are features of a remarkable mathematical solution of Einstein’s equations studied by physicist Gérard Clément.
Clément examined a three-parameter stationary solution of the four-dimensional vacuum Einstein equations. In simple terms, this solution describes a very unusual spacetime structure: a Lorentzian wormhole containing two antipodal, co-rotating black holes.
The solution is important because it combines several strange features of general relativity in one model. It includes a wormhole, two rotating black holes, cosmic strings, closed timelike curves, and, for almost all values of its parameters, a naked ring singularity.
What Is a Lorentzian Wormhole?
A wormhole is a theoretical tunnel connecting two different regions of spacetime. Instead of travelling through the normal distance between two places, a traveller might, in theory, pass through a shortcut created by the shape of spacetime itself.
A Lorentzian wormhole is a wormhole described within the framework of ordinary relativistic spacetime, where space and time are connected. Such wormholes are solutions of Einstein’s field equations, which describe how matter, energy, space, and time interact.
However, keeping a wormhole open is not easy. Many theoretical wormhole solutions require unusual forms of matter. This type of matter is often called exotic matter because it can violate the weak energy condition. This condition is related to the idea that observers should measure a reasonable, non-negative amount of energy.
One of the simplest examples is the well-known Ellis-Bronnikov wormhole. This solution is spherically symmetric and is connected with a self-gravitating phantom scalar field. The unusual properties of this field help keep the wormhole open.
But not all wormhole solutions require obvious exotic matter. Some can arise from the vacuum Einstein equations or from the Einstein-Maxwell equations. In these cases, unusual structures such as line singularities or cosmic strings can appear.
The Strange Kerr Wormhole
Another important example comes from the famous Kerr solution. The Kerr solution describes a rotating black hole.
Under normal conditions, a rotating black hole has an event horizon that hides its central singularity. But when the rotation becomes extremely large, satisfying the condition
[
a^2 > M^2,
]
the usual event horizon disappears.
The result is an over-rotating Kerr solution with a naked ring singularity. The extended geometry of this solution can be interpreted as a wormhole connecting different regions of spacetime.
Other wormhole-like solutions have also been found. Some are connected with cosmic string loops, while others involve straight structures known as Misner-Dirac strings.
Gérard Clément’s work adds another fascinating possibility to this family of solutions.
Two Black Holes Inside One Wormhole
The most unusual feature of Clément’s solution is that the wormhole contains two black holes.
The black holes are located opposite each other, which is why they are described as antipodal. They are also co-rotating, meaning that both black holes rotate in the same direction.
This arrangement is very different from many other two-black-hole solutions in general relativity.
The two black holes are not connected to each other by a strut. A strut is a mathematical structure that can sometimes be used to prevent two massive objects from moving together under their mutual gravitational attraction.
They are also not connected by a Misner string.
Instead, each black hole is connected to one of the wormhole’s other spacelike infinities by a semi-infinite cosmic string.
Cosmic Strings Connect the Black Holes to Another Infinity
Cosmic strings are theoretical one-dimensional objects that may have formed during the early history of the universe. Although they are extremely thin, they could have enormous energy and produce powerful gravitational effects.
In Clément’s solution, two semi-infinite cosmic strings extend from the black holes toward the “other” spacelike infinity of the wormhole.
These strings may have either positive or negative tension, depending on the values of the parameters used in the solution.
In simple terms, the picture is something like this: two rotating black holes are placed opposite each other inside a wormhole, while cosmic strings extend from them toward another distant region of spacetime.
This gives the solution a structure that is both mathematically complex and physically unusual.
A Wormhole Without Misner Strings
The solution studied by Clément was obtained by analytically continuing another solution known as the gravimagnetic dipole solution.
That original solution describes two black holes connected by a Misner string. A Misner string is a type of singular structure that can appear in certain solutions of Einstein’s equations. It is sometimes compared with the Dirac string that appears in the mathematical description of magnetic monopoles.
However, after the analytical continuation used to obtain the wormhole solution, the Misner strings disappear.
This is one of the most interesting aspects of the model. The two black holes are not connected to one another by a Misner string. Instead, they are connected to the other end of the wormhole through cosmic strings.
Closed Timelike Curves
The solution also contains another strange feature: closed timelike curves.
In general relativity, a timelike curve represents a possible path through spacetime for an object moving slower than the speed of light. A closed timelike curve is a path that eventually returns to its starting point.
This means that, at least mathematically, an observer could follow a path through spacetime and return to an earlier event.
In Clément’s solution, closed timelike curves exist inside a limited, bounded region. Their presence creates serious questions about the nature of time and causality.
Although Einstein’s equations allow such mathematical possibilities, many physicists believe that a complete theory of nature may eventually prevent these strange situations from occurring physically.
The Naked Ring Singularity
For almost all real values of the parameters (m), (a), and (d), the solution also contains a naked ring singularity.
A singularity is a region where the mathematical description of spacetime becomes extreme and breaks down. In ordinary black holes, the singularity is hidden behind an event horizon.
A naked singularity, however, is not hidden by such a horizon. This means that the singular region could, in principle, be visible from the outside.
The naked ring singularity in this solution is therefore an important feature. It makes the spacetime much more unusual than an ordinary black hole.
The Special Case (d = 1)
A dramatic simplification occurs when the dimensionless parameter (d) takes the special value
[
d = 1.
]
In this case, the tension of the cosmic strings becomes zero. The strings therefore disappear.
At the same time, geodesics no longer end on the equatorial ring located at (\rho = a).
The mathematical expressions describing the metric also simplify because terms containing the complex functions (r_\pm) cancel between the numerators and denominators.
After these cancellations, the metric reduces to the familiar Kerr solution, which describes a rotating black hole.
This special case is particularly important because it shows a direct connection between the complicated wormhole solution and the well-known Kerr black-hole geometry.
Why This Work Is Important
Gérard Clément’s solution demonstrates the extraordinary variety of structures allowed by Einstein’s theory of gravity.
The equations of general relativity can describe much more than ordinary stars and black holes. They can also produce wormholes, cosmic strings, rotating black-hole systems, naked singularities, and even regions where the normal rules of cause and effect become complicated.
Clément’s solution is probably not a realistic model of an object that exists in the universe. The presence of a naked singularity and closed timelike curves creates major physical problems.
Nevertheless, such solutions are extremely valuable to theoretical physics. They help scientists explore the limits of Einstein’s equations and understand how gravity behaves under extreme conditions.
The idea of a Lorentzian wormhole containing two co-rotating black holes is especially remarkable. These black holes are not connected by a strut or a Misner string. Instead, cosmic strings connect them to another spacelike infinity.
For most parameter values, the geometry also contains a naked ring singularity and closed timelike curves. Yet, when (d=1), the complicated structure simplifies dramatically and becomes the familiar Kerr solution.
This extraordinary result shows just how rich and surprising the geometry of spacetime can be. Einstein’s equations allow mathematical worlds that are far stranger than anything we experience in everyday life—and Clément’s wormhole with two rotating black holes is one of the most fascinating examples.
Reference: Gérard Clément, "A wormhole with two black holes", Arxiv, 2026. https://arxiv.org/abs/2607.19145
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