Wormholes are one of the most fascinating ideas in modern physics. They are often described as tunnels through space that connect two distant regions of the universe. In science fiction, wormholes allow instant travel between stars or even time travel. In real physics, wormholes come from Einstein’s theory of general relativity.
For a long time, wormholes were seen as beautiful but unrealistic solutions of Einstein’s equations. Many physicists believed they could never exist in the real universe. The main reason was simple: most known wormholes either collapse immediately or require strange “exotic matter” that has never been observed.
However, recent research has brought new life to this topic. One important direction is the study of degenerate wormholes, which are very different from the wormholes usually discussed in textbooks. These wormholes are made purely of curved space and contain no matter at all.
In this article, we explain in simple and easy language the work of Juri Dimaschko, who studied the dynamics of a degenerate wormhole in empty space, using the Klinkhamer wormhole as an example. His work shows that such a wormhole behaves like a real physical object: it has mass, it moves, and it can collapse under its own gravity. Surprisingly, its motion follows the same rules as a particle falling into a black hole.
2. Gravity Without Matter: An Old but Powerful Idea
Most people think that gravity is always caused by matter: stars, planets, and galaxies. But Einstein’s theory says something deeper. Gravity is actually caused by curved spacetime. Matter bends space, but curved space can also exist on its own.
In 1935, Albert Einstein and Nathan Rosen showed this clearly. They discovered a solution of Einstein’s equations that describes an empty universe with a strong gravitational field. This solution connected two identical regions of space through a narrow bridge. Today, we call this structure the Einstein–Rosen bridge, or simply a wormhole.
This wormhole had an important feature:
It was empty (no matter, no energy).
It was static (it did not change with time).
It still produced gravity.
This was a revolutionary idea: gravity can exist without matter.
3. What Does “Degenerate” Mean?
The Einstein–Rosen wormhole had a strange mathematical property. At the throat of the wormhole, the determinant of the spacetime metric became zero. Such a metric is called degenerate.
Why is this important? Because the usual Einstein equations do not work when the metric determinant is zero. Many quantities in general relativity depend on dividing by this determinant, which becomes impossible.
Einstein and Rosen solved this problem in a clever way. They rewrote the equations in a regularized (polynomial) form, where no division by the determinant was needed. This allowed degenerate metrics to be treated consistently.
So, from the very beginning, degenerate wormholes were part of general relativity—but they required special care.
4. Why Wormholes Were Ignored for So Long
For many decades after 1935, wormholes were mostly of theoretical interest. They were seen as mathematical curiosities with no clear physical meaning.
This changed in 1988, when Morris and Thorne introduced the idea of traversable wormholes. These wormholes could, in principle, be crossed by humans without being destroyed.
Their approach was practical:
Choose a wormhole geometry.
Calculate what kind of matter is needed to support it.
Unfortunately, this approach revealed a serious problem. To keep a wormhole open, one must violate the null energy condition (NEC). This requires exotic matter with negative energy density.
Since no such matter is known to exist, traversable wormholes were considered unrealistic.
5. A New Direction: Back to Degenerate Wormholes
Recently, physicists began to reconsider degenerate wormholes. One key contribution came from Frans Klinkhamer, who proposed a new class of wormhole solutions.
The Klinkhamer wormhole has several important properties:
It is degenerate, like the Einstein–Rosen wormhole.
It is empty—there is no matter at all.
Its throat radius can be larger than the gravitational radius, making it traversable.
This was a major step forward. It suggested that traversable wormholes might exist without exotic matter, if we allow degenerate metrics and use regularized Einstein equations.
6. A Key Question: Can a Matter-Free Wormhole Be Stable?
The Klinkhamer wormhole produces gravity. That means it has gravitational mass. According to the equivalence principle, gravitational mass and inertial mass are the same.
If the wormhole has inertial mass, then it should behave like any other massive object:
It should respond to forces.
It should move.
It should feel its own gravity.
This leads to a simple but deep question:
Can a matter-free wormhole remain forever stable, or will it collapse under its own gravity?
This is the main question studied by Juri Dimaschko.
7. The Problem of Dynamics
To answer this question, one must study the dynamics of the wormhole—how it changes with time. In a spherically symmetric wormhole, the most important quantity is the radius of the throat, usually written as ( b(t) ).
At first glance, the Einstein equations seem to allow any function ( b(t) ). This led to criticism.
In particular, Feng (2023) argued that:
The time evolution of the Klinkhamer wormhole is not uniquely determined.
The initial-value problem is ill-posed.
Therefore, the wormhole has no clear physical meaning.
This criticism was serious and required a careful response.
8. Why Einstein’s Equations Are Not the Whole Story
Dimaschko agreed with one part of the criticism:
Yes, the Einstein equations alone do not fix ( b(t) ).
But this does not mean the theory is inconsistent.
The key insight is that a wormhole is not fully described by the local metric
alone. A wormhole also has global and topological properties.
The complete description is given by:
-
The metric
-
The global structure of the spacetime manifold
In earlier work, Dimaschko showed that when the principle of least action is applied correctly, it produces:
Einstein’s equations for the metric
Additional equations for global parameters, such as the wormhole throat radius
Thus, the dynamics of ( b(t) ) must come from physics beyond the local field equations.
9. Extending the Equivalence Principle
To find the correct equation for ( b(t) ), Dimaschko used a powerful idea: an extended equivalence principle.
The usual equivalence principle says that:
All objects fall the same way in a gravitational field, regardless of their composition.
Dimaschko extended this idea to include matter-free objects that generate gravity, such as degenerate wormholes.
The idea is simple:
If a wormhole has gravitational mass,
then it should move like any other massive object.
This allows us to treat the wormhole throat as if it were a particle moving in a gravitational field.
10. A Beautiful Result: Wormhole Motion Equals Particle Fall
Using the extended equivalence principle, Dimaschko showed something remarkable:
The radial motion of the Klinkhamer wormhole throat is exactly the same as the motion of a test particle falling in a Schwarzschild gravitational field.
This is a huge simplification. Instead of solving complicated wormhole equations, we can use well-known results from classical general relativity.
This correspondence:
Fixes the equation for ( b(t) )
Makes the Cauchy (initial-value) problem well-posed
Gives a unique evolution for any initial conditions
This completely resolves Feng’s objection.
11. What Happens to the Wormhole?
Once the correct equation of motion is known, the fate of the wormhole becomes clear.
The analysis shows that:
Any bound state of the Klinkhamer wormhole is unstable.
Over time, the throat radius decreases.
Eventually, the wormhole collapses.
The final state is a non-traversable Einstein–Rosen wormhole. The tunnel still exists mathematically, but nothing can pass through it.
This collapse is not caused by matter, radiation, or quantum effects. It is caused purely by gravity acting on itself.
12. Is the Collapse Fast or Slow?
One might think that such a wormhole would collapse almost instantly. Surprisingly, this is not the case.
Dimaschko’s estimates show that:
The Klinkhamer wormhole can be long-lived.
Its lifetime can be very large compared to many physical timescales.
This means that although the wormhole is not truly stationary, it can exist long enough to be physically meaningful.
In this sense, the Klinkhamer wormhole realizes the idea of a traversable wormhole—at least for a long time.
13. Why This Work Is Important
Dimaschko’s work is important for several reasons:
It gives a clear dynamical picture of a degenerate wormhole.
It resolves a major criticism about the consistency of the Klinkhamer wormhole.
It extends a fundamental principle of physics to a new class of objects.
It shows that exotic matter is not always required for traversable wormholes.
More broadly, it deepens our understanding of gravity itself.
14. A Deeper Lesson About Space and Gravity
One of the most profound messages of general relativity is that space is not just a passive background. It is an active physical entity.
Dimaschko’s results reinforce this idea:
Space can have mass.
Space can move.
Space can collapse.
A wormhole made of nothing but geometry behaves like a self-gravitating object. In a very real sense, empty space can fall in on itself.
15. Conclusion: When Nothing Becomes Something
The study of degenerate wormholes brings physics full circle, back to the ideas of Einstein and Rosen. By carefully combining geometry, topology, and fundamental principles, Juri Dimaschko has shown that a wormhole made of empty space is not just a mathematical curiosity.
It is a physical object with dynamics, a lifetime, and a clear fate.
The Klinkhamer wormhole may not live forever—but for a long time, it stands as a remarkable example of how rich and surprising Einstein’s theory of gravity still is.
Sometimes, the most interesting structures in the universe are made of nothing at all.
Reference: Juri Dimaschko, "Gravitational collapse of a degenerate wormhole", Arxiv, 2025. https://arxiv.org/abs/2512.16933
Technical Terms
1. Wormhole
A wormhole is a theoretical tunnel in space that connects two separate regions of the universe. Instead of traveling the long way through space, a wormhole would allow a shortcut.
Simple idea:
Imagine folding a sheet of paper and poking a hole through it. The hole connects two distant points instantly—that’s a wormhole.
2. Traversable Wormhole
A traversable wormhole is a wormhole that can be safely crossed by light, signals, or even people.
Simple idea:
Some wormholes collapse too fast or are blocked. A traversable wormhole stays open long enough for something to pass through.
3. Einstein–Rosen Bridge
The Einstein–Rosen bridge is the first wormhole solution discovered in 1935 by Albert Einstein and Nathan Rosen.
Simple idea:
It is a tunnel made entirely of curved space, with no matter inside it. However, it cannot be crossed—it closes too quickly.
4. Degenerate Metric
A metric describes how distances and time are measured in space. A degenerate metric means that this description becomes “singular” or incomplete at some point.
Simple idea:
It’s like trying to measure distance with a broken ruler at one spot—it works everywhere except at a special point.
5. Metric Determinant
The determinant of the metric is a mathematical value that helps describe the volume and shape of spacetime.
Simple idea:
It tells us how “healthy” spacetime is. When it becomes zero, normal equations stop working.
6. Regularized (Polynomial) Einstein Equations
The regularized Einstein equations are a modified version of Einstein’s original equations that still work when the metric becomes degenerate.
Simple idea:
Einstein and Rosen rewrote the equations in a safer form so they wouldn’t break when spacetime becomes special or extreme.
7. Gravitational Field
A gravitational field describes how gravity pulls objects toward each other.
Simple idea:
It’s the invisible influence that makes apples fall and planets orbit stars.
8. Matter-Free (Vacuum) Solution
A matter-free solution means a spacetime geometry that contains no matter or energy.
Simple idea:
Gravity exists even when nothing is there—space itself can curve and create gravity.
9. Equivalence Principle
The extended equivalence principle states that not only ordinary matter, but also objects made purely of curved space, such as wormholes, follow the same laws of motion in a gravitational field. This allows matter-free gravitational objects to be treated like massive particles.
10. Extended Equivalence Principle
The extended equivalence principle applies the equivalence principle not only to matter, but also to objects made purely of gravity.
Simple idea:
Even a wormhole made of empty space should move like a massive object.
11. Schwarzschild Gravitational Field
The Schwarzschild gravitational field describes the gravity around a simple, non-rotating object like a star or black hole.
Simple idea:
It’s the most basic and well-understood gravity model in physics.
12. Test Particle
A test particle is an object so small that it does not affect gravity itself.
Simple idea:
It’s like a dust grain falling toward Earth—it feels gravity but doesn’t change it.
13. Radial Dynamics
Radial dynamics describe motion toward or away from the center of an object.
Simple idea:
It’s motion straight in or straight out, not sideways.
14. Wormhole Throat
The throat is the narrowest part of a wormhole.
Simple idea:
It’s the middle of the tunnel—the tightest spot.
15. Throat Radius ( b(t) )
The throat radius describes how wide the wormhole opening is, and ( b(t) ) means it can change with time.
Simple idea:
It tells us whether the tunnel is opening, closing, or staying the same size.
16. Gravitational Collapse
Gravitational collapse happens when gravity pulls something inward so strongly that it shrinks.
Simple idea:
It’s what happens when an object cannot resist its own gravity and caves in.
17. Einstein Equations
The Einstein equations describe how spacetime bends in response to gravity.
Simple idea:
They tell space how to curve and matter how to move.
18. Null Energy Condition (NEC)
The null energy condition is a rule that normal matter follows in general relativity.
Simple idea:
It says energy must behave in a reasonable, positive way. Wormholes usually break this rule.
19. Exotic Matter
Exotic matter is matter that violates normal energy rules.
Simple idea:
It behaves in ways no known matter does—like having negative energy.
20. Phase Portrait
A phase portrait is a graph that shows how a system evolves over time.
Simple idea:
It’s like a map showing all possible futures of a system.
21. Cauchy Problem (Initial-Value Problem)
The Cauchy problem asks whether knowing the starting conditions is enough to predict the future.
Simple idea:
If you know where something starts and how fast it’s moving, can you predict what happens next?
22. Manifold
A manifold is the mathematical name for spacetime.
Simple idea:
It’s the “stage” on which space and time exist.
23. Topology
Topology describes the overall shape and connectedness of space.
Simple idea:
A coffee mug and a donut have the same topology because both have one hole.
24. Bound State
A bound state means something is trapped and cannot escape.
Simple idea:
A planet orbiting a star is in a bound state.
25. Long-Lived State
A long-lived state exists for a very long time before changing.
Simple idea:
It’s not permanent, but it lasts long enough to matter physically.
26. Klinkhamer wormhole
A Klinkhamer wormhole is a theoretical wormhole made only from curved space, without any matter. It is based on a degenerate spacetime metric and can be traversable if its throat is large enough. It acts as a gravitational object and slowly evolves over time.
Comments
Post a Comment