Can wormhole spacetimes in Unimodular Gravity be supported by ordinary matter? A general proof of the exotic matter requirement

This paper establishes a general no-go theorem proving that traversable wormholes in Unimodular Gravity, like those in General Relativity, fundamentally require exotic matter that violates the Null Energy Condition due to the geometric flaring-out condition at the throat, regardless of specific metric functions or equations of state.

The Gist

The Big Question: Can We Build a "Normal" Wormhole?

Imagine you want to build a tunnel through a mountain to get from one side to the other instantly. In physics, this tunnel is called a wormhole. For a long time, scientists have known that to keep this tunnel open so a spaceship (or a person) can pass through, you need a very strange kind of fuel.

This fuel is called "exotic matter." Think of it like "anti-gravity" or "negative energy." It's the opposite of the stuff we see everywhere (like rocks, stars, and you). Exotic matter pushes outward instead of pulling inward, acting like a structural pillar that stops the tunnel from collapsing.

For decades, the rule in our standard theory of gravity (General Relativity) was: You cannot build a traversable wormhole without this exotic, "magic" fuel.

Recently, some scientists claimed they found a way to build these tunnels using ordinary matter (like normal gas or stars) by using a different version of gravity called Unimodular Gravity. They thought this new theory was flexible enough to let the tunnel stay open without the "magic" fuel.

This paper says: "No, you can't."

The authors (Mauricio Cataldo, Norman Cruz, and Patricio Salgado) have proven that even in this new theory of gravity, you still need exotic matter. It is impossible to build a wormhole with just ordinary stuff.

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The Analogy: The "Flaring-Out" Rule

To understand why this is impossible, imagine the wormhole throat (the narrowest part of the tunnel) as a funnel or a hourglass.

1. The Shape Requirement: For a wormhole to work, the tunnel has to get wider as you move away from the narrowest point. In physics, this is called the "flaring-out condition."

   • Imagine a funnel: The narrow neck is the throat. As you go up or down, the funnel gets wider.
   • The Math: The paper proves that for the funnel to get wider (flare out), the geometry of space itself demands a specific push.

2. The "Ordinary" Problem: Ordinary matter (like a rock or a star) always pulls things together (gravity). It acts like a heavy blanket trying to crush the funnel shut.

   • If you try to use ordinary matter to hold the funnel open, the weight of the matter will crush the throat, and the tunnel will collapse into a black hole.

3. The "Exotic" Solution: To keep the funnel open, you need something that pushes outward. This is exotic matter. It acts like a rigid, invisible scaffold holding the walls apart against the crushing weight of gravity.

The New Theory: Unimodular Gravity

The authors looked at Unimodular Gravity. You can think of this as a "remix" of Einstein's original gravity rules.

• Einstein's Rules (General Relativity): The universe is like a flexible sheet that can stretch and shrink in volume.
• Unimodular Rules: The universe is like a rigid box. The total volume of space is fixed and cannot change. The rules are slightly different, and the equations are "under-determined" (meaning there are fewer rules to follow, so you have more freedom to choose how to solve them).

The Hope: Because Unimolar Gravity has fewer rules and more freedom, the authors of the previous claim thought, "Maybe with all this extra freedom, we can arrange the geometry so that ordinary matter can hold the tunnel open without needing exotic stuff."

The Verdict: Geometry Wins

The authors of this paper did the math to check if that freedom helps. They found that geometry is the boss.

Here is the simple breakdown of their proof:

• They looked at the "throat" of the wormhole.
• They applied the "Flaring-Out Condition" (the rule that the tunnel must get wider).
• They discovered that this geometric rule automatically forces the math to say: "The energy density plus the pressure must be negative."

In everyday terms:
No matter how you twist the rules of Unimodular Gravity, no matter what shape you give the tunnel, and no matter what kind of ordinary matter you try to stuff inside it, the shape of the tunnel itself demands that you use "negative" or "exotic" energy to keep it from collapsing.

It's like trying to build a house of cards on a windy day. You can change the color of the cards, the type of glue, or the table they sit on (changing the gravity theory), but if the wind is too strong (the geometric requirement to flare out), you must use a heavy weight (exotic matter) to hold them down. You can't do it with just air.

Why This Matters

1. It Closes a Loophole: It shuts down the hope that a slight tweak to Einstein's equations (Unimodular Gravity) would let us build wormholes with normal stuff.
2. It's a Universal Law: The paper shows that the need for exotic matter isn't a mistake in our calculations; it's a fundamental law of the universe. Whether you use Einstein's old rules or the new Unimodular rules, the "Flaring-Out" geometry always wins.
3. The "No-Go" Theorem: They call this a "No-Go Theorem." It's a signpost that says, "Do not proceed." If you want a wormhole, you must find a way to create exotic matter. If you can't create exotic matter, you can't build a wormhole, regardless of which gravity theory you use.

Summary

The paper is a definitive "No." Even in the alternative gravity theory called Unimodular Gravity, you cannot build a wormhole that people can travel through using only ordinary matter. The shape of the wormhole itself requires "exotic" fuel to stay open, just like it does in our standard understanding of the universe. The geometry of the tunnel makes the need for magic fuel unavoidable.

Technical Summary

1. Problem Statement

The central problem addressed is whether Unimodular Gravity (UG)—an alternative theory to General Relativity (GR) where the metric determinant is fixed ($\sqrt{-g} = \epsilon_0$)—allows for traversable wormholes supported solely by ordinary matter (matter satisfying standard energy conditions).

Recent work by Agrawal et al. (2023) claimed to have found traversable wormhole solutions in UG supported by ordinary matter, suggesting UG could evade the "exotic matter" requirement inherent in GR. A previous comment by the authors (Cataldo & Cruz, 2025) refuted a specific barotropic solution proposed by Agrawal et al., but left open the question of whether other less restrictive configurations in UG might succeed. This paper aims to provide a definitive, general answer to that question.

2. Methodology

The authors employ a rigorous geometric analysis based on the flaring-out condition required for traversability, independent of specific matter models or equations of state.

• Metric and Framework: They utilize the standard static, spherically symmetric wormhole metric:
  $$ds^2 = -e^{2\Phi(r)}dt^2 + \frac{dr^2}{1 - b(r)/r} + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$
  where $\Phi(r)$ is the redshift function and $b(r)$ is the shape function.
• Field Equations: They apply the trace-free field equations of Unimodular Gravity:
  $$R_{\mu\nu} - \frac{1}{4}R g_{\mu\nu} = \kappa \left(T_{\mu\nu} - \frac{1}{4}T g_{\mu\nu}\right)$$
• System Analysis: The authors note that the UG field equations for this metric result in an underdetermined system: there are only 2 independent equations for 5 unknown functions ($\Phi, b, \rho, p_r, p_t$), compared to 3 independent equations in GR. This necessitates additional constraints (equations of state or functional forms) to find specific solutions.
• Proof Strategy: Instead of solving for specific solutions, the authors derive general expressions for the energy density ($\rho$) and radial pressure ($p_r$) at the throat ($r_0$) directly from the field equations. They then evaluate the Null Energy Condition (NEC), defined as $\rho + p_r \geq 0$, specifically at the throat.
• Case Studies: The proof is applied to two distinct configurations:
  1. Zero-tidal-force wormholes: Where $\Phi(r) = \text{const}$.
  2. Tidal-force wormholes: Where $\Phi(r)$ is a variable function.

3. Key Contributions

• General No-Go Theorem: The paper establishes a universal theorem proving that all traversable wormhole configurations in Unimodular Gravity require exotic matter. This generalizes previous results that were limited to specific equations of state.
• Independence from Modeling Choices: The authors demonstrate that the violation of energy conditions is independent of:
  • The specific form of the shape function $b(r)$.
  • The specific form of the redshift function $\Phi(r)$.
  • The choice of equation of state (barotropic or otherwise).
  • The specific closure scheme used to solve the underdetermined system.
• Geometric Origin of Exotic Matter: The paper identifies the geometric flaring-out condition ($b'(r_0) \leq 1$) as the sole cause of the energy condition violation, rather than a specific artifact of the gravitational theory's field equations.

4. Key Results

The core mathematical result is derived by evaluating the sum of energy density and radial pressure at the throat ($r = r_0$).

• The Flaring-Out Condition: For a traversable wormhole, the shape function must satisfy $b(r_0) = r_0$ and $b'(r_0) \leq 1$.
• The NEC Violation:
  Regardless of whether the redshift function is constant or variable, the authors derive the following expression at the throat for Unimodular Gravity:
  $$\rho(r_0) + p_r(r_0) = \frac{b'(r_0) - 1}{\kappa r_0^2}$$
  Since the traversability condition requires $b'(r_0) \leq 1$, it follows inevitably that:
  $$\rho(r_0) + p_r(r_0) \leq 0$$
• Implications:
  • This inequality represents a violation of the Null Energy Condition (NEC).
  • Consequently, the Weak Energy Condition (WEC) and Strong Energy Condition (SEC) are also violated.
  • This result holds for both tidal and zero-tidal-force configurations.
  • The mathematical form of this violation is identical to that found in General Relativity, despite the structural differences in the field equations (trace-free vs. Einstein tensor).

5. Significance

• Refutation of UG as an Exotic-Matter-Free Solution: The paper definitively closes the door on the possibility that Unimodular Gravity allows for traversable wormholes supported by ordinary matter. It corrects the interpretation of recent literature (Agrawal et al.) by showing that the violation is not a modeling error but a fundamental geometric necessity.
• Comparison with Other Modified Theories: The authors contrast UG with other modified gravity theories (e.g., $f(R)$ gravity, Einstein-Cartan theory, Brane-world models) where higher-order curvature terms or extra dimensions can sometimes support wormholes with ordinary matter. UG fails to provide this escape because it lacks these additional geometric degrees of freedom; it only reduces the number of independent equations without altering the fundamental geometric constraint of the throat.
• Theoretical Distinction: The result reinforces the view that while UG and GR share classical field equations under certain conservation assumptions, they are distinct frameworks. However, regarding the specific physics of traversable wormholes, they share the same fundamental constraint: exotic matter is required.
• Universality of the Exotic Matter Requirement: The study suggests that the need for exotic matter is a robust feature of traversable wormholes, arising from the geometry of spacetime itself (the flaring-out condition) rather than the specific formulation of gravity (GR vs. UG). This implies that evading the exotic matter requirement would likely require a fundamentally different geometric structure beyond simple modifications to the field equations.

In conclusion, the paper establishes that Unimodular Gravity shares the same fundamental "no-go" theorem for traversable wormholes as General Relativity: the geometric requirement for a traversable throat inevitably forces a violation of the Null Energy Condition, necessitating the presence of exotic matter.