Null Raychaudhuri Equation and the Impossibility of Traversable Wormholes in Unimodular Gravity

This paper demonstrates that in unimodular gravity, traversable wormholes are impossible to support with ordinary matter because the theory's preservation of standard null focusing properties necessitates a violation of the null energy condition.

The Gist

Imagine the universe as a vast, flexible fabric. In this fabric, there are theoretical tunnels called wormholes that could, in theory, connect two distant points, allowing travelers to zip through instantly. For a long time, physicists have known a major problem: to keep these tunnels open and safe for travel, you need a very strange, "exotic" type of matter that pushes outward, acting like a cosmic anti-gravity. Normal matter, like stars, planets, or even you and me, only pulls inward (gravity), which would cause the tunnel to collapse.

Recently, some scientists proposed a new theory called Unimodular Gravity. They suggested that maybe, just maybe, this new theory allows for wormholes to stay open using only "normal" matter, without needing that weird exotic stuff.

This paper says: No, that's not possible.

Here is the simple breakdown of why, using some everyday analogies:

1. The Traffic Rule (The Raychaudhuri Equation)

Think of a beam of light traveling through space as a group of cars driving on a highway.

• Normal Gravity: If you drive through a normal gravitational field (like near a planet), the "cars" (light rays) naturally want to bunch up and get closer together. This is called focusing.
• The Tunnel Requirement: For a wormhole to work, the "cars" must do the opposite at the narrowest point (the throat). They must spread out or defocus so they don't crash into each other and block the path.

The paper uses a mathematical rule called the Raychaudhuri Equation. Think of this as the "Traffic Law of the Universe." It says that for light rays to spread out (defocus) at the narrowest point of a tunnel, something specific must happen: the "gravity" at that spot must be pushing outward.

2. The "Unimodular" Twist

The authors looked at Unimodular Gravity, a theory that changes how we calculate the total "volume" of space but keeps the local rules of how space bends and how light travels exactly the same as Einstein's original theory.

Imagine Unimodular Gravity as a different accounting method for the universe's budget. It changes how the total money is tallied up at the end of the year, but it doesn't change how a single dollar buys a loaf of bread in the grocery store.

The paper proves that while Unimodular Gravity changes the "accounting" (the big picture equations), it does not change the local traffic laws. The rule that says "light rays naturally want to bunch up" remains exactly the same.

3. The Impossible Shortcut

Because the local traffic laws haven't changed, the requirement for a wormhole to stay open is still the same:

• To keep the tunnel open, the light rays must spread out.
• To make light rays spread out, you need a force that pushes them apart.
• In physics, this "pushing apart" force requires exotic matter (matter that violates the "Null Energy Condition").

The paper argues that even in Unimodular Gravity, if you try to build a wormhole with only "normal" matter (which only pulls in), the light rays will bunch up, the tunnel will pinch shut, and the wormhole will become impassable.

The Verdict

The authors conclude that Unimodular Gravity cannot save us from the need for exotic matter.

If someone claims to have found a wormhole solution in this new theory that uses only normal matter, the paper suggests they are likely looking at a trick of the math or a misunderstanding of what "traversable" actually means. If the tunnel is truly passable for light (and therefore for anything else), it must violate the standard rules of energy, meaning it still needs that "exotic" stuff.

In short: You can change the rules of the universe's accounting, but you can't change the fact that to keep a tunnel open, you need something to push it apart. Normal matter can't do that, and Unimodular Gravity doesn't give it the ability to do so.

Technical Summary

Technical Summary: Null Raychaudhuri Equation and the Impossibility of Traversable Wormholes in Unimodular Gravity

Problem Statement
Recent literature has suggested that Unimodular Gravity (UG), a modification of General Relativity (GR) where the metric determinant is fixed, might admit traversable wormhole solutions supported by ordinary matter satisfying the Null Energy Condition (NEC). This possibility arises because UG alters the trace sector of Einstein's field equations, potentially allowing vacuum energy contributions to emerge as integration constants rather than direct sources. While some claims suggest UG could evade the standard GR prohibition on traversable wormholes without exotic matter, other critiques have questioned these conclusions specifically within the Morris–Thorne metric framework. The central problem addressed is whether the geometric obstruction to traversability—traditionally linked to NEC violation—persists in UG when analyzed from a purely local and covariant perspective, independent of specific metric ansätze.

Methodology
The authors formulate the condition for wormhole traversability not through specific solutions or effective stress-energy tensors, but as a purely local and covariant condition derived from the behavior of null geodesic congruences. The methodology proceeds in three stages:

1. Geometric Foundation: The analysis relies on the Null Raychaudhuri equation, a purely geometric identity governing the evolution of the expansion scalar ($\theta$) of a null geodesic congruence. The authors emphasize that this equation depends only on the affine structure of spacetime and the Riemann curvature tensor, remaining independent of the specific gravitational field equations.
2. Traversability Criterion: Traversability is defined as the ability of null signals to pass through a wormhole throat without encountering caustics (conjugate points). Geometrically, this requires the throat to be a local minimum of the cross-sectional area ($A$) of the congruence. This implies the expansion scalar must vanish ($\theta = 0$) at the throat and the rate of change of expansion must be positive ($d\theta/d\lambda > 0$), a condition termed "defocusing" or the "flare-out" condition.
3. Application to Unimodular Gravity: The authors apply this geometric criterion to UG. They demonstrate that while UG modifies the trace of the field equations, it preserves the local affine and causal structure of GR. Consequently, the Null Raychaudhuri equation remains identical to that in GR. The authors then contract the UG field equations with the null tangent vector $k^a$ to relate the curvature term ($R_{ab}k^a k^b$) to the matter content.

Key Contributions and Results

• Invariance of the Raychaudhuri Equation: The paper establishes that the Null Raychaudhuri equation is a geometric identity that holds unchanged in Unimodular Gravity. Since UG shares the same affine structure as GR, the kinematical evolution of null geodesics is identical in both theories.
• Equivalence of Curvature-Matter Coupling: By contracting the traceless field equations of UG with a null vector $k^a$, the authors show that the trace terms (including any cosmological constant-like integration constants) vanish identically because $g_{ab}k^a k^b = 0$. This yields the relation $R_{ab}k^a k^b = 8\pi T_{ab}k^a k^b$, which is identical to the relation derived from Einstein's equations in GR.
• The No-Go Theorem: Combining the defocusing requirement ($d\theta/d\lambda > 0$) with the Raychaudhuri equation and the UG field equations, the authors derive the necessary condition $T_{ab}k^a k^b < 0$ at the throat. This proves that any genuinely traversable wormhole in UG, supported by twist-free null generators, necessarily violates the Null Energy Condition.
• Generalization Beyond Specific Metrics: Unlike previous analyses restricted to the Morris–Thorne metric, this result is formulated covariantly. It applies to any traversable wormhole configuration regardless of the specific metric ansatz, provided the throat is defined by the behavior of null geodesic congruences.
• Clarification of "Ordinary Matter" Claims: The paper argues that claims of traversable wormholes in UG supported by ordinary matter must either involve a redefinition of the effective stress-energy tensor (shifting the NEC violation to geometric terms) or fail to satisfy the covariant traversability criterion derived from the Raychaudhuri equation.

Significance and Claims
The paper establishes a local no-go theorem for traversable wormholes in Unimodular Gravity supported by ordinary matter. The authors claim that the geometric obstruction to traversability is robust against the dynamical modifications introduced by UG. Specifically:

• The connection between traversability, null defocusing, and NEC violation is not an artifact of Einstein's equations but a consequence of the local causal structure of spacetime, which UG preserves.
• The presence of integration constants (acting as cosmological constants) in UG does not aid traversability because they do not contribute to the contraction $R_{ab}k^a k^b$ along null directions.
• The paper concludes that UG does not provide a mechanism to evade the Raychaudhuri obstruction. Any claim of a traversable wormhole in UG without exotic matter must be interpreted with the understanding that it either violates the standard NEC or fails to meet the strict geometric definition of traversability for null signals.

The authors note that while "timelike-only" traversability (allowing massive particles but not light to pass) is a theoretical possibility, it would violate standard causal ordering and is not considered physically admissible within a consistent relativistic framework. Thus, the result stands as a definitive geometric constraint on wormhole physics in Unimodular Gravity.