⚛️ general relativity

On the geometrical and dynamical distinction between Unimodular and General Relativistic wormholes

This paper demonstrates that while traversable wormholes in Unimodular Gravity and General Relativity share identical geometric and geodesic structures, they are dynamically distinguished by their source sectors, where Unimodular Gravity requires either restricted equations of state or an effective inhomogeneous vacuum contribution to sustain the same geometry.

Original authors: Marco Bosquez, Erick Pastén, Mauricio Cataldo, Norman Cruz

Published 2026-07-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Marco Bosquez, Erick Pastén, Mauricio Cataldo, Norman Cruz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, stretchy trampoline. In the most famous theory of gravity, General Relativity, the way this trampoline bends depends entirely on what you put on it. If you place a bowling ball (a star) in the middle, the fabric curves down, and marbles (planets) roll around it. This theory has been our best guide for over a century, but it has a weird quirk: it treats the "cosmological constant"—a mysterious energy that makes the universe expand—as a fixed setting you have to dial in before you start the experiment.

Now, imagine a different version of this trampoline called Unimodular Gravity. It's like the original trampoline, but with a strict rule: the total amount of fabric must stay the same size, no matter how much you stretch or squish it. Because of this rule, the "cosmological constant" isn't a dial you set; it pops out naturally as a result of the math, like a surprise gift at the end of a puzzle. Scientists are fascinated by this because it might solve some of the biggest headaches in physics regarding why the universe is expanding the way it is. But here's the big question: If you build a weird, tunnel-like shortcut through space (a wormhole) using the old rules versus the new rules, does the tunnel actually feel different to a traveler? Or is the only difference just in the invisible "stuff" holding the tunnel open?

This paper dives into that exact mystery. The authors, Marco Bosquez and his team, decided to build a specific type of wormhole—a tunnel connecting two distant parts of the universe—using both General Relativity and Unimodular Gravity. They wanted to see if the two theories create different kinds of tunnels or if they just use different "ingredients" to build the same shape.

Here is what they found: The tunnels are geometrically identical. If you were a spaceship flying through one, you wouldn't notice any difference in how the tunnel curves, how fast you fall, or how the stars look outside. The path you take (the geodesic) is exactly the same in both theories because the shape of the tunnel is the same.

However, the ingredients holding the tunnel together are completely different. In the old theory (General Relativity), the tunnel needs a very specific, rigid recipe of "exotic matter" (a strange type of fuel that pushes outward instead of pulling in) to stay open. If you try to use the Unimodular recipe in the old theory, the tunnel collapses unless you force the ingredients to follow a strict rule.

But if you let the Unimodular theory do its own thing, it allows for a much wider variety of recipes. The "exotic matter" doesn't have to be uniform; it can change as you move through the tunnel. The authors discovered that this flexibility creates a hidden, invisible "vacuum energy" that shifts and flows inside the tunnel. It's as if the Unimodular tunnel is held up by a fluid that changes its density as you travel, whereas the General Relativity tunnel is held up by a rigid, unchanging block.

Crucially, the paper rules out the idea that Unimodular Gravity gets rid of the need for "exotic matter" entirely. Even with the new rules, you still need that strange, repulsive stuff to keep the wormhole from pinching shut. The difference isn't that one theory is "better" at avoiding the weird physics; the difference is that Unimodular Gravity allows for a more dynamic, shifting source of energy to do the job.

The team also found that this shifting energy acts a bit like a cosmological constant that changes its mind as you get further away from the center of the tunnel. Near the tunnel's narrowest point (the throat), the energy might be very different from what it is far away in deep space. This suggests that the "cosmological constant" we see in the universe might actually be the result of these local, shifting effects from cosmic structures, rather than a fixed number set at the beginning of time.

In short, the paper concludes that while the "road" (the geometry) looks the same in both theories, the "engine" (the source of gravity) driving it is fundamentally different. Unimodular Gravity doesn't change the map; it just changes the fuel. This helps physicists understand that two different theories of gravity can produce the exact same universe we see, but they might be telling us very different stories about what's actually powering it.

Technical Summary: Geometrical and Dynamical Distinction Between Unimodular and General Relativistic Wormholes

Problem Statement
The central question addressed is the physical distinction between Unimodular Gravity (UG) and General Relativity (GR) when applied to traversable wormhole geometries. While UG modifies the dynamical structure of gravity by restricting the metric determinant to unity (reducing diffeomorphism invariance to volume-preserving transformations) and promoting the cosmological constant to an integration constant, it is unclear whether these theoretical differences manifest in observable spacetime properties. Specifically, the authors investigate whether wormhole solutions in UG are merely geometrically equivalent to GR solutions with different source interpretations, or if they possess distinct kinematical or dynamical features. Previous literature suggested UG might avoid the need for exotic matter, but recent analyses indicate that violations of the Null Energy Condition (NEC) persist. The paper seeks to clarify if the differences between the theories are geometric, kinematical, or purely dynamical (related to the source sector).

Methodology
The authors analyze a class of static, spherically symmetric Morris–Thorne wormholes described by the line element:
$ds^2 = dt^2 - \frac{dr^2}{1 - b(r)/r} - r^2(d\theta^2 + \sin^2\theta d\phi^2)$
where the shape function $b(r)$ is parameterized by a barotropic equation of state with radial pressure $p_r = \alpha \rho$ and tangential pressure $p_t = \beta p_r$ .

The investigation proceeds through three main analytical steps:

Geodesic Analysis: The authors derive the equations of motion for timelike and null geodesics to determine if the kinematical behavior of test particles differs between UG and GR for a fixed metric.
Source Sector Comparison: They compare the matter content required to sustain the same geometry in both theories by analyzing the energy-momentum conservation laws. In UG, the covariant divergence of the energy-momentum tensor is not necessarily zero ( $\nabla_\mu T^{\mu\nu} = J^\nu$ ), whereas in GR, it must vanish.
Strategy Classification: The authors define three strategies to map UG solutions to GR:
- Strategy A: Imposing local energy-momentum conservation ( $\nabla_\mu T^{\mu\nu} = 0$ ) in UG.
- Strategy B: Interpreting the non-conservation current in UG as an effective, inhomogeneous vacuum contribution ( $\Lambda_{int}(r)$ ) within a GR framework.
- Strategy C: Verifying consistency by solving the full Einstein equations in GR under the same barotropic assumptions.
Exoticity Quantification: The Volume Integral Quantifier (VIQ) is employed to calculate the total amount of NEC-violating matter required to support the wormhole, analyzing the convergence of the integral over the parameter space $(\alpha, \beta)$ .
Asymptotic Analysis: The radial profile of the effective vacuum term $\Lambda_{int}(r)$ is derived to understand its asymptotic behavior and its relation to the cosmological constant.

Key Contributions and Results

Geometrical and Kinematical Equivalence: The analysis confirms that for a fixed metric, the geodesic structure (particle trajectories, turning points, and traversability conditions) is identical in both UG and GR. The differences between the theories are not encoded in the kinematics of test particles but are entirely dynamical, residing in the source sector.
Constraints on the Equation of State (Strategy A): When local energy-momentum conservation is imposed on the UG wormhole solution, the two-parameter family of solutions $(\alpha, \beta)$ collapses to a one-parameter subset defined by the condition $\beta = \xi/2$ (where $\xi$ is the exponent of the shape function). This subset corresponds exactly to the unique solution obtained by solving the full Einstein equations in GR with the same barotropic ansatz.
Effective Inhomogeneous Vacuum (Strategy B): If the conservation condition is relaxed, the UG solution can be embedded into GR by introducing an effective, radially dependent vacuum energy density $\rho_\Lambda(r)$ and pressure $p_\Lambda(r) = -\rho_\Lambda(r)$ . This effective fluid accounts for the non-conservation current $J^\nu$ . While the geometry remains unchanged, the matter content in the GR interpretation becomes position-dependent, contrasting with the original barotropic fluid.
Exoticity and the VIQ: The study demonstrates that UG does not eliminate the need for exotic matter; the NEC is violated throughout the spacetime in all valid configurations. The VIQ analysis partitions the parameter space $(\alpha, \beta)$ $(α, β)$ into three regions:
1. White: No wormhole throat exists ( $\xi \geq 0$ ).
2. Yellow: A geometric throat exists, but the total exotic matter content diverges ( $-1 \leq \xi < 0$ ).
3. Green: The physically admissible sector where the throat exists and the total exotic matter is finite ( $\xi < -1$ ).
  The VIQ is found to be insensitive to the non-conservation current because the effective vacuum contribution cancels out in the NEC combination $(\rho + p_r)$ . Thus, the integrated exotic matter content is identical for a given geometry regardless of whether it is interpreted via UG or the GR-with-vacuum approach.
Asymptotic Vacuum Structure: The effective vacuum term $\Lambda_{int}(r)$ generated by the non-conservation current exhibits a power-law decay toward an asymptotic constant value $\Lambda_\infty$ at spatial infinity. The deviation from this asymptotic value at the throat, $\Delta \Lambda$ , is directly proportional to the non-conservation current. The condition $\Delta \Lambda = 0$ recovers the exact GR limit (Strategy A).

Significance and Claims
The paper concludes that Unimodular Gravity is geometrically equivalent to General Relativity; identical spacetime geometries can be generated by both theories. The distinguishing features of UG emerge solely in the dynamical interpretation of the source sector. Specifically:

UG allows for a broader class of solutions (a two-parameter family) where energy-momentum is not conserved, which can be reinterpreted in GR as a standard barotropic fluid plus an inhomogeneous vacuum energy.
The requirement of energy-momentum conservation in GR restricts the solution space to a specific curve within the UG parameter space.
The cosmological constant in UG can be interpreted not only as an integration constant (when conservation is imposed) but also as the asymptotic limit of an inhomogeneous vacuum structure generated by compact objects (wormholes) when conservation is relaxed.

The authors assert that their results reinforce the view that different gravitational theories may yield identical spacetime geometries while requiring fundamentally different source configurations to sustain them. The wormhole serves as a laboratory to disentangle kinematical (metric-dependent) from dynamical (source-dependent) aspects of gravitational theories, showing that spacetime geometry alone does not uniquely determine the physical content of a gravitational system.

Technical Summary: Geometrical and Dynamical Distinction Between Unimodular and General Relativistic Wormholes

More like this