Traversable Wormholes

• Traversable wormholes are nontrivial spacetime bridges defined by metrics like the Morris–Thorne line element, creating a throat connecting distant regions.
• Their construction requires precise geometric constraints and energy conditions, often involving exotic matter or alternative frameworks to ensure stable traversal.
• Modified gravity models and quantum corrections can suppress exotic matter needs, offering practical insights into astrophysical applications like lensing and shadow effects.

Traversable wormholes are topologically nontrivial solutions to the gravitational field equations representing spacetime channels that allow matter, light, and information to pass between remote regions. While foundational Morris–Thorne constructs in general relativity require exotic stress–energy violating the null energy condition (NEC), multiple frameworks—including modified gravity models, quantum backreaction, multimetric theories, dimensional extensions, and specific topological defects—can admit traversable wormhole solutions with suppressed or even absent exoticity. These solutions have rigorous definitions of traversability, explicit geometric constraints, and quantifiable energy condition properties.

1. Geometric Construction and Traversability Criteria

The prototypical traversable wormhole metric is the Morris–Thorne line element,

$$ds^2 = e^{2\Phi(r)} dt^2 - [1-b(r)/r]^{-1} dr^2 - r^2(d\theta^2 + \sin^2\theta\,d\phi^2)$$

with $\Phi(r)$ the redshift function (finite $\forall r$ to avoid horizons) and $b(r)$ the shape function. Traversability demands:

• Existence of a throat $r = r_0$ with $b(r_0) = r_0$
• Flaring-out condition $b'(r_0)<1$
• Asymptotic flatness $b(r)/r\to 0$ as $r\to\infty$
• Absence of event horizons ($e^{2\Phi(r)} > 0$ everywhere)
• Finite tidal accelerations for plausible travelers, typically $$|R_{\hat{t}\hat{i}\hat{t}\hat{j}}|\,|\xi^i| \lesssim g_\oplus$$ for some spatial vector $\xi^i$ (Sahoo et al., 2020, Konoplya et al., 2021)

The stress–energy tensor threading the throat is generally anisotropic, $$T^{\mu}{}_\nu = \mathrm{diag}(\rho, -p_r, -p_t, -p_t)$$, with $\rho$ energy density, $p_r$ radial pressure, $p_t$ tangential pressure.

2. Energy Conditions and Exoticity

Canonical general relativity solutions demand violation of at least the NEC at the throat, i.e., $\rho + p_r < 0$, interpretable as "exotic matter". Precise classification employs:

• Null Energy Condition (NEC): $\rho + p_r \geq 0$, $\rho + p_t \geq 0$
• Weak Energy Condition (WEC): $\rho \geq 0$ and NEC
• Dominant Energy Condition (DEC): $\rho \geq |p_r|,\,|\rho| \geq |p_t|$
• Strong Energy Condition (SEC): $\rho + p_r + 2p_t \geq 0$ plus WEC

In unimodular gravity, for example, explicit analytic families satisfy $\rho,\,\rho+p_r,\,\rho+p_t \geq 0$ everywhere for barotropic fluids, bypassing the need for NEC violation (Agrawal et al., 2022). Modified gravity and quantum corrections can further soften or localize the exotic sector, reducing the integrated violation (see the volume integral quantifier $I_V$ below).

Exotic Matter Quantification

The total "exoticity" supporting the wormhole can be measured by the volume integral quantifier (VIQ),

$I_V = \int_{r_0}^\infty (\rho + p_r) 4\pi r^2 dr$

which may be made arbitrarily small in specific models by tuning coupling constants or profiles (Sahoo et al., 2020, Cruz et al., 2024, Garattini, 2019).

3. Modified Gravity and Matter Sector Mechanisms

Multiple approaches relax or circumvent the exotic matter requirement:

3.1. Traceless $f(R,T)$ Gravity

The traceless $f(R,T)$ model replaces $R$ in the Einstein–Hilbert action with $f(R,T)=R+2\lambda T$, yielding field equations that, for $\lambda<-4\pi$ and fluid equation of state $p_r = \omega\rho$ with $\omega > -1$, satisfy all classical energy conditions except SEC, enabling arbitrarily small $I_V$ (Sahoo et al., 2020).

3.2. Unimodular Gravity

Unimodular gravity imposes the traceless Einstein equations. Power-law solutions—with barotropic anisotropic fluid $p_r = \alpha\rho$, $p_t = \beta p_r$—fulfill all energy conditions for wide parameter ranges (e.g., $\alpha=0.8$, $\beta=-1.1$) and admit macroscopic, traversable wormholes without exotic matter (Agrawal et al., 2022).

3.3. Multimetric Gravity

For $N\ge 2$ metric sectors with repulsive cross-metric coupling ($G_{\text{eff}}<0$), classical traversable wormhole solutions exist with matter satisfying all energy conditions. The construction in (Hohmann, 2013) yields massless, traversable wormholes with vanishing ADM mass for asymptotic observers, but practical assembly is impeded by sectoral repulsion and lack of non-gravitational communication.

3.4. Loop Quantum Gravity (LQG)

LQG regularizes the throat with quantum-corrected effective stress–energy from self-dual regular black holes. The polymeric parameter $\mathcal{P}$ and minimum area $a_0$ control the violation of the NEC, and increasing $\mathcal{P}$ can drive $I$ (the exotic-matter integral) to zero (Cruz et al., 2024). NEC violations are localized near the throat and diminish as quantum corrections become strong.

3.5. Loop Quantum Cosmology with Dark Matter

Combined LQC corrections and realistic dark matter profiles (NFW, pseudo-isothermal, perfect fluid types) source traversable solutions whose energy condition violations can be minimized or eliminated depending on the equation of state and parameters. LQC effects allow the shadow size to potentially mimic observed black hole shadows (Silva et al., 2024).

3.6. Gravitational Decoupling and Trace-Free Gravity

Minimal geometric deformation (MGD) in trace-free gravity allows analytic control over the seed fluid and "exotic" sector through a deformation function $f(r)$, localizing NEC violation to a subdomain via small parameter $\delta$. Embedding and lensing properties can be computed for each deformation (Panyasiripan et al., 2024).

4. Quantum Backreaction and Casimir Source Models

Quantum field backreaction offers alternative negative energy sources:

4.1. Quantum-Improved Gravity

Functional renormalization group (FRG) methods for asymptotic safety provide running Newton constants $G(\chi)$ whose antiscreening corrections $X_{\mu\nu}$ can source the required repulsive geometry near the throat. For pseudospherical (hyperbolic) slices, non-exotic matter suffices within explicit ranges $(\omega, \zeta)$, while spherical slices remain non-traversable without exoticity (Moti et al., 2020).

4.2. Casimir Energy

Negative Casimir energy from quantized fields inside suitable geometries allows special constructions:

• For spacetime dimension $D>3$, massless fields with $\rho_{\text{Casimir}}(r) = -\lambda_D/r^D$ provide explicit traversable wormhole solutions, with precise constraints on throat size and profile (Oliveira et al., 2021).
• Multi-mouth wormholes combining Casimir energy with quantum fields (e.g., massless charged fermions) yield entangled throat networks with fundamental group $F_n$, and the negative averaged null energy condition (ANEC) is satisfied locally, maintaining traversability even upon successive mouth insertion (Emparan et al., 2020).

4.3. Bulk Fermions in Black Hole Quotients

Bulk Weyl or Dirac fermions in non-contractible quotient backgrounds induce backreacted negative null stress-energy, enabling perturbatively and sometimes eternally traversable Planckian wormholes for proper boundary conditions, with explicit expressions for the time advance $\Delta V$ in terms of field parameters and horizon data (Marolf et al., 2019).

5. Exoticity Suppression and Topological/Tensor Mechanisms

Specific geometric or topological conditions can eliminate or suppress the need for exotic matter.

5.1. Spacetime Defects

A localized three-dimensional spacetime defect (hypersurface of vanishing metric determinant) can stabilize the wormhole throat in vacuum, leading to solutions in general relativity without exotic matter. The defect functions as a geometric substitute for negative energy (Klinkhamer, 2023).

5.2. Higher-Dimensional Extensions

Addition of extra spatial dimensions can shift the locus of energy condition violation: in a five-dimensional extension, NEC violation is carried by the extra dimension, while the four-dimensional observable matter at the throat can satisfy standard energy conditions. The effect depends on warp and redshift gradients in the extra coordinate (Kuhfittig, 2018).

5.3. Thin-Shell and Polyhedral Geometries

Non-spherically symmetric and thin-shell constructions (e.g., cubical or polyhedral wormholes) can confine exotic matter to lower-dimensional loci (edges or shells) or avoid traverser contact with negative energy regions altogether (0809.0907).

6. Traversability, Physical Realizability, and Astrophysical Implications

Traversable wormhole viability requires precise tuning of throat radius, mass/charge parameters, and matter sector control. Generally, practical traversability demands:

• No horizon or singularity (as verified by finiteness of curvature invariants, e.g., the Kretschmann scalar)
• Controlled tidal accelerations (constrained by curvature eigenvalues and throat geometry)
• Acceptable traversal time (integrability of proper distance and signal propagation)
• Minimized total NEC violation (quantified by $I_V$ or analogous integrals)

In Planckian limit solutions (quantum self-sustained, gravity's rainbow, noncommutative regularization), traversability is only formal, with unacceptably high tidal forces and microscopic throat size (Garattini, 2015).

Astrophysical applications include light deflection and lensing signatures. For LQC/dark matter models, computed shadow radii can match black hole observations such as the M87 shadow for certain parameter values (Silva et al., 2024). Multi-mouth wormholes and AdS/CFT generalizations exhibit nontrivial entanglement phase transitions associated with extremal surfaces (Liu et al., 4 Jan 2025).

7. Stability and Engineering Considerations

Most models lack comprehensive perturbative or nonlinear stability analyses. Multimetric wormholes are likely unstable under matter sector perturbations (Hohmann, 2013). Thin-shell or defect-based constructions depend critically on the physical existence of negative-tension states or degenerate metrics (Klinkhamer, 2023, 0809.0907). Feasibility of natural formation or laboratory assembly is generally negative except in speculative scenarios involving quantum gravity phase transitions (Horowitz et al., 2019).

Engineering multi-metric traversable wormholes would require coordinated assembly of identical matter sectors, with cross-sector gravitational repulsion and minimal direct coupling. Communication barriers arise due to non-overlapping matter sectors (Hohmann, 2013).

Table: Traversable Wormhole Solution Classes

Model/Framework	Exotic Matter Required	Key Parameter Constraints
Unimodular gravity (Agrawal et al., 2022)	No	$(\alpha, \beta)$ controls NEC/WEC
Traceless $f(R,T)$ (Sahoo et al., 2020)	No/Arbitrarily little	$\lambda < -4\pi,\; \omega > -1$
Multimetric gravity (Hohmann, 2013)	No	$G_{\text{eff}}<0$ (sector repulsion)
LQG (Cruz et al., 2024)	Diminished	$\mathcal{P}, a_0$ slope $I\rightarrow0$
Quantum improvement (Moti et al., 2020)	Avoidable (hyperbolic)	$0<\omega \lesssim 0.05,\; 0<\zeta\lesssim 0.01$
Casimir energy (Oliveira et al., 2021)	Yes	$D>3$; $r_0$ bounded below
Spacetime defect (Klinkhamer, 2023)	No	$\lambda^2\geq b_0^2$
Extra dimension (Kuhfittig, 2018)	Shifted to $l$ sector	$\partial_r H(r_0,l)<2/r_0$
Thin-shell/polyhedral (0809.0907)	Localized	Geometric surgery

• (Sahoo et al., 2020) Traversable wormholes in the traceless $f(R,T)$ gravity
• (Hohmann, 2013) Traversable wormholes without exotic matter in multimetric repulsive gravity
• (Agrawal et al., 2022) Unimodular Gravity Traversable Wormholes
• (Cruz et al., 2024) Traversable Wormholes from Loop Quantum Gravity
• (Moti et al., 2020) Traversability of quantum improved wormhole solution
• (Oliveira et al., 2021) Traversable Casimir Wormholes in D Dimensions
• (Klinkhamer, 2023) New Type of Traversable Wormhole
• (Kuhfittig, 2018) Traversable wormholes sustained by an extra spatial dimension
• (0809.0907) Traversable wormholes: Some simple examples
• (Emparan et al., 2020) Multi-mouth Traversable Wormholes
• (Silva et al., 2024) Traversable Wormholes Sourced by Dark Matter in Loop Quantum Cosmology
• (Liu et al., 4 Jan 2025) Traversable Wormhole in AdS and Entanglement
• (Garattini, 2015) Traversable Wormholes in Distorted Gravity
• (Panyasiripan et al., 2024) Traversable Wormholes in Minimally Geometrical Deformed Trace-Free Gravity using Gravitational Decoupling
• (Horowitz et al., 2019) Creating a Traversable Wormhole
• (Garattini, 2019) Traversable Wormholes and Yukawa Potentials
• (Marolf et al., 2019) Simple Perturbatively Traversable Wormholes from Bulk Fermions
• (Bakopoulos et al., 2021) Traversable wormholes in beyond Horndeski theories

Traversable wormholes occupy a rich intersection of gravitational physics, quantum effects, geometric analysis, and topological surgery, providing rigorous laboratories for energy condition violations, causal structure, and nontrivial spacetime topology. Constraints on exoticity and traversability metrics are sharply determined by each underlying model, with ongoing research focused on stability, astrophysical phenomenology, and the realization of large-scale traversable structures.