Non-exotic traversable wormholes in Einstein-Chern-Simons gravity

Abstract: We investigate traversable wormhole solutions in five-dimensional Einstein-Chern-Simons (EChS) gravity, a gauge-theoretic extension of General Relativity that introduces a higher-curvature correction parametrized by the combination $αl<sup>2$, where $α$ is a dimensionless coupling and $l$ is a length scale. Working with the Morris-Thorne metric and an anisotropic fluid, we derive exact expressions for the energy density, radial pressure, and lateral pressure at the wormhole throat. We show that the radial Null Energy Condition (NEC) is satisfied at the throat if and only if $αl<sup>2\leq-r_0<sup>2$, being saturated at $αl<sup>2=-r_0<sup>2$ and strictly satisfied for $αl<sup>2&lt;-r_0<sup>2$, independently of the shape function. In the latter regime, the energy density is strictly positive and the full NEC, Weak, Strong, and Dominant Energy Conditions can be simultaneously satisfied. These results are established analytically for arbitrary shape functions and illustrated concretely for the power-law family $b(r)=r_0(r_0/r)<sup>n$, $Φ=0$, for which all four standard energy conditions are satisfied at the throat for all $n&gt;0$ when the EChS coupling is sufficiently negative. We further show that the radial pressure is negative and geometrically fixed at the throat, independently of $α$, $b(r)$, and $Φ(r)$. Using the Volume Integral Quantifier (VIQ), we establish that the EChS correction reduces the magnitude of the negative integrated NEC contribution relative to GR, and derive an exact closed-form critical coupling $αl^2_{\rm crit}$ at which the VIQ vanishes. The hierarchy $αl^2_{\rm crit}&lt;-r_0<sup>2&lt;0$ shows that NEC satisfaction at the throat and a non-negative global VIQ are compatible but distinct conditions, both achievable within EChS gravity. All results reduce continuously to those of GR in the limit $l\to0$, confirming the consistency of the framework.

• The paper constructs five-dimensional Morris–Thorne wormholes in the torsionless Einstein–Chern–Simons sector and finds that the physical matter satisfies the radial NEC when the coupling obeys αl² < −r₀², independent of the shape function.
• The analysis shows that suitable negative coupling can make the full WEC, SEC, and DEC hold at the throat, although these conditions generally fail farther from the throat and do not yet establish globally non-exotic solutions.
• For power-law wormholes, the volume integral quantifier becomes non-negative at αl²crit = −2r₀²(n+1)/(n−1), demonstrating that integrated NEC balance and pointwise throat NEC satisfaction are related but distinct requirements.

Overview

This paper constructs traversable wormhole solutions within five-dimensional Einstein–Chern–Simons (EChS) gravity, a gauge-theoretic extension of General Relativity obtained via S-expansion of the AdS algebra (2608.16723). The theory's action contains the Einstein–Hilbert term plus a Gauss–Bonnet-like curvature-squared correction controlled by the dimensionless coupling $\alpha$ and a length scale $l$, entering only through the combination $\alpha l^2$. In the limit $l \to 0$ at fixed effective Newton constant, standard five-dimensional GR is recovered. The authors work with the Morris–Thorne metric, an anisotropic fluid source, and—crucially—evaluate energy conditions on the physical matter stress-energy tensor $\hat T_{ab}$ obtained by varying the matter action with respect to the vielbein, rather than on an effective tensor that absorbs geometric contributions. This distinguishes their approach from much of the modified-gravity wormhole literature, where NEC satisfaction is typically achieved only for the effective combination of matter and curvature terms.

Field equations and throat data

In the torsionless, spinless sector ($T^a = 0$, $k^{ab} = 0$), the vielbein field equation takes the form

$$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$

Substituting the five-dimensional Morris–Thorne metric yields exact expressions for $\rho(r)$, $p_r(r)$, and $l$0, each decomposing into an Einstein piece plus an $l$1 correction. Evaluated at the throat $l$2 (with $l$3), these reduce to remarkably simple forms:

$l$4

$l$5

where primes denote evaluation at $l$6. Two structural results follow immediately. First, the radial pressure is strictly negative and universally fixed: $l$7, independent of $l$8, $l$9, $\alpha l^2$0, and even of the gravitational correction itself. This is a purely geometric consequence of the throat field equations—the matter must exert radial tension to sustain the geometry—and cannot be evaded by any choice of matter model or coupling. Second, these throat values provide complete initial data $\alpha l^2$1 for numerical outward integration once a closure condition is supplied.

Energy conditions at the throat

The central analytical result is the sharp threshold governing the radial Null Energy Condition:

$\alpha l^2$2

Since the flare-out condition enforces $\alpha l^2$3, the sign is fixed entirely by $\alpha l^2$4: the radial NEC is violated for $\alpha l^2$5 (including the Einstein limit $\alpha l^2$6), saturated at $\alpha l^2$7, and strictly satisfied for $\alpha l^2$8—independently of the shape function. In the non-exotic regime $\alpha l^2$9, the energy density is automatically positive, and the lateral NEC holds whenever $l \to 0$0 or when a stated upper bound on $l \to 0$1 is met. Consequently, the full WEC, SEC, and DEC can be simultaneously satisfied by ordinary matter for suitable choices of $l \to 0$2 and $l \to 0$3; at the critical value $l \to 0$4, the NEC, WEC, and SEC hold automatically, with the DEC requiring only $l \to 0$5.

The paper contrasts this with cosmology: negative pressure in FLRW spacetimes (dark energy, quintessence) does not force NEC violation because it is isotropic, whereas the wormhole's negative radial pressure acts precisely along the direction where the NEC is most stringently tested. The EChS correction compensates this through its contribution to $l \to 0$6, playing a role analogous to dark energy but tied to local curvature rather than acting uniformly.

Power-law family and parameter space

Specializing to $l \to 0$7 with $l \to 0$8, all four standard energy conditions are satisfied at the throat for every $l \to 0$9 provided $\hat T_{ab}$0 is sufficiently negative. For $\hat T_{ab}$1, the condition $\hat T_{ab}$2 suffices for all four; for $\hat T_{ab}$3, the lateral DEC imposes the stronger bound $\hat T_{ab}$4. Notably, the SEC trace combination $\hat T_{ab}$5 remains positive throughout the exterior, not merely at the throat, when $\hat T_{ab}$6. Graphical analysis confirms, however, that NEC$\hat T_{ab}$7, $\hat T_{ab}$8, and DEC$\hat T_{ab}$9 generally change sign farther from the throat: the non-exotic property is established analytically at the throat but does not generically extend globally—a limitation the authors state explicitly.

The parameter-space analysis shows that no point satisfying the radial NEC at the throat can have negative energy density there, since $T^a = 0$0 whenever $T^a = 0$1. Tracing each condition as $T^a = 0$2 decreases from zero reveals the threshold ordering

$T^a = 0$3

with individual thresholds shifting under different choices of $T^a = 0$4 and $T^a = 0$5 while the universal radial NEC threshold remains fixed.

Isotropic pressure

Imposing $T^a = 0$6 at the throat reduces the free local data from three parameters to two, fixing $T^a = 0$7 algebraically in terms of $T^a = 0$8 and $T^a = 0$9. Two exact results emerge: isotropy with $k^{ab} = 0$0 forces $k^{ab} = 0$1, violating flare-out regardless of $k^{ab} = 0$2; and demanding global isotropy with $k^{ab} = 0$3 yields the unique solution $k^{ab} = 0$4, which violates both flare-out and asymptotic flatness. Hence no globally isotropic wormhole exists for $k^{ab} = 0$5 in EChS gravity; any viable isotropic solution requires a non-trivial redshift function, whose construction is left as an open problem requiring integration of the resulting differential relation between $k^{ab} = 0$6 and $k^{ab} = 0$7.

Volume Integral Quantifier

Using the Visser–Kar–Dadhich Volume Integral Quantifier $k^{ab} = 0$8 for the power-law family with $k^{ab} = 0$9 and $$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$0, the paper derives the closed-form result

$$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$1

linear in $$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$2 with slope $$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$3 independent of $$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$4. For $$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$5, the GR contribution diverges logarithmically while the EChS shift remains finite. The critical coupling at which the VIQ vanishes is

$$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$6

and the hierarchy $$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$7 establishes that pointwise NEC satisfaction at the throat and a non-negative integrated NEC balance are compatible but distinct conditions. At the NEC threshold, the magnitude of the negative VIQ is already reduced relative to GR by approximately 17% for $$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$8 and 41% for $$\frac{1}{8}\epsilon_{abcde}\left(2R^{bc}e^d e^e + \alpha l^2 R^{bc}R^{de}\right) = -\kappa\,\hat T_{ba}\star e^b.$$9. The authors caution that a non-negative VIQ does not imply pointwise NEC satisfaction throughout the exterior, consistent with the graphical findings.

Limitations and open questions

The paper is explicit about several restrictions. All results are confined to the torsionless, spinless sector with $\rho(r)$0 and vanishing sources coupled to $\rho(r)$1; the analysis is local to the throat except where noted, and the energy-condition satisfaction demonstrated there does not persist throughout the exterior for generic configurations. The stability of these solutions, their behavior under the full EChS constraint system (including the equation constraining $\rho(r)$2), and observational viability of the required negative coupling $\rho(r)$3 are not addressed. Globally complete non-exotic solutions—with appropriate matter closure conditions and boundary data—remain unconstructed, and the isotropic case with $\rho(r)$4 is identified as the most compelling target for future analytical and numerical work.

Conclusion

This work demonstrates that in five-dimensional Einstein–Chern–Simons gravity, the higher-curvature correction parametrized by $\rho(r)$5 permits traversable Morris–Thorne wormholes whose physical matter satisfies the full set of standard energy conditions at the throat, governed by the universal threshold $\rho(r)$6. The results are exact, shape-function-independent at the level of the radial NEC, and reduce continuously to the exotic-matter requirement of GR as $\rho(r)$7. The quantitative VIQ analysis refines this picture by separating local NEC satisfaction from the global integrated balance, showing both are achievable but require distinct couplings. The framework thus provides a clean setting in which the burden of sustaining a wormhole is transferred from exotic matter to gravitational dynamics, while leaving the construction of globally regular, energy-condition-satisfying solutions as the principal outstanding task.