Wormhole Solutions in f(Q,T) Gravity

• Wormhole solutions are spacetime configurations featuring a throat, defined by Morris–Thorne geometry with flaring‐out and asymptotic flatness criteria.
• The f(Q,T) approach, incorporating nonmetricity and an effective viscous pressure, redistributes geometric and material support to avoid exotic matter.
• Anisotropic fluids, modified field equations, and precise parameter tuning ensure the weak energy condition holds throughout the traversable wormhole geometry.

Searching arXiv for the cited wormhole paper and closely related work to ground the article in current literature. A wormhole solution is a spacetime configuration with a throat that connects separated regions while remaining free of horizons at the throat and, in many constructions, asymptotically flat. In the modern literature, the term covers both the classical Morris–Thorne class and a broad set of modified-gravity, teleparallel, Lorentz-violating, cosmological, and defect-supported geometries. A recent representative example is the viscous $f(Q,T)$ construction in symmetric teleparallel gravity, where the gravitational sector is written in terms of the nonmetricity scalar $Q$ and the stress-energy trace $T$, and where the weak energy condition can be satisfied throughout space for specific parameter ranges, so that the supporting matter is “near normal” rather than exotic (Sadatian et al., 2024).

1. Geometric definition and throat criteria

The canonical static, spherically symmetric wormhole geometry is the Morris–Thorne ansatz

$$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$

with redshift function $\phi(r)$ and shape function $b(r)$ (Sadatian et al., 2024). The throat is located at $r=r_0$ such that $b(r_0)=r_0$, and the flaring-out condition is

$b'(r_0)<1.$

Traversability further requires $\phi(r)$ finite everywhere, so that no horizon forms, and asymptotic flatness is usually imposed through

$Q$0

(Sadatian et al., 2024).

This geometric definition is common across otherwise disparate frameworks. It underlies exact or approximate solutions in symmetric teleparallel $Q$1 gravity with viscosity (Sadatian et al., 2024), in $Q$2 gravity with strange quark matter and a radial dependent bag parameter (Tayde et al., 2022), in Unimodular Gravity (Agrawal et al., 2022), in bumblebee gravity (Övgün et al., 2018), in modified teleparallel–Rastall gravity (Nazavari et al., 2023), and in several non-Riemannian or defect-based constructions (Klinkhamer, 2023). By contrast, some solutions are deliberately non-traversable despite having a wormhole interpretation, such as the de Sitter configuration built from two antipodal Schwarzschild–de Sitter black holes matched by a shell (Dai et al., 2018).

2. Symmetric teleparallel $Q$3 formulation

In the symmetric teleparallel framework, also called nonmetric gravity, gravitational effects are encoded in the nonmetricity scalar $Q$4, constructed from the nonmetricity tensor $Q$5 (Sadatian et al., 2024). The $Q$6 extension used for the viscous wormhole solution is defined by the action

$Q$7

where $Q$8 is the trace of the matter stress-energy tensor (Sadatian et al., 2024). Variation with respect to the metric and the connection yields modified field equations involving $Q$9, $T$0, the symmetric-teleparallel superpotential $T$1, and the hypermomentum density $T$2 (Sadatian et al., 2024).

For the static, spherically symmetric wormhole metric, the nonmetricity scalar is

$T$3

(Sadatian et al., 2024). The matter sector is taken with $T$4, where $T$5 is the isotropic pressure, giving

$T$6

Under this choice, the $T$7 coupling induces non-conservation of ordinary matter, $T$8, with source terms depending on $T$9 and on the adopted matter Lagrangian (Sadatian et al., 2024).

The specific model adopted for the viscous solution is

$$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$0

with

$$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$1

(Sadatian et al., 2024). This functional form is non-linear in $$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$2 and linear in $$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$3. The paper’s qualitative interpretation is that the direct $$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$4–$$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$5 coupling provides an effective extra source term in the field equations, and, together with viscosity, can support wormhole geometries without the standard GR requirement of exotic matter (Sadatian et al., 2024). This suggests a redistribution between geometric and material support, although the paper formulates that effect through the modified field equations rather than through an effective-fluid reformulation.

3. Matter sector, viscosity, and explicit wormhole ansatz

The wormhole is threaded by an anisotropic fluid

$$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$6

with trace

$$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$7

(Sadatian et al., 2024). Bulk viscosity is incorporated phenomenologically through

$$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$8

with

$$ds^2=e^{2\phi(r)}dt^2-\left(1-\frac{b(r)}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta\,d\varphi^2\right),$$9

in units $\phi(r)$0 (Sadatian et al., 2024). In this setup, viscosity acts as a uniform pressure shift.

To ensure regularity and asymptotic fall-off, the explicit wormhole functions are chosen as

$\phi(r)$1

with $\phi(r)$2, $\phi(r)$3 arbitrary, and $\phi(r)$4 (Sadatian et al., 2024). These choices give $\phi(r)$5, enforce asymptotic flatness, and yield

$\phi(r)$6

for any $\phi(r)$7 (Sadatian et al., 2024).

The paper then substitutes $\phi(r)$8, $\phi(r)$9, $b(r)$0, and $b(r)$1 into the modified field equations and, instead of solving differential equations, evaluates the resulting algebraic expressions for $b(r)$2, $b(r)$3, and $b(r)$4 pointwise (Sadatian et al., 2024). For analytical transparency, the expressions are truncated at order $b(r)$5, producing

$b(r)$6

$b(r)$7

$b(r)$8

(Sadatian et al., 2024).

The same study also introduces a phenomenological equation of state with viscosity,

$b(r)$9

and derives additional expressions for $r=r_0$0, $r=r_0$1, and $r=r_0$2 in terms of $r=r_0$3 (Sadatian et al., 2024). According to the numerical discussion in that paper, decreasing $r=r_0$4 to more negative values, together with appropriate $r=r_0$5, helps maintain the weak energy condition.

4. Weak energy condition and parameter domain

For anisotropic matter, the standard pointwise energy conditions used in the paper are

$r=r_0$6

$r=r_0$7

$r=r_0$8

(Sadatian et al., 2024). The analysis concentrates on the WEC.

Using the truncated expressions, the key combinations become

$r=r_0$9

$b(r_0)=r_0$0

$b(r_0)=r_0$1

(Sadatian et al., 2024). Evaluated at the throat $b(r_0)=r_0$2, the paper quotes

$b(r_0)=r_0$3

(Sadatian et al., 2024).

From these expressions and from the numerical profiles, the WEC-satisfying parameter windows are reported as

$b(r_0)=r_0$4

or

$b(r_0)=r_0$5

equivalently

$b(r_0)=r_0$6

or

$b(r_0)=r_0$7

(Sadatian et al., 2024). The paper further states that the weak energy condition is established in the whole space for those intervals, with representative plots based on $b(r_0)=r_0$8, $b(r_0)=r_0$9, $b'(r_0)<1.$0, and $b'(r_0)<1.$1 (Sadatian et al., 2024).

The central conclusion is that the model admits wormhole configurations without exotic matter and that the supporting matter is “near normal” (Sadatian et al., 2024). In the language of the paper, the $b'(r_0)<1.$2–$b'(r_0)<1.$3 coupling and the viscous pressure shift together provide the needed support while preserving $b'(r_0)<1.$4, $b'(r_0)<1.$5, and $b'(r_0)<1.$6 throughout space (Sadatian et al., 2024).

5. Traversability, regularity, and caveats

The traversability criteria are geometric. For the viscous $b'(r_0)<1.$7 solution, the absence of horizons follows from $b'(r_0)<1.$8 being finite everywhere, and asymptotic flatness follows from $b'(r_0)<1.$9 in

$\phi(r)$0

(Sadatian et al., 2024). The throat remains traversable because $\phi(r)$1 is automatically satisfied for $\phi(r)$2 (Sadatian et al., 2024).

The paper notes that the scalar $\phi(r)$3 is finite away from the throat, while near $\phi(r)$4 its denominator contains $\phi(r)$5, which vanishes at the throat. The adopted geometry still satisfies the flaring-out condition, and the wormhole throat is treated as traversable within the symmetric teleparallel framework (Sadatian et al., 2024). No explicit Ricci scalar $\phi(r)$6 or Kretschmann invariant is computed, but the authors state that the chosen Morris–Thorne functions ensure the absence of curvature singularities in the Morris–Thorne sense if the throat conditions are satisfied (Sadatian et al., 2024).

Human traversability is discussed only qualitatively. The usual tidal bounds are stated as

$\phi(r)$7

and the paper observes that small $\phi(r)$8 and $\phi(r)$9 suppress $Q$00 and $Q$01 away from the throat (Sadatian et al., 2024). A quantitative tidal-force analysis is not carried out. Likewise, the paper does not perform a full sound-speed or TOV stability analysis, although it states that the smooth, positive profiles in the WEC-satisfying branches suggest that causality-compliant sound speeds may be obtained by tuning $Q$02 (Sadatian et al., 2024).

Several limitations are explicit. The analytic WEC argument is based on an $Q$03 truncation; viscosity is introduced phenomenologically through a constant $Q$04; NEC, DEC, and SEC are not comprehensively studied; and curvature invariants are not explicitly computed (Sadatian et al., 2024). This suggests that the solution is best regarded as a mathematically viable branch within a specified approximation scheme rather than as a complete phenomenological model.

6. Position within the wormhole-solution literature

The recent literature shows that “wormhole solution” now denotes a family of constructions rather than a single paradigm. The main frameworks represented in the cited work are summarized below.

Framework	Support mechanism	Energy-condition statement
$Q$05 with viscosity (Sadatian et al., 2024)	$Q$06–$Q$07 coupling plus bulk viscosity	WEC satisfied for $Q$08 or $Q$09
$Q$10 with MIT bag matter (Tayde et al., 2022)	Strange quark matter, embedding procedure	WEC and SEC satisfied; NEC tangentially satisfied and radially partially violated near the throat
Unimodular Gravity (Agrawal et al., 2022)	Trace-free UG field equations with anisotropic EoS	NEC, WEC, DEC, and SEC can all hold
Kalb–Ramond background (Lessa et al., 2020)	Lorentz-symmetry-breaking tensor VEV	For $Q$11, NEC, WEC, DEC, and SEC are all satisfied
Bumblebee gravity (Övgün et al., 2018)	Lorentz-violating vector coupled to curvature	Under specific conditions, normal matter supports the geometry
Modified teleparallel–Rastall gravity (Nazavari et al., 2023)	Rastall-modified conservation and torsion coupling	NEC and WEC valid at the throat and through spacetime
Bopp–Podolsky electrodynamics (Frizo et al., 2022)	Nonminimal electromagnetic couplings	NEC and WEC violated near the throat; shadow remains viable versus Sgr A*
Defect-supported GR wormhole (Klinkhamer, 2023)	$Q$12 spacetime defect or hypermomentum	Ideal configuration has zero negative-energy requirement

Several contrasts are especially sharp. In GR, standard Morris–Thorne traversable wormholes require NEC violation at the throat. That feature remains explicit in some constructions, such as the Bopp–Podolsky wormhole, where $Q$13 in the radial direction (Frizo et al., 2022), or in cosmological wormholes embedded in FLRW backgrounds, where the NEC is violated at the throat independently of the cosmological model (Pérez et al., 2023). By contrast, Unimodular Gravity, Kalb–Ramond backgrounds, bumblebee gravity, and modified teleparallel–Rastall gravity each provide explicit regimes in which classical energy conditions hold at the throat or throughout the spacetime (Agrawal et al., 2022).

There are also solutions whose significance is not primarily the avoidance of exotic matter. In Palatini-modified gravity, wormhole cores can replace black-hole singularities while preserving geodesic completeness even when curvature divergences occur at the throat (Olmo et al., 2016). In Wyman’s massless-scalar solution, a throat exists and particles may traverse it, but the geometry is not humanly traversable because acceleration, tidal-force, and travel-time constraints cannot be met simultaneously (Formiga et al., 2014). In the de Sitter construction with antipodal Schwarzschild–de Sitter black holes and a positive-energy shell, the exterior region remains causally connected, but the wormhole is non-traversable because the throats lie behind black-hole horizons (Dai et al., 2018).

A plausible implication of this broader literature is that the status of “exotic matter” depends strongly on the gravitational framework and on what is treated as material rather than geometric support. In the viscous $Q$14 solution, that distinction is encoded in the direct $Q$15–$Q$16 coupling and in the pressure shift produced by viscosity (Sadatian et al., 2024). In other models, the same role is played by unimodular trace-free dynamics, Lorentz-symmetry-breaking condensates, torsion–matter couplings, or codimension-one defects (Klinkhamer, 2023). The common invariant across these approaches is the throat geometry; what varies is the mechanism by which the field equations permit it.