This paper presents a comprehensive analysis of traversable wormhole geometries within the framework of f(Q, T) gravity, where Q represents the non-metricity scalar and T denotes the trace of the energy-momentum tensor. We investigate four distinct functional forms: linear Model I (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f = \alpha Q + \beta T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mi>α</mi> <mi>Q</mi> <mo>+</mo> <mi>β</mi> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>), quadratic Model II (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f = \alpha Q + \gamma Q^2 + \beta T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mi>α</mi> <mi>Q</mi> <mo>+</mo> <mi>γ</mi> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>β</mi> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>), product Model III (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f = \alpha Q^n T^m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mi>α</mi> <msup> <mi>Q</mi> <mi>n</mi> </msup> <msup> <mi>T</mi> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>), and exponential Model IV (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f = \alpha Q e^{\beta T}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mi>α</mi> <mi>Q</mi> <msup> <mi>e</mi> <mrow> <mi>β</mi> <mi>T</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>). Two novel shape functions are introduced: an exponential relaxation form <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$b(r) = r_0 + r_0 \epsilon [1 - \exp (-(r-r_0)/\sigma )]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>r</mi> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>0</mn> </msub> <mi>ϵ</mi> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>-</mo> <mo>exp</mo> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>-</mo> <msub> <mi>r</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and an oscillatory corrected form <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$b(r) = r_0 (r/r_0)^{\eta } + \lambda r_0 \sin ^2(\pi r_0/r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>r</mi> <mn>0</mn> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">/</mo> <msub> <mi>r</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mi>η</mi> </msup> <mo>+</mo> <mi>λ</mi> <msub> <mi>r</mi> <mn>0</mn> </msub> <msup> <mo>sin</mo> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <msub> <mi>r</mi> <mn>0</mn> </msub> <mo stretchy="false">/</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Our analysis reveals that for negative coupling parameters (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\beta < -0.3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo><</mo> <mo>-</mo> <mn>0.3</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\gamma \approx 0.12$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>≈</mo> <mn>0.12</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n \approx 1.45$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≈</mo> <mn>1.45</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$m \approx -0.42$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≈</mo> <mo>-</mo> <mn>0.42</mn> </mrow> </math></EquationSource> </InlineEquation>), the null energy condition is satisfied at the throat, eliminating the need for exotic matter. The Volume Integral Quantifier yields minimal exotic matter content with <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$I_V = 0.08$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>V</mi> </msub> <mo>=</mo> <mn>0.08</mn> </mrow> </math></EquationSource> </InlineEquation> for Model II, compared to <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$I_V = 1.45$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>V</mi> </msub> <mo>=</mo> <mn>1.45</mn> </mrow> </math></EquationSource> </InlineEquation> for positive coupling. Stability analysis via the modified TOV equation confirms dynamical equilibrium with force deviations <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$< 0.002$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo><</mo> <mn>0.002</mn> </mrow> </math></EquationSource> </InlineEquation>. Sound speeds remain within the causal limit (<InlineEquation ID="IEq14"> <EquationSource Format="TEX">$0 \le v_s^2 \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <msubsup> <mi>v</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>). Junction conditions with phantom energy (<InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\omega = -1.2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo>=</mo> <mo>-</mo> <mn>1.2</mn> </mrow> </math></EquationSource> </InlineEquation>) yield stable thin-shell configurations at <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$a = 2.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>=</mo> <mn>2.5</mn> </mrow> </math></EquationSource> </InlineEquation>. Gravitational lensing reveals distinctive deflection angles up to <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\hat{\alpha }_{\text {max}} = 1.8\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi>α</mi> <mo stretchy="false">^</mo> </mover> <mtext>max</mtext> </msub> <mo>=</mo> <mn>1.8</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> for Model II, compared to <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$1.2\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1.2</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> for Schwarzschild black holes. Quasinormal mode frequencies differ by <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$2-6\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>-</mo> <mn>6</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> from black hole values. Model parameters are constrained using Pantheon+ supernovae, Planck CMB, and BAO data, yielding best-fit values <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\beta = -0.48 \pm 0.14$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>=</mo> <mo>-</mo> <mn>0.48</mn> <mo>±</mo> <mn>0.14</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\gamma = 0.12 \pm 0.04$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.12</mn> <mo>±</mo> <mn>0.04</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$n = 1.45 \pm 0.13$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1.45</mn> <mo>±</mo> <mn>0.13</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$m = -0.42 \pm 0.09$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mo>-</mo> <mn>0.42</mn> <mo>±</mo> <mn>0.09</mn> </mrow> </math></EquationSource> </InlineEquation> at 95% confidence.

Traversable Wormholes in f(Q, T) Gravity: Quadratic Corrections, Exponential Relaxation Shape Functions, and Gravitational Lensing Signatures

期刊论文
发表日期：

2026年6月18日

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Uzma Gul,
Hina Zahir,
Sumaira Saleem Akhtar,
Jamshed Khan,
Adnan Malik

摘要

This paper presents a comprehensive analysis of traversable wormhole geometries within the framework of f(Q, T) gravity, where Q represents the non-metricity scalar and T denotes the trace of the energy-momentum tensor. We investigate four distinct functional forms: linear Model I ( $f = \alpha Q + \beta T$ $f = α Q + β T$ ), quadratic Model II ( $f = \alpha Q + \gamma Q^2 + \beta T$ $f = α Q + γ Q^{2} + β T$ ), product Model III ( $f = \alpha Q^n T^m$ $f = α Q^{n} T^{m}$ ), and exponential Model IV ( $f = \alpha Q e^{\beta T}$ $f = α Q e^{β T}$ ). Two novel shape functions are introduced: an exponential relaxation form $b(r) = r_0 + r_0 \epsilon [1 - \exp (-(r-r_0)/\sigma )]$ $b (r) = r_{0} + r_{0} ϵ [1 - exp (- (r - r_{0}) / σ)]$ and an oscillatory corrected form $b(r) = r_0 (r/r_0)^{\eta } + \lambda r_0 \sin ^2(\pi r_0/r)$ $b (r) = r_{0} {(r / r_{0})}^{η} + λ r_{0} {sin}^{2} (π r_{0} / r)$ . Our analysis reveals that for negative coupling parameters ( $\beta < -0.3$ $β < - 0.3$ , $\gamma \approx 0.12$ $γ \approx 0.12$ , $n \approx 1.45$ $n \approx 1.45$ , $m \approx -0.42$ $m \approx - 0.42$ ), the null energy condition is satisfied at the throat, eliminating the need for exotic matter. The Volume Integral Quantifier yields minimal exotic matter content with $I_V = 0.08$ $I_{V} = 0.08$ for Model II, compared to $I_V = 1.45$ $I_{V} = 1.45$ for positive coupling. Stability analysis via the modified TOV equation confirms dynamical equilibrium with force deviations $< 0.002$ $< 0.002$ . Sound speeds remain within the causal limit ( $0 \le v_s^2 \le 1$ $0 \leq v_{s}^{2} \leq 1$ ). Junction conditions with phantom energy ( $\omega = -1.2$ $ω = - 1.2$ ) yield stable thin-shell configurations at $a = 2.5$ $a = 2.5$ . Gravitational lensing reveals distinctive deflection angles up to $\hat{\alpha }_{\text {max}} = 1.8\pi $ ${\hat{α}}_{max} = 1.8 π$ for Model II, compared to $1.2\pi $ $1.2 π$ for Schwarzschild black holes. Quasinormal mode frequencies differ by $2-6\%$ $2 - 6 %$ from black hole values. Model parameters are constrained using Pantheon+ supernovae, Planck CMB, and BAO data, yielding best-fit values $\beta = -0.48 \pm 0.14$ $β = - 0.48 \pm 0.14$ , $\gamma = 0.12 \pm 0.04$ $γ = 0.12 \pm 0.04$ , $n = 1.45 \pm 0.13$ $n = 1.45 \pm 0.13$ , and $m = -0.42 \pm 0.09$ $m = - 0.42 \pm 0.09$ at 95% confidence.