Ellis--Bronnikov wormhole in Quasi-topological Gravity

We construct higher-dimensional traversable wormholes in quasi-topological gravity (QTG) supported by a phantom scalar field. Using a static, spherically symmetric ansatz, we numerically analyze how quasi-topological gravity corrections affect the geometry and physical properties of the wormhole solutions. The resulting wormhole solutions are symmetric about the throat. Negative mass can arise for certain choices of parameters. For certain parameter ranges, the scalar charge $\mathcal{D}$ of the phantom field rapidly decreases with increasing the higher-curvature coupling parameter $α$ and approaches zero. Moreover, by changing $α$, the overall level of the Kretschmann scalar is also lowered. Finally, for sufficiently large $α$, $-g_{tt}$ becomes close to zero near the throat, exhibiting a ``horizon’’-like structure.

💡 Research Summary

In this work the authors investigate traversable wormhole solutions of the Ellis‑Bronnikov type within the framework of quasi‑topological gravity (QTG), a higher‑curvature modification of general relativity that remains second‑order for static, spherically symmetric backgrounds. The model consists of a D‑dimensional gravitational sector described by the Lagrangian
( \mathcal{L}g = R + \sum{n=2}^{\infty}\alpha_n Z_n )
where (Z_n) are the n‑th order quasi‑topological curvature invariants and (\alpha_n) are coupling constants, together with a minimally coupled phantom scalar field (\Phi) whose Lagrangian is ( \mathcal{L}m = \frac12 g^{\mu\nu}\partial\mu\Phi\partial_\nu\Phi). The scalar field has a negative kinetic term, which is the standard way to keep the Ellis‑Bronnikov throat open in general relativity.

The authors adopt a static, spherically symmetric ansatz for the metric, ( ds^2 = -N^2(l)f(l)dt^2 + \frac{dl^2}{f(l)} + l^2 d\Omega_{D-2}^2), and assume the scalar depends only on the radial coordinate, (\Phi=\phi(l)). By varying the total action they obtain three coupled ordinary differential equations (ODEs) for (N(l)), (f(l)) and (\phi(l)). An integration constant (\mathcal{D}) emerges from the scalar equation and can be interpreted as the scalar charge; it also serves as a diagnostic of numerical accuracy.

To avoid a coordinate singularity in the original radial gauge, the authors introduce a new radial coordinate (r) and rewrite the line element as
( ds^2 = -e^{(D-3)A(r)}dt^2 + p(r)e^{A(r)}dr^2 + (r^2+r_0^2)d\Omega_{D-2}^2).
In this gauge the field equations reduce to two coupled ODEs for the functions (A(r)) and (p(r)). Asymptotic flatness on both sides of the throat imposes the boundary conditions (A(\pm\infty)=0) and (p(\pm\infty)=1).

For the numerical integration the infinite domain is compactified via (x = \frac{2}{\pi}\arctan r), mapping (r\in(-\infty,\infty)) to (x\in(-1,1)). The authors employ a uniform grid of 10 000 points and enforce a relative error below (10^{-4}). They work in dimensionless units with (4\pi G=1) and fix the throat radius (r_0=1). The primary control parameter is the higher‑curvature coupling (\alpha), taken to scale all (\alpha_n) uniformly; the truncation order (N) (the highest power of curvature retained) and the spacetime dimension (D) are varied to explore their influence.

Two families of coefficient choices are examined: (i) a simple geometric progression (\alpha_n = \alpha_{n-1}) (all orders included) and (ii) an alternating series (\alpha_n = 1-(-1)^n 2\alpha_{n-1}) (only odd orders survive). For each family the authors compute the ADM mass (M), the scalar charge (\mathcal{D}), the metric component (-g_{tt}=e^{(D-3)A(r)}), and the Kretschmann scalar (\mathcal{K}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}).

Key findings are:

1. Mass behavior – For positive (\alpha) the total mass starts near zero and grows monotonically with (\alpha), reaching values of order (10^3) for large couplings. Increasing the truncation order (N) tends to lower the mass at fixed (\alpha), indicating that higher‑order curvature terms soften the effective gravitational attraction. For negative (\alpha) the mass can become negative; the sign and magnitude depend sensitively on (N). In the all‑order ((N\to\infty)) case the mass remains negative throughout the (\alpha<0) branch, approaching zero asymptotically for large (|\alpha|).

2. Scalar charge – The scalar charge (\mathcal{D}) measures the amount of exotic phantom matter required. For (\alpha>0) and finite truncations (\mathcal{D}) typically shows a non‑monotonic dip: it first decreases, then rises again. In the infinite‑order limit (\mathcal{D}) decreases monotonically and tends to zero as (\alpha) grows, implying that the higher‑curvature corrections can essentially eliminate the need for exotic matter. For (\alpha<0) the charge drops to zero much more rapidly, suggesting that negative higher‑curvature couplings further reduce the exotic matter content.

3. Metric component (-g_{tt}) – As (\alpha) increases (positive branch) the minimum of (-g_{tt}) at the throat falls from unity toward zero. For sufficiently large (\alpha) the function becomes extremely small near the throat, mimicking a horizon‑like “quasi‑black‑hole” structure while the spacetime remains globally regular. In the negative‑(\alpha) branch, (-g_{tt}) can exceed unity in a region around the throat, correlating with the appearance of negative‑mass solutions.

4. Kretschmann scalar – The overall magnitude of (\mathcal{K}) is reduced as (\alpha) grows, confirming that the higher‑curvature terms smooth out curvature concentrations, analogous to results obtained for regular black holes in QTG.

5. Dimensional dependence – Comparing (D=5) and (D=6) shows the same qualitative trends, but higher dimensions slightly lower the absolute values of mass and scalar charge and broaden the variation of (-g_{tt}). This indicates that the impact of QTG becomes more pronounced in higher dimensions.

Overall, the study demonstrates that quasi‑topological gravity can support traversable Ellis‑Bronnikov wormholes with markedly reduced exotic‑matter requirements. The higher‑curvature coupling (\alpha) acts as a control knob: positive (\alpha) can generate horizon‑like features while still preserving regularity, whereas negative (\alpha) can drive the scalar charge to zero, effectively eliminating the phantom field. These results broaden the landscape of viable wormhole solutions beyond general relativity and suggest that modified gravity theories with well‑behaved higher‑order terms may provide a natural setting where energy‑condition violations are mitigated. Future work is suggested to address dynamical stability, possible observational signatures (e.g., gravitational‑wave echoes), and extensions to rotating or charged configurations.