Accretion Flow onto Ellis-Bronnikov Wormhole

Abstract: Study of accretion onto wormholes is rather rare compared to that onto black holes. In this paper, we consider accretion flow of cosmological dark energy modeled by barotropic fluid onto the celebrated Ellis--Bronnikov wormhole (EBWH) built by Einstein minimally coupled scalar field $φ$, violating the null energy condition. The accreting fluid is assumed to be phantom, quintessence, dust and stiff matter. We begin by first pointing out a mathematical novelty showing how the EBWH can lead to the Schwarzschild black hole under a complex Wick rotation. Then, we analyze the profiles of fluid radial velocity, density and the rate of mass variation of the EBWH due to accretion and compare the profiles with those of the Schwarzschild black hole. We also analyze accretion to the massless EBWH that has zero ADM mass but has what we call nonzero Wheelerian mass (``mass without mass''), composed of the non-trivial scalar field, that shows gravitational effects. Our conclusion is that the mass of SBH due to phantom and non-phantom accretion increases consistently with known results, while, in contrast, the mass of EBWH decreases. Accretion to massless EBWH (i.e., to nonzero Wheelerian mass) shares the same patterns as those of the massive EBWH; hence there is no way to distinguish massive and massless cases by means of accretion flow. The contrasting mass variations due to phantom accretion could be a reflection of the distinct topology of the central objects.

• The paper derives exact Michel-type velocity, density, pressure, and mass-flow profiles for phantom, quintessence, dust, and stiff fluids accreting onto massive and massless Ellis–Bronnikov wormholes.
• The analysis finds that wormhole mass decreases under both phantom and non-phantom accretion, unlike Schwarzschild black holes, whose mass grows with ordinary matter and shrinks with phantom energy.
• The results show that massive and massless wormholes have qualitatively identical accretion behavior, while the sign of mass evolution could help distinguish wormhole throats from black-hole horizons, subject to stability and backreaction limitations.

Overview

This paper studies steady, spherically symmetric accretion of barotropic fluids onto the Ellis–Bronnikov wormhole (EBWH), an exact vacuum solution of general relativity sourced by a minimally coupled ghost scalar field that violates the null energy condition. The analysis follows the Michel-type framework formalized for general compact objects by Bahamonde and Jamil (2606.26628), and compares accretion profiles of phantom ($\omega < -1$), quintessence ($-1<\omega<-1/3$), dust ($\omega=0$), and stiff matter ($\omega=1$) fluids between the EBWH and the Schwarzschild black hole (SBH). The central result is a sign reversal: while non-phantom accretion increases SBH mass and phantom accretion decreases it — consistent with Babichev et al. (2606.26628) — the EBWH mass decreases under both phantom and non-phantom accretion.

Mathematical relation to the Schwarzschild solution

The authors first establish that the massive EBWH metric in isotropic coordinates,

$$d\tau^2_{\text{EBWH}} = -P(r)dt^2 + Q(r)[dr^2 + r^2 d\Omega^2],$$

with $P(r) = \exp[2\epsilon + 4\gamma\tan^{-1}(2r/m)]$, $$Q(r) = (1+m^2/4r^2)^2\exp[2\zeta - 4\gamma\tan^{-1}(2r/m)]$$, ADM mass $M = m\gamma$, and constraint $2\delta^2 = 1+\gamma^2$, maps exactly onto the Schwarzschild metric under the combined transformation

$$r \to -\frac{m^2}{4r}, \qquad \gamma \to -i, \qquad m \to im,$$

using the identity $-1<\omega<-1/3$0. Under this map the throat radius $-1<\omega<-1/3$1 becomes the Schwarzschild horizon $-1<\omega<-1/3$2. This correspondence is not merely decorative: all accretion expressions derived for the EBWH reduce to their Schwarzschild counterparts upon setting $-1<\omega<-1/3$3, providing an internal consistency check on the entire calculation.

Accretion onto the massive EBWH

The accreting fluid is modeled as a perfect fluid with barotropic equation of state $-1<\omega<-1/3$4, flowing radially on the positive (attractive) mouth only, with no interaction assumed between the background ghost scalar $-1<\omega<-1/3$5 and the fluid. The energy-momentum conservation law, continuity equation, and mass flux equation yield integration constants $-1<\omega<-1/3$6, from which closed-form solutions follow for the radial velocity $-1<\omega<-1/3$7, density $-1<\omega<-1/3$8, pressure $-1<\omega<-1/3$9, and mass variation rate

$\omega=0$0

Crucially, since $\omega=0$1 with $\omega=0$2 and $\omega=0$3 for real wormhole parameters, the sign of $\omega=0$4 is fixed negative regardless of whether the fluid satisfies or violates the NEC. This contrasts with the SBH case where $\omega=0$5 flips the sign, reproducing the standard result that phantom energy shrinks black holes while ordinary matter grows them. The implication is direct: the sign of the mass evolution encodes the topology of the central object, offering a potential discriminator between wormholes and black holes through accretion signatures alone.

At the phantom divide $\omega=0$6, the velocity diverges and the density vanishes. The authors correctly identify this as the known pathology of divergent adiabatic sound speed at the divide [Kunz & Sapone], noting that crossing requires either alternative frames with finite $\omega=0$7 or scale-factor-dependent equations of state $\omega=0$8.

Accretion onto the massless EBWH and Wheelerian mass

Setting $\omega=0$9 yields the massless EBWH with zero ADM mass but nonzero "Wheelerian mass" $\omega=1$0 — the integrated energy of the nontrivial scalar field. The Misner–Sharp quasi-local mass enclosed within the throat evaluates to $\omega=1$1 at $\omega=1$2, explaining why this object still gravitates, scatters waves, and lenses light despite zero ADM mass.

Recomputing the accretion profiles ab initio for the massless metric gives

$\omega=1$3

which shares the same qualitative patterns as the massive case. The consequence is significant: accretion flow cannot distinguish massive from massless EBWHs, since both exhibit identical velocity, density, and mass-evolution behavior.

Profile comparisons

For matched central masses ($\omega=1$4 across all three objects), the numerical profiles show:

• Phantom fluid velocities are consistently higher near the EBWH than near the SBH at all radii, converging asymptotically.
• Fluid density near the EBWH throat exceeds that near the SBH horizon; lowering $\omega=1$5 lowers the density profile.
• For quintessence, dust, and stiff matter, the massless EBWH exhibits the highest accreting-fluid velocities and the SBH the lowest, with increasing $\omega=1$6 suppressing velocity in the massive case.
• All profiles bunch together far from the source, as expected in the asymptotically flat regime.

Limitations and open questions

Several caveats bear directly on the results. First, the EBWH is known to be linearly unstable to perturbations [Shinkai & Hayward; González et al.]; the accretion scenario is therefore physically relevant only to observers who, per observer-dependent stability arguments [Nandi et al.], would classify the wormhole as stable. Second, the analysis assumes the background scalar field does not interact with the accreting fluid and that flow occurs only on one mouth — assumptions that simplify but also restrict the physical scope. Third, backreaction of the accreted mass on the geometry is neglected, consistent with treating the metric as static throughout. Fourth, Hawking radiation is omitted; combining evaporation with accretion would require an additional term in the mass-evolution equation. Finally, whether the contrasting $\omega=1$7 signature survives in more realistic disk-accretion or nonspherical settings remains open, as does observational distinguishability given that lensing observables already show partial mimicry between wormholes and black holes.

Conclusion

The paper establishes a complex Wick-rotation connection between the EBWH and Schwarzschild metrics, derives exact analytical accretion profiles for barotropic fluids onto both massive and massless EBWHs, and demonstrates a robust topological signature: EBWH mass decreases under both phantom and non-phantom accretion, opposite to SBH behavior. The equivalence of massive and massless accretion patterns implies accretion cannot discriminate between them, while the sign reversal of $\omega=1$8 relative to black holes suggests accretion diagnostics may complement lensing as probes of horizon versus throat topology.