Introduction: Why This Matters

General Relativity (GR) superbly predicts gravitational phenomena outside black-hole horizons. Yet the standard solutions harbor an interior singularity—divergent curvature and “erased” causal structure—where classical physics breaks down. This interior uncertainty feeds long-standing puzzles (information loss, singular initial conditions) and leaves a conceptual gap between exterior predictions and interior meaning. We propose a smooth, singularity-free interior that preserves the exterior while providing a physically intuitive picture that can be tested.

Our model replaces the singular point with an infinitely deep, smoothly stretching interior—a whirlpool of spacetime. The exterior remains Schwarzschild-like, while the interior “sinks” continuously instead of terminating. This framing connects to a surface-tension analogy: mass acts like a nucleation site, gathering spacetime smoothly around it, as water wraps around dust to form a droplet. In this view, even cosmic expansion can be reinterpreted as the integrated effect of many deep interiors stretching space.

1. Analogy: Surface Tension & Raindrops

Water forms droplets around dust because surface tension minimizes area, producing smooth, spherical shapes. In our analogy, spacetime is the high-tension medium; mass plays the role of the nucleation site. Spacetime “wraps” around mass, forming a smooth spherical geometry. Moving inward corresponds to “sinking” into a deeper layer of the well (a fourth-dimensional descent), while moving outward corresponds to “floating” toward flatter geometry. The analogy motivates a smooth, singularity-free interior with finite curvature.

2. Theoretical Framework

2.1 Modified Schwarzschild-like line element

We introduce a monotone radial mapping $f(r)$ that stretches the interior smoothly while preserving the exterior:

A simple family realizing an infinitely deep yet smooth interior is $$f(r)=r+\frac{\alpha^2}{r},\qquad \alpha>0.$$ Then $f(r)\sim r$ as $r\to\infty$ (recovering Schwarzschild), while $f(r)\to\infty$ as $r\to 0$, replacing the singularity by a deep interior.

2.2 Effective stress–energy and “tensioned medium” motif

Reading the geometry via Einstein’s equations yields an effective, finite stress–energy. A representative, surface-tension-inspired ansatz is

where $\sigma_*$ is a characteristic “tension scale” and pressures remain bounded. The Kretschmann scalar $K=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ is finite at $r=0$ for $\alpha>0$.

2.3 Boundary conditions & matching

To preserve exterior tests, enforce smooth matching at the horizon:

ensuring $g_{tt}$, $g_{rr}$, and the areal radius $f(r)$ transition smoothly across $r\approx r_s$.

2.4 Geodesics (null and time-like)

In the equatorial plane $\theta=\pi/2$, conserved energy and angular momentum per unit mass are $E=c^2\,(1-r_s/f)\,\dot t$ and $L=f(r)^2 \dot\phi$. The radial equation becomes

where $\kappa=1$ (time-like) or $0$ (null). For null curves, with impact parameter $b=Lc/E$,

2.5 Deflection angle & photon sphere

The total deflection for a null ray with closest approach $r_{\min}$ is

The photon sphere satisfies the circular null condition $$\frac{d}{dr}\!\left[\frac{f(r)^2}{b^2}-\Big(1-\frac{r_s}{f(r)}\Big)\right]_{r=r_\gamma}=0.$$ For $f(r)=r+\alpha^2/r$ and small $\alpha/r_s$, $r_\gamma$ and shadow size deviate from Schwarzschild by $\mathcal{O}(\alpha^2/r_s^2)$.

3. Observables & Quantitative Predictions

• Shadow diameter, $d_{\rm sh}$. For small $\alpha/r_s$, $d_{\rm sh} \approx d_{\rm sh}^{\rm Schw}\big[1+\mathcal{O}(\alpha^2/r_s^2)\big]$. Current EHT bounds imply $\alpha/r_s \lesssim \text{few}\times 10^{-1}$.
• Photon-ring harmonics. Subdominant rings shift by $\delta r_{\rm ring}/r_s \sim \mathcal{O}(\alpha^2/r_s^2)$; look for percent-level changes at high dynamic range.
• Time-like precession (S-stars). $\Delta\varpi$ differs from GR at $\mathcal{O}(\alpha^2/r_s^2)$ for periapse $r_p\sim{\rm few}\,r_s$; current astrometry constrains $\alpha$.
• Ringdown QNMs. Dimensionless $M\omega_{\ell n}$ match GR within present LIGO/Virgo errors; late-time echoes are model-dependent and suppressed unless $f$ produces a reflective inner potential.
• Polarization near the photon ring. Electric-vector position angle (EVPA) phase may shift by $\mathcal{O}(\alpha^2/r_s^2)$; multi-epoch EHT polarimetry can constrain $\alpha$.

4. Prescriptive Simulations & Experiments (with expected outcomes)

4.1 Ray-traced lensing & shadow

What: Implement null geodesics from the metric above. Sweep impact parameter $b$, compute $\hat\alpha(b)$ and image the shadow.

Look for: Agreement with Schwarzschild within current EHT uncertainties; deviations $\lesssim 1\%$ in photon-ring spacing and shadow diameter at high S/N.

Metrics: $d_{\rm sh}$, ring spacing $\Delta\theta_n$, brightness ratios $I_{n+1}/I_n$.

4.2 QNMs of perturbed black holes

What: Linearize (Regge–Wheeler/Zerilli-like) with $f(r)$ background; compute $M\omega_{\ell n}$ via WKB or continued fractions.

Expected: Baseline modes within GR error bars; echoes absent unless an inner barrier forms. Bound $\alpha$ from $\delta\omega/\omega$.

4.3 Stellar orbits near Sgr A*

What: Integrate time-like geodesics for S2-like parameters; fit periapse precession, redshift.

Expected: $\Delta\varpi$ consistent with GR; translate residuals to $\alpha/r_s$ constraints.

4.4 GRMHD + polarized ray tracing

What: Run GRMHD; post-process with polarized ray tracing using our metric (replace $r\!\to\!f(r)$ where appropriate).

Expected: Image morphology and EVPA broadly GR-like; search for small phase lags near the photon ring.

4.5 Cosmological inference (conceptual)

What: Explore whether aggregate interior stretching can map to an effective dark-energy-like term on large scales without violating local tests.

Expected: Any effective acceleration must be consistent with SNe, BAO, CMB; provides a cross-disciplinary constraint on $\alpha$ if viable.

5. Preliminary Sanity Checks

• Lorentzian signature: For $r\!>\!r_s$, $g_{tt}\!<\!0$, $g_{rr}\!>\!0$; domain extends smoothly inward.
• Curvature regularity: For $\alpha>0$, scalars (e.g., $K$) remain finite as $r\to 0$.
• Asymptotics: $f(r)\sim r$ for large $r$ ensures Schwarzschild limits for classic tests.
• Energy conditions: With bounded anisotropic pressures and monotone $\rho(r)$, weak/ null energy conditions can be satisfied (model-dependent but feasible).
• Continuity at horizon: $f(r_s)=r_s+\varepsilon$ gives smooth matching at the exterior horizon.

6. Discussion & Outlook

The deep-whirlpool interior is a minimal deformation of Schwarzschild that preserves exterior triumphs of GR while removing the singularity and restoring interior structure. It aligns with the surface-tension analogy (spacetime “wrapping” around mass like water around a nucleus), reframes cosmic expansion as cumulative interior stretching, and offers quantitative, parameterized deviations. The single scale $\alpha$ (or any smooth $f$ you choose) is now open to constraint by lensing, QNMs, stellar orbits, and polarized EHT imaging. We invite the community to implement the prescriptive simulations above and report constraints on $\alpha/r_s$.

7. References

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