Tested horizon · active interior researchStrong field and
Strong field and
global structure
The finite horizon result is repeatedly reproduced. The current diagonal centre extrapolation has divergent invariants, while a separately derived SSZ interior or boundary completion remains an open research path.
K
Strongest secure result
Finite horizon time dilation
\[\displaystyle \Xi(r_s)=1-e^{-\varphi}\approx0.801711847,\qquad D(r_s)=\frac1{2-e^{-\varphi}}\approx0.555027709\]
This establishes that the declared diagonal SSZ metric does not have the Schwarzschild-coordinate zero/infinity pair at r_s. It is a precise model result. Whether the surface is a global event horizon requires the complete causal geometry, not just local coefficients.
Local versus coordinate physics
Static frames, proper distance and radial light
The strong field amplifies the cost of confusing chart quantities with local observables. A static orthonormal frame makes the distinction explicit.
Static orthonormal frame
\[\displaystyle e_{\hat0}=\frac1{Dc}\partial_t,\qquad e_{\hat r}=D\,\partial_r,\qquad e_{\hat\theta}=\frac1r\partial_\theta,\qquad e_{\hat\phi}=\frac1{r\sin\theta}\partial_\phi\]
Components carrying hats are measured in this local frame. The construction requires a physically admissible static observer.
Null coordinate slope
\[\displaystyle \frac{dr}{dt}=\pm cD^2(r),\qquad \Delta t=\frac1c\int_{r_1}^{r_2}\frac{dr}{D^2(r)}\]
This describes the selected coordinates. The local frame measures the light ray at \(c\).
Proper radial length
\[\displaystyle \ell(r_1,r_2)=\int_{r_1}^{r_2}\frac{dr}{D(r)}\]
It is a spatial-slice distance—not the elapsed proper time of an infalling observer.
Static redshift
\[\displaystyle \frac{\nu_o}{\nu_e}=\frac{D(r_e)}{D(r_o)},\qquad 1+z=\frac{D(r_o)}{D(r_e)}\]
Doppler motion, transfer through plasma and atmospheric line formation must be added for real spectra.
Geodesic diagnostics
Photon candidates, circular matter orbits and stability
Strong-field radii are solutions of differential conditions in a specified metric version. They are not universal constants that can be copied between historic SSZ variants.
Null effective potential
\[\displaystyle \dot r^2=\frac{E^2}{c^2}-\frac{A(r)L^2}{r^2},\qquad \frac d{dr}\!\left(\frac{A}{r^2}\right)=0\]
A candidate circular photon orbit also needs accessibility and an instability check from the second derivative.
Critical impact parameter
\[\displaystyle b_{\rm crit}=\frac{r_{\rm ph}}{\sqrt{A(r_{\rm ph})}}\]
This geometric proxy is not yet a synthetic EHT image. Rotation, inclination, plasma emissivity and instrumental response change an observed shadow.
Timelike circular orbit
\[\displaystyle L^2=\frac{c^2r^3A'}{2A-rA'},\qquad E^2=\frac{2c^4A^2}{2A-rA'}\]
The expressions follow for the declared static areal-radius metric and convention. A denominator failure marks a domain boundary, not a numerical nuisance.
Marginal stability
\[\displaystyle V'_{\rm eff}=0,\qquad V''_{\rm eff}=0,\qquad \Omega^2=\frac{c^2A'}{2r}\]
An ISCO result must report the branch, root-finding interval, tolerance and comparison clock.
Current published-engine diagnostic: for the canonical strong–C²–weak continuation, the shared engine finds a bridge-localised stationary maximum of \(A/x^2\) near \(x=2.130175\), a static critical-impact proxy near \(2.656527\,r_s\), and a nearby minimum of circular-timelike \(L^2\) near \(x=2.199652\). These belong to this exact matching prescription—not universal SSZ constants or measurements.
| Quantity | Condition | Current diagnostic | Before observational use |
|---|---|---|---|
| Null stationary point | \(d(A/r^2)/dr=0\) | \(r\approx2.130175\,r_s\) | accessibility, stability, metric-version lock |
| Critical impact proxy | \(b=r/\sqrt A\) | \(b\approx2.656527\,r_s\) | rotation, inclination, plasma and transfer |
| Timelike stability candidate | minimum of \(L^2=r^3A'/(2A-rA')\) | \(r\approx2.199652\,r_s\) | endpoint sensitivity and full stability audit |
Current portal rule: the bridge can introduce an interior stationary point of \(A/r^2\). It is reported as a metric candidate, never silently relabelled as an observationally confirmed photon sphere.
60-decimal reference computation
Numerical certificates and blend sensitivity
The portal now records the equation, branch boundaries, bridge family, root interval, precision, residual, curvature test, source hash and limitations together with each value. The chart shows structured variations of both transition boundaries and a higher-order C²-preserving perturbation.
| x₀ | x₁ | C² perturbation ε | stationary radius | impact proxy |
|---|---|---|---|---|
| Loading certificates… | ||||
Method, provenance and interpretation boundary
Derived model observables
Photon sphere, ISCO, shadow, redshift and ringdown
Each quantity needs its own differential problem. None can be read from D(r_s) alone.
| Observable | Required calculation | Current evidence type | Main limitation |
|---|---|---|---|
| Photon sphere | Extremum of the null effective potential | repository calculations and tests | branch/metric conventions must be explicit |
| ISCO | Timelike circular orbit plus marginal stability | geodesic/effective-potential code | not inferable from photon sphere |
| Shadow radius | critical null impact parameter plus emission model | model comparison | observed image depends on plasma and instrument response |
| Static redshift | emitter/observer clock ratio | direct D proxy | astrophysical kinematics and atmosphere matter |
| Lensing | null geodesic integration | weak and strong lensing pipelines | PPN weak limit differs from strong path calculation |
| QNM/ringdown | linear perturbation equation and boundary conditions | proxy/model-lock studies | full SSZ field dynamics is not yet derived |
| Sgr A* / M87* | forward imaging and orbital data model | exploratory comparisons | systematics and degeneracies dominate percent-scale differences |
Interpretation rule: a numerically stable critical orbit is a result of the implemented effective metric. It is not automatically an observed feature of nature.
Diagnostic—not fundamental matter
Einstein tensor and effective energy conditions
The geometry can be inserted into the Einstein tensor to ask what GR-style effective source would support it. This does not derive SSZ dynamics.
Effective source
\[\displaystyle G_{\mu\nu}=R_{\mu\nu}-\frac12Rg_{\mu\nu},\qquad T_{\mu\nu}^{\rm eff}=\frac{G_{\mu\nu}}{8\pi G}\]
Density and principal pressures must be projected into an orthonormal frame before energy-condition statements are made.
Condition ladder
\[\displaystyle \mathrm{WEC}:\ \rho\ge0,\ \rho+\frac{p_i}{c^2}\ge0;\qquad \mathrm{DEC}:\ \rho\ge\frac{|p_i|}{c^2}\]
Historic tests reporting finite transition proxies or bounded effective quantities do not override the P0 invariant divergence at \(r\to0\).
Central P0 correction
SSZ Interior and Global Structure
The formal strong branch approaches Ξ→1 and D→1/2 as r→0. Finite field and metric coefficients do not make the areal centre regular.
Metric coefficient
A(r)→1/4
A finite non-unit constant multiplying time and its inverse multiplying radius leaves a solid-angle/areal-radius mismatch at the centre.
Ricci scalar
R(r)~3/(2r²)
The scalar curvature diverges quadratically as the areal radius vanishes.
Kretschmann scalar
K(r)~9/(4r⁴)
The quartic invariant divergence cannot be removed by a coordinate relabelling.
Scope conclusion: the present diagonal extrapolation is not by itself a regular global interior. This does not rule out an SSZ solution using a new inner metric, matching surface, minimal sphere or defined boundary geometry; those possibilities require their own derivation and invariant tests.
Unfinished extension
Rotation, frame dragging and perturbations
Rotating metric
A complete axisymmetric solution must determine \(g_{t\phi}\), horizon/ergosurface structure, multipoles and the static limit from one set of dynamics. A Kerr-inspired analogy is not yet that derivation.
Linear perturbations
QNM frequencies require a well-defined perturbation operator, gauge choice, potentials and ingoing/outgoing boundary conditions. Static \(D(r)\) alone is insufficient.
Waveform inference
The LIGO repository explicitly labels current strain corrections as a V0 proxy. Likelihood differences from that pipeline are engineering and exploratory evidence, not a locked SSZ waveform detection.
\[\displaystyle \Delta t=-\frac2c\oint\frac{g_{0i}}{g_{00}}\,dx^i\]
The familiar \(4A\Omega/c^2\) result follows only after specifying the rotating-loop limit. Potential-dependent, direction-symmetric terms cancel differently from genuine direction-odd terms.
What a completion must supply
Requirements for a global solution
New interior solution
- Areal-centre regularity or an explicitly non-central topology
- Finite curvature invariants
- Well-defined stress-energy and energy conditions
- Matching of induced metric and extrinsic curvature
- Stable perturbations and causal evolution
Or a complete boundary geometry
- A precise inner boundary instead of r=0
- Boundary conditions for fields and geodesics
- Proof of geodesic and causal completeness or a clear failure mode
- Conservation laws and a well-posed initial-boundary value problem
- Observable consequences distinct from an arbitrary cutoff
Action
A fundamental or effective action must specify which variables are dynamical and how Ξ couples to matter and geometry.
Field equations
Equations must determine solutions rather than merely define a metric profile, and their constraint propagation must be shown.
Global causal structure
Penrose structure, trapped surfaces, horizons, extensions, and endpoints of causal curves require a full spacetime analysis.
Falsifiability
How nature could reject SSZ
Static compact-object redshift
A preregistered SSZ atmosphere-plus-geometry forward model that systematically fails accurate neutron-star line or clock measurements would reject that branch. A simple “+13%” slogan is not sufficient without source modelling.
Shadow and photon ring
If a complete SSZ ray-traced image predicts a robust diameter or subring structure outside observational confidence regions while GR remains viable, the corresponding metric model is rejected.
Pulsar timing and Shapiro delay
Binary timing can constrain deviations only after orbital dynamics, propagation, plasma, and parameter covariance are fitted jointly.
Ringdown
A locked SSZ perturbation spectrum inconsistent with high-SNR multimode ringdowns would reject that dynamical completion. Current proxy shifts are not yet enough.
Internal mathematical failure
No well-posed field equations, unavoidable ghosts, unstable modes, or impossible matching conditions could rule out proposed completions before observation.
Weak-field regression
Any canonical change that breaks established PPN, clock, lensing, or orbital constraints would immediately invalidate that revision.
Required precision: it cannot be stated as one universal percentage. It is observable- and instrument-specific and must include astrophysical nuisance parameters, calibration, model covariance, and look-elsewhere effects.
Research roadmap
Priority-ordered open work
P0 — interior and equations
Construct or rule out a regular interior/boundary completion; derive the governing action and field equations; establish constraints and well-posedness.
P1 — rotation and perturbations
Develop a non-perturbative rotating solution and derive linear perturbation equations, QNM spectra, and stability from the same dynamics.
P2 — forward observations
Build uncertainty-aware shadow, spectrum, timing and gravitational-wave pipelines with blinded model comparison.
P3 — independent replication
Freeze data, code, environments and predictions before external groups reproduce or challenge them.