Exterior/strong static profile
The piecewise public profile supplies \(\Xi(r)\), \(D(r)\) and the static spherical metric in its declared branches.
Tested horizon · interior construction programmeSSZ Interior and
SSZ Interior and
Global Structure
Finite horizon coefficients are repeatedly reproduced. The centre asymptotics diagnose the presently studied diagonal extrapolation, while a physical SSZ inner solution or boundary completion remains open for construction and testing.
r→0
Metric: canonical static diagonal continuation •Radius: areal \(r\) •Status: present extrapolation diagnosed •Alternative SSZ completion: open
Reviewer scope: the present continuation has divergent curvature invariants. This identifies requirements for the next inner solution; it does not exclude a regular SSZ core, boundary geometry or different topology once explicitly derived and tested.
Definitions before conclusions
Three geometries must not be conflated
Continue the same formula to \(r\to0\)
This is the object for which the P0 invariant asymptotics are stated. It is mathematically inspectable but not thereby a physical stellar or black-hole interior.
Physical inner solution
A new metric and matter/dynamical solution with explicit matching, regularity and causal analysis has not yet been supplied.
Local statements
The Schwarzschild scale and the areal centre are different loci
At \(r=r_s\)
\[\Xi(r_s)=1-e^{-\varphi}\approx0.801711847,\qquad D(r_s)\approx0.555027709\]
Within the declared static chart, the time factor is finite and nonzero. This is the strongest secured theoretical result of the current static programme.
As \(r\to0^+\)
\[A(r)=D^2(r)\to\frac14,\qquad 4\pi r^2\to0\]
A finite temporal coefficient is only the first rung of a regularity analysis. Because \(r\) is areal radius, invariant curvature at \(r=0\) must also be checked.
Invalid inference: finite \(D(r_s)\Rightarrow\) no global horizon, no trapped surfaces, regular centre, or complete spacetime. None of those implications follows from one local coefficient.
Coordinate-independent diagnostics
Curvature of the present diagonal continuation
Ricci scalar
\[R(r)\sim\frac{3}{2r^2}\]
Diverges as \(r^{-2}\).
Kretschmann scalar
\[K(r)=R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\sim\frac{9}{4r^4}\]
Diverges as \(r^{-4}\).
Ricci-tensor square
A complete certificate must publish \(R_{\mu\nu}R^{\mu\nu}\) with its exact leading coefficient and source derivation. The present portal does not invent that coefficient.
calculation requiredP0 conclusion: the canonical diagonal continuation is not regular at the areal centre. Older claims of complete singularity freedom are superseded.
Completeness is a trajectory theorem
Radial and non-radial geodesics
Required calculation
\[\frac{d^2x^\mu}{d\lambda^2}+\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\lambda}\frac{dx^\beta}{d\lambda}=0\]
For null curves \(\lambda\) is affine; for timelike curves proper time may parametrise the path. One must prove whether every inextendible path has unbounded parameter or reaches a genuine boundary.
Current evidence boundary
Selected numerical trajectories and conserved-quantity checks test implementations over finite domains. They do not prove radial or non-radial geodesic completeness of a maximal spacetime.
Open-interior Claim-IDGlobal questions
Null congruences, trapping and maximal extension
| Question | Required object | Current status | Acceptance criterion |
|---|---|---|---|
| Null expansions | \(\theta_+\), \(\theta_-\) for future-directed radial null congruences | open | Sign map across a declared global chart |
| Trapped surfaces | Closed two-surfaces with both expansions negative | open | Invariant calculation, not coefficient inspection |
| Causal completeness | All causal curves and spacetime boundary | open | Maximal extension and boundary classification |
| Global hyperbolicity | Cauchy surfaces and causal diamonds | open | Proof for the completed geometry |
| Collapse/formation | Time-dependent field equations plus matter | open | Well-posed evolution producing the proposed state |
Possible research directions, not established solutions
What a completion could look like
Regular material core
Introduce an interior stress-energy model with finite density/pressure invariants, solve its equations and match to the SSZ exterior.
Boundary geometry
Treat the inner limit as a genuine boundary and specify admissible causal curves, boundary data and conservation laws.
Different topology or areal map
A nonzero minimal sphere or other topology would require a new metric definition. It cannot be inferred by relabelling the current areal coordinate.
Matching requirements
Junction conditions and effective source
No thin shell
For a standard non-null matching surface, continuity of the induced metric and extrinsic curvature is required to avoid a distributional surface layer.
\[[h_{ab}]=0,\qquad[K_{ab}]=0\]
With a shell
If \([K_{ab}]\neq0\), the surface stress tensor must be calculated, interpreted and tested against energy conditions. A smooth-looking plot is insufficient.
\[S_{ab}\propto [K_{ab}]-h_{ab}[K]\]
Missing foundation: without a complete SSZ action or field equations, \(T^{\rm eff}_{\mu\nu}=G_{\mu\nu}/(8\pi G)\) is a diagnostic reconstruction, not a derived fundamental matter law.
Falsifiable completion programme
Concrete next tasks
| Priority | Task | Deliverable | Negative result retained |
|---|---|---|---|
| P0 | Independent symbolic invariant package | Exact \(R\), \(R_{\mu\nu}R^{\mu\nu}\), \(K\) plus 50-digit limits | Any divergence remains public |
| P0 | Radial/null geodesic certificate | Affine-parameter analysis and extension boundary | Finite affine termination |
| P1 | Candidate regular interior | Metric, source, equations, regularity and matching | No admissible matching or energy-condition failure |
| P1 | Null-expansion map | \(\theta_\pm\), trapped/marginal surfaces | Trapped region found |
| P2 | Dynamical formation | Well-posed collapse/evolution model | Instability or non-formation |