Strong
\[\Xi_{\rm strong}(x)=1-e^{-\varphi/x},\quad x<1.8\]
Canonical static sector · interactive formula atlas
The SSZ metric
Definitions, branches, coefficients and limits are linked to one synchronized radius probe. Animation illustrates mathematical dependence; it is not an astrophysical simulation.
Shared radius probe
Move through the metric
All diagrams below use x = r/rs. Play moves logarithmically from the present centre diagnostic toward the asymptotic region.
Animation active
active branchstrong
Ξ(x)—
D(x)—
A(x)—
B(x)—
A·B1
Definition
Static diagonal ansatz
\[ds^2=-A(r)c^2dt^2+B(r)dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)\]
\[D=(1+\Xi)^{-1},\qquad A=D^2,\qquad B=D^{-2},\qquad AB=1\]
Geometric reading
The animated local chart shows how the time and radial coordinate scales change with the selected radius. The angular sphere retains areal radius r.
Interact: drag horizontally over any diagram, use the mouse wheel, or focus a canvas and press ←/→.
Coordinate statement: changing coordinate slopes does not change the locally measured value of c.
The coordinate r is the areal radius: a sphere at fixed r has area 4πr2. This is an effective static spherical geometry and a concrete basis for further dynamical development.
Piecewise definition
Canonical branches and C² bridge
The moving marker and shaded bridge make the routing and its derivative matching visible.
C² bridge
\[\Xi_{\rm bridge}(x)=\sum_{k=0}^{5}a_k t^k,\quad t=\frac{x-1.8}{0.4}\]
Six endpoint constraints match value, first derivative and second derivative.
Weak
\[\Xi_{\rm weak}(x)=\frac{1}{2x},\quad x>2.2\]
Ξ′(x), numerical—
Ξ″(x), numerical—
distance to bridge—
Algebraic identities
From Ξ to D, A and B
Each bar is evaluated from the same selected radius. The reciprocal product is recalculated rather than hard-coded.
\[D=\frac{1}{1+\Xi},\qquad g_{tt}=-D^2,\qquad g_{rr}=D^{-2},\qquad |g_{tt}|g_{rr}=1\]
Tested kinematic identity
Escape–fall duality
The public corpus defines a complementary fall scale vfall. It is an algebraic kinematic dual, not a locally measured material 3-velocity.
Dual velocity pair
\[\frac{v_{\rm esc}}{c}=\sqrt{\frac{r_s}{r}}=\frac1{\sqrt{x}},\qquad \frac{v_{\rm fall}}{c}=\sqrt{\frac{r}{r_s}}=\sqrt{x}\]
\[v_{\rm esc}\,v_{\rm fall}=c^2\]
The product identity is mass-independent after normalization and is tested across multiple radii in the repository suite.
Do not mix two meanings
The dual vfall can exceed c for r > rs because it is a reciprocal kinematic scale. It does not describe a local object overtaking light. A separate metric-derived infall-speed proxy sometimes used in the corpus is c√(1−D2); it must be labelled separately.
vesc/c—
dual vfall/c—
vescvfall/c²1
metric infall proxy/c—
Tested horizon · open completion
Horizon result and areal-centre scope
The test corpus repeatedly verifies finite Ξ and D at r = rs. Explore the tested horizon, the present centre extrapolation and possible completion requirements separately.
At r = rs: repeatedly tested
\[\Xi(r_s)\approx0.801711847,\qquad D(r_s)\approx0.555027710\]
Repository tests repeatedly reproduce these finite values and the associated metric identities. This establishes finite static-chart time scaling at the tested radius; a software test is still not an empirical observation or a proof of global causal structure.
As r → 0: completion remains open
\[A\to\frac14,\qquad R\sim\frac{3}{2r^2},\qquad K\sim\frac{9}{4r^4}\]
These asymptotics diagnose the current diagonal extrapolation to the areal centre. They do not rule out a different SSZ inner solution, a non-areal boundary construction or another mathematically complete continuation. Whether SSZ supplies such a completion remains an open research question.
A asymptotic diagnostic—
R·x² proxy1.5
K·x⁴ proxy2.25
Epistemic boundary: the curvature curves visualize the leading asymptotic terms of the presently declared diagonal continuation. They neither prove a regular global interior nor prove that no alternative SSZ completion can resolve the centre question.
Reading compass
How to read Metric
This page separates the declared definition, its computation or visualisation, the evidence supporting it, and the conclusions that remain outside its scope.
Start with the formula or control, identify its domain and inputs, then follow the linked implementation and evidence record.
A passing identity, numerical limit, plot or comparison supports only the stated relation under its recorded assumptions and provenance.
It is not automatically an independent experiment, a complete physical theory, or a proof beyond the explicit claim boundary.
Foundational synthesis
Definitions, evidence and limitations stay linked
The portal keeps current canonical locks above historical descriptions and keeps software verification separate from empirical confirmation.
Canonical source
Use the current P0, JIF or mathematical lock for the formula and domain shown on this page.
Evidence class
Read tests, convergence, reference compatibility, dataset-conditioned comparisons and independent replication as different evidence classes.
Boundary
Every result retains its assumptions, limits and explicit non-claim so a visual or passing assertion is not over-promoted.