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.