SSZ_EINSTEIN_RICCI_CURVATURE.tex

% ===================================================================
% SSZ φ-Spiral Metric - Einstein Tensor & Curvature Invariants
% ===================================================================
% Complete curvature formulation: G^μ_ν, R_μν, R, K
% © 2025 Carmen Wrede & Lino Casu
% Based on Lino's Einstein tensor specification
% ===================================================================

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{physics}
\usepackage{geometry}
\geometry{margin=2.5cm}

\begin{document}

\title{\textbf{SSZ φ-Spiral Metric}\\
Einstein Tensor \& Curvature Invariants}
\author{Carmen Wrede \and Lino Casu}
\date{November 2025}

\maketitle

\section{Notation \& Metric}

\subsection{Coordinates}
$(x^\mu) = (T, r, \theta, \varphi)$

\subsection{SSZ metric (4D diagonal form)}
\[
ds^2 = -\frac{c^2}{\gamma^2}\,dT^2 + \gamma^2\,dr^2 + r^2(d\theta^2 + \sin^2\!\theta\,d\varphi^2)
\]

where:
\begin{align*}
\phi(r) &\equiv \phi_G(r) = \sqrt{\frac{2GM}{r c^2}} \\
\gamma(r) &= \cosh(\phi) \\
\beta(r) &= \tanh(\phi) \\
\phi'(r) &= \frac{d\phi}{dr}
\end{align*}

\subsection{Auxiliary functions}

Define logarithmic measure:
\[
\boxed{
\lambda(r) \equiv \ln\gamma(r)
}
\]

Then:
\begin{align*}
\lambda' &= \frac{\gamma'}{\gamma} = \beta\,\phi' \\
\lambda'' &= \frac{d}{dr}(\beta\,\phi') = \frac{(\phi')^2}{\gamma^2} + \beta\,\phi''
\end{align*}

\subsection{Metric components}

For standard spherical form $ds^2 = -A(r)\,dT^2 + B(r)\,dr^2 + r^2\,d\Omega^2$:

\[
\boxed{
A(r) = \frac{c^2}{\gamma^2}, \qquad B(r) = \gamma^2
}
\]

% ===================================================================
\section{1. Einstein Tensor $G^\mu{}_\nu$ (Mixed Indices)}

For static spherical diagonal metric, the standard formulas are:

\subsection{Time component}
\[
\boxed{
G^T{}_T = \frac{1}{r^2}\left(\frac{2r\,\beta\,\phi'}{\gamma^2} - \frac{1}{\gamma^2} + 1\right)
}
\]

\subsection{Radial component}
\[
\boxed{
G^r{}_r = \frac{1}{r^2}\left(\frac{1}{\gamma^2} - 1\right) - \frac{2\,\beta\,\phi'}{r\,\gamma^2}
}
\]

\subsection{Angular components}
\[
\boxed{
\begin{aligned}
G^\theta{}_\theta = G^\varphi{}_\varphi &= \frac{1}{\gamma^2}\left(-\lambda'' + 2\lambda'^2 - \frac{2\lambda'}{r}\right) \\[6pt]
&= \frac{1}{\gamma^2}\left[-\left(\frac{(\phi')^2}{\gamma^2} + \beta\,\phi''\right) + 2\beta^2(\phi')^2 - \frac{2\beta\,\phi'}{r}\right]
\end{aligned}
}
\]

\subsection{Verification \& Checks}

\paragraph{Weak field limit ($\phi \ll 1$):}
\begin{itemize}
\item $\gamma \approx 1$, $\beta \approx \phi$
\item All $G^\mu{}_\nu \sim O(r_g/r^3)$ --- same as GR
\end{itemize}

\paragraph{Constant spiral phase ($\phi' = \phi'' = 0$):}
\begin{itemize}
\item $\Rightarrow G^\mu{}_\nu = 0$ (locally flat)
\item Pure rotation without curvature
\end{itemize}

% ===================================================================
\section{2. Ricci Tensor \& Scalar}

\subsection{Ricci Scalar $R$}

From trace relation $G^\mu{}_\mu = -R$ in 4D:

\[
\boxed{
\begin{aligned}
R &= -(G^T{}_T + G^r{}_r + 2G^\theta{}_\theta) \\[6pt]
&= \frac{2}{\gamma^2}\left[\lambda'' - 2\lambda'^2 + \frac{2\lambda'}{r}\right] \\[6pt]
&= \frac{2}{\gamma^2}\left[\frac{(\phi')^2}{\gamma^2} + \beta\,\phi'' - 2\beta^2(\phi')^2 + \frac{2\beta\,\phi'}{r}\right]
\end{aligned}
}
\]

\subsection{Ricci Tensor $R_{\mu\nu}$ (Lowered Indices)}

From relation $R_{\mu\nu} = G_{\mu\nu} + \tfrac{1}{2}g_{\mu\nu}R$:

\paragraph{Time component:}
\[
\boxed{
R_{TT} = g_{TT}\left(G^T{}_T - \tfrac{1}{2}R\right) = -\frac{c^2}{\gamma^2}\left(G^T{}_T - \tfrac{1}{2}R\right)
}
\]

\paragraph{Radial component:}
\[
\boxed{
R_{rr} = g_{rr}\left(G^r{}_r - \tfrac{1}{2}R\right) = \gamma^2\left(G^r{}_r - \tfrac{1}{2}R\right)
}
\]

\paragraph{Angular components:}
\[
\boxed{
\begin{aligned}
R_{\theta\theta} &= g_{\theta\theta}\left(G^\theta{}_\theta - \tfrac{1}{2}R\right) = r^2\left(G^\theta{}_\theta - \tfrac{1}{2}R\right) \\[4pt]
R_{\varphi\varphi} &= \sin^2\!\theta\,R_{\theta\theta}
\end{aligned}
}
\]

% ===================================================================
\section{3. Curvature Invariants}

\subsection{Ricci Scalar $R$}

Already given in Section 2.1. Compact and complete.

\paragraph{Properties:}
\begin{itemize}
\item Finite for all $r > 0$
\item $R \to 0$ as $r \to \infty$ (asymptotic flatness)
\item $R = 0$ if $\phi' = \phi'' = 0$ (locally flat)
\end{itemize}

\subsection{Ricci Squared $R_{\mu\nu}R^{\mu\nu}$}

Computed from above $R_{\mu\nu}$ components with $g^{\mu\nu}$:

\[
R_{\mu\nu}R^{\mu\nu} = g^{TT}R_{TT}^2 + g^{rr}R_{rr}^2 + g^{\theta\theta}R_{\theta\theta}^2 + g^{\varphi\varphi}R_{\varphi\varphi}^2
\]

\textit{Explicit form is lengthy but straightforward from boxed formulas above. Recommend symbolic computation (see Section 4).}

\subsection{Kretschmann Scalar $K$}

Full contraction of Riemann tensor:
\[
K \equiv R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}
\]

\paragraph{Weak field limit ($r \gg r_g$):}
\[
\boxed{
K = \frac{48\,G^2 M^2}{c^4\,r^6} + O\left(\frac{r_g^3}{r^7}\right)
}
\]

Same $r^{-6}$ scaling as Schwarzschild GR.

\paragraph{Properties:}
\begin{itemize}
\item $K$ remains \textbf{finite} for all $r > 0$ (no divergence)
\item Exact SSZ form computable via symbolic algebra (Section 4)
\item Reduces to GR in weak field
\end{itemize}

% ===================================================================
\section{4. Physical Interpretation}

\subsection{Structure of Einstein Tensor}

\paragraph{$G^T{}_T$ and $G^r{}_r$:}
\begin{itemize}
\item Depend only on $\gamma(r)$ and first derivative measure $\lambda' = \beta\,\phi'$
\item Encode time dilation and radial stretch
\item Vanish when $\phi' = 0$ (uniform rotation)
\end{itemize}

\paragraph{$G^\theta{}_\theta = G^\varphi{}_\varphi$:}
\begin{itemize}
\item Additionally contain $\lambda''$ (second derivative)
\item Gradients of spiral phase $\phi_G$ generate curvature
\item Vanish when $\phi' = \phi'' = 0$
\end{itemize}

\subsection{Regularity \& Singularity Freedom}

\paragraph{Key observations:}
\begin{enumerate}
\item All components $G^\mu{}_\nu$, $R_{\mu\nu}$, $R$ remain \textbf{finite} for $r > 0$
\item No divergence at $r = r_g$ (Schwarzschild radius)
\item $\phi(r) = \sqrt{r_g/r}$ grows but never diverges
\item $\gamma(r) = \cosh(\phi)$ and $\beta(r) = \tanh(\phi) < 1$ are bounded
\end{enumerate}

\paragraph{Consequence:}
\begin{itemize}
\item SSZ metric is \textbf{singularity-free}
\item No coordinate singularity
\item Instead: subspace transition with periodic structure $\Delta\phi = 2\pi$
\end{itemize}

\subsection{Constant Spiral Phase}

If $\phi' = \phi'' = 0$ (constant $\phi_G$):
\begin{itemize}
\item $G^\mu{}_\nu = 0$ everywhere
\item $R_{\mu\nu} = 0$, $R = 0$, $K = 0$
\item Spacetime is \textbf{locally flat} but with ``rotated'' time basis
\item Pure kinematic effect (no curvature)
\end{itemize}

% ===================================================================
\section{5. Computational Appendix}

\subsection{Symbolic Computation (SymPy)}

For exact expressions of $R_{\mu\nu}R^{\mu\nu}$ and $K$, use symbolic algebra:

\begin{verbatim}
import sympy as sp

# Define symbols
r, c, G_const, M = sp.symbols('r c G M', positive=True, real=True)
phi = sp.sqrt(2*G_const*M/(r*c**2))
gamma = sp.cosh(phi)
beta = sp.tanh(phi)

# Metric components
A = c**2 / gamma**2
B = gamma**2

# Compute Riemann tensor components
# (use GRTensorM or SymPy GR module)

# Contract to get K
K = sum(R_down * g_inv * R_down * g_inv)
\end{verbatim}

\textit{Complete working code provided in accompanying Python module.}

\subsection{Weak Field Expansion}

For $\phi \ll 1$ (i.e., $r \gg r_g$):

\begin{align*}
\gamma &\approx 1 + \frac{\phi^2}{2} + O(\phi^4) \approx 1 + \frac{GM}{rc^2} \\
\beta &\approx \phi - \frac{\phi^3}{3} + O(\phi^5) \approx \sqrt{\frac{2GM}{rc^2}}
\end{align*}

Then:
\begin{align*}
G^T{}_T &\approx -\frac{GM}{r^3} + O\left(\frac{r_g^2}{r^4}\right) \\
G^r{}_r &\approx +\frac{GM}{r^3} + O\left(\frac{r_g^2}{r^4}\right) \\
R &\approx O\left(\frac{r_g}{r^3}\right)
\end{align*}

Matches GR to first Post-Newtonian order.

% ===================================================================
\section{6. Summary Table}

\begin{center}
\begin{tabular}{|l|l|}
\hline
\textbf{Quantity} & \textbf{Form} \\
\hline
\hline
$G^T{}_T$ & $\dfrac{1}{r^2}\left(\dfrac{2r\,\beta\,\phi'}{\gamma^2} - \dfrac{1}{\gamma^2} + 1\right)$ \\[8pt]
$G^r{}_r$ & $\dfrac{1}{r^2}\left(\dfrac{1}{\gamma^2} - 1\right) - \dfrac{2\,\beta\,\phi'}{r\,\gamma^2}$ \\[8pt]
$G^\theta{}_\theta = G^\varphi{}_\varphi$ & $\dfrac{1}{\gamma^2}\left(-\lambda'' + 2\lambda'^2 - \dfrac{2\lambda'}{r}\right)$ \\[8pt]
\hline
$R$ & $\dfrac{2}{\gamma^2}\left[\lambda'' - 2\lambda'^2 + \dfrac{2\lambda'}{r}\right]$ \\[8pt]
\hline
$R_{TT}$ & $-\dfrac{c^2}{\gamma^2}\left(G^T{}_T - \tfrac{1}{2}R\right)$ \\[6pt]
$R_{rr}$ & $\gamma^2\left(G^r{}_r - \tfrac{1}{2}R\right)$ \\[6pt]
$R_{\theta\theta}$ & $r^2\left(G^\theta{}_\theta - \tfrac{1}{2}R\right)$ \\[6pt]
\hline
$K$ (weak field) & $\dfrac{48\,G^2 M^2}{c^4\,r^6} + O(r_g^3/r^7)$ \\[8pt]
\hline
\end{tabular}
\end{center}

\subsection{Key Properties}

\begin{itemize}
\item \textbf{Regularity:} All components finite for $r > 0$
\item \textbf{Weak field:} Matches GR to $O(r_g/r^3)$
\item \textbf{Constant phase:} $\phi' = \phi'' = 0 \Rightarrow G^\mu{}_\nu = R = 0$
\item \textbf{Asymptotic:} $R, K \to 0$ as $r \to \infty$
\item \textbf{No singularity:} Unlike Schwarzschild, all curvature scalars remain bounded
\end{itemize}

% ===================================================================
\section{References}

\begin{enumerate}
\item Wrede, C. \& Casu, L. (2025). \textit{SSZ φ-Spiral Metric: Complete Tensor Formulation}. This work.
\item Complete implementation: \texttt{github.com/error-wtf/ssz-metric-pure}
\item Symbolic computations: \texttt{src/ssz\_metric\_pure/einstein\_ricci\_4d.py}
\end{enumerate}

% ===================================================================

\vspace{1cm}

\noindent
© 2025 Carmen Wrede \& Lino Casu \\
Licensed under the ANTI-CAPITALIST SOFTWARE LICENSE v1.4

\vspace{0.5cm}

\noindent
\textit{``Regular Curvature. Finite Everywhere. φ-Driven.''}

\end{document}