Metric Split Ansatz

By Nusa February 10, 2026
We introduce a single scalar that controls how the ambient geometry breathes along unobserved directions.

What if spacetime’s hidden structure is not curled up, but breathing? In this post, we introduce the simplest but most consequential ansatz in the GU setup: the split metric on the ambient manifold $Y$. This ansatz respects the decomposition $TY|_X \simeq TX \oplus N_\iota$ and introduces a single scalar field $\sigma(x)$ that governs the internal geometry’s effective size.

We are not doing compactification. There is no separate “internal manifold.” Instead, the 10 normal directions to $X$ in $Y$ are fully present—but we scale their influence using $\sigma(x)$, the breathing mode. This not only preserves full covariance but also ensures that dimensionful quantities and induced terms (like cosmological constants) scale predictably when traced over internal directions.

Crucially, this ansatz is compatible with the $\mathrm{Spin}(7,7)$ structure. The normal bundle remains flat, orthogonal, and geometrically meaningful. The scalar $\sigma(x)$ acts as a dynamical filter: it doesn’t hide directions; it weights them.

Definitions / Notation used

  • $g_Y$: ambient metric on $Y$
  • $g_X := \iota^* g_Y$: induced metric on $X$
  • $TY|_X \simeq TX \oplus N_\iota$: tangent/normal decomposition
  • $\sigma(x)$: scalar field on $X$ (breathing mode)
  • ${n^a}$: orthonormal coframe for the normal bundle $N_\iota$
  • $\delta_{ab}$: standard flat metric on $N_\iota$ (Euclidean signature)

Metric split ansatz: $$ g_Y \simeq g_X \oplus \sigma^2(x)\cdot\delta_{ab}\cdot n^a \otimes n^b $$

The Technical Heart: Breathing the Normal Directions

The ansatz introduces no new degrees of freedom besides $\sigma(x)$. All internal curvature, spinors, and torsion components still live on the 14D manifold $Y$, but now the metric on $Y$ is weighted differently in tangential vs normal directions:

  • Along $TX$: $g_Y$ matches $g_X$ exactly.
  • Along $N_\iota$: the internal metric is scaled by $\sigma^2(x)$, a function that varies over $X$ but not in the normal directions.
  • Cross terms vanish: the decomposition is orthogonal by construction.

This block-diagonal form simplifies many calculations:

  • Contractions involving normal directions gain an explicit $\sigma$ dependence.
  • The Hodge star $*_Y$ acquires a $\sigma^{-10}$ scaling when saturating normal directions.
  • The induced cosmological term from $\Theta_E$ in the Shiab operator picks up a $\sigma^{-10}(x)$ factor (from volume suppression in normal directions).

This is not arbitrary. The form of $\Theta_E$ is fixed (we’ll get to that), and it saturates all normal directions. So when we evaluate integrals like:

$$ I_1(\omega) = \int_Y \langle T, *_Y (\bullet_\varepsilon F_B - \kappa_1 T)\rangle $$

the normal directions contribute multiplicatively via $\sigma(x)$. That means $\sigma(x)$ can dynamically modulate effective cosmological terms, without breaking geometric consistency.

One Diagram in Words

Picture $Y^{14}$ as a cylinder of space whose cross-section can breathe at every point $x \in X$. The breathing mode $\sigma(x)$ rescales the internal geometry’s influence. Pulling back to $X$, you see a fixed 4D slice—but the weight of normal contributions changes based on $\sigma(x)$. This is not shrinking dimensions—it’s dynamically weighting them.

Assumptions vs Consequences

Assumptions (Ansatz):

  • The ambient metric $g_Y$ takes the form: $$g_Y \simeq g_X \oplus \sigma^2(x)\cdot\delta_{ab}\cdot n^a \otimes n^b$$
  • The normal bundle $N_\iota$ is flat and orthogonal to $TX$.
  • $\sigma(x)$ is a scalar field on $X$, positive and smooth.

Consequences (Operational):

  • Any contraction over normal indices introduces $\sigma$-dependent scaling.
  • The internal volume element contributes $\sigma^{10}(x)$ in integrals, leading to $\sigma^{-10}$ in dualizations via $*_Y$.
  • The effective coupling of torsion to curvature (in $\Upsilon_\omega$) becomes $\sigma$-dependent.
  • $\sigma(x)$ becomes a natural candidate for a dynamical field influencing cosmological or mass-scale phenomena.

Why This Matters

  • Hodge star manipulations in the action $I_1(\omega)$ depend critically on this $\sigma(x)$ structure; it regulates how normal modes scale.
  • Chirality selection via axial torsion relies on normal-mode overlaps, which are $\sigma$-weighted.
  • Cosmological terms derived from $\Theta_E$ pick up $\sigma^{-10}(x)$ factors—linking $\sigma$ directly to vacuum energy and scalar curvature.
  • Normal-mode quantization of spinors depends on Hermite–Gaussian weighting, tied to $\sigma(x)$.

Key Takeaway

The breathing mode $\sigma(x)$ doesn’t hide extra dimensions—it controls how strongly they influence observed physics. This scalar is the dial that sets the weight of internal geometry in the induced 4D picture.

Technical Takeaway

  • $g_Y \simeq g_X \oplus \sigma^2(x)\cdot\delta_{ab}\cdot n^a \otimes n^b$
  • $*_Y$ applied to $\Theta_E$-saturated forms yields $\sigma^{-10}(x)$ scaling
  • $\sigma(x)$ modulates induced cosmological and fermionic effects
#geometric unity #gu #physics #science #metric
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