• Definitions / Notation used
• The single technical move: double-disease cancellation
• Why perturbation fails
• Why this matters
• Key takeaway
• Technical takeaway

General Relativity sets torsion to zero by definition. GU does the opposite: it refuses to define geometry without it. This is not because torsion is exotic, but because without it, gauge-covariant transport has no tensorial foothold. The key is not “torsion vs no torsion,” but which torsion transforms cleanly.

§Definitions / Notation used

• $X = X^4$ (observed spacetime), $Y = Y^{14}$ (ambient/observerse space); immersion $\iota: X \to Y$; pullback $\iota^*$.
• Tangent/normal split along $\iota(X)$: $TY|_X \simeq TX \oplus N_\iota,$ with $\mathrm{rank}(N_\iota)=10;$ indices $\mu,\nu$ for $TX;$ $a,b$ for $N_\iota;$ $M,N$ for $Y.$
• Metric: $g_X := \iota^* g_Y;$ metric split ansatz $g_Y \simeq g_X \oplus \sigma(x)^2 \delta_{ab} n^a \otimes n^b;$ Hodge stars $*_X$ and $*_Y.$
• Gauge structure: $H$ gauge group; $\mathcal{N} := \Omega^1(Y, \mathrm{ad}(P_H));$ inhomogeneous group $\mathcal{G} := H \ltimes \mathcal{N};$ $\omega = (\varepsilon, \phi);$ background connection $A_0;$ rotated connection $B_\omega := A_0 \cdot \varepsilon;$ curvature $F_B$.
• Shiab operator $\bullet_\varepsilon;$ $\Upsilon_\omega := \bullet_\varepsilon(F_B) - \kappa_1 T;$ action $I_1(\omega) := \int_Y \langle T,$ $*_Y \Upsilon _{\omega} \rangle.$
• Augmented torsion $T := \phi - \varepsilon^{-1} d_{A_0} \varepsilon$.

§The single technical move: double-disease cancellation

Connections do not transform tensorially. Differences of connections do.

Let $A_0$ be a fixed background connection. Any other connection can be written as

$$A = A_0 \cdot \varepsilon + \phi$$

Individually:

• $\phi$ is not covariant (it shifts under gauge change).
• $\varepsilon^{-1} d_{A_0} \varepsilon$ is not covariant (it is the inhomogeneous gauge term).

But their difference

$$T := \phi - \varepsilon^{-1} d_{A_0} \varepsilon$$

is covariant:

$$T \mapsto h^{-1} T h \quad \text{for } h \in H$$

This is not a choice; it is a cancellation. Two non-tensorial objects subtract to yield a tensorial one. There is no alternative covariant “torsion-like” variable in this setting.

§Why perturbation fails

If torsion were treated as a small deformation of a torsionless background, one would expand around a variable that does not transform cleanly. Gauge covariance would be broken order-by-order.

In GU, torsion is not a correction to geometry; it is the coordinate on the affine space of connections. Setting $T=0$ is a special point, not a preferred background.

§Why this matters

The Shiab contraction requires a covariant source term. $T$ is the only object available. Without it, one cannot write a gauge-covariant Einstein-like equation. Treating torsion perturbatively would destroy the symmetry GU is built to preserve.

§Key takeaway

Torsion in GU is not optional and not small. It is the only covariant way to parametrize deviation from a reference connection.

§Technical takeaway

$$T := \phi - \varepsilon^{-1} d_{A_0} \varepsilon$$

transforms as $T \mapsto h^{-1} T h$. Neither $\phi$ nor $\varepsilon^{-1} d_{A_0} \varepsilon$ does.