Fibre-integrated stress

Geometric and dynamical assumptions

Choose a fixed local product description of nested spatial submanifolds. Coordinates $x^\mu$ comprise the shared time and smaller spatial sector; $y^a$ comprise the additional ordinary spatial directions. They all describe the same physical manifold. The coordinate grouping is relative to the smaller sector and gives the containing particle no inherent division among its spatial dimensions.

Assume a block metric

$$G_{AB}= \begin{pmatrix}g_{\mu\nu}(x)&0\\0&h_{ab}(x,y)\end{pmatrix}, \qquad \sqrt{|G|}=\sqrt{|g|}\sqrt h,\qquad h=\det(h_{ab})>0.$$

The integration domain $F$ in $y$ is fixed. The base metric is independent of $y$. These hypotheses specify the projection being calculated; a general embedding, moving integration domain, or off-diagonal metric requires its corresponding geometric terms.

Let $T^{AB}=T^{BA}$ be the full source satisfying $\nabla_A^{(G)}T^A{}_B=0$. One sufficient origin is a diffeomorphism-invariant matter action with all its participating fields on shell. In the convention

$$\delta S_{\rm m}=\frac12\int\sqrt{|G|}\,T^{AB}\delta G_{AB}\,dx\,dy,$$

the metric change under a compactly supported vector field $\xi$ is $\delta G_{AB}=2\nabla_{(A}\xi_{B)}$. On-shell invariance and integration by parts give $0=-\int\sqrt{|G|}(\nabla_AT^{AB})\xi_B$, hence conservation. This is the standard action identity derived in Carroll's GR notes, §5, equations 5.35–5.38. All relevant interaction contributions to $T$ must be retained.

Exact integrated balance

Define the tangential stress integrated over the full spatial fibre:

$$\bar T^\mu{}_\nu(x)=\int_F\sqrt h\,T^\mu{}_\nu(x,y)\,d^py.$$

Expanding covariant conservation for a symmetric tensor gives the coordinate identity

$$\partial_A\!\left(\sqrt{|G|}T^A{}_\nu\right) =\frac12\sqrt{|G|}T^{AB}\partial_\nu G_{AB}.$$

To see the factor $1/2$, the connection contribution with the lower index is $\Gamma^C_{A\nu}T^A{}_C$. After lowering the contracted index, symmetry of $T^{AB}$ cancels the two oppositely signed metric derivatives in the Christoffel symbol, leaving $T^{AB}\partial_\nu G_{AB}/2$. The upper-index connection trace supplies the volume derivative on the left.

Integrate the identity over $F$ and divide by $\sqrt{|g|}$. The term containing $\partial_\nu g_{\alpha\beta}$ combines with the base volume derivative into $\nabla^{(g)}_\mu\bar T^\mu{}_\nu$. Consequently,

$$\boxed{ \nabla^{(g)}_\mu\bar T^\mu{}_\nu =\frac12\int_F\sqrt h\,T^{ab}\partial_\nu h_{ab}\,d^py -\mathcal F_\nu, \qquad \mathcal F_\nu=\int_{\partial F}n_aT^a{}_\nu\,dS_h. }$$

The normal and surface measure are those of the fibre metric at fixed $x$. For a noncompact fibre, $\mathcal F_\nu$ means the limiting flux through an exhaustion by bounded domains. Require convergence of the displayed integrals and permission to differentiate them, for example by locally uniform integrable bounds on the integrands and their base derivatives. The same derivation can be formulated weakly against compactly supported base test fields.

If $h$ is independent of $x$ and the boundary flux vanishes, then $\nabla^{(g)}_\mu\bar T^\mu{}_\nu=0$. Finite polynomial combinations of oscillator eigenmodes give useful decaying test sources when the remaining coefficients are controlled. The actual stress tensor determines which decay estimates are needed.

A varying fibre metric

The volume factor belongs inside $\bar T$. The metric term on the right measures exchange with a spatial metric that varies along the base. For a concrete check, take

$$ds^2=-dt^2+dx^2+e^{2ax}dy^2,\qquad -L\leq y\leq L, \qquad T^A{}_B=P\delta^A_B,$$

with constants $a,L>0$ and $P$. This tensor is conserved by metric compatibility. Its tangential boundary flux is zero, while

$$\bar T^x{}_x=2LP e^{ax},\qquad \partial_x\bar T^x{}_x=2LaP e^{ax}.$$

Since $T^{yy}=Pe^{-2ax}$ and $\partial_xh_{yy}=2ae^{2ax}$, the metric term is also $2LaPe^{ax}$. Thus a conserved full tensor can have a nonzero integrated tangential divergence supplied exactly by the varying spatial geometry. This example checks the balance law without requiring an infinite fibre volume or a particle assignment.

Sampling a finite portion

For a smooth scalar weight $w(x,y)$, define $\bar T_w^\mu{}_\nu=\int_F\sqrt h\,w T^\mu{}_\nu\,dy$. Multiplying the coordinate identity by $w$ and applying the product rule gives

$$\begin{aligned} \nabla^{(g)}_\mu\bar T_w^\mu{}_\nu ={}&\frac12\int_F\sqrt h\,wT^{ab}\partial_\nu h_{ab}\,dy -\int_{\partial F}w n_aT^a{}_\nu\,dS_h\\ &+\int_F\sqrt h\left[ (\partial_\mu w)T^\mu{}_\nu+(\partial_a w)T^a{}_\nu \right]dy. \end{aligned}$$

Even a time-independent aperture can exchange tangential momentum through its spatial gradient. For an explicit flat example use the symmetric spatial tensor

$$T^{xx}=f(x)q''(y),\qquad T^{yx}=T^{xy}=-f'(x)q'(y),\qquad T^{yy}=f''(x)q(y).$$

Both divergences vanish by commuting derivatives. With $q=e^{-y^2}$ and $w=e^{-\beta y^2}$, $\beta>0$, the full boundary flux is zero but

$$\partial_x\bar T_w^x{}_x =-\frac{2\beta\sqrt\pi}{(1+\beta)^{3/2}}f'(x) =\int_{\mathbb R}w'(y)T^{yx}(x,y)\,dy.$$

The sampling term is therefore essential even for a smooth, rapidly decaying source.

Factorized normalization

For an $x$-independent fibre metric and the specified factorization $T^\mu{}_\nu=|\chi(y)|^2 t^\mu{}_\nu(x)$ with $\int\sqrt h|\chi|^2dy=1$, direct integration gives $\bar T^\mu{}_\nu=t^\mu{}_\nu$. This is a normalization identity. Kinetic and interaction terms must actually permit that factorization. For a finite coherent mode expansion, a bilinear expression integrates with the Gram matrix $\int\sqrt h\,\chi_i^*\chi_jdy$; orthonormality removes those cross terms only when the operator acting on the modes permits it.

These formulas provide conservation and normalization tests for an effective source. A metric field equation and its coupling coefficient still come from the specified gravitational action.

Verification and related

spatial_profile_checks.py checks the Christoffel identity, the varying-metric example, finite-boundary flux, weighted sampling, and mode normalization.

• Orbit current projection — the companion vector-current construction includes a moving spatial readout.
• Interaction operator tools — navigation to the surrounding projection and action calculations.