Curvature traces obey identities without containing every tidal component

Use Rμν=Rρμρν with the Riemann convention already declared. Metric compatibility and the Levi-Civita curvature symmetries make this Ricci tensor symmetric. A further contraction with the inverse metric gives the Ricci scalar. The inverse matters: in angular coordinates its entries supply inverse length squared, so a dimensionless angular Ricci component can still contribute to a scalar in m⁻². The Einstein tensor subtracts one half the scalar times the metric. These are tensor constructions, not operations tied to one special coordinate chart.

On a round two-sphere, Rθθ=1 and Rφφ=sin²θ. Contracting with the inverse radius-squared metric gives R=2/a². In two dimensions the Einstein tensor vanishes identically despite this nonzero curvature; a four-dimensional interpretation must not be imported into that dimensional identity. In four dimensions, the Weyl part retains curvature not specified by Ricci traces. Schwarzschild vacuum has Rμν=0 but a nonzero Riemann-square invariant 48m²/r⁶, where m=GM/c² is a length. Thus even a vanishing Ricci scalar is far from a proof of flatness.

The differential Bianchi identity is a cyclic relation among covariant derivatives of Riemann curvature. Contract it twice and use metric compatibility to obtain ∇μRμν equal to one half the gradient of R. The Einstein tensor therefore has zero covariant divergence for every sufficiently smooth metric with its Levi-Civita connection. No matter equation or Einstein equation was required. When the gravitational field equation relates this tensor to matter stress, the identity provides a consistency condition with covariant matter conservation.

Covariant conservation does not automatically provide a globally conserved ordinary energy in an arbitrary spacetime. The extra free tensor index means one cannot integrate components at different points as though their bases were the same. Symmetry, boundary conditions, and suitable currents are needed for a global charge. A cosmological term Λgμν has vanishing divergence when Λ is constant. Treating Λ as an arbitrary position-dependent scalar while keeping an otherwise unchanged conserved source would violate the same local consistency condition unless additional dynamical contributions are included.

For a vector current the covariant divergence is the coordinate derivative of √|g| times the current, divided by √|g|. Integrating with the invariant density converts it to a directed boundary flux by Stokes’ theorem. Orientation fixes the signs of initial and final pieces, and induced measures must be used. A non-null boundary can be expressed using a normalized normal with the appropriate Lorentzian orientation sign. A null boundary needs a separate surface-element construction because its normal is also tangent; setting a unit normal there is invalid. Boundary terms vanish only under stated boundary conditions.

The function lab illustrates the sphere scalar’s inverse-radius scaling. Reset and set both controls to a=2,x=0.5, predicting y=0.5. Reset and set a=2,x=1, predicting y=2. Assign x to an inverse radius in chosen reference-length units and y to the corresponding dimensionless scalar 2x². Halving radius quadruples this curvature. The lab computes neither tensor traces nor a boundary flux; the interpretation depends on the analytic sphere metric, not on the parabola alone.