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∂j(U2ϕ,i)=U2ϕ,iϕ,j+U2ϕ,ij\partial_j(U^2\phi_{,i}) = U^2\phi_{,i}\phi_{,j} + U^2\phi_{,ij}

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What must therefore vanish, identically, is not the slope but the correctly defined curvature-frame component built from it. For any static metric of the form ds2s^2 = -U(x\mathbf x)^2\,dx02x_0^2 + δij\delta_{ij}\,dxix^i dxjx^j — flat spatial slices, all the position dependence carried by U — the only nonzero Christoffel symbols are Γ0i0\Gamma^0_{0i} = ∂i\partial_i ln⁡\ln U = ϕ,i\phi_{,i} and Γ00i\Gamma^i_{00} = U2U^2ϕ,i\phi_{,i} , both following directly from the metric’s definition. Substituting into the Riemann tensor’s defining combination for the mixed component RiR^i{}_{0j0} , every term built from purely spatial Christoffels vanishes because the spatial slices are exactly flat, and the two surviving pieces are…

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ϕ,i\phi_{,i}

Symbol phi_,i

phi_,i is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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ϕ,ij\phi_{,ij}

Symbol phi_,ij

phi_,ij is one of the signed contributions combined to compute the quantity on the left.

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Published contexts (1)

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∂j(U2ϕ,i)=U2ϕ,iϕ,j+U2ϕ,ij\partial_j(U^2\phi_{,i}) = U^2\phi_{,i}\phi_{,j} + U^2\phi_{,ij}

Equation 85 · Evolutionary Physics

The Bend an Elevator Cannot Fake

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

What must therefore vanish, identically, is not the slope but the correctly defined curvature-frame component built from it. For any static metric of the form ds2s^2 = -U(x\mathbf x)^2\,dx02x_0^2 + δij\delta_{ij}\,dxix^i dxjx^j — flat spatial slices, all the position dependence carried by U — the only nonzero Christoffel symbols are Γ0i0\Gamma^0_{0i} = ∂i\partial_i ln⁡\ln U = ϕ,i\phi_{,i} and Γ00i\Gamma^i_{00} = U2U^2ϕ,i\phi_{,i} , both following directly from the metric’s definition. Substituting into the Riemann tensor’s defining combination for the mixed component RiR^i{}_{0j0} , every term built from purely spatial Christoffels vanishes because the spatial slices are exactly flat, and the two surviving pieces are…

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