Equation 85 · Part 1 · The Bend an Elevator Cannot Fake

Symbol j

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

j

What this part means

j is part of the quantity the equation computes from the expression on the right.

Its job in the formula

j is part of the quantity the equation computes from the expression on the right.

Full expressionSymbol jArticle meaning

The passage around this formula

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 d $s^2$ = -U( $\mathbf x$ )^2\,d $x_0^2$ + $\delta_{ij}$ \,d $x^i$ d $x^j$ — flat spatial slices, all the position dependence carried by U — the only nonzero Christoffel symbols are $\Gamma^0_{0i}$ = $\partial_i$ $\ln$ U = $\phi_{,i}$ and $\Gamma^i_{00}$ = $U^2$ $\phi_{,i}$ , both following directly from the metric’s definition. Substituting into the Riemann tensor’s defining combination for the mixed component $R^i$ {}_{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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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the article section

[14] General Relativity ↗

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