Equation 78 · Part 3 · The Bend an Elevator Cannot Fake

Symbol U

ds^2 = -U(\mathbf x)^2\,dx_0^2 + \delta_{ij}\,dx^i dx^j

U

What this part means

U is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

U is one of the signed contributions combined to compute the quantity on the left.

Full expressionSymbol UArticle meaning

The passage around this formula

…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…

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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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