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A relationship between the components of the metric of spacetime and those of the spatial 3-metric are defined.
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We can look at the (ADM) form of the metric to see this:
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ds2 = - a2dt2+gij(dxi+bidt)(dxj+b jdt),
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where the functions a and bi are called the lapse and shift functions, and gij is the 3-metric
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a determines the lapse in proper time, and bi describes the shift in the spatial coordinates, and ti are the tangents to a coordinate observer (moving through the spacetime moves timelike direction by a dt, and spacelike direction by b i).
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