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In standard units, each term on the left has units of 1/length 2. The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime.
But if one requires an exact solution or a solution describing strong fields, the evolution of both the metric and the stress–energy tensor must be solved for at once. To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation (to determine the evolution of the stress–energy tensor):
If pressure did not contribute equally to spacetime curvature as does mass–energy, the moon would not be in the orbit predicted by classical mechanics. They used their measurements to tighten the limits on any discrepancies between active and passive mass to about 10 −12 . [ 22 ]
In this form pressure is selected as the vertical coordinate and the horizontal coordinates are written for the Cartesian tangential plane (i.e. a plane tangent to some point on the surface of the Earth). This form does not take the curvature of the Earth into account, but is useful for visualizing some of the physical processes involved in ...
If k = −1, then (loosely speaking) one can say that i · a is the radius of curvature of the universe. a is the scale factor which is taken to be 1 at the present time. k is the current spatial curvature (when a = 1). If the shape of the universe is hyperspherical and R t is the radius of curvature (R 0 at the present), then a = R t / R 0
The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
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Here σ is the surface tension, n, t and s are unit vectors in a local orthogonal coordinate system (n,t,s) at the free surface (n is outward normal to the free surface while the other two lie in the tangential plane and are mutually orthogonal). The indices 'l' and 'g' denote liquid and gas, respectively and K is the curvature of the free surface.