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The Einstein field equations (EFE) may be written in the form: [5] [1] + = EFE on the wall of the Rijksmuseum Boerhaave in Leiden, Netherlands. where is the Einstein tensor, is the metric tensor, is the stress–energy tensor, is the cosmological constant and is the Einstein gravitational constant.
The kappa curve has two vertical asymptotes. In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letter ϰ (kappa).The kappa curve was first studied by Gérard van Gutschoven around 1662.
A closely related notion of curvature comes from gauge theory in physics, where the curvature represents a field and a vector potential for the field is a quantity that is in general path-dependent: it may change if an observer moves around a loop. Two more generalizations of curvature are the scalar curvature and Ricci curvature. In a curved ...
is the equation of motion for the metric field. The right hand side of this equation is (by definition) proportional to the stress–energy tensor, [4]:= = +. To calculate the left hand side of the equation we need the variations of the Ricci scalar and the determinant of the metric.
It follows that the Einstein field equations are a set of 10 quasilinear second-order partial differential equations for the metric tensor. The contracted Bianchi identities can also be easily expressed with the aid of the Einstein tensor: ∇ μ G μ ν = 0. {\displaystyle \nabla _{\mu }G^{\mu \nu }=0.}
Albert Einstein believed that the geodesic equation of motion can be derived from the field equations for empty space, i.e. from the fact that the Ricci curvature vanishes. He wrote: [ 5 ] It has been shown that this law of motion — generalized to the case of arbitrarily large gravitating masses — can be derived from the field equations of ...
These amount to only 14 equations (10 from the field equations and 4 from the continuity equation) and are by themselves insufficient for determining the 20 unknowns (10 metric components and 10 stress–energy tensor components). The equations of state are missing. In the most general case, it's easy to see that at least 6 more equations are ...
A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane. The curvature and the torsion of a helix are constant. Conversely, any space curve whose curvature and torsion are both constant and ...