The Principle of Relativity · Albert Einstein
(A)
A_{μν} = _g__{μα} _g__{νβ} A^{αβ} (complement to the co-variant or contravariant tensors) and B_{μν} = _g__{μν} _g_^{αβ} A_{αβ} We can call B_{μν} the reduced tensor related to A_{μν}. Similarly B^{μν} = _g_^{μν}_g__{αβ}A^{αβ}. It is to be remarked that _g_^{μν} is no other than the “complement” of _g__{μν} for we have,— _g_^{μα}_g_^{νβ}_g__{αβ} = _g__{μα}δ^{ν}_{α} = _g_^{μν}. § 9. Equation of the geodetic line (or of point-motion). As the “line element” _ds_ is a definite magnitude independent of the co-ordinate system, we have also between two points P₁ and P₂ of a four dimensional continuum a line for which ∫_ds_ is an extremum (geodetic line), _i.e._, one which has got a significance independent of the choice of co-ordinates. Its equation is (20) δ{ ∫^{P₂}_{P₁} _ds_ } = 0 From this equation, we can in a wellknown way deduce 4 total differential equations which define the geodetic line; this deduction is given here for the sake of completeness. Let λ, be a function of the co-ordinates _x__{ν}; this defines a series of surfaces which cut the geodetic line sought-for as well as all neighbouring lines from P₁ to P₂. We can suppose that all such curves are given when the value of its co-ordinates _x__{ν} are given in terms of λ. The sign δ corresponds to a passage from a point of the geodetic curve sought-for to a point of the contiguous curve, both lying on the same surface λ.
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