The Principle of Relativity

by Albert Einstein; H. Minkowski

Similarly

B^μν = g^μνg_αβA^αβ.

It is to be remarked that g^μν is no other than the “complement” of g_μν for we have,—

g^μαg^νβg_αβ = g_μαδ^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.