I was reading "The foundation of the general theory of relativity" by Albert Einstein. I faced some difficulty to understand his derivation of "geodetic line", stated in "The equation of the geodetic line. The motion of particle" headed section. I will try to describe my problem clearly and in short.(adsbygoogle = window.adsbygoogle || []).push({});

Einstein first stated that along geodetic line [itex]\int ds[/itex] is stationary, which means equation of geodetic should be,

[itex]δ\int_P^{P'}ds=0[/itex]between two points P and P' of four dimensional continuum, where [itex]ds[/itex] means linear element. Solving thisvariationequation would give 4 new differential equations describing the geodetic line. This is clear and simple and no difficulty arises here.

However, then he replaced this equation with a parametric version,..

[itex]\int_{λ1}^{λ2}δwdλ=0[/itex]

where, [itex]w^2=g_{μν}\frac{dx_μ}{dλ}\frac{dx_ν}{dλ}[/itex]

Einstein, described λ as follows,

Thus, lines, joining P and P', seems to become function of this new "family of surfaces" λ, as mentioned by him in this way, Let λ be a function of the co-ordinates [itex]x_ν[/itex], and let this define a family of surfaces which intersect the required geodetic line as well as all the lines in immediate proximity to it which are drawn through the points P and P'.

Upto this point, things are not yet quite that difficult, although some elaboration of λ would be useful. But next he reinterpret δ as follows, Any such line may then be supposed to be given by expressing its co-ordinates [itex]x_ν[/itex] as functions of λ.

My question is what is this transition? Is it change of Let the symbol δ indicate the transition from a point of the required geodetic to the point corresponding to the same λ on a neighbouring line.line-length? Moreover, how the δ sign will cross the integral ∫ sign, as it happened in the 2nd, replaced, equation, involving λs? although, i guess, knowing the meaning of δ and transition will answer the 2nd question.

I have stressed my brain neurons quite a bit to understand what thistransitionmeans and run through some pages of calculus of variations, but still it is hiding at large from me. An enlightenment from you will re-energize my brain cells.

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# Derivation of Geodetic line by Einstein

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