- #1
sujoykroy
- 18
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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.
Einstein first stated that along geodetic line [itex]\int ds[/itex] is stationary, which means equation of geodetic should be,
However, then he replaced this equation with a parametric version,..
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,
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,
My question is what is this transition? Is it change of 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 this transition means 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.
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 this variation equation 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]
where, [itex]w^2=g_{μν}\frac{dx_μ}{dλ}\frac{dx_ν}{dλ}[/itex]
Einstein, described λ as follows,
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'.
Thus, lines, joining P and P', seems to become function of this new "family of surfaces" λ, as mentioned by him in this way,
Any such line may then be supposed to be given by expressing its co-ordinates [itex]x_ν[/itex] as functions of λ.
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,
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.
My question is what is this transition? Is it change of 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 this transition means 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.