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This is the equation is in the "mathematical form" section of the following wikipedia article: http://en.wikipedia.org/wiki/Geodesics_in_general_relativity More specifically, the "Full geodesic equation": $$ {d^2 x^\mu \over ds^2} =- \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds} $$

My question is: on the right hand side of the equation, the indices alpha and beta are repeated, thus the summation convention applies (as specifies in the article). But are there any cross-terms? If the right hand side of the equation is expanded, will there be any terms where alpha equals one and beta equals two, and so on? OR does the summation automatically mean alpha equals beta, and there are thus no cross terms? Or does the previous statement apply only if a Kronecker delta is present?

Also, the Wikipedia page states that "s is a scalar parameter of motion, ex.: proper time." What other parameters are a "sacalar parameter of motion"?

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    $\begingroup$ There are cross-terms if $\Gamma^{\mu}_{\alpha\beta} \neq 0$ for $\alpha \neq \beta$. $\endgroup$ – oscarafone May 13 '16 at 17:09
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$$\Gamma^\mu{}_{\alpha\beta} y^\alpha y^\beta$$ is defined to mean $$\sum_{\alpha = 0}^3 \sum_{\beta = 0}^3 \Gamma^\mu{}_{\alpha\beta} y^\alpha y^\beta $$ that is, each repeated index is summed over independently.

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  • $\begingroup$ So there will be all the possible cross-terms? (And of course alpha-equals-beta-terms), right? $\endgroup$ – Investor May 13 '16 at 17:09
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    $\begingroup$ Just to add... the double sum has $4\times 4=16$ terms in total. 4 of them have $\alpha=\beta$ and the remaining $12$ have $\alpha\neq \beta$. Those 12 terms may be paired to 6 pairs - where both members of the pair are the same because they are $\alpha\leftrightarrow\beta$ symmetric. But that doesn't matter, one still sums them twice. So the sum is equivalent to the sum over $\alpha=\beta$ plus twice the sum over $\alpha\lt \beta$. In the Einstein sum rule, one never tries to "suppress redundancies" by hand. It doesn't matter that some terms are equal to other terms. $\endgroup$ – Luboš Motl May 13 '16 at 17:10
  • $\begingroup$ So the expansion of the geodesic equation has, in total, 16 terms? $\endgroup$ – Investor May 13 '16 at 17:11
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    $\begingroup$ Each one, yes, but there are four equations, so 64 in total $\endgroup$ – m4r35n357 May 13 '16 at 17:33
  • $\begingroup$ And what is a "scalar parameter of motion"? The s in the equation. Wikipedia says "ex. Proper time " what else? $\endgroup$ – Investor May 14 '16 at 12:57

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