A Taylor Expansion of Metric Tensor: Troubles & Logic

[mertcan]

Hi, my question is related to taylor expansion of metric tensor, and I have some troubles, I would like to really know that why the RED BOX in my attachment has g_ij (t*x) instead of g_ij(x) ? I really would like to learn the logic...

 

[mertcan]

Hi, I have not received any responses for my first question for a long time, so I would like to ask in different way and share a NEW different attachment, by the way my question is related to taylor expansion of metric tensor, and I have some troubles, I would like to really know how "t" terms vanish while we are proceeding to equation 4.25 ( equation in attachment) from equation 4.24 or proceeding to 4.29 from 4.28.I really would like to learn the logic ?

 

[strangerep]

In "NEW ATTACHMENT".pdf ...

I guess it's because $g$ is being used here as a distance function, and these are homogeneous. See this Wiki page, in particular the section titled "Metrics on vector spaces" which gives the homogeneity condition.

I also guess that $R$, as a curvature 2-form is also assumed to be curvature-homogeneous, satisfying a similar definition of homogeneity.

Then, since $\alpha$ involves $1/t$, you end up with a $1/t^2$ outside the $g_0$, canceling the $t^2$ in eq(4.24), because $t^2 = |t|^2$.

Re your 1st post, it's a similar thing -- you've just got to track through the definitions. E.g., $f$ is an inner product of 2 $J$'s, hence involves $t$'s. But the inner product is taken at a point $tx$, hence that appears as argument to the metric.

I hope that helps.

 

[mertcan]

Hi everyone, first of all I would like you to take a look at my NEW ATTACHMENT 2 only the pages between 18 and 24. Those pages include mostly the taylor expansion of metric, I also would like you to look at the link about jacobi field https://en.wikipedia.org/wiki/Jacobi_field. After I read and compared them, I have some issues because I see some contradictions. In NEW ATTACHMENT 2 and part 3.3 (jacobi field part) $$\gamma$$ is a scalar function in terms of "s" and "t" variables, and when you look at wikipedia link "t" variable is defined as if it is $$\theta$$ in spherical coordinates and "s" variable is replaced by $$\tau$$, and because of the fact that wikipedia link says the geodesics through the North pole are great circles and separated by an angle $$\tau$$, so $$\tau$$ is defined as if it is $$\phi$$ in spherical coordinates. In NEW ATTACHMENT 2 equation 3.17, jacobi field is partial derivative of $$\gamma$$ function with respect to "s", actually in terms of the wikipedia link jacobi field is partial derivative of $$\gamma$$ function with respect to $$\tau$$ which means derivative with respect to $$\phi$$ in spherical coordinates. But if you look at equation 3.27 in NEW ATTACHMENT 2, it says jacobi field equals "t" multiplied by $$\beta$$, so I consider that if we want a derivative of scalar function to be vector then we should use directional derivative because if we use ordinary derivative for scalar then we obtain again scalar value, but jacobi field is vector so we should use directional derivative with respect to "s" variable to obtain vector value which means we should use directional derivative with respect to $$\tau$$ or $$\phi$$ (because geodesics through the North pole are great circles and separated by an angle $$\tau$$). Besides, if we use directional derivative with respect to $$\phi$$ or "s" variable for $$\gamma$$ scalar function then like the gradient in spherical coordinates we should have some extra terms like "1/r" for $$\theta$$ direction and "1/sin($$\theta$$)" for $$\phi$$ direction as well as ordinary derivative of $$\gamma$$ function. But jacobi field in NEW ATTACHMENT 2 has only ordinary derivative for scalar $$\gamma$$ function which means actually no vector structure.

So, I hope I am explicit and I ask could you help me about that problem, or enlighten me about the part I miss?
Also I would like to express that I really looking forward to see your replies here, I really tried to dig valuable things out of internet or my other sources, but nothing helps me, I feel as if I am in impasse. Therefore I really wonder your valuable responses...

 

[PeterDonis]

@mertcan, attachments showing equations and other things you want people to respond to are not allowed on PF. Please use the PF LaTeX feature. You can find help on that here:

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