Christoffel symbols Puns

https://preview.redd.it/l9lfx5sec2n41.png?width=1280&format=png&auto=webp&s=e61ed0bc461739460b9c6b1b824fcc429f746a6d

Every student of General Relativity learns it at the beginning of the course.

https://imgur.com/a/QjLbQ2b

I'm trying to learn tensor calculus on my free time. I've mainly followed this (https://www.youtube.com/playlist?list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx) video series of eigenchris. But I don't understand one thing in the derivation of the Christoffel symbols (in intrinsic geometry).

In the image I am going from what I am pretty sure of is still correct (I've used sigma summation to make sure I am not missing any summations).

Then I brought the 2 to the right side and turned it into a single expression to keep a better overview.

And then I did the next step according to the internet (the expression on the right). But also according to the internet should the left side of the expression become a sum over L all of the sudden... But I don't know how/why. And I can't find anything about it on the internet (or I don't know what to look for).

Ps: Sorry for the long story just wanted to make sure I wasn't missing anything.

I'm putting myself through a general relativity crash course, and am all good with the concepts, but in order to understand the maths, I just need to understand what a christoffel symbol actually is.

Through research, I've seen loads of weird notation which I've managed to understand, but I can't seem to understand the significance of upper and lower indicies, why one can't contract vectors of both lower indicies, and why it is that a tensor of upper indicies seems to act as a reciprocal to a tensor of upper indicies when manipulating equations (such as 'moving' a tensor to 'the other side' of an equation).

I also need some help in understanding the notation of a christoffel symbol. Cheers

Because I think we're making affine connection!

(Sorry if I unintentionally stole this from someone, let me know and I'll give credit!)

Hello all! I've been practicing for my upcoming final exam in GR, but it seems I never actually got a hang of calculating Christoffel symbols. I'm not sure where exactly I'm going wrong, but it seems like I definitely am doing something very wrong, so I would appreciate if someone could correct me here:

I start with a metric:

[; dS^2 = r^2(d\theta^{2}+\sin^{2}(\theta)d\phi^{2}) ;]

My metric elements are:

[; g_{11} = r^2, g_{22} = r^2\sin^{2}(\theta), g^{11} = r^{-2}, g^{22}=(r\sin(\theta))^{-2} ;]

According to the solution, the only nonzero symbols are [; \Gamma^{1}_{22}, \Gamma^{2}_{12};], so I try to calculate the first one:

[; \Gamma^{\delta}_{\beta \gamma} = 1/2g^{\alpha \delta}(\partial_{\gamma} g_{\alpha \beta} + \partial_{\beta} g_{\alpha \gamma} -\partial_{\alpha} g_{\beta \gamma}) ;]

[; \Gamma^{1}_{22} = 1/2 g^{11}(\partial_{2} g_{12} +\partial_{2} g_{12} - \partial_{1} g_{22}) ;]

[; \Gamma^{1}_{22} = 1/2 r^{-2}(0 + 0 - \partial_{r}r^2\sin(\theta)) = \sin^{2}(\theta)/r ;]

while the answer is:

[; \Gamma^{1}_{22} = -\sin(\theta)\cos(\theta) ;]

Similarly, the Christoffel symbols that are supposed to be zero, some of them I get non-zero, for example:

[; \Gamma^{1}_{11} = 1/2 g^{11}(\partial_{1} g_{11} +\partial_{1} g_{11} - \partial_{1} g_{11}) ;]

[; \Gamma^{1}_{22} = 1/2 r^{-2}(\partial_{r}r^{2}) = 1/r ;]

Can someone direct me where I'm going wrong?

I tried to prove that the Christoffel symbol (of the second kind) is not a tensor by seeing how it transforms across coordinate systems. I am not sure if I did it right. I feel a bit weird about a part where I used the chain rule which resulted in a jacobian. Did I do it right? If not, or regardless, Does anyone know where I can find the proof?
work: http://i.imgur.com/A8Wtf84.jpg

http://imgur.com/qObg2RJ

Any help and hints would be appreciated.

Whilst researching and trying to learn general relativity for fun, I ran across the Ricci tensor, and I want to fully understand what it means.

So far, I am aware that:

1. Its components R_uu can be seen as (given an orthogonal basis of dimension D) the sum of the sectional curvatures along d/dx^u with every other vector being a separation vector for adjacent geodesics.

2. It’s components are used in determining the volume change due to curvature of volume formed by any linearly dependent vectors travel along a geodesic (-R_uv T^u T^v).

3. That it is the only meaningful contraction of the Riemann curvature tensor.

However, what I don’t get is the actual meaning of let’s say R_01. For every other object, e.g. Christoffel symbols, if I was given gamma^1_23 I would know that it is the 1st component of the vector representing the covariant derivative of d/dx^2 with respect to d/dx^3. However, I only understand the Ricci curvature where the indices are the same, e.g. R_11 for the reasons I described above. Could someone please help explain the geometrical interpretation of the off diagonal elements of R_uv? If possible, please do go easy on the differential geometry, because I am an engineer who has basically no background in all of it.

Additionally, I also want to understand how lowering or raising an index affects the geometric/physical meaning of a tensor. For example, let’s bring up the Riemann curvature tensor components R^a_bcd. If I lowered it by doing R_ibcd = g_ia R^a_bcd, would the physical meaning of the curvature tensor change in any way? Because to me, multiplying in a set of numbers is bound to mess with the meaning of the tensor somehow, especially when an object turns from covariant to contra variant or vice versa.

And this brings me to my final question - even if I ignore the question above (but it would apply to the Einstein tensor the most actually), I still have no idea what the Einstein tensor represents. By adding together the Ricci tensor to a weird mix of the Ricci scalar and the metric, how can we even be sure if this even remotely represents the curvature of space time anymore? So how can we try to equate this to the stress energy tensor?

I know these are quite long questions, so I would really appreciate any input on this. Thank you :)

In general relativity, there seems to be a whole plethora of notation and conventions. I have several questions, why are there so many different symbols used for indexes in Christoffel symbols, often in different texts. Also, does change the metric change anything?

Phil

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