Let `Christoffel symbols` be renamed `Kurisutoffel symbols`

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.

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πŸ‘€︎ u/Ozawachen
πŸ“…︎ Mar 16 2020
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Tensor Calculus For Physics Ep. 12: Christoffel Symbols [March 15, 2020 at 08:48PM] youtube.com/watch?v=OJ-Mo…
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πŸ‘€︎ u/airboy1021
πŸ“…︎ Mar 16 2020
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Don't understand this step in the derivation of the Christoffel symbols.

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.

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πŸ‘€︎ u/MiDaDa
πŸ“…︎ Apr 30 2019
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Christoffel Symbols = Laplacian
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πŸ‘€︎ u/Ewie_Is_Dead
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What is the difference between upper and lower indicies in tensor analysis? Also, I need help with christoffel symbols?

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

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πŸ‘€︎ u/bolognie1
πŸ“…︎ Oct 22 2016
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Girl, are you a Christoffel Symbol of the second kind?

Because I think we're making affine connection!

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

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πŸ‘€︎ u/mitch251
πŸ“…︎ Nov 02 2016
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Calculating Christoffel symbols by hand?

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?

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πŸ‘€︎ u/zoreroz
πŸ“…︎ May 09 2018
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Christoffel symbol not a tensor proof

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

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πŸ‘€︎ u/quasar_tree
πŸ“…︎ May 15 2017
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Differential Geometry - Christoffel symbols and geodesics

http://imgur.com/qObg2RJ

Any help and hints would be appreciated.

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πŸ‘€︎ u/diffgeo
πŸ“…︎ Feb 12 2013
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So....what's a Christoffel Symbol? (A serious question + a funny video) youtube.com/watch?v=vkas4…
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πŸ‘€︎ u/narfarnst
πŸ“…︎ Apr 06 2011
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Geometric interpretation of the Ricci and Einstein Tensor

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 :)

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πŸ‘€︎ u/EmailsAreHorrible
πŸ“…︎ Jun 28 2021
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General Relativity Signs and Notations

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?

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πŸ‘€︎ u/kengye
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I hate it when this happens.
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What do you call a man with no arms or legs in a hole?

Phil

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πŸ‘€︎ u/Rumblebully
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DAD JOKES ARE NOT DIRTY.

Go post NSFW jokes somewhere else. If I can't tell my kids this joke, then it is not a DAD JOKE.

If you feel it's appropriate to share NSFW jokes with your kids, that's on you. But a real, true dad joke should work for anyone's kid.

Mods... If you exist... Please, stop this madness. Rule #6 should simply not allow NSFW or (wtf) NSFL tags. Also, remember that MINORS browse this subreddit too? Why put that in rule #6, then allow NSFW???

Please consider changing rule #6. I love this sub, but the recent influx of NSFW tagged posts that get all the upvotes, just seem wrong when there are good solid DAD jokes being overlooked because of them.

Thank you,

A Dad.

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Rational
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πŸ‘€︎ u/mainhoonmadrasi
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My wife blocked me on Facebook because I post too many bird puns.

Well, toucan play at that game.

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πŸ‘€︎ u/Zayan-ali
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There are only two white people in the movie Black Panther

Martin Freeman, and Andy Serkis.

They also play roles in Lord of the Rings.

I guess that makes them the Tolkien white guys.

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πŸ‘€︎ u/jzagri
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My 3 yr old daughter made her first pun today and I almost cried. She was eating an apple and I asked her if she liked apples.

She said apple-lutely

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I'd ruther not say
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πŸ‘€︎ u/Un_FaZed211
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What did the left eye say to the right eye when they got married?

'Eye-do'

This is my first post pls don't kill me lol.

The people in the comment section is why I love this subreddit!!

Cred once again my sis wants credit lol

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What was a very common name in the middle ages?

I heard parents named their children lance a lot.

First post please don't kill me

Edit: i went to sleep and now my inbox is dead, thank you kind strangers for the awards!

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Apologies if it's a bit corn-y
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Gottemm
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so many choices
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Well that didn't cross my mind
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He knows too much
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where do pirates get their hooks?

second hand stores!

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I've just deleted all the German names off my pre owned iPhone..

it's Hans free now..

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πŸ‘€︎ u/HugoZHackenbush2
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I've spent all morning trying to think of a quality pun, just to come up with THIS otter rubbish.
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πŸ‘€︎ u/awesome_smokey
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My 6 year old wanted me to share his joke with you all. What is a horses favorite store?

Old Neeeeiiiiighvy

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3.14159265387279 snek
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Did you know 10+10 and 11+11 are the same?

10+10 is twenty and 11+11 is twenty too

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Rock bottom
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How much does it cost for a pirate to get his ears pierced?

A buck-an-ear!

I Thank ye kind Matey for the booty! I be truly overwhelmed! Thank you!

Holy cow! Thank you everyone for the upvotes and awards! I wasn’t expecting this!

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My Son Ate a Bunch of Scrabble Tiles. My Wife is Scared but I'm not...

He should have a good vowel movement. His next diaper change could spell disaster though.

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