Determining Christoffel Symbols: Subscripts Explained

[atomqwerty]

My question is just,

How can I determinate the Christoffel Symbols?

I know that they're given by

http://img263.imageshack.us/i/17f2df132717bfc32dc2ce3.png/"

but, what does this mean? The subscripts I mean.

thank you very much! :)

 

[tiny-tim]

hi atomqwerty!

… what does this mean? The subscripts I mean.

each index has to take all the values 0 1 2 3 (or 1 2 3 4)

(and then sum over all repeated indices … the Einstein summation convention)

and yes, it does take a long time … but it helps that most of the derivatives are zero tongue2)

 

[atomqwerty]

hi atomqwerty!


each index has to take all the values 0 1 2 3 (or 1 2 3 4)

I see, but, what is for example g12 for a given g?

thanks!

 

[atomqwerty]

It is $$\frac{g_{mk}}{x_{l}}= \frac{\partial}{\partial x_{l}} \frac{\partial g_{m}}{\partial x_{k}}$$?

thanks

 

[tiny-tim]

I see, but, what is for example g12 for a given g?

thanks!

g is the metric …

g₁₂ is the coefficient of dx₁dx₂

(and g₁₁ is the coefficient of dx₁² etc)

 

[atomqwerty]

g is the metric …

g₁₂ is the coefficient of dx₁dx₂

(and g₁₁ is the coefficient of dx₁² etc)

With $$dx_{1}dx_{2}$$ you mean $$dx_{1}\otimes dx_{2}$$, right?

 

[atomqwerty]

The whole expression it's a sum, right? So for a metrics in R2, there will be... 24 different addends (12 for each k)?? :O

EDIT: Automessage- It's not a sum, they are Symbols! My fault :S

 

[tiny-tim]

With $$dx_{1}dx_{2}$$ you mean $$dx_{1}\otimes dx_{2}$$, right?

ooh, that's rather technical, we don't normally bother with that in physics

The whole expression it's a sum, right? So for a metrics in R2, there will be... 24 different addends (12 for each k)?? :O

EDIT: Automessage- It's not a sum, they are Symbols! My fault :S

not following you

 

[atomqwerty]

ooh, that's rather technical, we don't normally bother with that in physics

It's for differential Geometry ;)

Thank you, it's been very helpfull!