Understanding Notation in Hawking and Ellis: V_{(c;d)} and V_{[c;d]}

[robousy]

Hey, I'm going through Hawking and Ellis and want to confirm I have understood some notation correctly.

Are the following correct?

$$V_{(c;d)}=\nabla_c V^d + \nabla_d V^c$$

and

$$V_{[c;d]}=\nabla_c V^d - \nabla_d V^c$$

Also, do these have specific names?

Thanks in advance!

Richard

 

[robphy]

Hey, I'm going through Hawking and Ellis and want to confirm I have understood some notation correctly.

Are the following correct?

$$V_{(c;d)}=\nabla_c V^d + \nabla_d V^c$$

and

$$V_{[c;d]}=\nabla_c V^d - \nabla_d V^c$$

Also, do these have specific names?

Thanks in advance!

Richard

$$V_{(c;d)}=\frac{1}{2!} \left( \nabla_d V_c + \nabla_c V_d \right)$$

and

$$V_{[c;d]}=\frac{1}{2!}\left( \nabla_d V_c - \nabla_c V_d \right)$$

The combinatorial factor is a convenient convention.
With it, you can call these the symmetric and antisymmetric parts of $$V_{c;d}$$.
You could call the antisymmetric part the "curl" of $$V_c$$.


Note that the operation
$${(something)}_{;d}$$ is the same as $$\nabla_d (something)$$

 

[robousy]

Ok Rob! Thanks a lot for clarifying that for me. Very much appreciated.