Semi-colon or Nabla Notation: Which is Correct for Calculating Nabla?

[pellman]

Is it

$$\nabla_\mu\nabla_\nu A^\alpha={A^\alpha}_{;\mu\nu}$$

or

$$\nabla_\mu\nabla_\nu A^\alpha={A^\alpha}_{;\nu\mu}$$

?

 

[dextercioby]

The second is the right one.

$$\nabla_\mu \nabla_\nu A^\alpha=\left(A^{\alpha}_{~;\nu}\right)_{;\mu}= A^{\alpha}_{~;\nu;\mu} = A^{\alpha}_{~;\nu\mu}$$

To avoid errors, my suggestion is not to use them both at the same time when making calculations.

 

[CompuChip]

This is one reason I prefer the nabla-notation. (Another is that I often get confused afterwards whether I that little dot over the comma is just a little ink drop or I actually meant to write a semi-colon.)

 

[pellman]

Thanks!