Quick question on notation.

  • Thread starter cristo
  • Start date
  • #1
cristo
Staff Emeritus
Science Advisor
8,107
73
Does anyone know what this means: [tex]\nabla_{[a}F_{bc]}[/tex]? I know that [tex]F_{(ab;c)}=\frac{1}{3}(F_{ab;c}+F_{bc;a}+F_{ca;b})[/tex], and presume that the first expression can be written thus [tex]F_{[bc;a]}[/tex], but am not sure what it means!

Can anyone help?
 

Answers and Replies

  • #2
422
1
Does anyone know what this means: [tex]\nabla_{[a}F_{bc]}[/tex]? I know that [tex]F_{(ab;c)}=\frac{1}{3}(F_{ab;c}+F_{bc;a}+F_{ca;b})[/tex], and presume that the first expression can be written thus [tex]F_{[bc;a]}[/tex], but am not sure what it means!

Can anyone help?
Any time you see square brackets around indices it means that you permute the indices, with even permutations receiving a plus sign and odd permutations receiving a minus sign. Thus,

[tex]\nabla_{[a}F_{bc]} = \frac{1}{3!}\sum_{\pi\in S(3)}\textrm{sign}(\pi)\nabla_{\pi(a)}F_{\pi(b)\pi(c)}[/tex]

where [itex]S(3)[/itex] is the symmetric group of order three, [itex]\pi[/itex] is a permutation, and sign[itex](\pi)[/itex] equals one for an even permutation of the elements and minus one for an odd permutation of the elements. In your case you can expand out the above definition to obtain

[tex]\nabla_{[a}F_{bc]} = \frac{1}{3!}(\nabla_aF_{bc} + \nabla_bF_{ca} + \nabla_cF_{ab} - \nabla_aF_{cb} - \nabla_bF_{ac} - \nabla_cF_{ba})[/tex]
 
Last edited:
  • #3
cristo
Staff Emeritus
Science Advisor
8,107
73
Any time you see square brackets around indices it means that you permute the indices, with even permutations receiving a plus sign and odd permutations receiving a minus sign. Thus,

[tex]\nabla_{[a}F_{bc]} = \nabla_aF_{bc} + \nabla_bF_{ca} + \nabla_cF_{ab} - \nabla_aF_{cb} - \nabla_bF_{ac} - \nabla_cF_{ba}[/tex]
Ahh ok, that makes sense. Thanks for the quick reply! In this case F is the electromagnetic field tensor, and so is antisymmetric. Would I be right in assuming that in this case the equation becomes [tex]\nabla_{[a}F_{bc]} = \frac{1}{3}\left(\nabla_aF_{bc} + \nabla_bF_{ca} + \nabla_cF_{ab}\right)[/tex]
 
  • #4
422
1
Ahh ok, that makes sense. Thanks for the quick reply! In this case F is the electromagnetic field tensor, and so is antisymmetric. Would I be right in assuming that in this case the equation becomes [tex]\nabla_{[a}F_{bc]} = \frac{1}{3}\left(\nabla_aF_{bc} + \nabla_bF_{ca} + \nabla_cF_{ab}\right)[/tex]
Yes. If [itex]F_{ab}[/itex] are taken as the components of the Maxwell tensor then [itex]\nabla_{[a}F_{bc]}=0[/itex] is essentially the Bianchi identity for the electromagnetic field.

Another way to think about it is to notice that [itex]\nabla_{[a}F_{bc]}=0[/itex] is precisely the same statement as [itex]d\mathbf{F}=0[/itex] where [itex]\mathbf{F}=d\mathbf{A}[/itex] is the Maxwell two-form. The identity [itex]d\mathbf{F}=0[/itex] is guaranteed since for any [itex]p[/itex]-form field [itex]\mathbf{A}[/itex] one has [itex]d\cdot(d\mathbf{A})=0[/itex].
 
Last edited:
  • #5
cristo
Staff Emeritus
Science Advisor
8,107
73
That's a good way to think about it. Thanks a lot for your help, shoehorn!
 

Related Threads on Quick question on notation.

  • Last Post
Replies
2
Views
2K
Replies
11
Views
853
  • Last Post
Replies
4
Views
1K
Replies
8
Views
902
  • Last Post
Replies
16
Views
3K
Replies
7
Views
4K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
8
Views
1K
Replies
10
Views
1K
Replies
2
Views
718
Top