Varying an action wrt a symmetric and traceless tensor

In summary: In the E-L equations, the variation w.r.t the deriviative of the field comes with the opposing sign, so it doesn't matter which is chose.
  • #1
binbagsss
1,254
11
Consider a Lagrangian, #L#, which is a function of, as well as other fields #\psi_i#, a traceless and symmetric tensor denoted by #f^{uv}#, so that #L=L(f^{uv})#, the associated action is #\int L(f^{uv}, \psi_i)d^4x #.

To vary w.r.t #f^{uv}# , I write:

#f^{uv}=f_{symm}^{uv}-\frac{1}{d}\eta^{uv} tr(f)#, (1)

where #tr(f)=\eta_{uv}f^{uv}#, where #\eta_{uv}# is the metric associated to the space-time, and #f_{symm}^{uv}= (1/2) (f^{uv}+f^{vu})#.In (1), why is it to subtract the trace, I suspect adding the trace term is just as valid? (and perhaps there is a convention as to which consistent with the signature of the metric)? In the E-L equations, ofc, the variation w.r.t the deriviative of the field comes with the opposing sign, so it doesn't matter which is chose? thanks
 
Physics news on Phys.org
  • #2
binbagsss said:
Consider a Lagrangian, #L#, which is a function of, as well as other fields #\psi_i#, a traceless and symmetric tensor denoted by #f^{uv}#, so that #L=L(f^{uv})#, the associated action is #\int L(f^{uv}, \psi_i)d^4x #.

To vary w.r.t #f^{uv}# , I write:

#f^{uv}=f_{symm}^{uv}-\frac{1}{d}\eta^{uv} tr(f)#, (1)

where #tr(f)=\eta_{uv}f^{uv}#, where #\eta_{uv}# is the metric associated to the space-time, and #f_{symm}^{uv}= (1/2) (f^{uv}+f^{vu})#.In (1), why is it to subtract the trace, I suspect adding the trace term is just as valid? (and perhaps there is a convention as to which consistent with the signature of the metric)? In the E-L equations, ofc, the variation w.r.t the deriviative of the field comes with the opposing sign, so it doesn't matter which is chose?thanks
Use double-# for equations
https://www.physicsforums.com/help/latexhelp/
 
  • #3
Adjusted:

Consider a Lagrangian, L which is a function of, as well as other fields ##\psi_i##, a traceless and symmetric tensor denoted by ##f^{uv}##, so that ##L=L(f^{uv})##, the associated action is ##\int L(f^{uv}, \psi_i)d^4x ##.

To vary w.r.t ##f^{uv}## , I write:

##f^{uv}=f_{symm}^{uv}-\frac{1}{d}\eta^{uv} tr(f)##, (1)

where ##tr(f)=\eta_{uv}f^{uv}##, where ##\eta_{uv}## is the metric associated to the space-time, and ##f_{symm}^{uv}= (1/2) (f^{uv}+f^{vu})##.In (1), why is it to subtract the trace, I suspect adding the trace term is just as valid? (and perhaps there is a convention as to which consistent with the signature of the metric)? In the E-L equations, ofc, the variation w.r.t the deriviative of the field comes with the opposing sign, so it doesn't matter which is chose?

--edit-- maybe I'm stupid, but why does the double # not work?
 
  • #4
binbagsss said:
Consider a Lagrangian, #L#, which is a function of, as well as other fields #\psi_i#, a traceless and symmetric tensor denoted by #f^{uv}#, so that #L=L(f^{uv})#, the associated action is #\int L(f^{uv}, \psi_i)d^4x #.

To vary w.r.t #f^{uv}# , I write:

#f^{uv}=f_{symm}^{uv}-\frac{1}{d}\eta^{uv} tr(f)#, (1)

where #tr(f)=\eta_{uv}f^{uv}#, where #\eta_{uv}# is the metric associated to the space-time, and #f_{symm}^{uv}= (1/2) (f^{uv}+f^{vu})#.In (1), why is it to subtract the trace, I suspect adding the trace term is just as valid? (and perhaps there is a convention as to which consistent with the signature of the metric)? In the E-L equations, ofc, the variation w.r.t the deriviative of the field comes with the opposing sign, so it doesn't matter which is chose?thanks
I'm not sure what you do here, but you subtract the trace because the trace transforms to itself. Usually, one can break up a tensor into a symmetric traceless part, an antisymmetric part and the trace. But since you don't give any reference and your tex is hard to read, I'm merely guessing.
 

Similar threads

  • Calculus
Replies
1
Views
905
  • Linear and Abstract Algebra
Replies
1
Views
2K
  • Special and General Relativity
Replies
22
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Special and General Relativity
Replies
9
Views
2K
  • Special and General Relativity
Replies
2
Views
92
Replies
1
Views
786
  • Advanced Physics Homework Help
Replies
1
Views
922
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
3K
Back
Top