Is a symmetric Lagrangian leads to a symmetric Stress-Energy Momentum？

[centry57]

Is a symmetric Lagrangian leads to a symmetric Stress-Energy Momentum ？

 

[blechman]

A "symmetric energy momentum tensor" obeys T_{\mu\nu}=T_{\nu\mu}.

A Lagrangian is a scalar, with no indices.

So what does one have to do with another?

 

[centry57]

I was adoubt if a symmetric stress-energy tensor 's lagrangian is symmetry .

Since {\cal L}= - \frac{1}{16\pi}F^{\mu\nu} F_{\mu\nu} is symmetry on \mu &\nu,the corresponding Stress-Energy Tensor \Theta^{\mu}\,_{\nu} = - \frac{1}{4 \pi} F^{\mu \alpha} \partial_{\nu}A_{\alpha} + \frac{1}{16\pi} \delta^{\mu}_{\nu} F^{\alpha\beta}F_{\alpha\beta} is also symmetry.

Is this the special one ?