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Can anyone please help me to grasp a minor detail in the derivation of the Belinfante-Rosenfeld version of the Stress-Energy Tensor (SET) ?

To save type, I refer to the wiki webpage http://en.wikipedia.org/wiki/Belinfante–Rosenfeld_stress–energy_tensor

Using the Noether Theorem, it is indeed easy to arrive at the conservation of the tensor M.

And yes, therefrom we indeed obtain

∂_{μ}S^{μ}_{γλ}= T_{λγ}- T_{γλ}

Now, the canonical SET can be expressed as a sum of its symmetrical and antisymmetrical parts:

T^{γλ}

= (T_{γλ}+ T_{λγ})/2 + (T_{γλ}-T_{λγ})/2

= T^{γλ}_{B}+ (T_{γλ}- T_{λγ})/2

As we have just seen, the antisymmetric part can be expressed through the spin tensor, whence we obtain:

T^{μγ}= T^{γλ}^{B}- ∂_{μ}S^{μ}_{γλ}/2

However, this is not what we see in the article in Wikipedia. There, two more terms are present:

T^{γλ}_{B}= T^{γλ}+ ∂_{μ}(S^{γλμ}+ S^{λγμ}- S^{μλγ})/2

I don't think this is in error, because I saw those extra two terms also in the Relativity book by M. Gasperini /which is a good book, except that sometimes the author skips parts of the proof, clearly overestimating the abilities of an average reader/.

Could someone please tell me how the two extra terms have shown up in the above formula?

Many thanks!

Michael

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# The Belinfante_rosenfeld tensor

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