# Why symmetrization of energy-momentum tensor doesn't add additional term to Lagrangian density? [duplicate]

I am self-studying the book “James H. Luscombe, Core Principles of Special and General Relativity”. In “CHAPTER 9 : Energy-momentum of fields” of the book, it starts by introducing Noether’s theorem and then shows that invariance under spacetime translations implies conservation of field energy and momentum: $$T^{μν}≡g^{μν} L-(∂^μ ϕ_α)\frac{∂L}{∂(∂_ν ϕ_α)}$$ $$∂_ν T^{μν}=0$$ in which $$T^{μν}$$ is canonical energy-momentum tensor, and then shows that invariance under Lorentz transformations implies conservation of angular momentum current density tensor: $$J^{ρτμ}≡x^ρ T^{τμ}-x^t T^{ρμ}+ϕ^τ \frac{∂L}{∂(∂_μ ϕ_ρ )} -ϕ^ρ \frac{∂L}{(∂(∂_μ ϕ_τ))}$$ $$∂_μ J^{ρτμ}=0.$$ Then the book groups terms of $$J^{ρτμ}$$ in two part

1. orbital momentum density $$L^{ρτμ}=x^ρ T^{τμ}-x^t T^{ρμ}$$ and

2. spin density $$S^{ρτμ}=\frac{∂L}{∂(∂_μ ϕ_ρ )} -ϕ^ρ\frac{∂L}{(∂(∂_μ ϕ_τ))}$$.

Then the book define an equivalent symmetric energy-momentum tensor $$T^{μν}→θ^{μν}≡T^{μν}+\frac{1}{2} ∂_λ (S^{μλν}+S^{νλμ}-S^{μνλ})$$ to write $$J^{ρτμ}$$ as $$J^{ρτμ}→J_θ^{ρτμ}≡x^ρ θ^{τμ}-x^t θ^{ρμ}=J^{ρτμ}+∂_λ (x^ρ ψ^{τλμ}-x^τ ψ^{ρλμ}).$$

My question is "Does symmetrization of energy-momentum tensor add additional term to Lagrangian density? Why?"

Simple answer is that it doesn’t change Lagrangian density because conservation laws remain unchanged. But I think this is not enough!

• Commented Mar 7 at 15:46
• Other possible duplicates: physics.stackexchange.com/q/119838/2451 and links therein. Commented Mar 7 at 17:18
• To my mind, the symmetrizing of the stress-energy tensor is just an exploitation of the freedom of choice generated by the requirement that $\partial_\nu T^{\mu\nu}=0$. The process is unrelated to modification of the Lagrangian. Commented Mar 7 at 17:34
• Yes, I read some textbooks and saw above links but I couldn't find my answer. During symetrizing energy-momentum tensor we show that by transforming $T^{\mu\nu}$ to $\theta^{\mu\nu}$, physical laws I.e. $\partial_{\mu}\theta^{\mu\nu}=0$ holded but how can we prove that we didn't change Lagrangian density of field. Commented Mar 7 at 17:49