Goldstein's derivation of Noether's theorem

Question

This is a followup to my previoucs question: Translation invariance Noether's equation

In Goldstein's derivation of the Noether's theorem in chapter 13,

we have the infinitesimal transformation $$x'^\mu \rightarrow x^{\mu} + \delta x^{\mu}$$ $$\eta_\rho'(x'^\mu)=\eta_\rho(x^\mu)+\delta\eta_\rho(x^\mu).$$ The Lagrangian becomes $$\tag{13.129} \mathcal{L}(\eta_\rho(x^\mu),\eta_{\rho,\nu}(x^\mu),x^\mu)\rightarrow \mathcal{L}'(\eta'_\rho(x'^\mu),\eta'_{\rho,\nu}(x'^\mu),x'^\mu)$$

My question is: are the two sides of 13.129 just equal?

If that is the case then, in $$\tag{13.133} \int_{\Omega}\mathcal{L}(\eta_\rho(x^\mu),\eta_{\rho,\nu}(x^\mu),x^\mu)d^4x=\int_{\Omega'}\mathcal{L}'(\eta'_\rho(x'^\mu),\eta'_{\rho,\nu}(x'^\mu),x'^\mu)d^4x'$$

Since both sides of 13.133 are equal, so all we need to satisfy 13.133 is that the Jacobian from $x^\mu$ to $x'^\mu$ is 1?

Answer 1

Yes, those are equal and moreover, you do not even need the Jacobian from $x^{\mu}$ to $x^{'\mu}$, because of the fact that the authors, when they write $dx^4$, they actually mean the invariant integration measure $\sqrt{|g|}dx^0dx^1dx^2dx^3$.

You can verify yourself that the aforementioned integration measure is invariant under general coordinate transformations.

So, the following two equations hold separately:

1. $\mathcal{L}' \Big(\phi'(x^{'\mu}),\partial'_{\rho}\phi(x^{'\mu}),x^{'\mu}\Big)= \mathcal{L} \Big(\phi(x^{\mu}),\partial_{\rho}\phi(x^{\mu}),x^{\mu}\Big)$

2. $dx^{'4}=\sqrt{g}dx^{'0}dx^{'1}dx^{'2}dx^{'3}=\sqrt{g}dx^{0}dx^{1}dx^{2}dx^{3}=dx^4$ where $\sqrt{g}=\sqrt{|\text{det}(g)|}$