I read a paper where there was written that "a Traceless Energy-momentum Tensor implies a massless field", so I did a bit of calculations but I seems not really true, is it true?
So now I'm considering 3 Lagrangian density:
- GR, with $T^{\mu \nu}=-\frac{1}{2}\sqrt{-g}\frac{\delta \mathcal{L}}{\delta g_{\mu \nu}}$
- Spin $1$ field (EM):
$$\mathcal{L}_1=-\frac{1}{4}F^{\rho \sigma}F_{\rho \sigma}\sqrt{-g}$$
$$\Rightarrow T_1^{\mu \nu}=F^{\mu \sigma}F_{\rho\sigma}g^{\rho\nu}-\frac{1}{4}g^{\mu \nu}F^{\rho \sigma}F_{\rho \sigma}$$
- Spin $\frac{1}{2}$ field (Massless Dirac):
$$\mathcal{L}_2=i\bar{\psi}\left(\gamma^{\rho}\partial_{\rho}\right)\psi\sqrt{-g}$$
$$\Rightarrow T_2^{\mu \nu}=ig^{\mu \nu}\bar{\psi}\left(\gamma^{\rho}\partial_{\rho}\right)\psi-i\bar{\psi}\left(\gamma^{\mu}\partial^{\nu}+\gamma^{\nu}\partial^{\mu}\right)\psi$$
- Spin $0$ field (Massless Real Klein-Gordon):
$$\mathcal{L}_3=\frac{1}{2}\partial_{\rho}\phi \partial^{\rho}\phi\sqrt{-g}$$
$$\Rightarrow T_3^{\mu \nu}=\frac{1}{2}g^{\mu \nu}\partial_{\rho}\phi \partial^{\rho}\phi - \partial^{\mu}\phi\partial^{\nu}\phi$$
In 4 Dimension the Trace $T=g_{\mu \nu}T^{\mu \nu}$ is : $$T_1=0$$ $$T_2=2i\bar{\psi}\left(\gamma^{\rho}\partial_{\rho}\right)\psi\neq0$$ $$T_3=\partial_{\rho}\phi \partial^{\rho}\phi\neq0$$
- Classical Field Theory, with $T^{\mu \nu}=\frac{\delta \mathcal{L}}{\partial\left(\partial_\mu \theta \right)}\partial^\nu \theta-\mathcal{L}g^{\mu \nu}$
- Spin $1$ field (EM) ($\theta=A^{\rho}$):
$$\mathcal{L}_1=-\frac{1}{4}F^{\rho \sigma}F_{\rho \sigma}$$
$$\Rightarrow \tilde{T}_1^{\mu \nu}=F^{\mu \sigma}F_{\rho\sigma}g^{\rho\nu}+\frac{1}{4}g^{\mu \nu}F^{\rho \sigma}F_{\rho \sigma}$$
But then in this case the energy-momentum tensor is no more traceless, in fact: $$\tilde{T}_1=2 F^{\rho \sigma}F_{\rho \sigma}$$ (For completeness this is the Simmetrized Belinfante Form, I could have not simmetrised it but I would have obtained $\tilde{T}_1=\frac{3}{2} F^{\rho \sigma}F_{\rho \sigma}\neq 0$ anyway).
- Final Questions:
So what's the meaning of the Trace $\tilde{T}_1$ not being zero in the Classical Field Theory case?
Why $\tilde{T}_1^{\mu \nu}\neq T_1^{\mu \nu}$? Shouldn't they be the same? Do they describe different physics?
So "a Traceless Energy-momentum Tensor implies a massless field" is it true only for the EM field in GR?
