Traceless energy-momentum tensor

Question

I don't think it is clear to me what exactly is the physical meaning of the energy-momentum tensor being traceless.

Answer 1

The physical significance of a traceless energy-momentum tensor or $\text{Tr}(T_{ab}) = 0$ means that the addition of the diagonal terms of the matrix is $0$.

Now, the energy momentum tensor carries its identity with:

$T_{00}$ = $\text{energy density}$, and ${T_{\alpha\alpha}}$ = $\text{pressure}$

And the sum of the diagonal terms is $0$. Or,

$$T_{00} + T_{11} + T_{22}+T_{22} = 0$$

as summation, means that internal energy of the system, and internal pressure, do not contribute to the bulk motion of the system, as they cancel out each other's contribution, which indeed is a very significant property of the conformal invariant object, where, if the property $Tr(T_{ab}) = 0$ holds, it holds true for any metric transformation of type

$$g_{ab} \rightarrow \Omega^2g_{ab}$$