An elementary definition of the stress-energy tensor $T_{\mu\nu}$ in terms of the Lagrangian for flat spacetime is $$T^{\mu\nu}=\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi)}(\partial^\nu\phi)-\eta^{\mu\nu}\mathcal{L}.$$ For translational symmetries of spacetime both $T^{\mu\nu}$ is conserved: $$\partial_\mu T^{\mu\nu}=0.$$
But $T_{\mu\nu}$ is not always automatically symmetric in the indices $\mu,\nu$. Therefore, one considers a modified stress-energy tensor $\Theta_{\mu\nu}$ defined as $$\Theta^{\mu\nu}:=T^{\mu\nu}+\partial_\kappa A^{\kappa\mu\nu}$$ where $A^{\kappa\mu\nu}$ is an arbitrary tensor antisymmetric in the indices $\kappa,\mu$. This too satifies $$\partial_\mu\Theta^{\mu\nu}=0.$$
What is the need of defining a symmetric stress-energy tensor? What happens if I continue to work with $T^{\mu\nu}$ instead of the modified tensor $\Theta^{\mu\nu}$?
It is also unclear how one would choose the tensor $A^{\kappa\mu\nu}$ uniquely to symmetrize $T^{\mu\nu}$ and find a unique $\Theta^{\mu\nu}$.
