How measure spins along the same axis with relativity Suppose that a pion decays into an electron and a positron. Now let the particles travel a large distance so that they are lightyears apart. The idea of the EPR paradox is that when you measure their spins along the same axis, then one will give result $0.5$ and the other one will give result $-0.5$. ‘Along the same axis’ would mean that the two axes (that is, the axis that we use to measure the electron and the axis that we use to measure the positron) are parallel. However, according to relativity, parallel lines are not really defined. So what is then the meaning of measuring along the same axis?
 A: In Quantum Mechanics, one can use Pauli matrices as a matrix representation of spin-1/2 operators. This is of course done in a standard Newtonian or special-relativistic setting. In a curved 3+1 dimensional space-time, we can think of Pauli matrices as a special case of Infeld-Van der Waerden symbols (${\sigma_{AA'}}^a$). Here, we have the isomorphism:
$$T_pM\cong_{iso}\mathbb{S}\otimes\mathbb{S}'$$for each $p\in M$, where $\mathbb{S}'$ and $\mathbb{S}$ are primed and un-primed spin space. The ${\sigma_{AA'}}^a$'s allow us to go from spin space to tangent vector space:
$$v^a={\sigma_{AA'}}^av^A\bar{v}^{A'}$$
$v^a\in T_pM$ being the null vector. Only if we choose a Minkowski tetrad at the point $p$, we will see that
$${\sigma_{AA'}}^a=\frac{1}{\sqrt{2}}(I,\vec{\sigma})$$
Thus $\sigma^i$'s correspond to the spatial directions in the tangent space only.
In flat space-time, we can also think of the spin operators $S^a$ as a linear super-position of Pauli-Lubanski spin vector and they form the little group of the full Poincare symmetry group. In curved space-time, we don't have the Poincare group as our isometry group, but we can still do similar calculations in the tangent space.
Overall we see that indices in spin operators have meaningful interpretation strictly in the tangent space. In practice, if space-time is reasonable flat (like on surface of earth), we can extend our local Minkowski co-ordinate to a larger surrounding and still comment about measurement of spin along the same axis.
