Image parity in gravitational lensing ($\mathrm{sgn}\det A$)

Image parity in gravitational lensing is the sign of the determinant of the source-from-image Jacobian matrix $A=\partial \mathbf{\beta }/\partial \mathbf{\theta }$. A positive sign preserves the local handedness of a small source pattern; a negative sign reverses it. Negative signed lensing magnification therefore means a reflected orientation, not negative received light. This is a two-dimensional property: the derivative of a signed radial equation alone does not determine it.

For a point-mass gravitational lens, differentiate $\mathbf{\beta }=(1-{\theta }_E^2/r^2)\mathbf{\theta }$, where $r=|\mathbf{\theta }|$. The radial and tangential eigenvalues of $A$ areConsequently the outer member of the two images of a point-mass lens has positive image parity in gravitational lensing; the inner member has negative image parity in gravitational lensing. At $r={\theta }_E$ the tangential eigenvalue vanishes: this is the tangential critical curve of an axisymmetric lens, and its gravitational-lensing caustic is the aligned source point. The signed lensing magnification is $\mu =1/\det A$, but measured total light uses $|{\mu }_{+}|+|{\mu }_{-}|$.