Wednesday, April 24, 2013

invariants of the lensing Jacobian

the lensing Jacobian $\mathcal{A}=\partial\vec{\beta}/\partial\vec{\theta}$ describes how the lensing deflection angle $\beta$ changes with position $\theta$. how many invariants (under orthogonal transformations) of $\mathcal{A}$ can you construct? what are they and what is their physical interpretation?

bonus question: can you show that $\exp(x)^{\sin(x)}$ is an even function and that the derivative is an odd function?


  1. CQW: the two invariants are the trace $\mathrm{tr}(\mathcal{A})$ and the determinant $\mathrm{det}(\mathcal{A})$ (which itself can be constructed in this case from the trace of the squared matrix). the trace corresponds to twice the lensing convergence $\kappa$ and the determinant is the inverse magnification $\mu$.

  2. bonus question: the function is even because it's equal to $\exp(x\sin(x))$, and $x\sin(x)$ is even.