# Gravitational Lenses in External Shear Fields

I am reading Massimo Meneghetti's notes on gravitational lenses, available here: http://www.ita.uni-heidelberg.de/~massimo/sub/Lectures/gl_all.pdf

On page 38 he begins discussing embedding a lens in an external shear field sourced by matter in the neighbourhood.

He introduces the following conditions on the potential $$\Psi_{\gamma}$$:

$$\gamma_1 = \frac{1}{2}(\Psi_{11} - \Psi_{22}) = \mathrm{const}$$

$$\gamma_2 = \Psi_{12} = \mathrm{const}$$

$$\kappa = \frac{1}{2}(\Psi_{11} + \Psi_{22}) = \mathrm{const}$$

where the notation $$\Psi_{ij} \equiv \frac{\partial^2 \Psi}{\partial x_i \partial x_j}$$

Since $$\Psi_{11} \pm \Psi_{22} = \mathrm{const}$$, they must each individually be constant. He then says it follows that

$$\Psi_{\gamma} = Cx_1^2 + C'x_2^2 + D x_1x_2 + E$$.

I don't understand how these conditions were used to arrive at this equation. What does this equation tell us about the potential of the external shear field?

Reading over my old questions, I realised I can answer this now (it was really rather simple):

The condition that

$$\Psi_{\gamma} = Cx_1^2 + C'x_2^2 + D x_1x_2 + E$$

follows from the conditions that $$\Psi_{11}$$, $$\Psi_{22}$$, and $$\Psi_{12}$$ are constant by the definition that $$\Psi_{ij} = \frac{\partial^2 \Psi}{\partial x_i \partial x_j}$$. The form of $$\Psi_{\gamma}$$ ensures that all of these components are constant.

$$\Psi_{11} = \frac{\partial^2 \Psi_{\gamma}}{\partial x_1^2} = 2C$$

$$\Psi_{22} = \frac{\partial^2 \Psi_{\gamma}}{\partial x_2^2} = 2C'$$

$$\Psi_{12} = \frac{\partial^2 \Psi_{\gamma}}{\partial x_1 \partial x_2} = D$$