Theoretical Methods

Derivation of Gravitational Lensing equation

Brief Overview of Special and General Relativity

We are now going to derive equation 2 that I mentioned in the introduction. To start, we first need to realize an important principle of General Relativity - In a locally inertial frame, Special Theory of Relativity(STR) applies.

In STR, a point is described using four coordinates - $(ct, x, y, z)$ where $t$ is the time and $(x,y,z)$are the position coordinates measured in the frame of that point and $c$ is the speed of light. Using tensor notation, any trajectory of a particle can be described by a four vector $\xi^\mu(\tau) \equiv \xi^\mu(ct,\vec{x})$, where $\tau$ is the proper time of the particle, which is a function of the coordinates. In particular, if this particle is a free particle, just like in Lagrangian mechanics, the equation of motion of the particle is described by

(3) $\frac{d^2 \xi^\mu(\tau)}{d\tau^2} = 0$

The trajectory of this particle in the $(ct,\vec{x})$ space is simply a straight line. Note that we are still in the regime of STR. We also take the flat space metric to have $\eta = diag(1,-1,-1,-1)$ convention. A metric is simply a measure of the distance between any two points in space time. The distance between two spacetime points $(ct, x, y, z)$ and $(ct+cdt, x+dx,y+dy,z+dz)$ is defined as:

(4) $ds^2 = \begin{pmatrix}

cdt & dx & dy & dz\\ \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} cdt\\ dx\\ dy\\ dz \end{pmatrix}$

(5) $ds^2 = c^2dt^2-dx^2-dy^2-dz^2$

This metric corresponds to a spacetime that is ‘flat’, i.e. one with no curvature. Einstein’s revelation was that a massive body can change the metric, leading to a different measure of distance, implying the spacetime has ‘curved’.

Einstein Crosses

The image displayed at the introduction is famously known as the Einstein Cross. They are formed when the intervening lens galaxy or galaxy cluster is dense at some point(s). In the case when the matter is distributed evenly, we observed long stretched arcs of the lensed object. Since this report is focused on Einstein crosses, we will model a metric which has a dense mass at some point(s).

Schwarzschild Metric

The metric around a point mass is called the Schwarzschild metric, and is given here without derivation, in polar coordinates, centered around the point mass:

(6) $ds^2 = \begin{pmatrix}

cdt & dr & rd\theta & rsin\theta d\phi\\ \end{pmatrix} \begin{pmatrix} (1-\frac{2GM}{c^2r}) & 0 & 0 & 0\\ 0 & -(\frac{1}{1-\frac{2GM}{c^2r}}) & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} cdt\\ dr\\ rd\theta\\ rsin\theta d\phi \end{pmatrix}$