## Deriving the massless Dirac equation using differential geometry

Beginning with the line element:

$$ ds^2 = (c dt)^2 - dr^2$$

Taking the square root of the metric and using the gamma matrices:

$$ ds = \gamma^0 c dt - \gamma^i \cdot dr_i$$

For time like geodesics: $ ds = 0$. Taking the dual basis:

$$ 0 = \frac{1}{c}\gamma^0 \partial _t - \gamma^i \cdot \nabla_i$$

To convice yourself of the above formula use $\langle dx^i | \partial_j \rangle = \delta^{i}_j $

Multiplying $i \hbar$ both sides and the wave function we obtain:

$$ 0=i\hbar \gamma^\mu \partial_\mu \psi $$

## Using the same technique for static spherical symmetrical line element:

$$ds^2 = -(1-\frac{2M}{r}) dt^2 + (1-\frac{2M}{r})^{-1} dr^2 $$

Taking square root and setting $ds = 0$ (as done above):

$$ 0 = \gamma^0 dt - \gamma^i \cdot (1-\frac{2M}{r})^{-1} dr $$

Again using the dual basis and multiplying $\psi$:

$$ 0 = \gamma^0 \partial_t \psi- \gamma^i \cdot (1-\frac{2M}{r}) \nabla \psi $$

Multiplying $i \hbar$ we have:

$$ i \gamma^0 \hat E \psi = \gamma^i\cdot (1-\frac{2M}{r}) \nabla \psi$$

### Question

I then saw the wikipedia entry and saw a different equation. Why is my method wrong?

P.S: I am a postgraduate who tried his hand at quantum field theory at curved spacetime and genuinely want to know why I'm wrong? (I am not advocating my position)