Deriving the massless Dirac equation using differential geometry
Beginning with the line element:
ds2=(cdt)2−dr2
Taking the square root of the metric and using the gamma matrices:
ds=γ0cdt−γi⋅dri
For time like geodesics: ds=0. Taking the dual basis:
0=1cγ0∂t−γi⋅∇i
To convice yourself of the above formula use ⟨dxi|∂j⟩=δij
Multiplying iℏ both sides and the wave function we obtain:
0=iℏγμ∂μψ
Using the same technique for static spherical symmetrical line element:
ds2=−(1−2Mr)dt2+(1−2Mr)−1dr2
Taking square root and setting ds=0 (as done above):
0=γ0dt−γi⋅(1−2Mr)−1dr
Again using the dual basis and multiplying ψ:
0=γ0∂tψ−γi⋅(1−2Mr)∇ψ
Multiplying iℏ we have:
iγ0ˆEψ=γi⋅(1−2Mr)∇ψ
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)