I have been studying the basics of general relativity with Hartle's Gravity. He presents the geodesic equation as
d2xμds2=−Γμαβdxαdsdxβds
However, in reading Padmanabhan's Gravitation, he says that the equation
0=kα∇αkβ
is the geodesic equation for a wave vector k defined as kα=∇αψ, where ψ is just a scalar function.
How do these two definitions of the geodesic equation represent the same thing? They do not look at all alike. In fact, if I try to work it out, I get
0=kα(∂kβ∂xα−Γδβγkδ)
0=kα∂kβ∂xα−Γδβγkδkα
For one thing, the xα is not supposed to be on the bottom of the derivative! If kα is supposed to satisfy the geodesic equation, I expected this to look like the first equation I wrote.
Answer
In short, the first equation you wrote is in component form where curves are parametrized as xμ(s), while the second equation is more general. Now let's flesh out the details.
What does it mean for a curve to be a geodesic? Intuitively it has to be straightest curve possible in curved spacetime. The way you do that is to propagate the tangent vector Tα of a curve C along itself! That yields the coordinate independent geodesic equation: Tα∇αTβ=0
Then you can write this tensor equation in component form, by choosing a coordinate system and its associated Christoffel symbol. The covariant derivative is expressed as: ∇αTβ=∂αTβ+ΓβαγTγ
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