- #1
Apashanka
- 429
- 15
For the simple case of a 2-D curve in polar coordinated (r,θ) parametrised by λ (length along the curve).
At any λ the tangent vector components are V1=dr(λ)/dλ along ##\hat r## and V2=dθ(λ)/dλ along ##\hat θ##.
The non-zero christoffel symbol are Γ122 and Γ212.
From covariant derivative
(1)∇1V1=∂rV1
(2)∇2V2=∂θV2+Γ212V1.
(3)∇2V1=∂θV1+Γ122V2
(4)∇1V2=∂rV2+Γ212V2
For this curve to be geodesic V1=1 and V2=0
And (1),(3) and (4) becomes 0 and (2)≠0.
But for a geodesic the covariant derivatives of tangent vectors are 0.
Am I missing something??
I am trying to interpret physically by taking these examples.
Thank you
At any λ the tangent vector components are V1=dr(λ)/dλ along ##\hat r## and V2=dθ(λ)/dλ along ##\hat θ##.
The non-zero christoffel symbol are Γ122 and Γ212.
From covariant derivative
(1)∇1V1=∂rV1
(2)∇2V2=∂θV2+Γ212V1.
(3)∇2V1=∂θV1+Γ122V2
(4)∇1V2=∂rV2+Γ212V2
For this curve to be geodesic V1=1 and V2=0
And (1),(3) and (4) becomes 0 and (2)≠0.
But for a geodesic the covariant derivatives of tangent vectors are 0.
Am I missing something??
I am trying to interpret physically by taking these examples.
Thank you