- #1
cozycoz
- 13
- 1
Riemann tensor is defined mathematically like this:
##∇_k∇_jv_i-∇_j∇_kv_i={R^l}_{ijk}v_l##
Using covariant derivative formula for covariant tensors and covariant vectors. which are
##∇_av_b=∂_av_b-{Γ^c}_{ab}v_c##
##∇_aT_{bc}=∂_av_{bc}-{Γ^d}_{ac}v_{db}-{Γ^d}_{ab}v_{dc} ##,
I got these:
##∇_k∇_jv_i=∂_k∂_jv_i-∂_k{Γ^m}_{ji}v_m-{Γ^m}_{ji}∂_kv_m-{Γ^l}_{ki}∂_lv_j+{Γ^l}_{ki}{Γ^m}_{lj}v_m-{Γ^l}_{kj}∂_lv_i+{Γ^l}_{kj}{Γ^m}_{li}v_m##(I'll call each term by its number : ##∂_k∂_jv_i## is "1" because it's the first term)
##∇_j∇_kv_i=∂_j∂_kv_i-∂_j{Γ^m}_{ki}v_m-{Γ^m}_{ki}∂_jv_m-{Γ^l}_{ji}∂_lv_k+{Γ^l}_{ji}{Γ^m}_{lk}v_m-{Γ^l}_{jk}∂_lv_i+{Γ^l}_{jk}{Γ^m}_{li}v_m##
Then 1, 3, 4, 6, 7 terms all vanish when we substract the lower from the upper' according to my professor.
Especially he noted that 3rd term vanishes because it's basically the same when you just exchange the indices j⇔k and add up all m's.
But then isn't it also the case for the second term?In 2nd
I see no reason why I can't apply the logic for 3rd to 2nd...could you help me please
##∇_k∇_jv_i-∇_j∇_kv_i={R^l}_{ijk}v_l##
Using covariant derivative formula for covariant tensors and covariant vectors. which are
##∇_av_b=∂_av_b-{Γ^c}_{ab}v_c##
##∇_aT_{bc}=∂_av_{bc}-{Γ^d}_{ac}v_{db}-{Γ^d}_{ab}v_{dc} ##,
I got these:
##∇_k∇_jv_i=∂_k∂_jv_i-∂_k{Γ^m}_{ji}v_m-{Γ^m}_{ji}∂_kv_m-{Γ^l}_{ki}∂_lv_j+{Γ^l}_{ki}{Γ^m}_{lj}v_m-{Γ^l}_{kj}∂_lv_i+{Γ^l}_{kj}{Γ^m}_{li}v_m##(I'll call each term by its number : ##∂_k∂_jv_i## is "1" because it's the first term)
##∇_j∇_kv_i=∂_j∂_kv_i-∂_j{Γ^m}_{ki}v_m-{Γ^m}_{ki}∂_jv_m-{Γ^l}_{ji}∂_lv_k+{Γ^l}_{ji}{Γ^m}_{lk}v_m-{Γ^l}_{jk}∂_lv_i+{Γ^l}_{jk}{Γ^m}_{li}v_m##
Then 1, 3, 4, 6, 7 terms all vanish when we substract the lower from the upper' according to my professor.
Especially he noted that 3rd term vanishes because it's basically the same when you just exchange the indices j⇔k and add up all m's.
But then isn't it also the case for the second term?In 2nd
##∂_j{Γ^m}_{ki}v_m##
##∂_k{Γ^m}_{ji}v_m## ,
And in 3rd##∂_k{Γ^m}_{ji}v_m## ,
##{Γ^m}_{ki}∂_jv_m##
##{Γ^m}_{ji}∂_kv_m##
(They are the same)
##{Γ^m}_{ji}∂_kv_m##
(They are the same)
I see no reason why I can't apply the logic for 3rd to 2nd...could you help me please
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