Generalized divergence of tensor in GR

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Although I've forgotten the proof (and cannot find it in, say, Carroll's book), the following formula holds for the covariant divergence in general relativity:



$$nabla_mu A^mu = frac1sqrt partial_mu left( sqrt A^muright),$$



where $g = det(g_alphabeta)$. I was wondering if this formula holds if $A^mu$ is replaced with a general rank $(n,m)$ tensor



$$T^mu mu_1mu_2 cdots mu_n-1_,,,,,,,,,,,,,,,,,,,nu_1cdots nu_m?$$



If not, could you point me to any references that have divergence formulas for higher rank tensors?










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    It does not. It holds when $A^mu$ is replaced with a general $p$-form (which is a totally antisymmetric $(p,0)$ tensor.
    – Prahar
    Oct 1 at 14:40














up vote
5
down vote

favorite
1












Although I've forgotten the proof (and cannot find it in, say, Carroll's book), the following formula holds for the covariant divergence in general relativity:



$$nabla_mu A^mu = frac1sqrt partial_mu left( sqrt A^muright),$$



where $g = det(g_alphabeta)$. I was wondering if this formula holds if $A^mu$ is replaced with a general rank $(n,m)$ tensor



$$T^mu mu_1mu_2 cdots mu_n-1_,,,,,,,,,,,,,,,,,,,nu_1cdots nu_m?$$



If not, could you point me to any references that have divergence formulas for higher rank tensors?










share|cite|improve this question



















  • 1




    It does not. It holds when $A^mu$ is replaced with a general $p$-form (which is a totally antisymmetric $(p,0)$ tensor.
    – Prahar
    Oct 1 at 14:40












up vote
5
down vote

favorite
1









up vote
5
down vote

favorite
1






1





Although I've forgotten the proof (and cannot find it in, say, Carroll's book), the following formula holds for the covariant divergence in general relativity:



$$nabla_mu A^mu = frac1sqrt partial_mu left( sqrt A^muright),$$



where $g = det(g_alphabeta)$. I was wondering if this formula holds if $A^mu$ is replaced with a general rank $(n,m)$ tensor



$$T^mu mu_1mu_2 cdots mu_n-1_,,,,,,,,,,,,,,,,,,,nu_1cdots nu_m?$$



If not, could you point me to any references that have divergence formulas for higher rank tensors?










share|cite|improve this question















Although I've forgotten the proof (and cannot find it in, say, Carroll's book), the following formula holds for the covariant divergence in general relativity:



$$nabla_mu A^mu = frac1sqrt partial_mu left( sqrt A^muright),$$



where $g = det(g_alphabeta)$. I was wondering if this formula holds if $A^mu$ is replaced with a general rank $(n,m)$ tensor



$$T^mu mu_1mu_2 cdots mu_n-1_,,,,,,,,,,,,,,,,,,,nu_1cdots nu_m?$$



If not, could you point me to any references that have divergence formulas for higher rank tensors?







general-relativity differential-geometry tensor-calculus definition differentiation






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edited Oct 1 at 16:05

























asked Oct 1 at 14:07









Dwagg

520110




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  • 1




    It does not. It holds when $A^mu$ is replaced with a general $p$-form (which is a totally antisymmetric $(p,0)$ tensor.
    – Prahar
    Oct 1 at 14:40












  • 1




    It does not. It holds when $A^mu$ is replaced with a general $p$-form (which is a totally antisymmetric $(p,0)$ tensor.
    – Prahar
    Oct 1 at 14:40







1




1




It does not. It holds when $A^mu$ is replaced with a general $p$-form (which is a totally antisymmetric $(p,0)$ tensor.
– Prahar
Oct 1 at 14:40




It does not. It holds when $A^mu$ is replaced with a general $p$-form (which is a totally antisymmetric $(p,0)$ tensor.
– Prahar
Oct 1 at 14:40










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No, this does not hold in general for higher-rank tensors. The general equation for the divergence of a completely contravariant tensor in terms of a coordinate derivative operator $partial_mu$ is
$$
nabla_mu T^mu nu_1 dots nu_n = partial_mu T^mu nu_1 dots nu_n + Gamma^mu _mu rho T^rho nu_1 dots nu_n + sum_i = 1^n Gamma^nu_i _mu rho T^mu nu_1 dots rho dots nu_n.
$$

We also have the fact that
$$
Gamma^mu _mu rho = frac1sqrt partial_mu sqrt.
$$

Thus,
$$
nabla_mu T^mu nu_1 dots nu_n = frac1sqrt partial_mu left( sqrt T^mu nu_1 dots nu_n right) + sum_i = 1^n Gamma^nu_i _mu rho T^mu nu_1 dots rho dots nu_n.
$$

This last sum will not vanish for a general tensor. However, some or all of the terms may vanish for tensors with a particular symmetry structure. In particular, if $T^mu nu_1 dots nu_n$ is antisymmetric in all of its indices, then any contraction of two of its indices with the symmetric indices of the Christoffel symbols automatically vanishes; and thus the entire sum goes away.



For references as to how to take the covariant derivative of a general tensor, see Chapter 5 of Schutz's A First Course in General Relativity for a coordinate-based approach, or Chapter 3 of Wald's General Relativity for a more general approach.






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  • I believe that for general Christoffel symbols and non-zero $T$, total antisymmetry is the only case where the extra sum vanishes.
    – Void
    Oct 1 at 15:06










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
8
down vote



accepted










No, this does not hold in general for higher-rank tensors. The general equation for the divergence of a completely contravariant tensor in terms of a coordinate derivative operator $partial_mu$ is
$$
nabla_mu T^mu nu_1 dots nu_n = partial_mu T^mu nu_1 dots nu_n + Gamma^mu _mu rho T^rho nu_1 dots nu_n + sum_i = 1^n Gamma^nu_i _mu rho T^mu nu_1 dots rho dots nu_n.
$$

We also have the fact that
$$
Gamma^mu _mu rho = frac1sqrt partial_mu sqrt.
$$

Thus,
$$
nabla_mu T^mu nu_1 dots nu_n = frac1sqrt partial_mu left( sqrt T^mu nu_1 dots nu_n right) + sum_i = 1^n Gamma^nu_i _mu rho T^mu nu_1 dots rho dots nu_n.
$$

This last sum will not vanish for a general tensor. However, some or all of the terms may vanish for tensors with a particular symmetry structure. In particular, if $T^mu nu_1 dots nu_n$ is antisymmetric in all of its indices, then any contraction of two of its indices with the symmetric indices of the Christoffel symbols automatically vanishes; and thus the entire sum goes away.



For references as to how to take the covariant derivative of a general tensor, see Chapter 5 of Schutz's A First Course in General Relativity for a coordinate-based approach, or Chapter 3 of Wald's General Relativity for a more general approach.






share|cite|improve this answer




















  • I believe that for general Christoffel symbols and non-zero $T$, total antisymmetry is the only case where the extra sum vanishes.
    – Void
    Oct 1 at 15:06














up vote
8
down vote



accepted










No, this does not hold in general for higher-rank tensors. The general equation for the divergence of a completely contravariant tensor in terms of a coordinate derivative operator $partial_mu$ is
$$
nabla_mu T^mu nu_1 dots nu_n = partial_mu T^mu nu_1 dots nu_n + Gamma^mu _mu rho T^rho nu_1 dots nu_n + sum_i = 1^n Gamma^nu_i _mu rho T^mu nu_1 dots rho dots nu_n.
$$

We also have the fact that
$$
Gamma^mu _mu rho = frac1sqrt partial_mu sqrt.
$$

Thus,
$$
nabla_mu T^mu nu_1 dots nu_n = frac1sqrt partial_mu left( sqrt T^mu nu_1 dots nu_n right) + sum_i = 1^n Gamma^nu_i _mu rho T^mu nu_1 dots rho dots nu_n.
$$

This last sum will not vanish for a general tensor. However, some or all of the terms may vanish for tensors with a particular symmetry structure. In particular, if $T^mu nu_1 dots nu_n$ is antisymmetric in all of its indices, then any contraction of two of its indices with the symmetric indices of the Christoffel symbols automatically vanishes; and thus the entire sum goes away.



For references as to how to take the covariant derivative of a general tensor, see Chapter 5 of Schutz's A First Course in General Relativity for a coordinate-based approach, or Chapter 3 of Wald's General Relativity for a more general approach.






share|cite|improve this answer




















  • I believe that for general Christoffel symbols and non-zero $T$, total antisymmetry is the only case where the extra sum vanishes.
    – Void
    Oct 1 at 15:06












up vote
8
down vote



accepted







up vote
8
down vote



accepted






No, this does not hold in general for higher-rank tensors. The general equation for the divergence of a completely contravariant tensor in terms of a coordinate derivative operator $partial_mu$ is
$$
nabla_mu T^mu nu_1 dots nu_n = partial_mu T^mu nu_1 dots nu_n + Gamma^mu _mu rho T^rho nu_1 dots nu_n + sum_i = 1^n Gamma^nu_i _mu rho T^mu nu_1 dots rho dots nu_n.
$$

We also have the fact that
$$
Gamma^mu _mu rho = frac1sqrt partial_mu sqrt.
$$

Thus,
$$
nabla_mu T^mu nu_1 dots nu_n = frac1sqrt partial_mu left( sqrt T^mu nu_1 dots nu_n right) + sum_i = 1^n Gamma^nu_i _mu rho T^mu nu_1 dots rho dots nu_n.
$$

This last sum will not vanish for a general tensor. However, some or all of the terms may vanish for tensors with a particular symmetry structure. In particular, if $T^mu nu_1 dots nu_n$ is antisymmetric in all of its indices, then any contraction of two of its indices with the symmetric indices of the Christoffel symbols automatically vanishes; and thus the entire sum goes away.



For references as to how to take the covariant derivative of a general tensor, see Chapter 5 of Schutz's A First Course in General Relativity for a coordinate-based approach, or Chapter 3 of Wald's General Relativity for a more general approach.






share|cite|improve this answer












No, this does not hold in general for higher-rank tensors. The general equation for the divergence of a completely contravariant tensor in terms of a coordinate derivative operator $partial_mu$ is
$$
nabla_mu T^mu nu_1 dots nu_n = partial_mu T^mu nu_1 dots nu_n + Gamma^mu _mu rho T^rho nu_1 dots nu_n + sum_i = 1^n Gamma^nu_i _mu rho T^mu nu_1 dots rho dots nu_n.
$$

We also have the fact that
$$
Gamma^mu _mu rho = frac1sqrt partial_mu sqrt.
$$

Thus,
$$
nabla_mu T^mu nu_1 dots nu_n = frac1sqrt partial_mu left( sqrt T^mu nu_1 dots nu_n right) + sum_i = 1^n Gamma^nu_i _mu rho T^mu nu_1 dots rho dots nu_n.
$$

This last sum will not vanish for a general tensor. However, some or all of the terms may vanish for tensors with a particular symmetry structure. In particular, if $T^mu nu_1 dots nu_n$ is antisymmetric in all of its indices, then any contraction of two of its indices with the symmetric indices of the Christoffel symbols automatically vanishes; and thus the entire sum goes away.



For references as to how to take the covariant derivative of a general tensor, see Chapter 5 of Schutz's A First Course in General Relativity for a coordinate-based approach, or Chapter 3 of Wald's General Relativity for a more general approach.







share|cite|improve this answer












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answered Oct 1 at 14:42









Michael Seifert

13.8k12651




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  • I believe that for general Christoffel symbols and non-zero $T$, total antisymmetry is the only case where the extra sum vanishes.
    – Void
    Oct 1 at 15:06
















  • I believe that for general Christoffel symbols and non-zero $T$, total antisymmetry is the only case where the extra sum vanishes.
    – Void
    Oct 1 at 15:06















I believe that for general Christoffel symbols and non-zero $T$, total antisymmetry is the only case where the extra sum vanishes.
– Void
Oct 1 at 15:06




I believe that for general Christoffel symbols and non-zero $T$, total antisymmetry is the only case where the extra sum vanishes.
– Void
Oct 1 at 15:06

















 

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