Higher order covariant derivative notation
$begingroup$
The covariant derivative of a tensor can be noted equivalently as
$$nabla_b X^a qquad mathrm{and} qquad X^a_{; ;b}$$
If we want to express a higher order covariant derivative
$$nabla_c nabla_b X^a,$$
here it is clear that $nabla_b$ acts first, in the other notation which one should write?
$$X^a_{; ;bc} qquad mathrm{or} qquad X^a_{; ;cb}$$
I guess is the first one, because $(X^a_{; ;b})_{;c}$ resembles more to $X^a_{; ;bc}$
Thank you.
differential-geometry tensors general-relativity
asked Dec 8 '18 at 16:32
AlexAlex
63
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add a comment |
$begingroup$
The covariant derivative of a tensor can be noted equivalently as
$$nabla_b X^a qquad mathrm{and} qquad X^a_{; ;b}$$
If we want to express a higher order covariant derivative
$$nabla_c nabla_b X^a,$$
here it is clear that $nabla_b$ acts first, in the other notation which one should write?
$$X^a_{; ;bc} qquad mathrm{or} qquad X^a_{; ;cb}$$
I guess is the first one, because $(X^a_{; ;b})_{;c}$ resembles more to $X^a_{; ;bc}$
Thank you.
differential-geometry tensors general-relativity
asked Dec 8 '18 at 16:32
AlexAlex
63
$endgroup$
$begingroup$
This first one, essentially for your reasoning.
$endgroup$
– Matt
Dec 11 '18 at 8:37
add a comment |
$begingroup$
The covariant derivative of a tensor can be noted equivalently as
$$nabla_b X^a qquad mathrm{and} qquad X^a_{; ;b}$$
If we want to express a higher order covariant derivative
$$nabla_c nabla_b X^a,$$
here it is clear that $nabla_b$ acts first, in the other notation which one should write?
$$X^a_{; ;bc} qquad mathrm{or} qquad X^a_{; ;cb}$$
I guess is the first one, because $(X^a_{; ;b})_{;c}$ resembles more to $X^a_{; ;bc}$
Thank you.
differential-geometry tensors general-relativity
asked Dec 8 '18 at 16:32
AlexAlex
63
$endgroup$
The covariant derivative of a tensor can be noted equivalently as
$$nabla_b X^a qquad mathrm{and} qquad X^a_{; ;b}$$
If we want to express a higher order covariant derivative
$$nabla_c nabla_b X^a,$$
here it is clear that $nabla_b$ acts first, in the other notation which one should write?
$$X^a_{; ;bc} qquad mathrm{or} qquad X^a_{; ;cb}$$
I guess is the first one, because $(X^a_{; ;b})_{;c}$ resembles more to $X^a_{; ;bc}$
Thank you.
differential-geometry tensors general-relativity
differential-geometry tensors general-relativity
asked Dec 8 '18 at 16:32
AlexAlex
63
asked Dec 8 '18 at 16:32
AlexAlex
63
asked Dec 8 '18 at 16:32
AlexAlex
63
asked Dec 8 '18 at 16:32
AlexAlex
63
asked Dec 8 '18 at 16:32
AlexAlex
63
63
$begingroup$
This first one, essentially for your reasoning.
$endgroup$
– Matt
Dec 11 '18 at 8:37
add a comment |
$begingroup$
This first one, essentially for your reasoning.
$endgroup$
– Matt
Dec 11 '18 at 8:37
$begingroup$
This first one, essentially for your reasoning.
$endgroup$
– Matt
Dec 11 '18 at 8:37
$begingroup$
This first one, essentially for your reasoning.
$endgroup$
– Matt
Dec 11 '18 at 8:37
add a comment |
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$begingroup$
This first one, essentially for your reasoning.
$endgroup$
– Matt
Dec 11 '18 at 8:37