Mixing
How do I write this multiple index tensor equation using Ricci calculus where the 4-gradient is acting on...
$begingroup$
In the context of special relativity I have this to show that this equation is correct: $$partial_mu F^{mu nu}=j^nu$$
To do that I'm trying to take this equation:
$$F^{mu nu}=partial^{mu}A^{nu}-partial^{nu}A^{mu}$$
and take the 4-gradient of it:
$$partial_mu F^{mu nu}=partial_mupartial^{mu}A^{nu}-partial_mupartial^{nu}A^{mu}$$
I'm not sure what to do at this point, I think my main problem is that I see these written here and I'm not sure I know what it means. From what I understand $$partial_mu=bigg({partialover partial t},-vecnablabigg),$$
but I don't know what $partial^{mu}$ or $partial^{nu}$ are equal to. What happens to $A^{nu}$ and $A^{mu}$ when they are acted on by $partial^{mu}$ and $partial^nu$, respectively? Is this even the right approach to prove that $partial_mu F^{munu}=j^nu$?
tensors index-notation
asked Jan 12 at 22:48
matryoshkamatryoshka
5541720
$endgroup$
add a comment |
$begingroup$
In the context of special relativity I have this to show that this equation is correct: $$partial_mu F^{mu nu}=j^nu$$
To do that I'm trying to take this equation:
$$F^{mu nu}=partial^{mu}A^{nu}-partial^{nu}A^{mu}$$
and take the 4-gradient of it:
$$partial_mu F^{mu nu}=partial_mupartial^{mu}A^{nu}-partial_mupartial^{nu}A^{mu}$$
I'm not sure what to do at this point, I think my main problem is that I see these written here and I'm not sure I know what it means. From what I understand $$partial_mu=bigg({partialover partial t},-vecnablabigg),$$
but I don't know what $partial^{mu}$ or $partial^{nu}$ are equal to. What happens to $A^{nu}$ and $A^{mu}$ when they are acted on by $partial^{mu}$ and $partial^nu$, respectively? Is this even the right approach to prove that $partial_mu F^{munu}=j^nu$?
tensors index-notation
asked Jan 12 at 22:48
matryoshkamatryoshka
5541720
$endgroup$
add a comment |
$begingroup$
In the context of special relativity I have this to show that this equation is correct: $$partial_mu F^{mu nu}=j^nu$$
To do that I'm trying to take this equation:
$$F^{mu nu}=partial^{mu}A^{nu}-partial^{nu}A^{mu}$$
and take the 4-gradient of it:
$$partial_mu F^{mu nu}=partial_mupartial^{mu}A^{nu}-partial_mupartial^{nu}A^{mu}$$
I'm not sure what to do at this point, I think my main problem is that I see these written here and I'm not sure I know what it means. From what I understand $$partial_mu=bigg({partialover partial t},-vecnablabigg),$$
but I don't know what $partial^{mu}$ or $partial^{nu}$ are equal to. What happens to $A^{nu}$ and $A^{mu}$ when they are acted on by $partial^{mu}$ and $partial^nu$, respectively? Is this even the right approach to prove that $partial_mu F^{munu}=j^nu$?
tensors index-notation
asked Jan 12 at 22:48
matryoshkamatryoshka
5541720
$endgroup$
In the context of special relativity I have this to show that this equation is correct: $$partial_mu F^{mu nu}=j^nu$$
To do that I'm trying to take this equation:
$$F^{mu nu}=partial^{mu}A^{nu}-partial^{nu}A^{mu}$$
and take the 4-gradient of it:
$$partial_mu F^{mu nu}=partial_mupartial^{mu}A^{nu}-partial_mupartial^{nu}A^{mu}$$
I'm not sure what to do at this point, I think my main problem is that I see these written here and I'm not sure I know what it means. From what I understand $$partial_mu=bigg({partialover partial t},-vecnablabigg),$$
but I don't know what $partial^{mu}$ or $partial^{nu}$ are equal to. What happens to $A^{nu}$ and $A^{mu}$ when they are acted on by $partial^{mu}$ and $partial^nu$, respectively? Is this even the right approach to prove that $partial_mu F^{munu}=j^nu$?
tensors index-notation
tensors index-notation
asked Jan 12 at 22:48
matryoshkamatryoshka
5541720
asked Jan 12 at 22:48
matryoshkamatryoshka
5541720
asked Jan 12 at 22:48
matryoshkamatryoshka
5541720
asked Jan 12 at 22:48
matryoshkamatryoshka
5541720
asked Jan 12 at 22:48
matryoshkamatryoshka
5541720
5541720
add a comment |
add a comment |
1 Answer
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You can lower and raise indices with the metric, which in the case of special relativity is
$$
eta^{mu nu} = pmatrix{1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1} tag{1}
$$
As an example
$$
partial^mu = eta^{mu nu}partial_nu tag{2}
$$
It is then easy to calculate
begin{eqnarray}
require{cancel}
partial^0 &=& eta^{0nu}partial_nu = cancelto{1}{eta^{00}}partial_0 = frac{partial}{partial t} \
partial^i &=& eta^{inu}partial_nu = -frac{partial}{partial x^i} ~~~mbox{for}~~ i = 1,2,3tag{3}
end{eqnarray}
Same applies for the field $F^{munu}$,
$$
F^{munu} = eta^{nubeta}F^{mu}_{;beta} = eta^{mualpha} eta^{nubeta}F_{alphabeta}
$$
answered Jan 12 at 23:40
caveraccaverac
14.6k31130
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You can lower and raise indices with the metric, which in the case of special relativity is
$$
eta^{mu nu} = pmatrix{1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1} tag{1}
$$
As an example
$$
partial^mu = eta^{mu nu}partial_nu tag{2}
$$
It is then easy to calculate
begin{eqnarray}
require{cancel}
partial^0 &=& eta^{0nu}partial_nu = cancelto{1}{eta^{00}}partial_0 = frac{partial}{partial t} \
partial^i &=& eta^{inu}partial_nu = -frac{partial}{partial x^i} ~~~mbox{for}~~ i = 1,2,3tag{3}
end{eqnarray}
Same applies for the field $F^{munu}$,
$$
F^{munu} = eta^{nubeta}F^{mu}_{;beta} = eta^{mualpha} eta^{nubeta}F_{alphabeta}
$$
answered Jan 12 at 23:40
caveraccaverac
14.6k31130
$endgroup$
add a comment |
$begingroup$
You can lower and raise indices with the metric, which in the case of special relativity is
$$
eta^{mu nu} = pmatrix{1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1} tag{1}
$$
As an example
$$
partial^mu = eta^{mu nu}partial_nu tag{2}
$$
It is then easy to calculate
begin{eqnarray}
require{cancel}
partial^0 &=& eta^{0nu}partial_nu = cancelto{1}{eta^{00}}partial_0 = frac{partial}{partial t} \
partial^i &=& eta^{inu}partial_nu = -frac{partial}{partial x^i} ~~~mbox{for}~~ i = 1,2,3tag{3}
end{eqnarray}
Same applies for the field $F^{munu}$,
$$
F^{munu} = eta^{nubeta}F^{mu}_{;beta} = eta^{mualpha} eta^{nubeta}F_{alphabeta}
$$
answered Jan 12 at 23:40
caveraccaverac
14.6k31130
$endgroup$
add a comment |
$begingroup$
You can lower and raise indices with the metric, which in the case of special relativity is
$$
eta^{mu nu} = pmatrix{1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1} tag{1}
$$
As an example
$$
partial^mu = eta^{mu nu}partial_nu tag{2}
$$
It is then easy to calculate
begin{eqnarray}
require{cancel}
partial^0 &=& eta^{0nu}partial_nu = cancelto{1}{eta^{00}}partial_0 = frac{partial}{partial t} \
partial^i &=& eta^{inu}partial_nu = -frac{partial}{partial x^i} ~~~mbox{for}~~ i = 1,2,3tag{3}
end{eqnarray}
Same applies for the field $F^{munu}$,
$$
F^{munu} = eta^{nubeta}F^{mu}_{;beta} = eta^{mualpha} eta^{nubeta}F_{alphabeta}
$$
answered Jan 12 at 23:40
caveraccaverac
14.6k31130
$endgroup$
You can lower and raise indices with the metric, which in the case of special relativity is
$$
eta^{mu nu} = pmatrix{1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1} tag{1}
$$
As an example
$$
partial^mu = eta^{mu nu}partial_nu tag{2}
$$
It is then easy to calculate
begin{eqnarray}
require{cancel}
partial^0 &=& eta^{0nu}partial_nu = cancelto{1}{eta^{00}}partial_0 = frac{partial}{partial t} \
partial^i &=& eta^{inu}partial_nu = -frac{partial}{partial x^i} ~~~mbox{for}~~ i = 1,2,3tag{3}
end{eqnarray}
Same applies for the field $F^{munu}$,
$$
F^{munu} = eta^{nubeta}F^{mu}_{;beta} = eta^{mualpha} eta^{nubeta}F_{alphabeta}
$$
answered Jan 12 at 23:40
caveraccaverac
14.6k31130
answered Jan 12 at 23:40
caveraccaverac
14.6k31130
answered Jan 12 at 23:40
caveraccaverac
14.6k31130
answered Jan 12 at 23:40
caveraccaverac
14.6k31130
14.6k31130
add a comment |
add a comment |
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