Mixing
tensor transformation
$begingroup$
I don't understand a passage about tensor transformations in the book I'm reading. A tensor obeys the transformation law:
If $A_{munu}' = b_{mu alpha }b_{tau alpha } A_{munu} $
Where the $b_{munu}$ terms come out considering a coordinate transformation between two coordinates system S and S':
$x_nu' = alpha_nu + sum_alpha b_{nu alpha } x_alpha$
$Delta x_nu' = sum_alpha b_{nu alpha } x_alpha$
and this equation holds:
$sum limits _{nu }b_{nu alpha }b_{nu beta }=delta _{alpha beta } quad(4)$
The books says that $delta _{alpha beta }$ is a tensor and that the following equation holds:
$A_{munu}' = b_{mu alpha }b_{tau alpha }=delta _{alpha beta } $
I don't understand how $A_{munu}$ has disappeared and how to derive it.
It is on paragraph 16: https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_1
linear-transformations tensors
asked Jan 27 at 9:43
Giuliano MalatestaGiuliano Malatesta
295
$endgroup$
add a comment |
$begingroup$
I don't understand a passage about tensor transformations in the book I'm reading. A tensor obeys the transformation law:
If $A_{munu}' = b_{mu alpha }b_{tau alpha } A_{munu} $
Where the $b_{munu}$ terms come out considering a coordinate transformation between two coordinates system S and S':
$x_nu' = alpha_nu + sum_alpha b_{nu alpha } x_alpha$
$Delta x_nu' = sum_alpha b_{nu alpha } x_alpha$
and this equation holds:
$sum limits _{nu }b_{nu alpha }b_{nu beta }=delta _{alpha beta } quad(4)$
The books says that $delta _{alpha beta }$ is a tensor and that the following equation holds:
$A_{munu}' = b_{mu alpha }b_{tau alpha }=delta _{alpha beta } $
I don't understand how $A_{munu}$ has disappeared and how to derive it.
It is on paragraph 16: https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_1
linear-transformations tensors
asked Jan 27 at 9:43
Giuliano MalatestaGiuliano Malatesta
295
$endgroup$
$begingroup$
That is not the transformation law for tensors. With or without the Einstein summation convention, it makes no sense. (And given that you explicitly sum elsewhere, I assume you are not using the summation convention.) You have subscripts $alpha$ and $tau$ on the right that do not appear on the left (the summation convention would sum over $alpha$, explaining its disappearence, but it would also sum over $mu$, making it appear only on the left). I suggest you go study the actual tensor transformation law before attempting this book.
$endgroup$
– Paul Sinclair
Jan 27 at 17:22
add a comment |
$begingroup$
I don't understand a passage about tensor transformations in the book I'm reading. A tensor obeys the transformation law:
If $A_{munu}' = b_{mu alpha }b_{tau alpha } A_{munu} $
Where the $b_{munu}$ terms come out considering a coordinate transformation between two coordinates system S and S':
$x_nu' = alpha_nu + sum_alpha b_{nu alpha } x_alpha$
$Delta x_nu' = sum_alpha b_{nu alpha } x_alpha$
and this equation holds:
$sum limits _{nu }b_{nu alpha }b_{nu beta }=delta _{alpha beta } quad(4)$
The books says that $delta _{alpha beta }$ is a tensor and that the following equation holds:
$A_{munu}' = b_{mu alpha }b_{tau alpha }=delta _{alpha beta } $
I don't understand how $A_{munu}$ has disappeared and how to derive it.
It is on paragraph 16: https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_1
linear-transformations tensors
asked Jan 27 at 9:43
Giuliano MalatestaGiuliano Malatesta
295
$endgroup$
I don't understand a passage about tensor transformations in the book I'm reading. A tensor obeys the transformation law:
If $A_{munu}' = b_{mu alpha }b_{tau alpha } A_{munu} $
Where the $b_{munu}$ terms come out considering a coordinate transformation between two coordinates system S and S':
$x_nu' = alpha_nu + sum_alpha b_{nu alpha } x_alpha$
$Delta x_nu' = sum_alpha b_{nu alpha } x_alpha$
and this equation holds:
$sum limits _{nu }b_{nu alpha }b_{nu beta }=delta _{alpha beta } quad(4)$
The books says that $delta _{alpha beta }$ is a tensor and that the following equation holds:
$A_{munu}' = b_{mu alpha }b_{tau alpha }=delta _{alpha beta } $
I don't understand how $A_{munu}$ has disappeared and how to derive it.
It is on paragraph 16: https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_1
linear-transformations tensors
linear-transformations tensors
asked Jan 27 at 9:43
Giuliano MalatestaGiuliano Malatesta
295
asked Jan 27 at 9:43
Giuliano MalatestaGiuliano Malatesta
295
edited Jan 28 at 8:27
Giuliano Malatesta
asked Jan 27 at 9:43
Giuliano MalatestaGiuliano Malatesta
295
asked Jan 27 at 9:43
Giuliano MalatestaGiuliano Malatesta
295
asked Jan 27 at 9:43
Giuliano MalatestaGiuliano Malatesta
295
295
$begingroup$
That is not the transformation law for tensors. With or without the Einstein summation convention, it makes no sense. (And given that you explicitly sum elsewhere, I assume you are not using the summation convention.) You have subscripts $alpha$ and $tau$ on the right that do not appear on the left (the summation convention would sum over $alpha$, explaining its disappearence, but it would also sum over $mu$, making it appear only on the left). I suggest you go study the actual tensor transformation law before attempting this book.
$endgroup$
– Paul Sinclair
Jan 27 at 17:22
add a comment |
$begingroup$
That is not the transformation law for tensors. With or without the Einstein summation convention, it makes no sense. (And given that you explicitly sum elsewhere, I assume you are not using the summation convention.) You have subscripts $alpha$ and $tau$ on the right that do not appear on the left (the summation convention would sum over $alpha$, explaining its disappearence, but it would also sum over $mu$, making it appear only on the left). I suggest you go study the actual tensor transformation law before attempting this book.
$endgroup$
– Paul Sinclair
Jan 27 at 17:22
$begingroup$
That is not the transformation law for tensors. With or without the Einstein summation convention, it makes no sense. (And given that you explicitly sum elsewhere, I assume you are not using the summation convention.) You have subscripts $alpha$ and $tau$ on the right that do not appear on the left (the summation convention would sum over $alpha$, explaining its disappearence, but it would also sum over $mu$, making it appear only on the left). I suggest you go study the actual tensor transformation law before attempting this book.
$endgroup$
– Paul Sinclair
Jan 27 at 17:22
$begingroup$
That is not the transformation law for tensors. With or without the Einstein summation convention, it makes no sense. (And given that you explicitly sum elsewhere, I assume you are not using the summation convention.) You have subscripts $alpha$ and $tau$ on the right that do not appear on the left (the summation convention would sum over $alpha$, explaining its disappearence, but it would also sum over $mu$, making it appear only on the left). I suggest you go study the actual tensor transformation law before attempting this book.
$endgroup$
– Paul Sinclair
Jan 27 at 17:22
add a comment |
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$begingroup$
That is not the transformation law for tensors. With or without the Einstein summation convention, it makes no sense. (And given that you explicitly sum elsewhere, I assume you are not using the summation convention.) You have subscripts $alpha$ and $tau$ on the right that do not appear on the left (the summation convention would sum over $alpha$, explaining its disappearence, but it would also sum over $mu$, making it appear only on the left). I suggest you go study the actual tensor transformation law before attempting this book.
$endgroup$
– Paul Sinclair
Jan 27 at 17:22