Is the variation of the metric ##\delta g_{\mu\nu}## a tensor?

[Kostik]

TL;DR

Two contradictory equations are shown. Where is the flaw?

From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by $g_{\alpha\beta}$ to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation $\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}$ should be a tensor; therefore, $$\delta g_{\beta\nu} = g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \quad\qquad(**)$$ Which equation is correct, $(*)$ or $(**)$?

 

[Kostik]

I think I found the (or an) answer. The relations $$\delta g^{\mu\nu} = -g^{\mu\rho}g^{\nu\sigma}\delta g_{\rho\sigma} \quad , \qquad \delta g_{\mu\nu} = -g_{\mu\rho} g_{\nu\sigma} \delta g^{\rho\sigma} \qquad(*)$$ appear to indicate that $\delta g_{\mu\nu}$ and $\delta g^{\mu\nu}$ are not tensors, since their indices are not raised and lowered in the usual way by the metric tensor. But this is not true; they are tensors, as can be seen by calculating $\delta g'_{\mu\nu}$ using $$\delta g_{\mu\nu} = \bar{g}_{\mu\nu} - g_{\mu\nu} \,\, .$$ $\qquad$ Given the variation $\delta V_\mu$, to find the variation of the contravariant vector $\delta V^\mu$, we must calculate $$\delta V^\mu = \delta (g^{\mu\alpha}V_\alpha) = g^{\mu\alpha}\delta V_\alpha + V_\alpha \delta g^{\mu\alpha} \,\, .$$ Notice that the last term "spoils" the usual rule for raising and lowering indices; the Leibniz rule requires that we account for the variation in $V_\alpha$ and the metric tensor. In the case of the metric tensor $g_{\alpha\beta}$, the Leibnitz rule gives $$\delta g^{\mu\nu} = g^{\mu\alpha} g^{\nu\beta} g_{\alpha\beta} + 2\delta g^{\mu\nu}$$ which gives $(*)$.

$\qquad$ In other words, the Leibniz rule is paramount; the “raising and lowering” rule does not apply across the “$\delta$”.

 

[PeterDonis]

the screenshot below

What reference is this screenshot from?

 

[Kostik]

@PeterDonis These are my own notes.

If I have made an error, please critique it!

 

[PeterDonis]

These are my own notes.

Then you should post them directly using LaTeX for equations, in accordance with the forum rules.

If I have made an error, please critique it!

We can't if all we can quote is an image. We need to be able to quote individual equations and words. That's why we need things posted directly using LaTeX for equations.

 

[Kostik]

That makes sense, I will endeavor to use LaTex in the future. Obviously it’s much faster for me to take a snapshot of my own notes (made with MSWord).

 

[PeterDonis]

I will endeavor to use LaTex in the future.

If you're not going to re-post your notes using LaTeX now, I'm going to close this thread. You asked for feedback; that can only be done if you post your content the way I described.

 

[PeterDonis]

Obviously it’s much faster for me to take a snapshot of my own notes (made with MSWord).

If you can't take the time to post using direct text and LaTeX, why should we take the time to read and answer?

Sorry to be blunt, but that's one of the reasons behind the PF rules on this: you need to be respectful of other posters' time, not just your own.

 

[Kostik]

The narrative in #2 has now been added in LaTex.

 

[anuttarasammyak]

TL;DR Summary: Two contradictory equations are shown. Where is the flaw?

From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by $g_{\alpha\beta}$ to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation $\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}$ should be a tensor; therefore, $$\delta g_{\beta\nu} = g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \quad\qquad(**)$$ Which equation is correct, $(*)$ or $(**)$?

Under variation of metric tensor, $g_{\mu\nu}+\delta g_{\mu\nu}$ is a tensor but its parts $g_{\mu\nu}$ and $\delta g_{\mu\nu}$ are not tensors. Its indeces up-down relation is
$$ g_{\mu\nu}+\delta g_{\mu\nu} =( g_{\alpha\nu}+\delta g_{\alpha\nu}) (g_{\mu\beta}+\delta g_{\mu\beta})( g^{\alpha\beta}+\delta g^{\alpha\beta} )$$
0-th order equation is usual one with no variation. The 1st order equation is
$$ \delta g_{\mu\nu}= g_{\mu\beta} g^{\alpha\beta} \delta g_{\alpha\nu} + g_{\alpha\nu}g^{\alpha\beta}
\delta g_{\mu\beta} + g_{\alpha\nu} g_{\mu\beta}\delta g^{\alpha\beta} $$
which gives $(*)$.

 

[haushofer]

TL;DR Summary: Two contradictory equations are shown. Where is the flaw?

From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by $g_{\alpha\beta}$ to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation $\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}$ should be a tensor; therefore, $$\delta g_{\beta\nu} = g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \quad\qquad(**)$$ Which equation is correct, $(*)$ or $(**)$?

Yes, it's a tensor, as one can easily see by taking the concrete example of the Lie derivative of the metric w.r.t. a vector field written with a covariant derivative.

That extra minus sign doesn't spoil the tensorial transformation properties, does it?

 

[haushofer]

Under variation of metric tensor, $g_{\mu\nu}+\delta g_{\mu\nu}$ is a tensor but its parts $g_{\mu\nu}$ and $\delta g_{\mu\nu}$ are not tensors. Its indeces up-down relation is
$$ g_{\mu\nu}+\delta g_{\mu\nu} =( g_{\alpha\nu}+\delta g_{\alpha\nu}) (g_{\mu\beta}+\delta g_{\mu\beta})( g^{\alpha\beta}+\delta g^{\alpha\beta} )$$
0-th order equation is usual and obvious. The 1st order equation is
$$ \delta g_{\mu\nu}= g_{\mu\beta} g^{\alpha\beta} \delta g_{\alpha\nu} + g_{\alpha\nu}g^{\alpha\beta}
\delta g_{\mu\beta} + g_{\alpha\nu} g_{\mu\beta}\delta g^{\alpha\beta} $$
which gives $(*)$.

I don't understand this comment or the equation.

 

[anuttarasammyak]

@haushofer The equation is the application of the rule for raising and lowering the indices in use of the (varied) metric tensor to the (varied) metric tensor itself.

That extra minus sign doesn't spoil the tensorial transformation properties, does it?

Then how can we distinguish which rule, (*) or (**) in OP, holds for each tensorial quantity of our interest ?

 

[haushofer]

@haushofer The equation is the application of the rule for raising and lowering the indices in use of the (varied) metric tensor to the (varied) metric tensor itself.


Then how can we distinguish which rule, (*) or (**) in OP, holds for each tensorial quantity of our interest ?

Do you know any more tensors which include that minus-sign? You just have to be careful if you vary the very object itself which enables you to lift and lower indices.

 

[anuttarasammyak]

Do you know any more tensors which include that minus-sign?

No, I don't know any case in tensors. That is one of the reasons that I think :

Under variation of metric tensor, gμν+δgμν is a tensor but its parts gμν and δgμν are not tensors.

I am afraid that you are against it in post #11, stating that one of them variation infinitesimal δgμν is a tensor. I should appreciate it if you could explain the validation of this minus-sign for tensor δgμν.

 

[haushofer]

No, I don't know any case in tensors. That is one of the reasons that I think :

I am afraid that you are against it in post #11, stating that one of them variation infinitesimal δgμν is a tensor. I should appreciate it if you could explain the validation of this minus-sign for tensor δgμν.

I don't understand that comment.

The validation of the minus-sign is because you consider the variation of the very metric tensor itself, not just some arbitrary tensor. I also don't understand this comment:

Under variation of metric tensor, $g_{\mu\nu}+\delta g_{\mu\nu}$ is a tensor but its parts $g_{\mu\nu}$ and $\delta g_{\mu\nu}$ are not tensors.

You say that the metric components g_{\mu\nu} don't constitute components of a tensor? And dito for $\delta g_{\mu\nu}$. which is the different of two tensors? Tensor form a vector space, so sums and differences of tensors are tensors. This just doesn't make any sense.

 

[haushofer]

@anuttarasammyak:
By the way, the answer is already given here below.

I think I found the (or an) answer. The relations $$\delta g^{\mu\nu} = -g^{\mu\rho}g^{\nu\sigma}\delta g_{\rho\sigma} \quad , \qquad \delta g_{\mu\nu} = -g_{\mu\rho} g_{\nu\sigma} \delta g^{\rho\sigma} \qquad(*)$$ appear to indicate that $\delta g_{\mu\nu}$ and $\delta g^{\mu\nu}$ are not tensors, since their indices are not raised and lowered in the usual way by the metric tensor. But this is not true; they are tensors, as can be seen by calculating $\delta g'_{\mu\nu}$ using $$\delta g_{\mu\nu} = \bar{g}_{\mu\nu} - g_{\mu\nu} \,\, .$$ $\qquad$ Given the variation $\delta V_\mu$, to find the variation of the contravariant vector $\delta V^\mu$, we must calculate $$\delta V^\mu = \delta (g^{\mu\alpha}V_\alpha) = g^{\mu\alpha}\delta V_\alpha + V_\alpha \delta g^{\mu\alpha} \,\, .$$ Notice that the last term "spoils" the usual rule for raising and lowering indices; the Leibniz rule requires that we account for the variation in $V_\alpha$ and the metric tensor. In the case of the metric tensor $g_{\alpha\beta}$, the Leibnitz rule gives $$\delta g^{\mu\nu} = g^{\mu\alpha} g^{\nu\beta} g_{\alpha\beta} + 2\delta g^{\mu\nu}$$ which gives $(*)$.

$\qquad$ In other words, the Leibniz rule is paramount; the “raising and lowering” rule does not apply across the “$\delta$”.

Yes. I guess that one could be pedantic and distinguish between \delta (V_{\mu}) and (\delta V)_{\mu}. The first one means lowering an index and then vary, while the second one means vary first and then lower an index on this variation. These two operations don't commute.

 

[haushofer]

So, to go back to the OP, if this confuses people, they should be careful and distinguish between \delta(g_{\mu\nu}) and (\delta g)_{\mu\nu}. But I guess this is also confusing, since g is also used for the determinant of the metric.

But I don't see why this raises the question whether these are "tensors". Just see how the components transform under a coordinate transformation. That marks a tensor. Not how we raise and lower indices on it.

 

[anuttarasammyak]

@haushofer Thanks for the teaching.

$$ g_{\mu\nu}+\delta g_{\mu\nu} =( g_{\alpha\nu}+\delta g_{\alpha\nu}) (g_{\mu\beta}+\delta g_{\mu\beta})( g^{\alpha\beta}+\delta g^{\alpha\beta} )$$

@haushofer The equation is the application of the rule for raising and lowering the indices in use of the (varied) metric tensor to the (varied) metric tensor itself.

Is it an incorrect equation though the result coincides with the equation (*) ?

 

[anuttarasammyak]

Under variation of metric tensor, gμν+δgμν is a tensor but its parts gμν and δgμν are not tensors.

@haushofer it is a good chance to learn from your comments.

May I say that in GR when tensor $A_{abc..}^{def...}$ is written arbitrary as $$A_{abc..}^{def...}=X_{abc..}^{def...}+(A_{abc..}^{def...}-X_{abc..}^{def...})$$
, are X and A-X also tensors ?

In GR any linear combination of tensors
$$ \sum_i a(i) T_{bcd...}^{efg..}(i) $$
are tensors ?

I would like to understand that non-linear feature of GR does not spoil these linear algebra relations.

 

[haushofer]

@haushofer it is a good chance to learn from your comments.

May I say that in GR when tensor $A_{abc..}^{def...}$ is written arbitrary as $$A_{abc..}^{def...}=X_{abc..}^{def...}+(A_{abc..}^{def...}-X_{abc..}^{def...})$$
, are X and A-X also tensors ?

In GR any linear combination of tensors
$$ \sum_i a(i) T_{bcd...}^{efg..}(i) $$
are tensors ?

I would like to understand that non-linear feature of GR does not spoil these linear algebra relations.

1) No, not necessarily; look e.g. at the definition of the covariant derivative. A partial derivative is not a tensor under gct's, and neither is the connection, but the inhomogeneous terms of both cancel out such that their sum is a tensor.

2) Yes, as long as you add/subtract tensors of the same type, of course.

 

[haushofer]

@haushofer Thanks for the teaching.

$$ g_{\mu\nu}+\delta g_{\mu\nu} =( g_{\alpha\nu}+\delta g_{\alpha\nu}) (g_{\mu\beta}+\delta g_{\mu\beta})( g^{\alpha\beta}+\delta g^{\alpha\beta} )$$

Is it an incorrect equation though the result coincides with the equation (*) ?

I don't understand what that equation says.

 

[anuttarasammyak]

I don't understand what that equation says.

In the world of metric tensor undertaking variation where the metric tensor is
$$ \bar{g}_{\mu\nu} := g_{\mu\nu}+\delta g_{\mu\nu} $$,
Tensor indeces up-down will be done as
$$ A_{\mu\nu}= \bar{g}_{\alpha\mu}\bar{g}_{\nu\beta}A^{\alpha\beta}$$
I consider the case that
$$ A_{\mu\nu}=\bar{g}_{\mu\nu}$$

[EDIT]
It would be better to wrilte indecies up-down equation as
$$ \bar{A}_{\mu\nu}= \bar{g}_{\alpha\mu}\bar{g}_{\nu\beta}\bar{A}^{\alpha\beta}$$
where with a bar on top means it is a tensor in the world of "new" metric under variation.

 

[anuttarasammyak]

1) No, not necessarily; look e.g. at the definition of the covariant derivative. A partial derivative is not a tensor under gct's, and neither is the connection, but the inhomogeneous terms of both cancel out such that their sum is a tensor.

Thanks. Even with this general caution, may we take it obvious that in tensor division of
$$ \bar{g}_{\mu\nu} := g_{\mu\nu}+\delta g_{\mu\nu} $$
all $\bar{g}_{\mu\nu}$ , $g_{\mu\nu}$ and $\delta g_{\mu\nu}$ are tensors in the world of metric undertaking variation ?

 

[JimWhoKnew]

Thanks. Even with this general caution, may we take it obvious that in tensor division of
$$ \bar{g}_{\mu\nu} := g_{\mu\nu}+\delta g_{\mu\nu} $$
all $\bar{g}_{\mu\nu}$ , $g_{\mu\nu}$ and $\delta g_{\mu\nu}$ are tensors in the world of metric undertaking variation ?

Ordinarily, $~T^{\mu\nu}~$ , $~T_{\mu\nu}~$ and $~T^\mu{}_\nu~$ are coordinate representation of the same coordinate-free object (tensor) $\mathbf{T}$ . In the present case we have the anomaly of 2 metrics. If we regard $g_{\mu\nu}$ as the "basic" metric, then $\bar{g}_{\mu\nu}$ and its reciprocal $\bar{g}^{\mu\nu}$ are not representations of the same coordinate-free object. That is also the case with $\delta g_{\mu\nu}$ and $\delta g^{\mu\nu}$ (and the reason why equation (**) in OP is wrong).

In equations: if $~\bar{g}_{\mu\nu}=(\mathbf{g}_1)_{\mu\nu}~$ then$$(\mathbf{g}_1)^{\mu\nu}=g^{\mu\rho}\bar{g}_{\rho\sigma}g^{\sigma\nu}\neq\bar{g}^{\mu\rho}\bar{g}_{\rho\sigma}\bar{g}^{\sigma\nu}=\bar{g}^{\mu\nu}$$

 

[weirdoguy]

Ordinarily, Tμν , Tμν and Tμν are coordinate representation of the same coordinate-free object (tensor) T

Are they? Tensors are mappings from cartesian products of vector spaces and their duals. What you wrote are coordinate representations of different coordinate-free objects, since they have different domains. Eg. $T^{\mu\nu}e_{\mu}\otimes e_{\nu}$, $T_{\mu\nu}\varepsilon^{\mu}\otimes \varepsilon^{\nu}$ where $e_i$ and $\varepsilon^i$ are bases in $V$ and its dual respectively, are different tensors.

 

[PeterDonis]

Ordinarily, $~T^{\mu\nu}~$ , $~T_{\mu\nu}~$ and $~T^\mu{}_\nu~$ are coordinate representation of the same coordinate-free object (tensor) $\mathbf{T}$

Only in the presence of a metric, so that you can use it to freely raise and lower indexes.

Otherwise those are three different tensors, a (2, 0), a (0, 2), and a (1, 1). Without a metric there is no natural mapping between those different vector spaces.

 

[PeterDonis]

Are they?

I think @JimWhoKnew was assuming the presence of a metric, which gives you a natural mapping between the different vector spaces that allows you to freely raise and lower indexes.

 

[weirdoguy]

which gives you a natural mapping between the different vector spaces

Yes, but still some people wouldn't call those tensors "the same object". Especially most mathematical physicists/mathematicians that I know, including myself. But yes, physicists with their sloppy use of language would

 

[anuttarasammyak]

If we regard gμν as the "basic" metric, then g¯μν and its reciprocal g¯μν are not representations of the same coordinate-free object.

Thank you @JimWhoKnew for the clear and easy-to-understand explanation for the "anomaly". I have preffered to regard "new" $\bar{g}_{\mu\nu}$ as "basic" then old $g_{\mu\nu}$ , $\delta{g}_{\mu\nu}$ and their reciprocals are not representations of the same coordinate free object. That is the resason why I said

Under variation of metric tensor, gμν+δgμν is a tensor but its parts gμν and δgμν are not tensors.