Geodesic coordinates and tensor identities

[haushofer]

Hi, I have a question about deriving tensor identities using geodesic coordinates ( coordinates in which one can put the connection to zero ). For the reference, I'm a master student physics and followed courses on general relativity and geometry. I'm busy with an article of Wald and Lee called "local symmetries and constraints", and that's where my question comes from.

In short, Wald defines a collection of tensor fields $$\phi$$:

$$\phi : M \rightarrow M^{'}$$

where M can be seen as our space-time, and M' is finite-dimensional. He then states the following: "In a sufficiently small neighbourhood U' of any point $$\phi_{0} \in M^{'}$$ we may choose coordinates for M' such that the map $$\phi$$ can be represented locally as a collection of scalar functions $$\phi^{a}$$ of the space-time point x. Note that a change of coordinates in U' corresponds to an x-independent field redefinition $$\psi^{a} = f^{a}(\phi^{b})$$. "

He uses this to write down the variation of the Lagrangian $$L(\phi^{a}, \nabla_{\mu}\phi^{a},\ldots,\nabla_{\mu_{1} }\cdots \nabla_{\mu_{k}}\phi^{a} ; \gamma^{b} )$$,
where $$\gamma^{b}$$ is any nondynamical background field ( like the minkowski-metric in special relativistic field theories ). He uses only the symmetric part of the derivatives, because every antisymmetric part can be rewritten in terms of the Riemann-tensor. The variation of the Lagrangian should be

$$\delta L = \frac{\partial L}{\partial \phi^{a}} \delta\phi^{a} + \frac{\partial L }{\partial (\nabla_{\mu}\phi^{a})}\delta(\nabla_{\mu}\phi^{a}) + \ldots$$

where the dots indicate higher derivatives. But it appears to me that he uses his specially chosen coordinates to write this variation as

$$\delta L = \frac{\partial L}{\partial \phi^{a}} \delta\phi^{a} + \frac{\partial L }{\partial (\nabla_{\mu}\phi^{a})}\nabla_{\mu}\delta\phi^{a} + \ldots$$

and then states that the result doesn't depend on the coordinates, so it holds in general. It's quite obvious that for an arbitrary tensor field the variation and covariant derivative don't commute; you are left with variations in the connection. For example, for the first derivative one has that

$$\delta\nabla\phi^{a} = \nabla\delta\phi^{a} + \sum\phi^{a}\delta\Gamma$$

in short-hand notation. As long as you treat the space-time metric as nondynamical background field everything is OK, but if we treat the metric as dynamical, this doesn't make any sense to me. Why is it possible to just say "let's choose special coordinates in which we can treat the tensor fields as scalar fields, so that we can commute the partial derivatives and variations, and then transform back to general coordinates and get the covariant derivative and variation commuting? Is this just simply chosing geodesic coordinates? In books like Inverno they use geodesic coordinates to get the identity for the variation of the Riemanntensor, which looks quite the same as this. What they do there, is:

*choose geodesic coordinates where the connection is zero
*write down the Riemann-tensor, where the connection terms are zero, but the derivatives are not
*induce a variation in the connection
*this induces a variation in the Riemann-tensor
*commute the partial derivative and variation
*observe that a tensor equation holds in every frame
*observe that the result holds in every coordinate frame

Could this be what Wald does in his article ?

I hope I have given enough information, because the last time I had the idea that I was pretty unclear in my questioning :) I have the idea that there are some fundamental things about geometry which aren't completely clear to me.

 

[Chris Hillman]

Terminological confusion?

I have a question about deriving tensor identities using geodesic coordinates ( coordinates in which one can put the connection to zero ).

You appear to be confusing several distinct concepts:

• the Levi-Civita connection vanishes in some neighborhood only for a manifold which is locally flat in that neighborhood,
• in Riemannian (Lorentzian) geometry, the term geodesic coordinates usually refers to a chart in which the integral curves of one of the coordinate vector fields, say $\partial_x$, are geodesic curves, with the coordinate playing the role of an arc-length parameter (examples include polar spherical chart on the ordinary sphere in which $\partial_\theta$ is a geodesic vector field with $\theta$ playing the role of arc length parameter),
• in Riemannian (Lorentzian) geometry, the term Riemann normal coordinates usually refers to a chart in which the connection vanishes at one point, and coordinate "lines" issuing from that point do represent geodesics issuing from that point (note that due to geodesic convergence creating "caustics", such charts are rarely global charts); indeed a common trick does involve proving theorems by assuming a Riemann normal chart with some "base point", proving that some tensor equation holds at the point, and inferring the desired result (see the discussion in the textbook by Sean Carroll, Spacetime and Geometry),
• in Lorentzian geometry, the term Fermi normal coordinates usually refers to a similar construction "centered" on a given geodesic rather than a given point.

(Pedantic warning: in Riemannian geometry, Gaussian normal coordinates is usually synonymous with Riemann normal coordinates, but in the context of gtr, a Gaussian chart is sometimes taken to mean something else entirely.)

For the reference, I'm a master student physics and followed courses on general relativity and geometry. I'm busy with an article of Wald and Lee called "local symmetries and constraints", and that's where my question comes from.

Since you are asking for advice at the research level (apparently), it seems to me that it would be appropriate to obey such conventions as proper citations of books/papers. I guess (but IMO shouldn't have to guess) that you mean this paper:

Lee, J., and R. M. Wald, "Local Symmetries and Constraints", Journal of Mathematical Physics 31 (1990): 725 – 743.

Please correct me if I my guess is wrong, and no, I haven't seen this, I simply found a citation which may or may not be the paper you have in mind.

I hope I have given enough information, because the last time I had the idea that I was pretty unclear in my questioning :) I have the idea that there are some fundamental things about geometry which aren't completely clear to me.

I didn't read the rest of your post because it seems to be based entirely upon a serious terminological confusion. Obvious question: shouldn't your professor be helping you read the literature? E.g. terminological questions certainly seem appropriate because as we see above one could easily guess wrong if an experienced researcher doesn't clue you in, and it probably best to obtain such crucial advice from faculty whenever possible.

But regardless, hope this helps!

 

[haushofer]

You appear to be confusing several distinct concepts:

• the Levi-Civita connection vanishes in some neighborhood only for a manifold which is locally flat in that neighborhood,
• in Riemannian (Lorentzian) geometry, the term geodesic coordinates usually refers to a chart in which the integral curves of one of the coordinate vector fields, say $\partial_x$, are geodesic curves, with the coordinate playing the role of an arc-length parameter (examples include polar spherical chart on the ordinary sphere in which $\partial_\theta$ is a geodesic vector field with $\theta$ playing the role of arc length parameter),
• in Riemannian (Lorentzian) geometry, the term Riemann normal coordinates usually refers to a chart in which the connection vanishes at one point, and coordinate "lines" issuing from that point do represent geodesics issuing from that point (note that due to geodesic convergence creating "caustics", such charts are rarely global charts); indeed a common trick does involve proving theorems by assuming a Riemann normal chart with some "base point", proving that some tensor equation holds at the point, and inferring the desired result (see the discussion in the textbook by Sean Carroll, Spacetime and Geometry),
• in Lorentzian geometry, the term Fermi normal coordinates usually refers to a similar construction "centered" on a given geodesic rather than a given point.

(Pedantic warning: in Riemannian geometry, Gaussian normal coordinates is usually synonymous with Riemann normal coordinates, but in the context of gtr, a Gaussian chart is sometimes taken to mean something else entirely.)

Ok, maybe that is part of the confusion. Thank you for clearing that up, obviously I was talking about Riemann normal coordinates ( Inverno calls this geodesic coordinates, or I am mixing things up ) :)

Since you are asking for advice at the research level (apparently), it seems to me that it would be appropriate to obey such conventions as proper citations of books/papers. I guess (but IMO shouldn't have to guess) that you mean this paper:

Lee, J., and R. M. Wald, "Local Symmetries and Constraints", Journal of Mathematical Physics 31 (1990): 725 – 743.

Please correct me if I my guess is wrong, and no, I haven't seen this, I simply found a citation which may or may not be the paper you have in mind.

Yes, that is the paper, I should have given the correct reference.

I didn't read the rest of your post because it seems to be based entirely upon a serious terminological confusion. Obvious question: shouldn't your professor be helping you read the literature? E.g. terminological questions certainly seem appropriate because as we see above one could easily guess wrong if an experienced researcher doesn't clue you in, and it probably best to obtain such crucial advice from faculty whenever possible.

But regardless, hope this helps!

Well, I have the idea that it is not only terminological confusion. Ofcourse I asked my professor for help, but he had some problems with this commutation too. What the author does in the paper, is simply commuting the variation and the covariant derivative. He states that in the field variations, it is understood that any nondynamical background field is held fixed.

A simple example:

$$L = L(g_{ab},F_{ab},\nabla_{c}F_{ab})$$

of which the variation yields

$$\delta L = \frac{\partial L}{\partial g_{ab}}\delta g_{ab} + \frac{\partial L}{\partial (\nabla_{c}F_{ab})}\partial_{c}\delta F_{ab} + \frac{\partial L}{\partial (\nabla_{c}F_{ab})}(\Gamma^{d}_{ca}F_{db} + \Gamma^{d}_{cb}F_{ad})$$

The problem lies in the construction; I can't see why one would leave out those connection-terms in the variation ( the last term on the RHS ).

But thank you for your comments !

 

[Chris Hillman]

Riemann normal coordinates ( Inverno calls this geodesic coordinates, or I am mixing things up ) :)

So he does, and how annoying! :grumpy: Well, the terminology appears to be more completely standardized in mathematical literature than the gtr literature, so I guess you need to check carefully when reading gtr papers that you correctly understand all the technical terms for special coordinate systems.

Of course I asked my professor for help, but he had some problems with this commutation too. What the author does in the paper, is simply commuting the variation and the covariant derivative. He states that in the field variations, it is understood that any nondynamical background field is held fixed.

OK, haven't read the paper myself, but Jack Lee is accessible so if you and your prof are both puzzled I think it would be OK to email him if after mulling the terminological confusion for a bit things still are not clear.

 

[samalkhaiat]

Metric variations & Field redefinitions

Ofcourse I asked my professor for help, but he had some problems with this commutation too. What the author does in the paper, is simply commuting the variation and the covariant derivative.

I have not seen the paper, so what I am going to say may not help clear away the confusion!

Let u be a multiplet of dynamical field (all indices suppressed) described by a Lagrangian $L(u,g)$, living in a curved spacetime with a (dynamical or backgroung) metric g.
Let $\bar{u} = f(u,g;h)$ be any smooth invertable transformations of the dynamical variables, which in general involve the metric tensor and a nondynamical tensor field h. Under such arbitrary field redefinitions, we set

$$L(u,g) = L \left( f^{-1}\left(\bar{u},g;h \right),g \right) \equiv \bar{L} \left( \bar{u},g;h \right) \ \ \ \ (1)$$

To find the (metric) variation of the LHS of Eq(1); $\delta_{g}L(u, \nabla u,g)$, we assume that the fundamental field is not affected by metric variations,$\delta_{g} u = 0$. For example, in electrodynamics, this is the case for the covariant vector potential (1-form) $\delta_{g}A_{a} = 0$, while $\delta_{g} A^{a} = A_{b}\delta g^{ab}$ is metric dependent.
Thus, in evaluating $\delta_{g}L$ we take into account only the explicit dependence of L on $g$ and $\Gamma$;

$$\delta_{g}L(u,g) = \frac{\partial L}{\partial g} \ \delta g + \frac{\partial L}{\partial ( \nabla u)} \ \delta_{g} ( \nabla u)$$

where
$$\delta_{g} (\nabla u) = \delta_{g} ( \partial u - u \Gamma ) = - u \delta_{g} \Gamma$$

Now, $\bar{L}(\bar{u},g;h)$ depends on the metric both explicitly (including the connection) and implicitly via $\bar{u} = f(u,g;h)$. Thus, its metric variation is determined by the total variation

$$\bar{\delta}_{g} \bar{L} \equiv \delta_{g} \bar{L} + \delta_{f} \bar{L}$$

Here, as befor, $\delta_{g} \bar{L}$ is the variation taking into account only the explicit metric dependence of $\bar{L}$;

$$\delta_{g} \bar{L}( \bar{u},g;h) = \frac{\partial \bar{L}}{\partial g} \ \delta g - \frac{\partial \bar{L}}{\partial ( \nabla \bar{u})} \ \bar{u} \ \delta_{g} \Gamma$$

But, $\delta_{f} \bar{L}$ takes into account the metric dependence of $\bar{L}$ via $\bar{u}$. We note that the metric variations of the new field are determined by the transformations;

$$\delta_{f} \bar{u} = \frac{\partial f}{\partial g} \ \delta g + \frac{\partial f}{\partial (\partial g)} \ \delta (\partial g)$$

Therefore

$$\delta_{f} \bar{L}( \bar{u},g;h) = \frac{\partial \bar{L}}{\partial \bar{u}} \ \delta_{f} \bar{u} + \frac{\partial \bar{L}}{\partial ( \nabla \bar{u})}\ \delta_{f} (\nabla \bar{u})$$

where

$$\delta_{f}(\nabla \bar{u}) = ( \partial - \Gamma ) \delta_{f} \bar{u} = \nabla \delta_{f} \bar{u}$$

So, it is $\delta_{f}$ that commutes with the covariant derivative. Notice that for L(u,g) the operators $\bar{ \delta}_{g}$ and $\delta_{g}$ coincide. Thus, the identity (1), which holds for ALL u, g and transformations f, implies;

$$\delta_{g} L(u,g) = \bar{ \delta}_{g} \bar{L}( \bar{u},g;h)$$

Please check the paper, were Lee & Wald calculating $\delta_{f} \bar{L}$( the variation in L brought about by the metric variations of the new (redefined) field)? If that was the case, then (I believe) the above does clear up the confusion!

regards

sam

 

[haushofer]

I have not seen the paper, so what I am going to say may not help clear away the confusion!

Let u be a multiplet of dynamical field (all indices suppressed) described by a Lagrangian $L(u,g)$, living in a curved spacetime with a (dynamical or backgroung) metric g.
Let $\bar{u} = f(u,g;h)$ be any smooth invertable transformations of the dynamical variables, which in general involve the metric tensor and a nondynamical tensor field h. Under such arbitrary field redefinitions, we set

$$L(u,g) = L \left( f^{-1}\left(\bar{u},g;h \right),g \right) \equiv \bar{L} \left( \bar{u},g;h \right) \ \ \ \ (1)$$

To find the (metric) variation of the LHS of Eq(1); $\delta_{g}L(u, \nabla u,g)$, we assume that the fundamental field is not affected by metric variations,$\delta_{g} u = 0$. For example, in electrodynamics, this is the case for the covariant vector potential (1-form) $\delta_{g}A_{a} = 0$, while $\delta_{g} A^{a} = A_{b}\delta g^{ab}$ is metric dependent.
Thus, in evaluating $\delta_{g}L$ we take into account only the explicit dependence of L on $g$ and $\Gamma$;

$$\delta_{g}L(u,g) = \frac{\partial L}{\partial g} \ \delta g + \frac{\partial L}{\partial ( \nabla u)} \ \delta_{g} ( \nabla u)$$

where
$$\delta_{g} (\nabla u) = \delta_{g} ( \partial u - u \Gamma ) = - u \delta_{g} \Gamma$$

Now, $\bar{L}(\bar{u},g;h)$ depends on the metric both explicitly (including the connection) and implicitly via $\bar{u} = f(u,g;h)$. Thus, its metric variation is determined by the total variation

$$\bar{\delta}_{g} \bar{L} \equiv \delta_{g} \bar{L} + \delta_{f} \bar{L}$$

Here, as befor, $\delta_{g} \bar{L}$ is the variation taking into account only the explicit metric dependence of $\bar{L}$;

$$\delta_{g} \bar{L}( \bar{u},g;h) = \frac{\partial \bar{L}}{\partial g} \ \delta g - \frac{\partial \bar{L}}{\partial ( \nabla \bar{u})} \ \bar{u} \ \delta_{g} \Gamma$$

But, $\delta_{f} \bar{L}$ takes into account the metric dependence of $\bar{L}$ via $\bar{u}$. We note that the metric variations of the new field are determined by the transformations;

$$\delta_{f} \bar{u} = \frac{\partial f}{\partial g} \ \delta g + \frac{\partial f}{\partial (\partial g)} \ \delta (\partial g)$$

Therefore

$$\delta_{f} \bar{L}( \bar{u},g;h) = \frac{\partial \bar{L}}{\partial \bar{u}} \ \delta_{f} \bar{u} + \frac{\partial \bar{L}}{\partial ( \nabla \bar{u})}\ \delta_{f} (\nabla \bar{u})$$

where

$$\delta_{f}(\nabla \bar{u}) = ( \partial - \Gamma ) \delta_{f} \bar{u} = \nabla \delta_{f} \bar{u}$$

So, it is $\delta_{f}$ that commutes with the covariant derivative. Notice that for L(u,g) the operators $\bar{ \delta}_{g}$ and $\delta_{g}$ coincide. Thus, the identity (1), which holds for ALL u, g and transformations f, implies;

$$\delta_{g} L(u,g) = \bar{ \delta}_{g} \bar{L}( \bar{u},g;h)$$

Please check the paper, were Lee & Wald calculating $\delta_{f} \bar{L}$( the variation in L brought about by the metric variations of the new (redefined) field)? If that was the case, then (I believe) the above does clear up the confusion!

regards

sam

Hey Sam, this helps a lot, but there are still some thing unclear. The variation what I'm talking about, is defined as follows. The authors define a smooth and one-parameter family of field configurations,

$$\phi(\lambda): M \rightarrow M^{'}$$

The first variation of the Lagrangian is then defined as

$$\delta L = \frac{d}{d\lambda}L |_{\lambda=0} = \frac{\partial L}{\partial \phi^{a}}\delta\phi^{a} + \frac{\partial L}{\partial \nabla_{\mu}\phi^{a}}\nabla_{\mu}\delta\phi^{a} + \cdots$$

about the field configuration $$\phi_{0} = \phi(\lambda=0)$$. The "a" indicates the particular dynamical field. The variation of phi is defined as

$$\delta\phi^{a} = \frac{\partial\phi^{a}(\lambda ; x)}{\partial \lambda} |_{\lambda = 0}$$

Is this lambda defining what you call "redefinition of the fields"?As I understand, with this lambda one can define a congruence of curves along which one can define variations of a tensor field. Could you clear up your explanation with an example like

$$L = L(g_{ab},R_{abcd},F_{ab},\nabla_{c}F_{ab})$$

? I mean, I should think that we are interested in the total variation of L, and that is

$$\delta L = \frac{\partial L}{\partial g_{ab}}\delta g_{ab} + \frac{\partial L}{\partial R_{abcd}}\delta R_{abcd} + \frac{\partial L}{\partial (\nabla_{c}F_{ab})}\partial_{c}\delta F_{ab} + \frac{\partial L}{\partial (\nabla_{c}F_{ab})}(\Gamma^{d}_{ca}F_{db} + \Gamma^{d}_{cb}F_{ad})$$


Anyway, I think your explanation makes clear how the different dependancies are split up, and I haven't seen an explanation like this before, so that's very useful !

 

[samalkhaiat]

Hey Sam, this helps a lot, but there are still some thing unclear. The variation what I'm talking about, is defined as follows. The authors define a smooth and one-parameter family of field configurations,

$$\phi(\lambda): M \rightarrow M^{'}$$

The first variation of the Lagrangian is then defined as

$$\delta L = \frac{d}{d\lambda}L |_{\lambda=0} = \frac{\partial L}{\partial \phi^{a}}\delta\phi^{a} + \frac{\partial L}{\partial \nabla_{\mu}\phi^{a}}\nabla_{\mu}\delta\phi^{a} + \cdots$$

about the field configuration $$\phi_{0} = \phi(\lambda=0)$$. The "a" indicates the particular dynamical field. The variation of phi is defined as

$$\delta\phi^{a} = \frac{\partial\phi^{a}(\lambda ; x)}{\partial \lambda} |_{\lambda = 0}$$

Is this lambda defining what you call "redefinition of the fields"?As I understand, with this lambda one can define a congruence of curves along which one can define variations of a tensor field. Could you clear up your explanation with an example like

$$L = L(g_{ab},R_{abcd},F_{ab},\nabla_{c}F_{ab})$$

Yes, If you like, you can rewrite my equations using your notations:

$$\bar{u} \rightarrow \phi^{a}(x;\lambda)$$

$$\delta_{f} \rightarrow \delta$$

$$\bar{L}(\bar{u},g) \rightarrow L(\phi^{a},g)$$

For example the equation

$$\delta_{f} ( \nabla \bar{u}) = \nabla (\delta_{f} \bar{u})$$

becomes

$$\delta \left( \nabla \phi^{a}(x;\lambda) \right) = \frac{\partial}{\partial \lambda} \nabla \phi^{a}|_{\lambda = 0} = \frac{\partial}{\partial \lambda}\left( ( \partial + \Gamma ) \phi^{a}\right)|_{\lambda = 0} \\ = (\partial + \Gamma ) \frac{\partial \phi^{a}}{\partial \lambda}|_{\lambda = 0} = \nabla ( \delta \phi^{a})$$

? I mean, I should think that we are interested in the total variation of L, and that is

$$\delta L = \frac{\partial L}{\partial g_{ab}}\delta g_{ab} + \frac{\partial L}{\partial R_{abcd}}\delta R_{abcd} + \frac{\partial L}{\partial (\nabla_{c}F_{ab})}\partial_{c}\delta F_{ab} + \frac{\partial L}{\partial (\nabla_{c}F_{ab})}(\Gamma^{d}_{ca}F_{db} + \Gamma^{d}_{cb}F_{ad})$$

In what sense do you call this "total variation"? What happened to the term $\frac{\partial L}{\partial F} \delta F$? and why is there NO variation symbol in the last term?
It helps if you think in terms of the algebraic properties of $\delta$; (it is a derivation), But then you also need to find the relation (if any) between $\delta \phi$ and $\delta g$ which is determined by the transformation law in the question.


regards

sam

 

[haushofer]

My mistake; indeed I wrote something silly in that last variation. It should read

$$\delta L = \frac{\partial L}{\partial g_{ab}}\delta g_{ab} + \frac{\partial L}{\partial R_{abcd}}\delta R_{abcd} + \frac{\partial L}{\partial (\nabla_{c}F_{ab})}\delta(\nabla_{c}F_{ab}) \ \ \ \ \ \ \ (1)$$

in which we could rewrite the last variation as

$$\delta(\nabla_{c}F_{ab}) = \nabla_{c}(\delta F_{ab}) - \delta\Gamma^{h}_{ca}F_{hb} -\delta\Gamma^{h}_{cb}F_{ah}$$

I call this the total variation in the sense one does if one wants to derive the equations of motion.

So, if we define the variation in the "lambda-way",
$$\delta\phi^{a} = \frac{\partial\phi^{a}(\lambda ; x)}{\partial \lambda} |_{\lambda = 0}$$
then taking variations of tensor fields and taking covariant derivatives commute. But this is not true for the kind of variations I used in my equation (1), for there we have

$$\delta\nabla\phi^{a} = \nabla\delta\phi^{a} - \sum\phi^{a}\delta\Gamma$$

where the tensor field has covariant indices. So now I'm a little confused :) It's clear that in taking these kind of variations, we can't commute variations and covariant derivatives; you always end up with variations in the connection. So what is the precise difference between the variation you take in deriving the equations of motion and identities like

$$\delta L = E \delta\phi + \nabla_{a}\Theta^{a}$$
( where E stands for the equations of motion and $$\Theta$$ is the so-called symplectic potential ) and the variation with the lambda?

 

[haushofer]

I believe I start to understand where the confusion comes from; the lambda notion of a variation can be regarded as a tangent vector in the configuration space, while the coordinate notion of a variation can be regarded as a vector in coordinate-space ( space-time ). Those are two completely different spaces, and in that light I understand why taking covariant derivatives and taking "lambda-variations" commute; you are considering two different spaces.

But in the coordinate space we have that varying our fields and taking covariant derivatives don't commute, due to the connection terms... Am I going into the right direction?

 

[samalkhaiat]

...
... So now I'm a little confused :) It's clear that in taking these kind of variations, we can't commute variations and covariant derivatives; you always end up with variations in the connection. So what is the precise difference between the variation you take in deriving the equations of motion and identities like

$$\delta L = E \delta\phi + \nabla_{a}\Theta^{a}$$
( where E stands for the equations of motion and $$\Theta$$ is the so-called symplectic potential ) and the variation with the lambda?

My 1st post contains al the information you need! Ok, let me put things in a terribly simplified form;

Let u be a matter field coupled to gravity, via $\nabla u = \partial u - \Gamma u$, with Lagrangian $L(u, \nabla u,g)$.
Let $\delta_{m}$ be an operator acting only on the matter field;

$$\delta_{m}u \equiv \delta u$$

$$\delta_{m} g = \delta_{m} \Gamma = \delta_{m}R = 0$$

From this we can write;

$$\delta_{m} \nabla u = \partial \delta_{m}u - \Gamma \delta_{m}u = \nabla \delta_{m}u \ \ \ (1)$$

Let $\delta_{g}$ be an operator acting only on the geometry;

$$\delta_{g}u = 0$$

$$\delta_{g} g^{ab} \equiv \delta g^{ab}$$

Then;

$$\delta_{g} \nabla u = - u \delta_{g} \Gamma \ \ \ (2)$$

For obvious reasons, we call the operator;

$$\delta_{t} = \delta_{m} + \delta_{g} \ \ \ (3)$$
$$\delta_{t} u = \delta_{m} u, \ \ \delta_{t}g^{ab} = \delta g^{ab} \ \ \ (4)$$

"total" variation.

From (1) & (2), we get;

$$\delta_{t}\nabla u = \nabla \delta_{m} u - u \delta_{g}\Gamma \ \ \ (5)$$

Let us now compute the total variation of L;

$$\delta_{t}L = \frac{\partial L}{\partial u} \delta_{t}u + \frac{\partial L}{\partial \nabla u} \delta_{t} \nabla u + \frac{\partial L}{\partial g^{ab}} \delta_{t} g^{ab}$$

Or, by using (3),(4) & (5),

$$\delta_{t}L = \left[ \frac{\partial L}{\partial u} \delta_{m}u + \frac{\partial L}{\partial \nabla u} \nabla \delta_{m}u \right] + \left[ \frac{\partial L}{\partial g^{ab}} \delta g^{ab} - \frac{\partial L}{\partial \nabla u} \ u \delta_{g}\Gamma \right]$$

The 1st square bracket can be put in the form;

$$\delta_{m}L = \left[ \frac{\partial L}{\partial u} - \nabla . \left( \frac{\partial L}{\partial \nabla u} \right) \right] \ \delta u + \nabla . \left( \frac{\partial L}{\partial \nabla u} \delta u \right)$$

or, if you like,

$$\delta_{m}L = E(u) \delta u + \nabla . \Theta$$

The 2nd square bracket in $\delta_{t}L$, can be written, using the definition of the energy-momentum tensor, as follows;

$$\delta_{g}L = \frac{1}{2} \left( g_{ab} L - T_{ab} \right) \delta g^{ab}$$

Hence, the total variation;

$$\delta_{t}L = \frac{1}{2}(g_{ab}L - T_{ab}) \delta g^{ab} + E(u) \delta u + \nabla . \Theta$$

regards

sam

 

[haushofer]

Thank you for all your effort, it is very much appreciated. I now have the feeling that I see the point here. I will write some cases with higher derivatives explicitly out, and try to get the hang of it.

This is not something which is absolutely crucial for my thesis, but I just wanted to have these things clear for myself. I find this kind of math quite a hard subject as you have noticed by my questions ( and my last topic, in which you also made some things clear for me ). All I can do now is to take some explicit examples, and ofcourse to wish you nice holidays ! Christmas greetings,

Haushofer.

 

[samalkhaiat]

I find this kind of math quite a hard subject as you have noticed by my questions ( and my last topic, in which you also made some things clear for me ).

Because of the similarity of the topics, I actually wanted to ask YOU to see my post in:

www.physicsforums.com/showthread.php?t=183784

I searched for that post only to realize that it was your question that I was answering there! well, that put a smile on my face! anyway merry chirstmas to you.

regards

sam