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**Question On Metric!!!**

DOES ANYONE KNOW HOW CAN PROVE

THIS

**METRIC/SPACE IS FLAT**?

[tex] ds^2 = (-dx^2 +dy^2)e^{ax+by} [/tex]

general what is flat space?

how i can find a transformation to make this metric in flat form?

:surprised

:zzz:

- Thread starter astronomia84
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- #1

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DOES ANYONE KNOW HOW CAN PROVE

THIS

[tex] ds^2 = (-dx^2 +dy^2)e^{ax+by} [/tex]

general what is flat space?

how i can find a transformation to make this metric in flat form?

:surprised

:zzz:

- #2

cristo

Staff Emeritus

Science Advisor

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Calculate the Riemann curvature tensor, and show that it vanishes.

- #3

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thanks!!!!Calculate the Riemann curvature tensor, and show that it vanishes.

- #4

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You don't actually need to prove that it's flat - just looking at it will tell you that it's flat. More specifically, the metric you've given above is conformal to two-dimensional Minkowski space, so it's said to beDOES ANYONE KNOW HOW CAN PROVE

THISMETRIC/SPACE IS FLAT?

[tex] ds^2 = (-dx^2 +dy^2)e^{ax+by} [/tex]

general what is flat space?

how i can find a transformation to make this metric in flat form?

:surprised

:zzz:

If you have some [itex](m\ge3)[/itex]-dimensional manifold with metric [itex]g_{ij}[/itex] then you can say that another metric [itex]\overline{g}_{ij}[/itex] is

[tex]\overline{g}_{ij} = \phi^{4/(m-2)}g_{ij}[/tex]

The scalar curvature derived from [itex]\overline{g}_{ij}[/itex] is related to that derived from [itex]g_{ij}[/itex] by a very famous relationship called the

[tex]\overline{R} = \phi^{-4/(m-2)}R - \frac{4(m-1)}{m-2}\phi^{-(m+2)/(m-2)}\Delta\phi,[/tex]

where [itex]\Delta=\nabla_m\nabla^m[/itex] is the Laplacian derived from the [itex]g_{ij}[/itex]-compatible covariant derivative. Using the Lichnerowicz equation is then a simple way to test for the flatness of a conformally related metric.

In your case, however, I suspect you'd be better off by thinking about exactly what type of metric you have. Can you think of anything special about two-dimensional manifolds? Is there some special (local) relationship between two-dimensional manifolds and flatness?

Hint: yes, there is.

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- #5

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Well, what would the "flat" form of the metric look like? Show us that you're willing to at least think about the problem and we'll help you out.thanks!!!!

how i can find a transformation to make this metric in flat form?

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INTERESTYou don't actually need to prove that it's flat - just looking at it will tell you that it's flat. More specifically, the metric you've given above is conformal to two-dimensional Minkowski space, so it's said to beconformally flat.

If you have some [itex](m\ge3)[/itex]-dimensional manifold with metric [itex]g_{ij}[/itex] then you can say that another metric [itex]\overline{g}_{ij}[/itex] isconformally equivalentto [itex]g_{ij}[/itex] if there exists some smooth, strictly positive function [itex]\phi[/itex] such that

[tex]\overline{g}_{ij} = \phi^{4/(m-2)}g_{ij}[/tex]

The scalar curvature derived from [itex]\overline{g}_{ij}[/itex] is related to that derived from [itex]g_{ij}[/itex] by a very famous relationship called theLichnerowicz equation:

[tex]\overline{R} = \phi^{-4/(m-2)}R - \frac{4(m-1)}{m-2}\phi^{-(m+2)/(m-2)}\Delta\phi,[/tex]

where [itex]\Delta=\nabla_m\nabla^m[/itex] is the Laplacian derived from the [itex]g_{ij}[/itex]-compatible covariant derivative. Using the Lichnerowicz equation is then a simple way to test for the flatness of a conformally related metric.

In your case, however, I suspect you'd be better off by thinking about exactly what type of metric you have. Can you think of anything special about two-dimensional manifolds? Is there some special (local) relationship between two-dimensional manifolds and flatness?

Hint: yes, there is.

I DO NOT HAVEWell, what would the "flat" form of the metric look like? Show us that you're willing to at least think about the problem and we'll help you out.

NOTE: SORRY FROM MY ENGLISH IS BAD.

- #7

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Note: Sorry For My English Is Bad.

- #8

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I will help you only if you appear to want to help yourself. I've already given you enough hints to solve this problem. If this is homework, I'm not going to do it for you.I DO NOT HAVEMORE ELEMENTSFOR THIS QUESTION.

Look, let me give you another hint. You have a metric

[tex]g_{ij} = \phi^2\eta_{ij}[/itex]

where [itex]\eta_{ij}[/itex] is the two-dimensional Minkowski metric and where

[tex]\phi^2 = e^{ax+by}[/itex]

for some constants [itex]a,b[/itex]. We've already told you that this metric is

i) Is the metric [itex]g_{ij}[/itex] flat?

ii) Can you find a clever choice of coordinates so that [itex]g_{ij}[/itex] becomes obviously flat?

You've already been told how to answer (i): simply calculate the components of the Riemann curvature tensor and show that they all vanish. This is easy; if you don't know how to do it, or if you don't actually know what the Riemann curvature tensor is, then you shouldn't be attempting to solve the question yet.

To answer (ii) I've given you a hint. I asked you the following question: If you had actually found some coordinates in which the metric is manifestly flat, what would the metric look like? As another hint, I'll tell you that if you had some coordinates [itex]U,V[/itex] for which the metric was flat, then you would be able to write it in the form

[tex]ds^2 = e^{ax+by}(-dx^2 + dy^2) = -dU^2 + dV^2[/tex]

So, what you have to do to solve this is to find some functions [itex]U(x,y)[/itex] and [itex]V(x,y)[/itex] so that the metric can be written in the above form. Can you do this?

- #9

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It is not correct. The conformal metric can be or can not be flat. The term 'conformally flat' does not have any sense. The flat metric can be only euclidian or pseudo-euclidean.You don't actually need to prove that it's flat - just looking at it will tell you that it's flat. More specifically, the metric you've given above is conformal to two-dimensional Minkowski space, so it's said to beconformally flat.

....

Sometimes looking on the metric, it is difficult to say that metric belongs to euclidean (pseudo-euclidian) type or not.

Trivial example

[tex] dl^2 = (dr)^2 + r^2 (d\phi)^2 [/tex];

This metric looks non flat (euclidian), but it is flat euclidian metric of the plane in polar coords. On the contrary, the metric

[tex] dl^2 = \frac{4}{(1+\frac{r^2}{R^2})^2}((dr)^2 + r^2 (d\phi)^2) [/tex], where R - number, is not flat (and not euclidian), because it is the metric of sphere with gaussian curvature K=1/R^2.

In such cases, the Riemannian curvature tensor helps. It equals always to zero on the flat metric.

There is the well known theorem that the ANY analytic metric of 2D surface can be expressed in conformal form. In other words, on any smooth surface it is always possible to find the orthogonal system (or even many systems) of coordinates with the conformal metric:

[tex] dl^2 = f(u,v)((du)^2 +(dv)^2) [/tex].

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- #10

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What, precisely, is not correct?It is not correct.

Your point being? I went to great lengths to point out that the curvature of a conformally related metric is governed by the Lichnerowicz equation; nowhere did I say that a conformally equivalent metric needs to be flat. Elements of a conformal equivalence class of metrics can of course generate different scalar curvatures - where did I suggest otherwise?gvk said:The conformal metric can be or can not be flat.

This is patent nonsense. Conformal flatness is a ubiquitous concept in differential geometry and appears in physics everywhere from electromagnetism to general relativity to Yang-Mills theories to string theory, and many places in between.gvk said:The term 'conformally flat' does not have any sense. The flat metric can be only euclidian or pseudo-euclidean.

''Conformally flat'' - 385,000 hits in Google

http://http://www.google.co.uk/search?hl=en&client=firefox-a&rls=org.mozilla%3Aen-GB%3Aofficial&hs=tE9&q=%22conformal+flatness%22&btnG=Search&meta= [Broken] - 22,000 hits in Google

Again, your point being? Did I not allude to this in the hints I gave to the OP?gvk said:There is the well known theorem that the ANY analytic metric of 2D surface can be expressed in conformal form. In other words, on any smooth surface it is always possible to find the orthogonal system (or even many systems) of coordinates with the conformal metric:

[tex] dl^2 = f(u,v)((du)^2 +(dv)^2) [/tex].

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- #11

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Well, it was not clear to me what do you mean saying:What, precisely, is not correct?

Looking at some metric. Sometimes it looks conformally, but obviously it will not tell you that it's flat.You don't actually need to prove that it's flat - just looking at it will tell you that it's flat.

Example which is similar to posted:

[tex] dl^2 = \frac{(dx)^2 + (dy)^2}{(1-(x^2 +y^2)^2)^2}[/tex];

Does it tell that it's flat?

Of course, not.

Let see again. The term "flat metric" means exactly the same as "euclidian" or "psuedo-euclidian" metric.This is patent nonsense. Conformal flatness is a ubiquitous concept in differential geometry and appears in physics everywhere from

electromagnetism to general relativity to Yang-Mills theories to string theory, and many places in between.

''Conformally flat'' - 385,000 hits in Google

''Conformal flatness'' - 22,000 hits in Google

IT's simply the same thing!

But euclidian or psuedo-euclidian metrics are ALWAYS conformal. What is the sense in using "Conformally flat" term?

May be we know "nonconformally flat" metric? It seems not.

If the people use "Conformally flat" billion times, it does not prove its correctness, from my prospective.

- #12

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Of course that metric isn't flat. Generally speaking, anyone worth his or her salt can look at a coordinate representation of a two-dimensional metric and, if given the ranges of the coordinates, tell immediately whether or not it's flat. In fact, looking at the metric you've given here, you can spot pretty much straight away that its scalar curvature is proportional to [itex]-1/r=-(x^2+y^2)^{-1}[/itex]. At least,Well, it was not clear to me what do you mean saying:

Looking at some metric. Sometimes it looks conformally, but obviously it will not tell you that it's flat.

Example which is similar to posted:

[tex] dl^2 = \frac{(dx)^2 + (dy)^2}{(1-(x^2 +y^2)^2)^2}[/tex];

Does it tell that it's flat?

Of course, not.

Quite seriously, you haven't got a clue what you're talking about. You want an example of how a metric can be "conformally non-flat"? Fine. A metric [itex]\overline{g}\in\mathscr{T}^0_2(\mathcal{M})[/itex] over an [itex]m[/itex]-dimensional manifold [itex]\mathcal{M}[/itex] is conformally non-flat if there does not exist some [itex]\phi\in\mathcal{F}^+(\mathcal{M})[/itex] such thatgvk said:Let see again. The term "flat metric" means exactly the same as "euclidian" or "psuedo-euclidian" metric.

IT's simply the same thing!

But euclidian or psuedo-euclidian metrics are ALWAYS conformal. What is the sense in using "Conformally flat" term?

May be we know "nonconformally flat" metric? It seems not.

[tex]\overline{g} = \phi^2 \cdot \delta[/tex]

or

[tex]\overline{g} = \phi^2 \cdot \eta[/itex],

where [itex]\delta,\eta[/itex] are the [itex]m[/itex]-dimensional Euclidean and Minkowskian metrics, respectively, and where the dot denotes pointwise conformal multiplication.

As I've said already, the idea of conformal flatness isgvk said:If the people use "Conformally flat" billion times, it does not prove its correctness, from my prospective.

And finally, this thread was started by somebody who asked for help with some homework. Don't you think you're doing more harm than good by butting in with comments which are, at best, distracting and at worst completely wrong?

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- #13

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I ask "nonconformally flat", you did "conformally non-flat".Quite seriously, you haven't got a clue what you're talking about. You want an example of how a metric can be "conformally non-flat"? Fine.

...

Do you see the difference?

- #14

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Yes. Conformally non-flat is gramatically correct, while "nonconformally flat" is not. But feel free to use "nonconformally flat" if you feel more comfortable with it. Regardless, I've provided ample evidence that your claims are untrue, so the name by which you refer to the problem is unimportant.I ask "nonconformally flat", you did "conformally non-flat".

Do you see the difference?

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- #15

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The knowing merely the gramatically correct name of something is the same as not knowingYes. Conformally non-flat is gramatically correct, while "nonconformally flat" is not. But feel free to use "nonconformally flat" if you feel more comfortable with it. Regardless, I've provided ample evidence that your claims are untrue, so the name by which you refer to the problem is unimportant.

anything at all about it. It seems, you did not get my point about "nonconformally flat".

I'll try to explain again and keep in mind that

Yes, "conformally flatness" is the wide using term. But, in the same time, its name contains ambiguity and the simple terms "conformal metric" or "conformally equivalent" metric sound more lucid. However, it's not a big deal. The most important that this term is usefull only for the (n>2)-dimensional manifolds, where are not too much conformal metrics and variety of others. It does not make sense to use "conformally flatness" for 1- and 2-dimensional manifolds, because 1- and 2-dimensional manifolds are always "conformally flat". There are no nonconformal manifolds (is that gramatically correct?). This was my point.

Now let's come back what Astronomia84 asked

There are 3 questions. First was answered byDOES ANYONE KNOW HOW CAN PROVE

THISMETRIC/SPACE IS FLAT?

[tex] ds^2 = (-dx^2 +dy^2)e^{ax+by} [/tex]

general what is flat space?

how i can find a transformation to make this metric in flat form?

:surprised

:zzz:

"Calculate the Riemann curvature tensor, and show that it vanishes."

The answer to the second question is:

Euclidian or psuedo-Euclidian are the only flat spaces.

Third one was unanswered.

Your hints for the third question are very vague and did not help.

Your deviation in the area of "a very famous relationship called the Lichnerowicz equation:" would be very interested for graduates, but nothing to do with those questions. By the way, you are not quite correct here too. The relation between scalar curvatures of two conformal spaces was received long before Lichnerowicz (1925) by Eisenhart.

And the last:

No, you can not. The scalar curvature is proportional toOf course that metric isn't flat. Generally speaking, anyone worth his or her salt can look at a coordinate representation of a two-dimensional metric and, if given the ranges of the coordinates, tell immediately whether or not it's flat. In fact, looking at the metric you've given here, you can spot pretty much straight away that its scalar curvature is proportional [itex]-1/r=-(x^2+y^2)^{-1}[/itex] At least, I can...

[itex] -(x^2+y^2)[/itex]

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- #16

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Quite honestly, you're determined to ignore my point that conformal flatness is a ubiquitous concept in physics. If you don't believe me, think about what you know about string theory. Consider the Brink-Di Vecchia-Howe action for a bosonic string:The knowing merely the gramatically correct name of something is the same as not knowing

anything at all about it. It seems, you did not get my point about "nonconformally flat".

I'll try to explain again and keep in mind thatastronomia84asked about specific 2-D metrics and deviations in the area of high dimentions can only be helpful if it demonstrates some general properties of the subject. Otherwise it's irrelevant, at best, and distracting for learning, at worst.

[tex]S_{\textrm{BDH}} =

-\frac{1}{4\pi\alpha}\int d\tau d\sigma

(-\gamma)^{1/2} \gamma^{ab}\partial_a X^\mu\partial_b X^\nu\eta_{\mu\nu}[/tex]

where [itex]X^\mu[/itex] are fields on the world-sheet, [itex]\eta_{\mu\nu}[/itex] is the [itex]D[/itex]-dimensional Minkowski metric, and [itex]\gamma_{ab}[/itex] is a metric on the world-sheet. The

[itex]\gamma_{ab} = e^{2\phi}\eta_{ab}[/itex]

Being able to choose this conformal gauge is then crucial in all of the nice results that you're familiar with in elementary bosonic string theory. Seriously, this is basic stuff.

No, no, no, no! There is no ambiguity in the name whatsoever. Conformal flatness is an implicit definition of an equivalence class of metrics, where the equivalence relation is defined by the existence of some smooth positive scalar function. This is not up for debate: the term is accepted by everyone I know and I havegvk said:Yes, "conformally flatness" is the wide using term. But, in the same time, its name contains ambiguity and the simple terms "conformal metric" or "conformally equivalent" metric sound more lucid.

You're correct: ultimately, it's not a big deal. However, for the purposes of the discussion at hand, itgvk said:However, it's not a big deal.

This was not your point at all. Your claim was that conformal flatness isgvk said:The most important that this term is usefull only for the (n>2)-dimensional manifolds, where are not too much conformal metrics and variety of others. It does not make sense to use "conformally flatness" for 1- and 2-dimensional manifolds, because 1- and 2-dimensional manifolds are always "conformally flat".

It doesn't matter whether it's gramatically correct or not since talking about "nonconformal manifolds" is meaningless. You can talk only about a conformal relationship betweengvk said:There are no nonconformal manifolds (is that gramatically correct?). This was my point.

On the contrary. My hints were perfectly obvious and, more importantly, in keeping with the guidelines of the forum about homework questions.gvk said:Now let's come back what Astronomia84 asked

There are 3 questions. First was answered bycristo:

"Calculate the Riemann curvature tensor, and show that it vanishes."

The answer to the second question is:

Euclidian or psuedo-Euclidian are the only flat spaces.

Third one was unanswered.

Your hints for the third question are very vague and did not help.

Again, no. What I presented was a specific form of the conformal factor, [itex]\phi^{4/(m-2)}[/itex] on an [itex]m[/itex]-dimensional manifold. The appearance of the [itex]m[/itex] in the exponent is crucial; Lichnerowicz chose a different, [itex]m[/itex]-independent expression for the conformal factor, hence my reference to the equation as the "Lichnerowicz" equation. If you compare it to eq. (28.8) in the 1949 printing of Eisenhart'sgvk said:Your deviation in the area of "a very famous relationship called the Lichnerowicz equation:" would be very interested for graduates, but nothing to do with those questions. By the way, you are not quite correct here too. The relation between scalar curvatures of two conformal spaces was received long before Lichnerowicz (1925) by Eisenhart.

Regardless, this thread has gone far enough off topic. Should you wish to argue the point more, please open a new thread.

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- #17

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Calculating all of the components neccesary to show that the Riemann tensor vanishes is a very cumbersome task. It'd make good practice if you are suicidal. :rofl:Calculate the Riemann curvature tensor, and show that it vanishes.

Pete

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