How do you derive general relativistic tensors in an orthonormal basis?

[chaos control]

^^^How do you do this (assuming that you already know your metric tensor and are trying to derive the rest of the tensors in the Einstein field equations in an orthonormal basis)?

I know how to do it in a coordinate basis. I also know the Christoffel symbol formula, the Riemann/Ricci tensor formulas and how to derive the curvature scalar. However, I don't know how to do this in an orthonormal basis.

Here is what I know thus far: I know that you must first derive 4 basis vectors that prove to be normalized when you dot product them with themselves and multiply this dot product by the corresponding element of the metric tensor. You must also make sure that all the basis vectors are orthogonal to each other (so that they equal 0 when you multiply an element of the metric tensor by the dot product of the basis vector and a different basis vector).

Curiously enough, when I did this, I found that if I squared the basis vectors then they would just equal elements of the inverse metric tensor.

Anyway, once I get the basis vectors, where do I go from there (or did I get the wrong information about basis vectors to begin with)? Please help me with this problem if this is your expertise or field of study.

P.S. If you are wondering why I am asking this here on NarutoBase, it is because nobody is telling me where I go from here on the physics forum. That is why I decided that it was worth a shot asking here. Scientific people read manga too after all.

 

[KillerbYo]

What the hell....Sorry I wish I knew, but it would likely take a lot to become smart enough to solve that problem.

 

[SupremeWater]

^^^How do you do this (assuming that you already know your metric tensor and are trying to derive the rest of the tensors in the Einstein field equations in an orthonormal basis)?

I know how to do it in a coordinate basis. I also know the Christoffel symbol formula, the Riemann/Ricci tensor formulas and how to derive the curvature scalar. However, I don't know how to do this in an orthonormal basis.

Here is what I know thus far: I know that you must first derive 4 basis vectors that prove to be normalized when you dot product them with themselves and multiply this dot product by the corresponding element of the metric tensor. You must also make sure that all the basis vectors are orthogonal to each other (so that they equal 0 when you multiply an element of the metric tensor by the dot product of the basis vector and a different basis vector).

Curiously enough, when I did this, I found that if I squared the basis vectors then they would just equal elements of the inverse metric tensor.

Anyway, once I get the basis vectors, where do I go from there (or did I get the wrong information about basis vectors to begin with)? Please help me with this problem if this is your expertise or field of study.

P.S. If you are wondering why I am asking this here on NarutoBase, it is because nobody is telling me where I go from here on the physics forum. That is why I decided that it was worth a shot asking here. Scientific people read manga too after all.

i wish i could discuss this matter with you sir

 

[90sKids98]

I feel like your just being a troll... Then again, eh. Either way idek

 

[Kai NB]

I admire your optimism in thinking there is someone on a manga forum who is capable and willing to assist you.

 

[chaos control]

i wish i could discuss this matter with you sir

Is it that you don't have time to discuss it or that you don't have the knowledge of this subject?

 

[xxsasukex]

WTF Dude. U Know What?

 

[KillerbYo]

Im gonna study a bit....

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[chaos control]

Im gonna study a bit....

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If you wish to self teach yourself as I did, here is a video playlist to help you get the mathematics of it.

You must be registered for see links

 

[Yusuke Urameshi]

I haven't learned anything about any of that, I don't think. I want to take a physics class soon. I like physics, I think.

 

[xxsasukex]

OHH so its Physics!

 

[Tennis Robot]

ResidentSleeper

 

[chaos control]

I haven't learned anything about any of that, I don't think. I want to take a physics class soon. I like physics, I think.

Well you may or may not learn about this depending on what the level of education your physics class is. This is either university undergraduate level or graduate school level. Quite frankly I am not 100% sure right now since I am in the 12th grade. If you are in high school like me (or lower), video playlists can be a huge help in self teaching if you want to learn this.

 

[Vapid]

First and foremost, you must multiply the quadrilateral proxy by the bi-product of x to the 10[sup]th[/sup] power. After this is done, take the relativity of the elasticity multiplied by the coordinate of y. The Gregorian theorem is used afterwards:

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The multiplicative, logistic anomaly is revealed with the usage of the above equation. Your congruent expletive directly circumvents the fifth variable revealed by finding the square root of the e. Then you will have your answer.

 

[chaos control]

First and foremost, you must multiply the quadrilateral proxy by the bi-product of x to the 10[sup]th[/sup] power. After this is done, take the relativity of the elasticity multiplied by the coordinate of y. The Gregorian theorem is used afterwards:

You must be registered for see images

The multiplicative, logistic anomaly is revealed with the usage of the above equation. Your congruent expletive directly circumvents the fifth variable revealed by finding the square root of the e. Then you will have your answer.

I appreciate your response, but do you have any references? This doesn't seem to apply to general relativity. For example, that exponential theorem doesn't even show up in the general relativistic equations (unless it is an element of one of your tensors). Furthermore, you mentioned nothing of the Christoffel symbol and its derivation and how the process you mentioned pertains to orthonormal bases.

 

[Slender Man]

Are you familiar with the solutions of the Dirac equation as “Fermions”? Just to clarify, this means that they are spin-1/2 particles. The spin operator in quantum mechanics can be determined semi-classically by using:

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But since the rotation operator yields:

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it is clear that there are two Eigenstates: ±1/2, as we knew there must be.

More generally, we can define a spin operator such that:

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It is clear that a spin-up particle

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which yields an Eigenvalue of +1/2 for spin. But of course! The particle really is right-handed. Likewise, u− yields −1/2. The v’s are reversed. Classically, these v’s produce a whole bunch of problems.

 

[chaos control]

Are you familiar with the solutions of the Dirac equation as “Fermions”? Just to clarify, this means that they are spin-1/2 particles. The spin operator in quantum mechanics can be determined semi-classically by using:

You must be registered for see images

But since the rotation operator yields:

You must be registered for see images

it is clear that there are two Eigenstates: ±1/2, as we knew there must be.

More generally, we can define a spin operator such that:

You must be registered for see images

It is clear that a spin-up particle

You must be registered for see images

which yields an Eigenvalue of +1/2 for spin. But of course! The particle really is right-handed. Likewise, u− yields −1/2. The v’s are reversed. Classically, these v’s produce a whole bunch of problems.

Thank you. While I am familiar with fermions and bosons and other sub-atomic particles, the field I am referring to is general relativity rather than quantum mechanics.

 

[Senju Bean]

Well, here is the solution.

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[Kai NB]

The copy and paste skills of these people are strong

 

[Migster257]

I'm in freshman honors physics I.
I have no idea what this means. Is this college level physics lol?