"6 Maxwell's and 12 Einstein 'tensor' equations"

"6 Maxwell's equations and 12 Einstein 'tensor' equations"

Who would have guessed that?

Please enlighten us all.

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This is sure to be a lost cause, but why do you say there are 12 Einstein equations? In 4 dimensions there are 10 independent components of the Ricci tensor and metric because they are rank 2 symmetric tensors (\(R_{\mu \nu} = R_{\nu \mu}\)). That's not quite the end of the story though, because you are free to choose what coordinates you use, reducing the number of independent components to 6. Maxwell's equations can be written as an equation for a rank 2 antisymmetric tensor (\(F_{\mu \nu} = -F_{\nu \mu}\)) which has 6 independent components.

 

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Dear Prometheus,

Thank you for at least trying.

Assume nothing, pretend you are talking to me, I big dummy, who knows nothing and prefers to look at pictures and diagram's.

Can you explain in simple terms, including all equations.

A teacher cannot blame his students, only his teaching methods.

We have all heard of and learnt about the famous '4' Maxwell's equations and '10' Einstein equations.

But it was Prof. Leonard Susskind, who said that there were "6 Maxwell's equations and 12 Einstein 'tensor' equations" and implied they were related.

Can someone write down the "6 Maxwell's equations" and the "12 Einstein 'tensor' equations" and show the connection?

Thanks in advance.

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Why pretend when that's exactly reality?

No, that's why most university courses take years to complete.

What exactly did he say and where exactly did he say it?

The connection between them is via Kaluza-Klein reduction, which Prometheus is aware of.

If you have a 5d metric \(h_{AB}\) such that \(ds^{2} = h_{AB}dx^{A}dx^{B} = g_{ab}dx^{a}dx^{b} + 2e^{\phi}A_{a}dx^{5}dx^{a} + e^{2\phi}dx^{5}dx^{5}\) where [/tex]g_{ab}[/tex] is a 4d metric then if you work out the Einstein Field Equations for h via \(G_{AB} = R(h)_{AB} - \frac{1}{2}R(h) h_{AB} = 8\pi T_{AB}\) and \(\nabla_{A}G^{AB} = 0\) then you find they split into the Einstein Field Equations for g, \(R(g)_{ab} - \frac{1}{2}R(g)g_{ab} = T_{ab}\) and the Maxwell equations for \(A_{a}\), \(\nabla_{a}F^{ab} = 0\), along with scalar \(\phi\) terms.

Thus KK methods provide a way to unify classical gravity and classical electromagnetism, but it costs you a scalar field \(\phi\) whose presence is not seen in experiments on the levels predicted by the theory. These methods are used in string theory which also recasts what \(\phi\) is and means that its not experimentally disproven.

 

Dear Alpha(Nob),

Please keep going, the equations must be written down somewhere on the internet.

Please show all workings and assumptions and definitions.

How did Theodor Kaluza explain it to Einstein?

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Do you think its an insult to call me names when you're also asking me to explain things to you which you don't understand? If I'm such a 'nob' then why do you want my help?

They are also written down in these things called 'books'. They're made of paper, have ink marks on each page which form letters, which combine to form words, which combine to form explanations and help convey information. They're those things other people read to learn.

If you want a full depth explanation of KK methods then buy a book. I'm not going to teach you everything you currently don't know but which is required to understand on a working quantitative level KK methods, as that would take years, even if I thought you were not too stupid to understand them.

Read the wikipedia pages on this stuff and if then you have specific questions I'll answer them. If you can't be bothered to find out for yourself I see no reason for me to be bothered to explain large quantities of physics to you. If you don't give a shit about your learning then neither do I.

Why does that matter? Its common in science for the original formulation of theories or ideas to be quite inelegant and only over time are better formulations developed. Maxwell originally did his work using quaternions, it wasn't until decades later people formulated them in terms of vectors or connections and curvatures of gauge bundles. Maxwell's original formulation is extremely unpleasant to learn from, modern approaches are much better. For instance the entirety of electromagnetism, all of Maxwell's equations, can be written as \(dF = d\ast F = 0\) where \(F = dA\). If you want the upgrade that to include any kind of gauge theory then you just use \(F = dA+A^{2}\).

Of course you have to understand differential forms to understand it but its certainly much easier to work with in that formulation.

For you the light might be on but no one is home....

 

Get off your tousch and go look up the original papers as written by Kaluza himself. Here's where you start: http://www.google.com

 

Counting from the (geometro) dynamical point of view:

In local coordinates, the Einstein field equations become 10 differential equations for the spacetime metric. Six of those involve second order time derivatives, and the remaining four do not. Therefore written in first order form, we have 12 dynamical equations and 4 constraint equations. Because of the divergence free property (i.e. the Bianchi identity) of the Einstein tensor, the 12 dynamical equations are not independent and the dynamics is so that the constraints are preserved. So there are really 8 dynamical degrees of freedom at each point in space. Finally, we have 4 degrees of gauge freedom, which means that in GR the physical degree of freedom at each point in space is 4. If we think of these 4 degrees of freedom as grouped by 2, i.e., that they are forming conjugate variables in the sense of Hamiltonian dynamics, then we say there are 2 dynamical degrees of freedom at each point in space. I heard that these correspond to polarizations of gravitons but I have a very cloudy understanding of quantum field theory. Perhaps somebody can clarify this.

A similar counting can be done for Maxwell's equations. I leave this nice exercise to Terry Giblin.

 

You can say that again, you all must have very clever grandmother's.....

The good news is that I managed to find a good version of the "Kaluza theory for dummies". - Thank you Viktor Toth.

What I am still not sure of, for definite, is the exact relationship between Maxwell's and Einstein's equations.

For the record,

Does this imply that from 6 (or less) Maxwell equations in 5 dimensions, we can derive the 12 Einstein equations in 4 dimensions?

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Did you read any of this thread Terry? I'll repeat my question from post #2:

Why do you say there are 12 Einstein equations?

 

So you've found an explanation for dummies and you still don't understand? What does that tell you Terry?

If you understood what I've said in previous posts you'd know how wrong you are.

The Einstein equations in 5D lead to the Einstein equations in 4D, Maxwell's equations in 4D and additional equations on a scalar field. Obviously all those videos you've been watching haven't helped you, as I said they wouldn't.

 

Dear Alpha(Nob),

Thank you for pointing out my mistake.

I hope Viktor Toth, will forgive me for misquoting him, he clearly states "Starting with empty 5-dimensional space, we can recover an equation of motion in four dimensions that contains the electromagnetic field tensor."

But I preferred your description even better,

Which immediately raises the obvious question, is it possible to work backwards, starting with Einstein equations and Maxwell equations in 4D, what is the relationship between them, which will allow us to combine them together, in 5D?

Did you read any of this thread prometheus?

But he does have a valid point, does anyone here know the answer?

Is it 10 or 12 Einstein equations?

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I suggest you stop Googling for links you don't bother to understand and instead learn some relativity and quantum field theory. The last question you asked was answered in your own link, thus demonstrating that you link to things you don't understand and have made no effort to understand. The quantitative snippets I've given were things you've already linked to so its not like my explanation was much different.

If you spent some time learning and less time deluding yourself about your understanding you might get somewhere. What use is you finding all these links when you don't try to understand anything they say? If you'd actually worked through a lecture course given by someone like Susskind you'd not need to ask the questions you are. There's no point in giving a detailed answer to you about Kaluza-Klein methods when you don't understand what a metric or gauge potential is?

Do you think you understand this link, which you've posted in other threads too? If you did you'd be able to answer your own questions. And that 'for dummies' page still assumes someone is competent at vector calculus and linear algebra, neither of which you can do. Things like 'Kaluza-Klein methods for dummies' isn't the same as 'Windows XP for dummies'. 'Windows XP for dummies' doesn't assume you have ever seen a computer before. On the other hand anyone writing explanations of KK methods for the purposes of students is going to assume the student is familiar with at least 3 years of a maths and/or physics degree, so the 'dummies' in this case are still final year or postgraduate students who have a firm grasp of about a dozen different areas of mathematical physics.

You most certainly don't so until you decide to put in a bit of effort even the 'for dummies' explanations are going to go over your head Terry.

 

Dear Alpha(Nob),

If you are thinking about a career change, you are a natural "barrister". - Who writes your speeches?

You never give, a straight truthful answer, only insults and false accusations.

When am I going to learn, "If you punch a bag of 'you', you get covered in 'you'."

Why am I arguing with a spotty faced, bag of 'you', with a few months 'work experience'?

I have met hundred's of students, post-doc's, numerous Prof.'s and several Nobel Laureates and I have never once had an argument, a disagreement or bad word spoken between us, until you appeared. I wish you would simply disappear or simply go away.

Can we now get back to the problem at hand.

"We have all heard of and learnt about the famous '4' Maxwell's equations and '10' Einstein equations.

But it was Prof. Leonard Susskind, who said that there were "6 Maxwell's equations and 12 Einstein 'tensor' equations" and implied they were related. "

So the question now is, is it "10 or 12 Einstein 'tensor' Equations"?

Answers on a post card.

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Its easy when you can think coherently.

The fact you don't like what I say doesn't make the things I say insults or false. What precisely was false in my previous post?

And yet you're the one asking me to explain things to you. What's it like to know less about physics than a bag of 'me'?

Because before I got my job I spent most of a decade learning these things, including several years working on precisely the topic at hand, Kaluza-Klein compactifications and their generalisations.

You complain I make half true comments but your "You've only got a few months work experience" is doing precisely the same, you're trying to paint the picture that I've only just started doing this stuff which is not the case. The job I have now is not doing KK research, the 'job' I had before was.

I doubt any of those people had any idea as to your claims about physics or your detachment from reality. If they'd spoken to you long enough to hear about your obsession with the double slit experiment and your deluded attitude about your ability they'd have reached similar views about your physics knowledge as I have. Whether they'd say as much in as direct a manner as me I don't know.

None of the students or postgrads on this forum or on any other forum I've seen you on have thought you are competent at physics.

I'd say the same things to anyone else (and I have) when it comes to learning physics, ie you're not going to do it by just watching video lectures, you have to put in some time and effort to try doing problems yourself else you can deceive yourself into thinking you understand something you don't. And by 'you' I don't just mean you, I mean anyone.

None of the people you've met would say watching video lectures is sufficient to get a working understanding of physics. None of the people you've met would make claims like "Sir Martin Rees was my mentor" just because they'd read some of his books. And if anyone else did I'd say to them what I've said to you, you're being dishonest.

Where precisely did he say it?

Besides, if you knew any relativity you'd know that the number of independent Einstein field equations depends on the dimensionality of space-time, as Prom commented. There's a single tensor equation relating to the Einstein field equations, \(G_{ab} = 8\pi T_{ab}\). G is symmetric so if you're in D dimensions you have \(\frac{D(D+1)}{2}\) different expressions in the entries of G but, as Temur explained, there's different ways of counting the unique equations which they reduce to. The associated D conservation equations are \(\nabla_{a}G^{ab} = 0\).

For the Maxwell tensor \(F_{ab}\) its antisymmetric so you get \(\frac{D(D-1)}{2}\) different entries. More specifically there's D-1 electric field components and \(\frac{(D-1)(D-2)}{2}\) magnetic field components. Its only in D=4, ie our space-time, that there's equal numbers of electric and magnetic field components. The components then satisfy conservation equations \(\nabla_{[a}F_{bc]} = 0\) and \(\nabla_{a}F^{ab}=0\). The number of independent constraints then depends on dimensionality and also the counting conventions Temur said.

If you find this explanation unsatisfactory or too lacking in detail then I suggest reading a book on tensors. Well actually I'd recommend you read a book on basic calculus, then one on basic linear algebra, then on vector calculus, then special relativity, then electromagnetism, then electrodynamics and then general relativity. Then you'll be able to answer your own question to as high a level of detail as you wish. I, on the other hand, am not going to teach you all of that ie the majority of a physics degree. After all, if you can't be bothered to help yourself why should anyone else be expected to help you?

 

The hypocrisy of this statement is overwhelming.

 

Do you know what 'hypocrisy' means? That statement would be hypocritical if I were asking people for help who I insult. Am I asking Terry for help? No. You? No. RJBerry? No.

Then its not hypocritical. When you're out buying a physics textbook buy yourself a dictionary too.

 

Yeh -

I do know what it means.

''Hypocrisy is the act of pretending to have beliefs, opinions, virtues, feelings, qualities, or standards that one does not actually have.''

Your opinions and beliefs are not reflected in how you want other people to behave. It is one thing to go around saying other people should not call you names, it's another one when you do the same back.

 

Guilty.

But only to Bullies.

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