Discussion : Magneto

The length of my post was due to the quantity of mistakes I saw. Ability is not determined by length of writing. Some of the greatest works of mathematics have been in papers less than half a dozen pages long.

I agree but not for the reasons you believe.

Please provide a list of resources you used. Doesn't have to be complete, a summary of textbooks and major papers you read.

Not at all. I think the world would be a vastly better place if everyone's science understanding were to be improved. All the things I've said in my first post were in reply to mathematics you posted. You posted equations, I have responded to them in kind. You can't accuse me of trying to play the "Dazzle them with maths" card when you're the one initiating mathematical discussions. Besides, you claim to be well versed in the relevant mathematics and I'm replying to you. As such the level of discourse should be higher than if I were talking to an admitted layperson.

I have no dialled up the mathematics beyond a level you've initiated. When you've given expressions for metric space-time intervals I've replied in kind. I have touched on things like differential forms and tangent co-bundles but these are the basic mathematical structures which underpin what we're discussing.

2) Please provide the metric units (i.e. distance -> m, time – >s, mass -> kg, force -> kgm/s^2) that describe the quantity or term that you are describing.[/quote]The choice of units has nothing to do with what any of us are talking about. The issue is tensor rank. I used the notion of correct units as an analogy. Mixing tensor structures is like mixing units, no matter what the values are its wrong. I agree that \(T_{ab}\) has units relating to energy and momentum, up to powers of c (hence why its often called the energy-momentum tensor). However, equating a scalar to a rank 2 tensor is wrong, regardless of whether or not the components of the tensor have the same units as one another and the units of the scalar. They are separate issues.

This simply isn't how physics is done. You're wanting closed analytic expressions for things. Most problems in physics don't have that.

For instance, while the space-time metric formed by a point mass (ie a black hole) can be computed analytically the space-time metric fr 2 point masses cannot. It's a well known result that some equations have solutions which cannot be written in 'simple' form. For instance, \(f(x) = \int_{-\infty}^{x} e^{-t^{2}}dt\), which is related to probabilities in Normal distributions, cannot be written in terms of basic expressions (polynomials, trig, hyperbolic, logs etc). It is known as the erf function.

This is why descriptions of space-times in GR are given as \(ds^{2} = \ldots\). It is because this is a differential expression which is always true. In order to find lengths or areas or volumes or paths you have to integrate this in some way or use it to derive the associated Lagrangian and solve its Euler-Lagrange equations of motion. Except in unbelievably rare instances this must be done using numerical methods or approximations, because no closed form exact solutions exist. This is seen in quantum field theory too, where it is almost always impossible to solve the equations which describe particle interactions exactly and instead perturbation series are done. This is why QED is well understood, the perturbation expansion is a nice one, and QCD is not so well understood, it doesn't have a perturbation expansion at low energies.

Hell, there's even more basic examples. For instance, the 3 body gravitational problem in Newtonian gravity. The force between two gravitationally interacting points goes like \(\frac{1}{r^{2}}\). If you consider 3 such objects interacting then except for very rare initial conditions its impossible to write down an expression for the points' positions later in time. In fact the system is chaotic. And when I say 'impossible' I don't mean "we haven't found the solution yet" but no such solution exists.

Thus if you restrict yourself to physics where you can solve equations in closed forms you might as well give up now because there isn't an area of physics where all problems have closed form solutions.

Fine, since my name isn't particularly secret I'll say who I am. Obviously my name itself won't mean anything to you but the following are papers I've written and got published

http://arxiv.org/abs/0910.4530
http://arxiv.org/abs/0811.2190
http://arxiv.org/abs/0910.3073

And my thesis is http://eprints.soton.ac.uk/161173/

You can still write a book on your own work, you aren't banned from continuing to publish material, commercial or otherwise, about your work. If you aren't in it for the money but want to advance understanding that shouldn't be a problem. My take on doing research is not to make a name for myself but to help chip away are problems in science and to help the community as a whole.

You still could have submitted the material you put into your books before to published it as a book. That way you get the development time and it is reviewed. There's zero excuse to publish a book before getting the material peer reviewed.

Those expressions are different to what you said. Before you were equating an entire metric, now you're equating individual components. Yes, you can equate \(gh_{\theta\phi}\) to a scalar, it is a component. That isn't what you originally stated.

You shouldn't use 'm' as units, you should be using a generic L as 'length', since the equation can be in metres, feet, furlongs, light years, whatever, so long as its a length. This is an example of poor notation.

I agree that \(ds^{2}\) has units of length squared. That has nothing to do with my comments about your confusion with \(d(s^{2})\) and \((ds)^{2}\).

Furthermore you can't just put in \(\theta \to \theta_{0}\). You have to define what that is and why you're doing it. You only put in specific values when you do the integration I discussed. You go through the derivation later, which I'll get to.

That's precisely what I said.

Do you understand what 'component' means in regards to tensors? A vector in N dimensional space has N components, the entries in the list which define it. What you're referring to is the notion in which you can turn a vector in say Cartesian coordinates into polar coordinates, so you have a modulus, r, and a direction defined by a unit vector \(\hat{n}\). In N dimensional space you need N-1 parameters to define \(\hat{n}\), the polar angles. These too are referred to as 'components' but now in polar coordinates.

I'm wondering if you're struggling with the notion of analogy, this is the second instance where you've failed to grasp it, after my analogy about units vs tensor rank. I gave a simple example, a 2d vector compared to a scalar. It was to illustrate my point. I know you didn't mention specifically 2d vectors but that is irrelevant to my point.

Furthermore 'hypocrisy' wouldn't apply even if your point were true.

I'm not confused, you go on to state a series of algebraic identities which agree with what I said. If you grasp the difference between vectors and scalars why do you mix them in your equations?

The 'PhD crowd' do not refer to the area of a sphere as 'metric of a sphere'. The metric of a sphere (or any manifold) defines infinitesimal lengths on the sphere. The metric can then be used to construct lengths of curves or areas of regions on the sphere (or any manifold) but the synonym you claim is used by the 'PhD crowd' is simply not true.

If you disagree with this then please provide a reputable reference which backs up your claim, ie states that the area of a sphere can be referred to as 'metric of a sphere'. Please ensure it doesn't just use the metric to compute the area, as that would not support your claim and would support what I've just said.

All you've done there is compute \(r^{2}\) in polar coordinates using its relationship to Cartesians. You haven't made reference to a metric or sphere or area. Polar coordinates are used on manifolds other than spheres.

This is completely wrong for the reasons I've previously discussed and you have acknowledged. You have worked under the impression that \(ds^{2} = d(s^{2})\), when it is actually \((ds)^{2}\).

You say you've familiarised yourself with the mathematics of GR but if you had you'd know what \(ds^{2} = (ds)^{2}\). This is because it is basically the extension of Pythagoras's theorem. I've previously pointed you to Riemannian geometry but I'll go over it here for completeness.

Riemannian Geometry

Suppose you want go know the length of a path from \(\mathbf{x} = (x^{1},x^{2},\ldots,x^{N})\) to \(\mathbf{y} = (y^{1},y^{2},\ldots,y^{N})\). Its difficult to compute lengths directly in curved space but we know how to compute the lengths of straight lines in flat space via Pythagoras's theorem. In that case we get

\(length^{2} = (x^{1}-y^{1})^{2} + \ldots + (x^{N}-y^{N})^{2} = \sum_{n=1}^{N} (x^{n}-y^{n})^{2}\)

This can be written in a nice way using the Kronecker delta and Einstein summation convention. We write s for length too.

\(s^{2} = \delta_{ab}(x^{a}-y^{a})(x^{b}-y^{b})\)

Fortunately curved space looks flat really close up and no matter the path you're considering very close up it looks straight too. Thus we split the path up into a great many little sections. We'll consider the first one, from \(\mathbf{x}\) to \(\mathbf{x+dx} = (x^{1}+(dx)^{1},\ldots,x^{N}+(dx)^{N})\). This path's length we'll call \(ds\). The curvature of space comes into play in the fact now all the \(x^{i}\) directions are weighted equally, some might be 'stretched' or 'compressed' a bit. To account for this we define an array of weightings \(g_{ab}\), in place of the equal weightings \(\delta_{ab}\). Putting these expressions into the Pytharorian formula we get

\((ds)^{2} = g_{ab}(dx)^{a}(dx)^{b}\)

This tells us that if we change our coordinate \(x^{1}\) by an amount \((dx)^{1}\), and likewise for the other coordinates, then we'll move a length \(ds\) where

\(ds = \sqrt{g_{ab}(dx)^{a}(dx)^{b}}\)

Now we have to sum up all the little sections of the path. If we parametrise the path by \(\lambda \in [0,1]\), use some calculus and note how in the limit of the summations going to infinitely many (because the sections get infinitesimal) then we have

\(s = \int ds = \int_{0}^{1} \sqrt{g_{ab}\dot{x}^{a}\dot{x}^{b}} d\lambda\)

where \(\dot{x}^{i} = \frac{\partial x^{i}}{\partial \lambda}\). From this construction it is clear that you do not arrive at the original metric by squaring and taking the change, ie

\(d(s^{2}) = d \left( \int_{0}^{1} \sqrt{g_{ab}\dot{x}^{a}\dot{x}^{b}} d\lambda \right)^{2} \neq g_{ab}(dx)^{a}(dx)^{b}\)

If you claim otherwise please provide the step by step calculation which does that.

Furthermore, even if we were to put this fatal error aside your expression involves \(\theta_{0}\). What is \(\theta_{0}\)? It should be \(\theta\), because the formula is true everywhere in the space, not at some specific value of \(\theta\). You can see this by considering just the spherical part of the metric. The \(\sin\theta\) term comes into play from the fact the length of paths defined by constant latitutude on a sphere depend on the value of the latitude. At the poles the path length is zero, ie \(\sin(\theta) = 0\) when \(\theta = 0,\pi\) and the maximum length is at the equator, \(\sin(\theta) = 1\) when \(\theta = \frac{\pi}{2}\). If you have a constant value of \(\theta\) in there then you are failing to correctly describe the behaviour of lengths, areas, whatever in the space. Hence even if your left hand side was \((ds)^{2}\) it would still be wrong.

Your constant value of \(\theta_{0}\) makes the expression false. You continue to confuse \((dx)^{2}\) and \(d(x^{2})\). As such that isn't a good mathematical form for a PhD or anything/anyone else.

I am not trying to confuse things, I am trying to be accurate. Einstein said things should be as simple as possible but no simpler. You have attempted to construct the SC metric using what you view as a simpler method. Your result is wrong in a number of ways. As such no matter how much simpler your approach is it is invalid.

To use another analogy (which I hope you don't also misconstrue) its much simpler to think everything is built from the 4 elements of earth, wind, fire and water, compared to the 110+ elements known to modern chemistry. Does this mean its superior? Of course not, simplicity without validity is worthless in science.

You would be picked apart for being wrong. This isn't a matter of nomenclature. Yes, some of it was due to your poor choice of notation but not all of it.

You had time to write a 3 volume book. As such that excuse just doesn't cut it. You've been in the game for 20 years and in all that time you couldn't familiarise yourself with the basic definitions of expressions central to relativity?

Actually the point of this was not for you to just answer what I said in my first post and then go but for us to have a back and forth. Unless you want to duck out early?

You completely misunderstood what Prom said. He was making the point I just made, that vectors in N dimensional space have N components, as you need N-1 to define a direction in polar coordinates. GR doesn't predict multiple dimensions, it has the ability to work for any number of dimensions. The fact you don't understand this and have made up your own term for this non-existent result, 'complex general relativity' doesn't do you any favours.

If I'm wrong about this then please provide a reputable reference which demonstrates that general relativity necessarily 'predicts multi dimension' (which itself needs to be defined). Working in arbitrary number of dimensions is different from necessarily requiring a particular amount of dimensions (almost the opposite it even!).

Because your expressions do not follow from the SC metric.

I explicitly stated I was considering the spherical metric, not the SC metric as a whole.

No, it wouldn't. s(L) is the length of a path. Hence why the integral is done along the path. The expression \( \int_{\Omega}d\Omega \) is the integral of an area element over an area. Thus they cannot be equal. That equation is simply wrong.

You copy the expression for arc length from Wikipedia yet you must not understand it because it doesn't gel with the methods/results you've previously claimed as your own.

The equations you've just quoted agree completely with what I said. For instance, when considering the path length I said "Parameterise it by \(\lambda\)". If you're interested in the length of a path on a sphere at constant latitude then you can choose \(\lambda = \phi\) and you get the result you've just quoted. The fact you're a constant latititude means you then set \(\theta = \theta_{0}\) but this is not a general result, it is the result for an arc lengh on a sphere at constant latitutude. You have carried this \(\theta_{0}\) through into your attempt to construct the SC metric. This suggests that not only do you not understand the SC metric and its derivation within general relativity but you don't understand the much simpler case of spherical metrics.

This seems to be a regular problem. You state something which isn't completely true and I correct you and walk you through the actual result. You then misunderstand and quote Wikipedia to me on precisely what I just told you. You're repeatedly demonstrating you don't understand the material because you fail to recognise that what I'm telling you is what Wikipedia is telling you. The fact you are able to do a copy and paste from Wikipedia doesn't imply you understand it or that you're correcting me.

It's called being intellectually honest and being well read. People generally look on my physics/maths posts favourably because I've demonstrated over a long period of time that I am competent and educated in the subjects I generally decide to chime in on. I don't go into the chemistry or biology forums much because I haven't got beyond high school education in those, unlike maths and physics.

Furthermore the other people here who have educations/experience similar to my own mean that they can verify that what I generally talk about is accurate, as well as being able to spot cranks themselves.

The general attitude people have to me here is earnt, just as my attitude to them is earnt. None of the people here I respect make claims about "I've written a book which will open the eyes of physicists!" or any other delusional nonsense. The people here who know their shit can demonstrate it in discussions, they don't need to make over the top nonsense claims. Those of us engaged in research don't post it all over the place, unlike the cranks. If someone posts their work on a forum, not in a journal, immediately red flags go up. This post is the first time I've linked to all of my work and the first time I've linked to my thesis at all. You link to your book incessantly. That sort of thing doesn't do you any favours.

The chip on your shoulder is showing. I've earn't a bit of respect because I can put my maths where my mouth is and you don't like it. Remember when you accused me of being jealous of you after I listed a bunch of mistakes you'd made (link can be provided)? Perhaps you're projecting? I've got a bit of respect and I can put my maths where my mouth is, both of which you seem to be lacking and perhaps want....

Its attitudes like that which make me question your claim you've been in academia and involved in physics lecturing. You wouldn't have such a naive and childish point of view if you'd been involved in actual research at universities.

You're just like another self publishing person here, Farsight. He keeps telling people like me I'll be teaching his work in a few years. At least he had the decency to send his work to journals (all of whom rejected him) before self publishing.

To convert me you're going to have to provide arguments for your claims and thus far all you've done is provide further damning evidence you're wrong.

We're now wondering into fantasy land. This post has taken around an hour to type. Weekdays I work 8 hours a day doing mathematics. Thursday night I worked into the evening and Friday night I play squash. As such wanting to sit down for an hour or more to reply to you, to point out mistakes you've repeatedly made, requires free time and a particular frame of mind.

Now that I've done it its clear that nothing you've said has made the slightest dent in my criticisms, in fact you've re-enforced them. Your little imagining of why I'm not at your beck and call says a lot about you. It adds further to my questioning about your supposed career, as your attitude/mindset shows quite a naivety about science in general, never mind the quantitative details.

I haven't shied away from it. It's just your posts contain so many errors its quite time consuming to reply. This 'wear your PhD on your chest' thing is a bit hypocritical. You regularly reference your book (commercial advertising is against the rules), you claim to have insight no one else does and the thread where I threw down this challenge was started by you claiming Kip Thorne's research group basically stole your work!

As I said, this is the first post I've ever linked to my PhD. The level of mathematics in this post is very low compared to what I'm capable of, its stuff I learnt in my 3rd year. I spent 8 years at university in total. Cranks here often think they stretch people like myself or Prom or BenTheMan by talking about relativity. You don't. Compared to the work we do day to day engaging in discussion with cranks requires us to go down several gears, if not put the hand break on entirely. Have a search for threads by Reiku, a well known crank here. He repeatedly made lengthy posts where he just spouted nonsense he'd made up, throwing in equations he'd copied from elsewhere or made up, claiming it to be his 'research'. I've never started such a thread, I've never discussed the specific details of my research, even with someone like BenTheMan who also has a PhD in Type II string theory inspired supergravity models. You once again show your naivety about just how far from the coal face of research you are.

 

Log in or Sign up to hide all adverts.

AlphaNumeric

Why not? It seems that would be a very interesting discussion to read for you both as well as others that could learn a thing or three. :shrug:

 

Log in or Sign up to hide all adverts.

Wow AlphaNumeric, you've always been pretty verbose but that post was 12 pages long! I don't think I care about anything enough to write 12 pages on it.

And Przyk, it's interesting to read your profiles on people, if not mildly stalkerish

Please Register or Log in to view the hidden image!

 

Log in or Sign up to hide all adverts.

I've got a long memory.

 

After I wrote that I realized that I have profiles built up on many people here as well. It might be something like a single word they use (e.g. "envisaging" is more British than American to me), or the time of day they post, that my brain involuntarily clues in on...

 

In regards to a discussion between say Ben and myself it's a mixture of 'talking shop' and the fact the exchange of detailed specifics is quite hard via text. A 20 page paper can take weeks to write, even when you know what you want to say, while its much easier to discuss things face to face. For non-string people it's a mixture of not feeling necessary to talk about it unless someone specifically asks and because my work was almost entirely in abstract mathematics pertaining to string theory. Some people's string theory PhDs involve attempts to get physical results out, like inflation rates or particle families. Mine involved examining structures in spaces which were only defined a few years ago using concepts less than a decade old.

General relativity uses a lot of differential geometry but only a specific area of differential geometry. A particular example is that GR doesn't involve torsion. Despite what Magneto said GR doesn't predict anything about the dimensionality of space-time but string theory does and you've got the extra dimensions. To curl them up you need to stablise their shape but to do that you need to turn on some fields akin to electromagnetism. But this introduces torsion to the geometry. Then you have to transform these fields under a series of dualities which introduces generalisations of torsion such that not only is the metric of the space very complicated, it isn't even defined!! The moduli mentioned in my PhD are variations in the space-time metric, the context of my entire PhD was variations in specific space-time intervals \(ds^{2}\) in string theory, so the stuff Magneto is talking about is very basic from my point of view. Under these dualities you go from a nice metric to one with torsion to one with 'twists' to one where you can't define a metric globally to one where a metric doesn't even make sense, yet the space still has well defined string dynamics in it. This is what a 'non-geometric space' is. It's extremely abstract, a concept which is very new and when I've given talks to my department on it most of the professors (who weren't string theorists) didn't get it. As such delving into anything more than the overview I've just given doesn't serve much purpose.

Actual theoretical physics research is generally so far from everyday knowledge/experience/concepts that to try to do a proper in depth discussion on a forum of laypersons would only lead to everyone going "Errr.... okay...." and stop replying. Crank 'research' never gets that advanced, its always stuff you can do blindfolded if you have a physics degree (or sometimes even high school physics), because cranks rarely are even that competent. Magneto claims to have been in the academic work for decades but his abuse of notation, misunderstanding of terminology and completely laughable attitude to science makes me question that claim.

Since I have nothing to hide I'll answer questions about my work if he or anyone else has any, it'd be a hypocritical of me to challenge his work and yet not defend my own if someone queried it. Bear in mind that if a full answer requires a 10 page essay on how to embed special Kahler manifolds inside quaternionic Kahler manifolds I reserve the right to skimp on the details.

Hence why I said it'd have to wait till the weekend. I didn't want to give Magneto the 'Reiku excuse', which is "Oh you didn't reply to that part of my post so you must accept it!". Unfortunately we're getting into the realm of Magneto doing a Gish gallop, where so many errors are thrown in that responding to all of them becomes a practical impossibility.

Depending on his reply it might be necessary to limit the scope of replies to a specific topic. I had a feeling this might happen since most of his 25 or so forum posts (as of the start of this thread) involved a lot of equations, most of them in error in some way and I basically went through them as they almost invariably pertained to his work (due to him trying to promote his book).

 

I think on crackpottery scale Reiku is an amateur compared to Magneto_1. But they are similar so I thought Reiku improved his "skills" instead of improving his skills in math and physics.

 

Reiku clearly didn't (or rather doesn't) know any maths or physics but he was certainly delusional about his capabilities. Unlike some cranks who know they are ignorant but want to appear knowledgable Reiku honestly believes he is a capable physicist. In one of his many hissy fits complaining about anyone and everyone he once posted a chat log between him and someone where they were talking about SciForums. He complained to said person that I'd mentioned I was stuck on some problem but that I hadn't asked him for help. He honestly wondered why I hadn't asked him for help. He reiterated this to me by PMs too. The fact I would never ask someone who can't multiply out (a+b)(c+d) properly didn't seem to register with him. While Reiku had a veneer of just youthful naivety I think his crankness went very deep.

As for Magneto I'm constrained by the rules of this discussion somewhat. A common method of cranks is to try to aim the discussion so high that it goes over everyone's heads. Reiku couldn't even aim above high school physics. Magneto's topic of discussion is at least undergraduate level and while its not common to end up discussing relativity with physics grads in day to day life you've got to aim a lot higher here to clear yours and my head

Please Register or Log in to view the hidden image!

 

I missed that. Damn.
Not that I actually needed any (more) confirmation that Reiku was nutty as a fruitcake, but it would have been amusing to read nonetheless.

 

I'm curious what AN's and przyk's assessments of me are..? There are really only about a half-dozen posters on this forum who have displayed an intellect deep enough to earn my respect. Don't get me wrong, I try to respect everyone, even the little people...haha:thankyou:

Someone made the case once that most cranks are very intelligent, because the simpletons can't even "get off the ground" with their strange ideas. Farsight is no dummy, but I also don't think his ideas are "his own" (and I don't think he would deny this). He has basically accumulated various alternative theories authored by others, and some of them I happen to have an affinity for. I don't know enough about Magneto, yet, but he has a couple of strikes against him. 1) Shameless promotion of his book as a valid, independent reference, rather than just saying "...as written in my book". 2) He recently pointed me to Wiki and said something along the lines of "I could answer your question easily, but you should learn the answer on your own." It smacked of insincerity, like he was trying to appear smarter than he was. Magneto, if you're reading this, and you're able, feel free to prove me wrong on this point and I'll apologize...

 

You seem to lurch from good, ie fairly rational and not uninformed over to 'bad', being ankle deep in crank denial. You're currently having a 'good day' in that regard.

Depends what you mean by 'off the ground'. Getting 'off the ground' in regards to theoretical physics research is being able to formalise your ideas and develop them somewhat clearly and coherently. Few, if any, cranks on forums manage that. Most of the time its just "I've got this idea about [something], it works by [wordy explanation]". Farsight doesn't use any equations which are beyond high school level, he can't model anything. Reiku couldn't model anything in his work, his equations were just copied from other people or pulled out of his backside. Magneto's aimed higher than most cranks but he's still made a slew of pretty basic (on the scale of GR) mistakes which completely destroy any result he claims to have.

I remember when I was in my 1st year of university and I looked at a problem sheet from 2nd year quantum mechanics. It was complete gibberish to me, I wondered how on Earth it worked. That stuff is as ingrained to me now as basic arithmetic was to me then (I'm 27 now, I was 19 then and so consider what I was doing when I was 11). Hence I can understand why cranks think they are doing something amazingly complicated when they talk about the Schrodinger equation.

Magneto's talking about area elements, line elements, changes in such things. This whole 'd' stuff is formalised in the mathematics of differential forms. While its okay on a simplistic level to consider 'd' as just 'change in' it has a very formal definition. Furthermore it plays such a pivotal role in differential geometry and differential forms are so useful to theoretical physics
that vast amounts of material exists examining them. While differential forms aren't something covered until 3rd, even 4th year of university that doesn't make what Magneto is doing 'advanced'. What he's actually doing with the 'd' or vectors etc is little more than trivial identities. Converting r into x,y,z expressions is just trigonometry. Likewise computing an area element on a sphere due to small changes in latitude and longitude. It's odd that he works with objects/formalisms which are advanced but yet he does almost nothing with them. I use differential forms, the derivative d and its generalisations extensively in my thesis, including to compute volumes of various dimensional spaces. Messing around with units is immaterial.

Which brings me to something I've just realised..... In my lengthy post I said the following :

In my original reply I basically said "Now you've explained it fine". I was wrong and I remembered something else I wanted to comment on.

While the \(Vol = \frac{4\pi}{3}r^{3}\) is okay, if a little dubious notation wise you're wrong about the units and you're demonstrating questionable understanding about dependencies. I'll talk about the latter first.

You write the angle as \(sin^2({\frac{\Psi({\theta},{\phi})}{2}})\), implying you think the argument is a function of \(\theta,\phi\) because the metric component is \(g_{\theta\phi}\). This is not true. The indices represent the effects a variation in both \(\theta\) and \(\phi\) have on \(ds\), not dependencies. For instance \(g_{tt}\) is not a function of t in the SC metric. In the SC metric \(g_{\theta\theta}\) is a function of only r.

Now for the units. I've previously mentioned to you (Magneto) that tensor coordinate transformation require the components to all have consistent units but it goes further than that for the metric. Consider \((ds)^{2} = g_{ab}dx^{a}dx^{b}\). ds and \(dx^{a}\) all have units of length, as they are literally lengths and as a result \(g_{ab}\) is unitless.

The reason you've got a different answer is because you've failed to realise that the form of the SC metric you're quoting is not in SI units. This is why both Rpenner and I commented on it being a bad idea to work in specific units. The SC metric you quote is given in natural units where c=1 and G=1. For instance, in more explicit units the SC radius is \(r_{S} = \frac{2GM}{c^{2}}\). The \(\frac{1}{c^{2}}\) factor acts as a huge supression factor, that's why the SC radius of a black hole is so small.

You should have been aware of this if you're familiar with GR, its a massive simplification to work in natural units and every textbook mentions it right at the start. You really should have noticed your mistake, particularly because the \(dt^{2}\) term in the metric doesn't have units of length so \(ds^{2} = dt^{2} + \ldots \) is meaningless unless there's a suppressed conversion factor. In non-c=1 units it should be \(ds^{2} = c^{2}dt^{2}\), ie distance ~ speed*time.

Rpenner explicitly pointed this out to you in regards to the SC metric here. Your reply, here, I somewhat skipped and didn't read because I was concentrating on what you said to me, not Rpenner but having looked over it there's more mistakes on it. I'll concentrate on what you say in response to Rpenner pointing out f(r) only makes sense if its dimensionless :

Citation needed.

This smacks of you having read a technical definition, not understood it and tried to rephrase it. Clearly Rpenner grasps the calculus, he's provided a lot of it. Instead of replying in kind you give an answer as if you're explaining it to a 5 year old. A 'field' in the GR sense has a much more precise definition than that. While such things as the object which describes the flow of fluid is an example of a vector field the expression you quote, \(g_{tt} = f(r)\), is not a vector field. It's a function which pertains to the gravitational potential and time dilation and I suspect it's a description pertaining to that which you've read and turned into "It's like a fluid going up and down a hill".

In a spherically symmetric system \(g_{tt}\) is associated, qualitatively, to the gravitational potential. In fact \(\Phi \propto \ln g_{tt}\) is the Newtonian gravity potential in the weak field limit. As such its often a term considered in research but it is not the 'forefront' of GR research, its just one of a huge number of interesting avenues.

Citation needed.

This appears to be you jumbling together terms in an attempt to say what I just said, that the term relates to the more familiar Newtonian gravitational potential. The jumbling suggests you don't know what you're trying to say.

No, it is a function of radius because the SC metric is constructed via solving a spherically symmetric static \(R_{ab} = M \delta(\mathbf{x})\) set of equations. Via Birkhoff's theorem there is one and only one solution, the SC metric. Thus the form of \(g_{tt}\) follows by solving a set of coupled PDEs which happen to have a unique solution. That is why its a function of r and r alone, staticity and spherical symmetry precludes \(\theta,\phi\) dependency on that term. That is how you provide the GR derivation of Newtonian gravity and its \(\frac{1}{r^{2}}\) force. Birkhoff's theorem explains \(g_{tt} = f(r)\) which explains Newtonian gravity, not the other way around.

Who are 'those people'? Certainly not GR researchers because anyone whose studied GR properly will be familiar with the notion of coordinate singularities, where the fact some things to go zero or infinity is not a sign of something physical but a bad choice of description. Change from polar coordinates to Kruskal coordinates and you can describe light or normal objects passing through the event horizon smoothly and without incident. That's why tensors are so important, they are valid in any coordinate system where the coordinates are valid.

By relying on particular units and not grasping the importance/role/meaning of tensors and coordinate transformations you've essentially blocked any real understanding of what GR is about.

Citation needed, A LOT! You're now just throwing out buzzwords. Where's your evidence for dark matter being an aether? It's not considered by physicists to be an aether. It is not really an ideal gas or fluid either. Where's your reasoning/derivation/evidence for any of that?

Why stop now, you've speculated on just about everything else.

 

Crank denial...ouch
I'd like to think that I'm ALWAYS rational, even if I'm misinformed about something. Do you have a specific example?

 

AlphaNumeric, thank you for being open about your work. I read though your papers. You are working in the field that I call Complex General Relativity; however you "Young Guys" have named the field "Super String Theory." Based on your work, and in my opinion, you are definitely a closet "Aether" and "Vacuum Energy" theorist.

I don't want to claim to truly understand the mathematics of your work. Because, I don't completely understand your work. But I have navigated through enough of those kinds of papers to get a general idea of what you are doing. The Branes could represent a kind of vacuum Energy medium, and your fluxes could represent the ultra faster than flow light speed and momentum of the particles in this medium.

Keep up the good work!

Now, back to this Tar and Feathering!

I am accusing you of Bu** Sh**!! And that is even more dubious than "Dazzling."

Having knowledge of math is not "Understanding" and understanding only comes with time; and twenty seven (27) is not enough time to mature. You are a kid, and even worse than that, a "Pompous Kid."

How about I list the resources that I have written, would that suffice?:

1) Super Principia Mathematica - The Rage to Master Conceptual Physics - The First Law of Motion, ISBN 978-0-9841518-0-6

2) Super Principia Mathematica - The Rage to Master Conceptual Physics - The Special Theory of Thermodynamics, ISBN 978-0-9841518-1-3

3) Super Principia Mathematica - The Rage to Master Conceptual Physics - The General Theory of Relativity, ISBN 978-0-9841518-2-0

Contained within the above reference is probably the most itemized summary of textbooks and major papers that I have read.

Good, "Clark Kent" keep your suit on there is no need to change into "Super Man", and there is no reason for me to start speaking "French" if you are speaking "English," or if you are speaking "German" and I am speaking "Spanish." We are only able to communicate by speaking the same language and sharing the same tongue. And Mathematics is the universal language of all tongues!!

I do not believe in anything such as a "Point Mass." And you would get infinities, probabilities in Normal distributions, cannot be written in terms of basic expressions (polynomials, trig, hyperbolic, logs etc) using point masses. A point mass is a "wrong" hypothetical model for mass, and nothing else!

Anything that has mass must occupy space is what the Schwarzschild Semi-Major Radius scalar term tells us!

Schwarzschild Semi-Major Radius - Scalar Term

\(r_{S} = \2 (\frac{m_{net}G} {c^2})\) \(-> m\)

Everything that I have presented has a closed analytic expression. What is your problem??

I agree with what you have stated above, however, you spill all of this technical dribble, and yet I have showed you that, when we adhere to metric units and maintain a uniform set of physical parameters pr terms of specific quantities (i.e. (r) length, (t) time, Area(v) velocity, (F) force, (E) energy, (P) Pressure, etc...) We have a cohesive structure for building the models that describe nature accurately.

This term denotes an infinitesimal "distance squared" or "Total Change Area" term, and this is why descriptions of space-times in GR are given as \(ds^{2} = \ldots\)

\(ds^{2} = (ds)^{2}\)\(-> m^2\)

\(ds^2 = \(\ d{\theta}^2 + \ d{\phi}^2 \sin^2(\theta_0) \) = \(d{\Omega}^2 \)\).\( -> m^2\)

\(ds^2 \) = (Area changes with distance + Area changes with direction) = Total Area Change.\( -> m^2\)

AlphaNumeric for clarity each term in the above equation is a line element which is squared, and the mathematical process is the Pythagorean’s Theorem

Taking the square root of the above term will produce a world line distance.

\(ds = \sqrt{(ds)^{2}}\)\(-> m\)

I don't agree, I do believe that this is possible. D'Alembert's principle comes close to showing us that it is possible.

I am however, aware that to date all physicist agree with you that this is impossible!

I believe that a level of understanding in the area of what I call Classical General Relativity is achievable in closed form solutions. For sure in the area that you are working in "Complex General Relativity" or "Super String Theory" you will come to the conclusion that you state!

I stated this before, when I posted originally, I had no Idea that I had to produce mathematical rigor worthy of a PhD. I was being sloppy because, I had no Idea that the Top Dog AlphaNumeric was the gate keeper of mathematical and physics truth!

This was the whole point of this post and your challenge, to clarify the mathematical statements I made in previous post. And I am doing this because the All Star Physics Text Reviewer, AlphaNumeric is not able to understand the subtle nuances of various authors of physics writings.

This sounds totally ridiculous!! Every academic physics text book that is used in every university or college today list the SI Metric units of: (length (m), time (s), mass (kg), energy (kgm^2/s^2)). I really can't believe that you just said this?? You should be a shamed of yourself!! This is how you keep the lay physics person in the dark. This way they can't discover your "Blatant Errors."

When, you continue re-hash that which, as you state "First Years" argue, what does that say about you?

Do you know why I made the distinction between \(\theta \to \theta_{0}\)?

The angle (\(\theta\)) represents the radian measure for any general arc of a curve, without any reference to an origin or beginning and or ending for that curve.

To denote an origin or a starting reference for the arc (\(\theta\)) of the curve we use (\(\theta_{0}\)) to represent the starting origin.

Initial Starting Origin or Reference Angle - \(\theta_{0}\)\(-> radians\)

Now what does \(d\theta\) represent? We use the differential to represent an infinitismal change in the arc of a curve; such that

For example:

\(\theta = \int d\theta = \int_{\theta_{0}}^{\theta_{final}} \ d\theta \)

Your post is so long, that I will get to the rest of your Crap!! later.

Also I am growing weary, of this challenge; because this is more like who has the biggest stick! And that would be me by the way. The big stick that I am using to wack your physics with!!

And, I don't see any profit in proving who has the biggest stick. Perhaps if the questions were more geared towards learning something, maybe we would both be served by the field of this endeavor. But if your goal is to prove that you are smarter than me, then you "WIN!!!"

 

AlphaNumeric, you are so "Dishonest" I provide a derived proof and you accuse me of blindly copying something from Wiki. The correct equation that you are describing I derived for you directly. Where you used some math trick to accomplish the same result.

My goal was to show you that with all of your complex mathematics, I could come to the same conclusion and even the more correct solution, using a simplier method. You are proving my point.

I am assuming \(g_{\theta\theta} = 1 = {r^2}\)
And, you are choosing \(g_{\theta\phi} = 0 = d{r}\)

\(ds^2 = \(d{r^2} + \ {r^2}\(d{\theta}^2 + \ d{\phi}^2 \sin^2(\theta_0) \) = \(d{r^2} + \ {r^2}\(d{\Omega}^2 \)\).\( -> m^2\)

So this becomes according to your condition:

\(ds^2 = \(\ d{\theta}^2 + \ d{\phi}^2 \sin^2(\theta_0) \) = \(d{\Omega}^2 \)\).\( -> m^2\)

and likewise

\(ds = sqrt{\(\ d{\theta}^2 + \ d{\phi}^2 \sin^2(\theta_0) \)} = \ d{\Omega} \).\( -> m\)

I don't think that your above equation, is technically correct, because of this condition (\(g_{\theta\phi} = 0 = d{r})\). I am claiming that you have made a slight technical mistake in your equation above, but you fixed the mistake in the next line. Your rationale produces the result below.

\(s(L) = \int_{L}ds = \int_{\Omega}d\Omega = \int_{\Omega}\sqrt{d\theta^{2} + \sin^{2}\theta d\phi^{2}}\)

I do however, agree with this mathematical "Trick" that you used!

\(s(L) = \int_{L}ds = \int_{L}\sqrt{\dot{\theta}^{2} + \sin^{2}\theta \dot{\phi}^{2}}d \lambda\)


AlphaNumeric, I am going to provide this derivation again, and this is not quoted from Wiki, but derived. And what the audience should note is that I will prove what you proved without all the complexity.

In differential form; if you integrate the equation below you will get the equation above:

This is conditional:
\( d\Omega^2 = \frac{dA_{{\theta}{\phi}}}{r^2} = (d\theta^{2} + \sin(\theta_{0})^{2} d\phi^{2}) = {3}(\frac{dg_{{\theta}{\phi}}}{r^3})\).\( -> radians^2\)

This is general:
\( d\Omega^2 = \frac{dA_{{\theta}{\phi}}}{r^2} = (d\theta^{2} + \sin(\theta)^{2} d\phi^{2}) = {3}(\frac{dg_{{\theta}{\phi}}}{r^3})\).\( -> radians^2\)


Let's review the conditional

Let \( d\Omega^2 = 0 \), meaning that \( \Omega^2 = constant \)

\(d\theta^{2} = - \sin(\theta_{0})^{2} d\phi^{2}\).\( -> radians^2\)

which becomes

\(\frac{d\theta^{2}}{d\phi^{2}} = - \sin(\theta_{0})^{2} \).\( -> unit less\)

Next using the Trigonometric Identity: \(1 = sin(\theta_{0})^{2} + cos(\theta_{0})^{2} \)

Then applying above yields,

\(\frac{d\theta^{2}}{d\phi^{2}} = ( \cos(\theta_{0})^{2} - 1) \).\( -> unit less\)

Next,

\((1 + \frac{d\theta^{2}}{d\phi^{2}}) = \cos(\theta_{0})^{2} = (\frac{ds}{dr})^2\).\( -> unit less\)

Next, we allow the cosine of the reference angle to be equal to the ratio of the change in the Arc Length to the change in the linear distance.

\(\frac{ds}{dr} = \sqrt{1 + \frac{d\theta^{2}}{d\phi^{2}}} = \cos(\theta_{0}) \).\( -> unit less\)

\(ds = (\sqrt{1 + \frac{d\theta^{2}}{d\phi^{2}}}){dr} \).\( -> m\)

\(s(L) = \int_{L}ds = \int_{L}(\sqrt{1 + \frac{d\theta^{2}}{d\phi^{2}}}){dr}\).\( -> m\)

See link for reference: Geodesic - Arc Length

AlphaNumeric, this is completely derived; it is "Dishonest" for you to claim otherwise!

 

1) I have already explained this, Self Promotion as "Spam" is not allowed in most forums. However, I feel if a book has an ISBN it is a valid reference to cite and or quote from, even if it is your own material. To get around the "Self Promotion" police, referencing your work in the third person is technically legal to do!

2) To resolve your problem, If I were you, I would get to know the equation below really well, and the "Friedmann Equations" that I suggested earlier.

The Schwarzschild metric:

\(-ds^{2} = -f(r)dt^{2} + f(r)^{-1}dr^{2} + r^{2}(d\theta^{2} + \sin(\theta)^{2}d\phi^{2})\) \( -> m^2 \).

Best.

 

Starting with the ISBN number, we trace to the publisher and a remarkable publishing record of perhaps only three books.

The very definition of self-publishing
http://www.superbookshop.net/index.php?page=publisher&pub=000503926

(from linked in: http://www.linkedin.com/pub/robert-kemp/31/582/b5 )
* Executive Engineering Director at Flying Car Publishing Company
* Faculty Professor at University of Phoenix - Southern California Campus

The "University of Phoenix" is a for-profit, barely accredited, degree-of-last-resort commercial enterprise with no research facilities to speak of. Consequently "Professor" is not apples-for-apples comparable with the same term as used by the University of California or Oxford or Cambridge. The "Creation Museum" in Tennessee similarly has no research facilities or even storage so it is not so much museum or art gallery, but a Halloween haunted house with a strictly one-way path past the "exhibits". It's a cargo-cult "museum" by people who wish to assume the modern authority of science without adopting the methodology and reliability. I feel similarly about the University of Phoenix (with its regular television and radio ads) and its unimpressive degrees.

There's also paid-for reviews from Gary R. Sorkin who is in business to provide glowing book reviews. Employing such a service is intellectually dishonest -- nay, corrupt even.
http://www.pacificbookreview.com/
http://www.pacificbookreview.com/About-Us.php
http://jetlib.com/news/tag/gary-r-sorkin/

Otto Roessler, a celebrated academic with at least one paper with over 1000 citations made to it, also tried this self-publishing of his less-worthy ideas on a journal where he had influence over its editorial process. He was rightly criticized for it. Authority in science and math derives from ideas passing review, not from vainglorious obtaining of a vanity press ISBN prefix, expenditure of money and assertion.

 

This is the first time I've seen this explanation, but it seems fair enough.

This is precisely as dodgy a response as the first one I was complaining about! I'm not here to join in on the dog-pile, I just want to know the answer (and it doesn't have to be from you, Magneto, but you seem to be "withholding" the answer for no good reason): How can a positive cosmological constant avoid breaking the universal conservation of energy as mass is accelerated ad infinitum relative to all other spatially-separated mass?

 

Which part, specifically, seems "fair enough" --
1) using the ISBN number block to enhance the illusion that a publisher made the economic decision that the book deserved to be printed when this is the work of a vanity press?
In fairness, an ISBN is required for the book to even be listed by many booksellers, but above it's being used as a signifier of merit..
https://www.myidentifiers.com/index.php?ci_id=1479

2) obfuscating that materials referred to are the work of the same author as the forum post, thus creating the illusion that independent thinkers agree on the same useful and communicable principles without actual evidence that the ideas are useful or communicable?
3) contorting the situation until it is deemed "technically legal" (in the potential offender's eye) regardless of what a competent ruling body might say (shades of John Yoo's pro-torture memo!)
4) some other point (I don't want to be accused of casting the situation in a false trilemma)

 

This is simply false and appears to be an attempt for you to twist actual mainstream work into somehow being in line with your work. You said 'complex general relativity' was GR which involved multi-dimensions. Firstly normal GR involves multiple dimensions. Now you're saying string theory is 'complex general relativity'. It is not. GR is a classical theory, it involves in no way quantum mechanics. String theory is a fundamentally quantum based theory which happens to predict general relativity. Aether has nothing to do with any of this and what a 'vacuum energy theorist' is I don't know. That's like saying to a quantum field theorist "You're an electron theorist". QED involves a description of the electron but it's more than that, just as GR involves a description of dark energy but its more than that or string theory involves a description of dark energy but it is way more than that.

Equivocation fallacies are fallacies, try not to make them, particularly when it comes to literally telling me my business.

'Ultra faster than flow light speed'? That's not even coherent. Your take on my work is wrong. You must be aware you don't know about this stuff and the fact we're talking about my PhD means I do. Thus to try and BS your way to pretending to understand it is extremely daft. Is it so hard to say "I don't understand that"? There's nothing wrong with saying that. I don't understand 99+% of physics, despite having a PhD in it. I don't understand 99.99+% of mathematics. The fields are vast. A complete inability to admit "I don't know that" is another massive red flag in regards to crankness.

I think it's a little unwise of you to call me pompous when you come out with comments like I'll be converted into an acolyte of yours and soon be spreading your 'gospel' or how you're beating me with a big physics stick. I am not claiming any more knowledge or capability than I have demonstrated to people. The same cannot be said for you.

As for knowledge and understanding of mathematics I have yet to see evidence you are in a position to tell me how it is. Rpenner's commented on your 'credentials' and they aren't very good. Further more its not like you're the first person whose worked at a university (supposedly) doing maths I've met. Obviously I met plenty during my degree and PhD, not to mention having a father whose a professor of mathematics. I didn't grow up in isolation of people who've been doing maths longer than I and I'm well aware that with time comes increased understanding. At present I have no reason to think you have spent much time at all doing mathematics.

Firstly I asked for 'reputable' sources. Self published books which haven't passed review do not count. The whole reason for this thread is that I dispute the scientific worth of those books and thus you cannot use them as back up for anything you say. Secondly I don't have those books and thus cannot check their references. If you have got all this stuff at your fingertips and in your brain is asking for you to explicitly state 2 or 3 references asking too much? Remember the conditions of this discussion, which are perfectly reasonable.

But that doesn't mean someone who has only just grasped their multiplication tables grasps tensor calculus.

Thanks for providing another reason for me to think you haven't got much of a maths education/background. Once again you fail to grasp an analogy. The integrated Gaussian I stated, ie the erf(x) function, has nothing to do with point masses. It was provided as an example of a function whose closed form cannot be written in terms of 'simple functions'. I mentioned it because the same is true for metrics constructed from multiple masses.

Besides, it doesn't have to be point masses. By Gauss's theorem the space-time metric above the surface of a spherically symmetry stationary ball (ie not a point mass) will be precisely the same as the metric due to a point mass of the same mass as the ball located at the same location as the ball's centre. This should be familiar to you if you've studied GR as its the reason the metric used to describe the Earth or Sun (in approximate isolation) is the SC metric.

Thus neither of your points are valid, but both serve to illustrate a large gap in your understanding and cognitive capabilities.

No, the SC radius tells us that if you put a given mass into too small of a region, ie a sphere of radius the SC radius, then space-time curvature becomes such that causal connection with that mass is lost, a black hole event horizon forms.

Many of the results you've given are incorrect. Even if they weren't that doesn't mean all possible things you could consider will have closed form solutions.

I gave the example of the 3 body problem. Is the fact we can't write down a closed form expression for the position of the three gravitationally interacting objects a sign the physics is wrong or not to be considered?

It isn't 'technical dribble', its accurate relevant corrections. If this is your attitude towards someone giving a damn about accuracy in science then you further demonstrate your naivety about science research.

This is simply denial on your part. I've shown fault with just about every attempted derivation you've done. The fact you got \((ds)^{2}\) confused with \(d(s^{2})\) shows you haven't done the reading you claimed (or at least didn't understand it as you claimed) and, as I've demonstrated, it leads to your results being completely wrong in numerous ways.

Until you can retort what I've said about your \(ds^{2}\) misunderstanding claiming you've got 'models that describe nature accurately' is unjustified and dishonest. Speaking of which....

I just spent considerable time explaining why that is wrong. Infinitesimal distance squared is \((ds)^{2}\). Change in area (if the area is line length squared) is \(d(s^{2})\). These are not the same, \((ds)^{2} \neq d(s^{2})\). In GR \(ds^{2}\) is the former, in your work is it the latter and you still get the SC metric wrong.

You have completely failed to heed anything I've said and having scrolled down through the rest of your post you don't address it, you just reiterate the same mistake.

You aren't retorting what I've said, you just repeat what you've said. Your derivation is wrong. You end up with something of the form \(d(s^{2})\) on the left, which means the right hand side cannot we written in the form \(g_{ab}dx^{a}dx^{b}\) and so it doesn't define a metric. Furthermore on the right hand side of all of your expressions all the \(dr^{2},d\theta^{2}\) etc are all \(d(r^{2}),d(\theta^{2})\), when they should be \((dr)^{2}.(d\theta)^{2}\), which makes the right hand sides wrong even if the left hand side was \((ds)^{2})\).

You have repeatedly quoted the arc length formula from Wikipedia but what you've failed to realise is that it contradicts your results. For instance, when you parameterise a path \(\theta \to \theta(\lambda)\) the arc length formula gets terms like \(\dot{\theta}^{2}\) in them. This is because \(d\theta^{2} = (d\theta)^{2} \to ( \frac{d\theta}{d\lambda} )^{2} = \dot{\theta}^{2}\). Thus the expressions must be \((d\theta)^{2}\) else you can't get \(\dot{\theta}^{2}\). If your expression were right you'd end up with \(d(\theta^{2}) \to \frac{1}{d\lambda} \frac{d(\theta^{2})}{d\lambda}\), which is meaningless, as its got 1 d on top and 2 on the bottom, the differential orders don't match. This simple bit of calculus should have told you your result was wrong, even if you didn't understand that \(ds^{2} = (ds)^{2}\). But instead you are just mindlessly quoting the formula at me and trying to pretend you understand it and I don't.

Evidence says otherwise. Then there's the \(\theta_{0}\) issue which you continue to fail to understand.....

This goes to illustrating you don't know what the metric means and how it relates to path lengths. The metric is not obtained by integration. The whole point is that to get lengths, areas etc you integrate the metric (or some functional of it). Thus the metric shouldn't contain any constants of integration.

You should have realised this from things you quoted from Wikipedia and things I've said. The length is

\(s(L) = \int_{L} \sqrt{g_{ab}dx^{a}dx^{b}}\)

Let's consider the example I've already covered, which addressed your \(theta_{0}\) term but which seems to be lost on you. Suppose you're on a 2-sphere, so we have

\(s(L) = \int_{0}^{1} \sqrt{\dot{\theta}^{2} + \sin^{2}\theta \dot{\phi}^{2}}d\lambda\)

This illustrates the point I just made, that the \(d\theta^{2}\) term must be \((d\theta)^{2}\) or else this parameterisation step is meaningless.

Right, so what paths do we want to consider? Let's consider constant latitude, say \(\theta = \theta_{0}\) and from \(\phi = \phi_{1}\) to \(\phi = \phi_{2}\). Therefore the parameterisation is \((\theta,\phi) = (\theta_{0},(1-\lambda)\phi_{1} + \lambda\phi_{2})\) so \((\dot{\theta},\dot{\phi}) = (0,\phi_{2}-\phi_{1})\) and so

\(s(L) = \int_{0}^{1} \sqrt{\dot{\theta}^{2} + \sin^{2}\theta \dot{\phi}^{2}}d\lambda = \int_{0}^{1} \sqrt{\sin^{2}\theta \dot{\phi}^{2}}d\lambda = \int_{0}^{1}(\phi_{2}-\phi_{1}) \sin\theta_{0}d\lambda = (\phi_{2}-\phi_{1})\sin\theta_{0}\)

If \(\theta_{0} = 0\) then the path is a point and s(L) = 0. If \(\theta_{0} = \frac{\pi}{2}\) then the path is an arc of the equator and the result is the usual arc length. That is how you get your \(\theta_{0}\) into expressions, it should NOT be in the metric itself. Suppose now its a path of constant longitude but different latitude, ie \(\theta \in [\theta_{1},\theta_{2}]\) and \(\phi = \phi_{0}\). By the same method as the previous example \((\dot{\theta},\dot{\phi}) = (\theta_{2}-\theta_{1},0)\) and thus

\(s(L) = \int_{0}^{1} \sqrt{\dot{\theta}^{2} + \sin^{2}\theta \dot{\phi}^{2}}d\lambda = \int_{0}^{1} \dot{\theta} d\lambda = \int_{\theta_{1}}^{\theta_{2}} d\theta = \theta_{2} - \theta_{1}\)

In these simple examples it doesn't matter whether you set \(\sin\theta[tex] to [tex]\sin \theta_{0}\) or not because it comes out the same. However if you computed the length of a path which isn't at constant \(\theta\) or constant \(\phi\) you'd find different results depending on whether you set \(\sin\theta[tex] to [tex]\sin \theta_{0}\) or not. The \(\sin\theta\) factor is that because a circle of constant \(\theta\) is larger near the equator than near the poles, so for the same change \(d\phi\) you get a larger contribution to \(ds\) if you are near the equator than near the poles. If you are varying your latitude then you vary this contribution, it cannot be constant.

What you, Magneto, have done is essentially put in an integration limit before you've done the integration! This only 'isn't wrong' (its not right in any situation) for trivial examples. The fact you don't see this suggests your experience with computing lengths, areas etc in GR is extremely limited. It's not too difficult to code up a numerical solver to compute geodesic lengths if you give it pairs of points on a sphere, particular if you have access to Mathematica which does numerical solving very easily. I suggest you give that a go and see for yourself how the answers are different when you consider non-trivial examples.

If you continue to contend this point then I suppose I could do this coding for you and provide examples but I think you'd learn more if you did it yourself. Alternatively, if you don't like coding but want to do a more analytic approach then you can use the SO(3) invariance of a 2-sphere. Pick any two points, use an SO(3) rotation to put them onto the equator and then compute their length. You'll find that unless there's a very particular relationship between their latitude and longitude coordinates the result you get from your 'metric' and the result you get from an actual 2-sphere metric are different.

That's a nice algebraic exercise and if you're as competent as you claim it should be well within your capabilities and thus you have no excuse for continuing this mistake. Well you should have no excuse at all if you've been doing physics academic work for decades but if Rpenner's assessment of your *academic* institution is right 'academic' is being damn generous.

This 'discussion' shouldn't have lasted very long if you could back up what you say. All you had to do was provide valid derivations of your claims but instead you've just given damn evidence you're grossly mistaken about your work and your abilities. This isn't about proving I'm smarter than you (though I think I am), it's about showing you you can't go around spouting BS and not expect to be called on it. If you want to be taken seriously by the academic community then people like me are the sorts of people you're going to have to convince. I really don't get why cranks are so indignant about people questioning them. You want to be taken seriously but you won't defend your work when questions by the very people you're trying to convince. Unless of course you're just trying to convince lay persons to scam money out of them.

You come to a science forum and repeatedly link to your book. Did you expect everyone to just go "OMG that's amazing, you're a genius!!"? Did you seriously think no one might raise a question (or ten)? It seems to me you came here to try to con lay persons into believing your nonsense and to get some money from of the more gullible people. That I find reprehensible, given you clearly know very little about relativity. I don't take kindly to loud mouth cranks because they basically try to infect lay persons, who might not know any better, with their ignorance and thus they might hinder or damage the learning of lay persons who honestly want to expand their understanding of science. Someone who knows nothing about GR might be taken in by your nonsense, pay you money and then get an utterly flawed understanding of relativity. I see that as a very bad thing because I want people to have a good understanding of science, not some crank's warped view of something they don't understand. Whether you had written a book or not that motivation for challenging you would remain. The fact you've written a book and charge people money for your clap trap is just utterly dishonest and you should be ashamed of yourself.

Tell you what. Rather than reply to anything else I've said restrict your reply specifically to the issue of the SC metric derivation. Can you provide a derivation of the SC metric which results in terms of the form \((dx)^{2}\) and not \(d(x^{2})\) and which has no \(\theta_{0}\) constants in it? In other words can you actually derive the SC metric and not something which superficially looks similar to it but is actually completely wrong?

All other points can be put aside. Let's forget your pointless obsession with units (which you also botched), your love of poor notation and equating different rank tensors. The crux of your work, or at least what you've posted here, seems to be that you claim to have a simpler derivation of the SC metric than currently used by the mainstream. Please provide an accurate correct one.

If it hasn't been peer reviewed then its not a valid reference for a scientific discussion, particularly if its been vanity published. If a publisher comes to you to write a science book then its a sign you're at least of commercial interest, people want to read your work. In the case of well known scientists this is generally because of the quality of your research. If you have to vanity publish then it means neither your research nor your qualitative ideas have garnered enough interest to be worth a publisher's time.

Anyone can publish anything as a book. Some people publish fiction as fiction, other people publish fiction as truth, such as religious people and science cranks. Farsight self published his book and you self published yours. You can't both be right so simply having a book doesn't make your ideas more valid or popular.

Right, I've made life easier for you by restricting the scope of what you have to do to convince me you have something worthwhile, a viable accurate derivation of the SC metric. Let's see you try.