Alternative Twins Paradox

Right, my proofs stand in this thread.

 

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Please either learn the correct context of the word 'proof' or stop using it.

And the fact you've managed to alienate anyone who knows any relativity because of your unending denials, lies and strawmen doesn't mean you've reached a logically valid conclusion. No one here who knows any relativity thinks you are anywhere close to valid in your claims. Engaging you in lengthy discussion is clearly pointless and you've said people such as myself are 'too primitive' so why keep asking us to respond?

If you think I'm too primitive to understand this stuff why do you want my opinions and comments? If you're going to ignore them anyway then what is the point of people discussing things with you? If you think we're too stupid to give viable responses why don't you submit your work to a journal?

Either you accept that Pete, myself etc know enough relativity to make our replies to you worth reading or you stick with your "you're too primitive" mindset and then fuck off to find someone less primitive to show your work so. I don't post my work here because I don't want the opinions/comments of people either too stupid or too ignorant to understand my work (a few people can and have provided useful discussions but they are in a minority). So if you think the same about anyone here who engages you in a discussion on relativity why do you want our replies? If we ever get close to nailing your shit to the wall you just change thread, start proclaiming no one responds to you or you just flat out deny anything we say. You can't have it both ways. Either you want our views because you believe we're capable of understanding them and may provide useful discussion or you believe our comments are ignorant nonsense and you simply want attention.

Either you state that you think we've got something worthwhile to say or you stop asking us to respond to your posts.

 

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Check out my first post in this thread. All you can do is pray to it.

What I offered is irrefutable proof SR is a farce.

You have done nothing to stop it. You can only trail behind me.

Listen I do not want any followers. It is not my way.

Othewise, take the first post on.

 

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Come on, are you really that far up your own backside? I'm having a hard time anyone can really come out with such a comment. But then perhaps I'm naive about deluded people, I don't tend to hang around them in real life.

The denial of what people have said to you doesn't mean it'll magically make you correct. And I've explained to you before that the mathematical consistency of SR is equivalent, through various relationships, to the mathematical consistency of basic geometry. You have essentially made the claim that pretty much everything in maths and physics is wrong. Too bad its not the case. I'd explain again but I know it would be lost on you given you can't grasp the meaning of 'proof'.

Flawed logic. I've already discussed relativity with you and its clear you don't want to listen. The fact I'm not particularly willing to waste another 5 pages of discussion and repeated explanations to you is hardly a sign you're right. You've been provided with retorts for all your threads, you just don't want to listen.

Again, can you really be believing what you're saying? You time and again seem to think that you're revealing new results or explaining things to me I've not heard or seen before. This is not the case. The fact you are naive about the length and breath of what I've studied doesn't mean you making up claims about it is a viable line of argument.

You certainly want attention.

So you expect me to do your bidding while you don't answer my direct questions? Nice hypocrisy. I asked you to tell me why you're so desperate to get the comments of myself when you've said you believe I odn't know any relativity, that I'm a poor student and that I'm too 'primitive'. If you can't explain why you want the comments of someone you deem ignorant and not particularly bright then it might seem like you're unwilling to admit you're simply trying to get attention. Why do you crave comments from people who you brand ignorant yet you're unwilling to submit to a journal review?

Either you want my comments because you think I can provide an informed critique of your 'work' or you want it because you want attention. Which is it? Please actually answer direct questions as its hard for you to simultaneously claim people won't talk to you about your claims while avoiding any and all attempts to make a discussion without you looking like a fraud.

If you've got noting to hide put some money on the table and submit your work to a journal. You can do it anonymously if you don't want the attention. You have no excuse other than you knowing you'll be rejected. You are scared to put up or shut up.

 

Gawd, it is comic book.

And you still cannot do anything?

Yes, I am going to do something, like I said, I must lower its level.

Really, even after I offered it as comic book you still submit?
Everyone wonders here why you claim to be so intelligent yet must submit to my findings.

 

I've previously engaged you in conversation on the matter. People only need to check the first page of this thread to see that. Whenever you're presented with something which requires you to admit a mistake you simply go into the denial. Its there for people to read. You keep saying "My proofs stand" which is just flat out denial.

Comic book? What are you talking about?

Why do you keep making completely unjustified non-sequitors?

Firstly I am not forced to 'submit' to your findings. Myself and others have engaged you in discussion and generally you fail to understand (or want to understand) the points anyone makes. When all your 'logic' boils down to "Because I say so" no one has to submit to any of your findings because you can't justify them. Besides, what has my intelligence got to do with whether or not I submit to your findings? If you were right then its impossible not to submit as the logic can't be argued with. If Person A makes a logically supported claim then Person B's intelligence has no bearing on whether Person A is right or not. Again, you seem to be unable to form a coherent and logical string of reasoning, And this isn't even difficult stuff, its just basic reasoning.

I replied to your original post and the discussion evolved from there where I went through some of your claims about special relativity and explained why you're incorrect. You obviously don't want to heed any of it else we'd not still be having this discussion. You ignored any and all points of contention or explanation provided by anyone and so I see no reason to need to go through it again. Even when someone has a valid point you ignore them, believing you've got superior understanding to the entire history of the maths and physics community.

You again ignored my direct question about why you crave comments from people you've previously dismissed as 'too primitive'. It would seem you can't admit to the answer, either you think I do understand relativity and my comments are worth listening to or you simply want attention. If you're unwilling to submit your work to a journal then why should anyone put in the effort to talk details to you? You demand I put in time walking you through material you've refused to read and which you refuse to submit for review to anyone whom you don't deem 'primitive'. If you can't be bothered to put in even that amount of effort why should anyone else?

 

This is all crap.

You never stopped anything I proved.

I do not think you could on your best day.

Simply admit it is beyond you.

You see, here is what I would have done if I can stop something.

I would post the link where I refuted it instead of typing all this nonsense.

Do you have said link?

 

You claimed therre exists a LT which maps a point on the light cone to a point not on the light cone. This is false. I proved so by using the definition of a LT, that is leaves the space-time interval measure \(ds^{2}\) invariant. If \(ds^{2} = 0\) is the definition of points on the light cone then by definition you have the LT coordinates \((ds')^{2} = ds^{2} = 0\). A point on the light cone in one frame is on the light cone in all other frames.

Once again you think that because I'm talking basic physics to you basic physics is all I can do. No, the topic is at such a low level because you're talking on such a low level. If you were to raise the level of discussion to general relativity or quantum field theory I'd be able to raise my discussion to.

Cranks always do this. They start a discussion on something very basic and then assume that people talking to them can't do anything more complicated or that they just parrot from textbooks. Flawed reasoning. They are pointed to textbooks because their misunderstandings as so simple they are covered in books. In your case we're stuck doing 1st year calculus because you're unable to get past even that. If you wanted to discuss Lorentz transformations and their role in general relativity I'd be happy to talk to you about veirbeins. If you wanted to discuss the role of Lorentz transformations in quantum field theory I'd be happen to talk to you about field quantisation.

You time and again make statements which imply you believe I'm only just struggling to the results or that I've never seen the algebra or physics you're talking about before. Your naivety as to my level of ability does not mean I don't have any additional knowledge. I'm dumbing my conversations down for you benefit, not mine. If you were able to raise your discussion to the level of fibre bundles then you'd grasp my comment that the mathematical consistency of Lorentz transformations is equivalent to the mathematical consistency of geometry itself. But you're unable to do that and so we're stuck at Calculus 101.

So you want me to link to a post of mine which you've already read to confirm to you that I've already said something you know I've said. This is a classic crank tactic. You know I've said it to you but you pander to an imaginary audience hoping that they haven't seen said posts and thus it might appear to them I've not said anything in retort to your claims. I went through precisely this already in this post. Each link is to a thread you've been in, to posts you've read and responded to. So you already know about them but you keep saying "So where are they?!" in other threads in order to make it seem like I've not responded.

If you have to resort to such dishonest tactics then it suggests you aren't as confident in your work as you try to imply.

 

You claimed you did not say there where multipe light spheres and I corrected you and proved you did.

As you can also see, I taught you that idea.

Yes, if you use exclusivly LT, then you can prove some nonsense.

But, for LT to be useful, it would need to preserve the light sphere from the stationary frame to the moving frame. The mapped sphere is not spherical as Einstein claimed.

I thought you knew all that there fancy geometry stuff.

Anyway, once you understand it is supposed to be spherical in the moving frame and LT cannot map it that way, intelligent folks then try to think about it.

So, we know the clock at the origin elapses r/c at the origin for the light sphere of the moving frame.

That point I developed, elapsed r/c in both frames. We then calculate where LT claims the positive x value is in the moving frame and then as a backup calculate that is r from O' since it is supposed to be a light sphere when viewed in the moving frame. A light sphere would need to have an equal radius in all directions. You know that right?

They do not match. SR is done as a result.

You have to be willing to wonder off into the land of facts if you are ever going to learn past the current failed SR theory.

Here you are with all this lecturing and the first post stands firm and unchallenged in this thread and brings down SR forcing it into a logical contradiction.

It would seem you would at least show some of this ability first before you think of yourself as superior.

Oh well.

 

I've said repeatedly that each and every frame has one and only one light sphere at any given moment. Each light sphere is obtained from the light sphere in another frame via a Lorentz transformation. Nothing I've said isn't basic well known standard knowledge you'd find in a textbook on the matter. You'd know if you'd ever read any.

The only non-trivial point you raise is that each frame will see a particular point in space-time as the centre of that sphere and they are not mapped to one another by the corresponding Lorentz transformation. Its non-trivial because its counter intuitive. However it is not a new result or even a problematic one. If you grasped coordinate transformations in non-Euclidean spaces you might understand but you simply don't want to listen to anyone who might have a bit more experience and understanding than you.

You keep trying to claim you've taught me or someone else something new. You demonstrate your naivety about special relativity, your naivety about my level of knowledge and a delusion that you're infallible. Also, you sound like you're trying to convince yourself more than anyone else.

So if I use the transformations which are defined to be the transformations which map between inertial frames in special relativity then I'll prove 'some nonsense'. You basically just said its wrong to use the defining transformations between inertial frames in special relativity. If I use nothing other than LT then I should be working entirely within special relativity since it is the physical study of the structure and geometry of Minkowski space-time. Lorentz transformations are the most general set of transformations consistent with this space-time by definition.

You have just demonstrated you're not working in special relativity or you aren't interested in what SR actually says. If you knew any more advanced mathematics you'd know that the Lorentz group, SO(3,1), is the group which defines the principle G bundle of the vector bundle defined by Minkowski space-time. The study of Lorentz transformations is entirely equivalent to the study of the geometry of Minkowski space-time. I have previously explained this to you several times. Once again we go around in circles because you're either unwilling or are incapable of understanding.

It is spherical. You seem to be confused by the fact the region inside the light sphere is distorted. The light cone maps to a light cone. I've proven this for you several times now using space-time intervals. All you have to do to see this is draw a 1+1 dimensional space-time diagram. The edge of the light cone is formed by the \(x = \pm t\) (for c=1) lines. Under a Lorentz transformation this line doesn't change despite the x direction being warped by the contraction in x. Its countered exactly by the contraction in t but only for the points on the [te]x= \pm t[/tex] lines. Any other line will change angle. This is covered in any good textbook on Lorentz transformations.

If you forget that the time direction is also changed then if you're only drawing a slice out of the light cone then yes, it will appear distorted but that is because you haven't accounted for the case the slice you take is also distorted! The slice is defined by a t=constant but t transforms too! You've not realised this and thus you have made the mistake in thinking the result is not on the light cone.

I've now explained your mistake using algebra (ie space-time intervals and null vectors), pictures (worldline diagrams) and wordy explanations. You are wrong. I see what mistake you've made, I understand why you've made it and I've explained why its not a problem with SR but with your understanding. All you do is jump to the conclusion it couldn't possibly be your fault. Its naive, ignorant and damn arrogant. But then that pretty much sums you up.

And I've provided you with it and you've never retorted it.

If a vector satisfies \(p^{\mu}p_{\mu}=0\) then its on the light cone. It Lorentz transforms to \(P^{\mu} = \Lamda^{\mu}_{\nu]p^{\nu}\) and by the defining properties of Lorentz transformations \(P^{\mu}P_{\mu} = 0\). Also on the light cone. Light cone is mapped to light cone. QED.

Do you honestly believe that in 100 years no one has ever applied a Lorentz transformation to the light cone? That no one has done computer simulations of these things?

I've already walked you through the \(p^{2} =0\) goes to \(P^{2}=0\). That's all you need. Light cone to light cone. Individual points move about the light cone but when you consider the entire image of a light cone under a Lorentz transformation you find the entire light cone maps to the entire light cone.

Which is completely immaterial and a strawman you have created because you dont' grasp coordinate geometry. You don't get it, I agree that different frames will point to different points when asked "Where is the centre of that light sphere" but it isn't a contradiction. It's a result of working in a non-Euclidean geometry. You're assuming Newtonian aka Euclidean properties apply to non-Euclidean geometry.

That point I developed, elapsed r/c in both frames. We then calculate where LT claims the positive x value is in the moving frame and then as a backup calculate that is r from O' since it is supposed to be a light sphere when viewed in the moving frame. A light sphere would need to have an equal radius in all directions. You know that right?

And you have to be willing to listen, learn and understand. You come here and do none of them. You lie and claim no one has retorted you when I'm currently replying to you in what must be 4 or 5 different threads with pictures, equations and explainations. Others have responded too. You're unwilling to put your money where your mouth is. You simultaneously call me 'too primitive' then whine if I don't reply, proving you're an attention whore. And all the while demonstrating you aren't interested in what SR actually says only what you believe it says or what you believe it should say. Either way all of your claims boil down to you not understanding some simple point covered in any elementary physics course. A course you obviously never took.

See this is simply flat out denial and lying. Your posts have been responded to by many people in many threads. Your errors demonstrated, problems explained. All you have now is just denial, lies and the naive belief you're doing something new or interesting. If all your posts are just "Oh look, no one has challenged my first post", ignoring any and all challenges to your claims then it demonstrates you're a desperate liar. I've provided a retort to your claims about distorted light spheres, different centres, intersecting spheres. It's all covered and explained. The problem is you're deaf to any thing which forces you to admit you're wrong. Is it so hard to accept you might not have a perfect grasp of a subject you don't know much about and which you have little experience with? Really?

I love how you try to take a pot shot at me for supposedly thinking myself superior when you're basically claiming to be superior to all physicists and mathematicians ever and you believe you're the first person to ever do something like apply a Lorentz transformation to a light cone! You demonstrate your naivety about physics. And your misplaced delusions of grandeur.

You have yet to demonstrate you have a working understanding of anything beyond basic 1st year algebra. Whenever you've wandered out of your preferred area of whining about special relativity you've been smacked down for being wrong. Your claims about set theory being such an example. All you do now is stuck with your flat out denial of any retorts people provide your claims, you don't venture into any other discussions such as helping people with homework or providing valid explanations for anything which can be checked and require you to have actual knowledge and ability. As I said previously, you strike me as someone who was never a particularly good student and you told yourself its because you're thinking so much further than anyone else that they don't understand you. Now you're out of the formal education system and don't set exams or tests you are free to delude yourself and whenever anyone points out your mistakes you just ignore them. You're 'that guy', the one who always has to top everyone and who eventually spins so much bullshit their lies because pathetically transparent.

In a formal set of exams I'm sure I can paste you in terms of ability. I've definitely got you beat in terms of level of education and amount of original work produced and published, that's a demonstrable fact. Go on, provide me with one post of yours where you show understanding and knowledge on a topic which would not be covered until 3rd year at university. Just one example. And obviously it has to be one where you weren't quickly corrected by someone else.

You like to talk big but the fact you're whining about SR on a forum and not publishing the work and that you're having to resort to repeated transparent lies says it all really. You like to pretend you're not another crank but just like all the other cranks its a self delusion brought on by ignorant and naivety.

 

Yes, LT does some mapping of some points to somewhere.

Let's see, each possible frame has its own center of this so called one light sphere.

So, first off, you must understand one sphere has multiple centers at different locations in space. This you agree.

What is this time t for each and every frame of when its personal light sphere reaches a distance r in the time coordinates of one selected frame.

If SR and LT is complete, it will provide an answer.

Now, what is the answer?

You made a bad mistake here. In any non-Euclidean space, you cannot map one point to many in that same space. There does not exists such a space or the distance formula would fail. That is required to map one center of the light sphere in the stationary frame to all possible frames light sphere centers.

So, why would I want to listen to someone that believes in the theory that requires the impossible to be true?

Yea, when you answer when in the time coords of a frame that all other frames will see the light sphere a distance r, then we can think about this correctly.

What I have correctly realized if LT cannot map to a time in the stationary frame of when light is a distance r in all directions of an arbitrary moving frame, then I do not have a transformation.

You see, your theory claims this time exists and is r/c. Now when in the stationary frame does this time occur. Your theory does include time, am I correct?

This twins contradiction is at post one killing your theory. Why do you refuse to confess the facts?
In fact, try to stop it.

Since you claim I am so basic, then this shoud be easy for you.

How long will this take?

 

I forgot to mention, there is not way to make it a light sphere in the moving frame.

I showed you a picture. It is length expanded in the moving frame by a factor of γ along the x-axis. You can play with the simultaneity of your closk all you want, but you cannot solve this problem.

Do not forget, if LT is true, it must preserve the light sphere is some way. But, that is impossible under LT.

So, it incorrectly maps the light sphere.

Your space time diagrams only account for the term vtγ and completely forget about the xγ term which is expanded in the moving frame taken as LT mapped stationary. It cannot be a sphere.

Your theory fails.

 

I also, on account of you, reworked my R of S argument/Einstein's Train/Embankment Experiment that refutes SR.

I included lots of pictures to make it more readily understandable. I think I did not word it well before and pictures help.

It is the third one down.

http://www.proofofabsolutemotion.com/srproblems.html

 

IT follows by applying a LT to the sphere. Ignore the centre because its obviously confusing you.

No, each change of coordinates maps one point to one point. They are all invertible, which immediately means its 1-1 and onto. If you'd ever studied coordinate transformations you'd know about injective and surjective. When its both its bijective. A non-singular coordinate transformation, ie \(det(\Lambda) \neq 0\), gives an invertible transformation. By definition \(det(\Lambda)=1\) for LT so its 1-1 and on-to. Yet another example of you not understanding stuff you should have covered if you're as educated as you claim.

Different coordinate transformations gives different images. This is not what you're saying.

I can't help but notice that you're not actually retorting anything I said. I just quote it and ask a question which may or may not have anything to do with what I've said.

Can you please respond to my mathematical demonstration that \(p^{\mu}p_{\mu}\) is mapped to \(P^{\mu}P_{\mu}\) which satisfies \(p^{\mu}p_{\mu} = P^{\mu}P_{\mu}\). This demonstrates a light cone goes to a light cone. Please respond. You keep asking for 'maths proofs' and when someone does the mathematics you don't respond.

You are simply claiming what you believe SR to be. Rather than attacking a strawman about what you believe SR to be why not actually find out what it is and examine that? For instance, rather than asking leading questions which are aimed at you making yet another strawman why don't you respond to the points I've made?

Why do you keep telling such pathetic lies? People, including myself, have replied to the posts. As my previous post demonstrates, I'll provide mathematics and explanations and you just refuse to respond to them. You demand 'math proofs' and then can't respond when someone does mathematics. 4-vector norms are invariant under Lorentz transformations by definition and the 4-vector norm of zero defines the light cone. I've said this several times in several threads and none of them have had a response from you which actually addresses what I said.

Ah the "Because I can't think of it its not possible" argument from incredulity.

If you're struggling with the picture then consider just the 4-norm of a given vector. If \(p^{\mu}\) is on the light cone and it maps to \(P^{\mu} = \Lambda^{\mu}_{\nu}p^{\nu}\) then what is the 4-norm of \(P^{\mu}\)? If you think its non-zero prove it. If you agree its zero then explain why its not on the light cone since that's the definition.

This is the simplest easiest way because you don't have to worry about appropriate time slices etc. All you are considering is a single point.

Still not submitting your work vto a journal. If you consider me 'too primitive' why do you want my views? I've asked you several times and every time you avoid the question. If you're so confident why are you avoiding simple questions? You ask for maths, I provide it. You ask for pictures, I provide it. I have explaned why your understanding of the mathematics is wrong. I've offered to raise the level of discussion to fibre bundles but you don't take me up on it. You want people to believe you're well educated but you can't provide any evidence. I asked you for one link, just one link, to a post of yours doing mathematics or physics which is covered towards the end of a degree. You ignored me. I'll provide links to me doing advanced stuff. I've got nothing to hide.

You're full of talk but when it comes to someone providing retorts suddenly you change ther subject or repeat some already exposed lie. Come on, actually retort the mathematics I've given. \(ds^{2}=0\) is pretty simple.

 

I get that impression too. His misuse of the word 'proof' is at odds with his claim that he's familiar with mathematical areas like set theory or logic. He tries to use the word 'decidable' yet he's unable to correctly use the word 'proof'. It's not possible for someone to learn about decidability and not learn what 'proof' means. He demands people provide mathematics to justify their claims and then doesn't respond to the maths when people provide it. He claims he's found a Lorentz transformation which maps a point on the light cone to a point not on the light cone but can't respond when I prove (and I'm using the word correctly here) that such a transformation isn't a Lorentz transformation then, as it violates the definition of the transformations.

Never the less for the purposes of being complete and to bring things like the results of this post all into one thread (else it'll scatter into many different threads and its silly to say the same stuff 4 times in different threads to the same person) I'll go through the maths in more detail here.

Definition : A Lorentz transformation is one which leaves the metric \(\eta\) of Minkowski space-time, hence for referred to as M, invariant.

Definition : Minkowski space-time is isotropic, homogeneous, flat and has 1 time-like direction and N space-like direction.

Given the tangent bundle TM the metric acts as \(\eta \,:\, TM \times TM \to \mathbb{R}\). If \(X,Y \in TM\) then \(\eta(X,Y) \in \mathbb{R}\). The dimension of TM is N+1.

Definition : The Lorentz group G is therefore defined as \(G = \{ \Lambda \in \textrm{GL}(N+1,\mathbb{R}) \,s.t.\, \eta(X,Y) = \eta(\Lambda \cdot X,\Lambda \cdot Y) \, \forall X,Y \in TM \} \equiv SO(1,N)\)

Given coordinates of M \((t,x_{n}) \equiv x^{\mu}\) the induced coordinates of TM are \((\partial_{t},\partial_{x_{n}}) \equiv e_{\mu}\). A general element of TM has components in these coordinates \(X = X^{\mu}e_{\mu}\). The components of \(\eta\) are then \(\eta_{\mu\nu} \equiv \eta(e_{\mu},e_{\nu})\).

The Lorentz transformations act on the components of vector as \(\Lambda : X^{\mu} \to \tilde{X}^{\mu} = \Lambda^{\mu}_{\nu}X^{\nu}\). Inserting this into the definition of the Lorentz transformation we have \(\eta(X,Y) = \eta(\Lambda \cdot X,\Lambda \cdot Y)\) becoming \(\tilde{X}^{\mu}\tilde{Y}^{\nu}\eta_{\mu\nu} = X^{\mu}Y^{\nu} \Lambda^{\rho}_{\mu} \Lambda^{\sigma}_{\nu} \eta_{\rho\sigma}\)

This is true for all X and Y and therefore we have the defining action of a Lorentz transformation in terms of coordinate induced components : \(\eta_{\mu\nu} = \Lambda^{\rho}_{\mu} \Lambda^{\sigma}_{\nu} \eta_{\rho\sigma}\).

Definition : All indices are raised and lowered by \(\eta\). \(X_{\mu} = \eta_{\mu\nu}X^{\nu}\) and \(\chi^{\mu} = \eta^{\mu\nu}\chi_{\nu}\) where \(\eta^{\mu\nu}\eta_{\nu\rho} = \delta^{\mu}_{\rho}\).

Definition : A vector \(p = p^{\mu}e_{\mu}\) is time-like if \(p^{\mu}p_{\mu} < 0\), space-like if \(p^{\mu}p_{\mu} > 0\) and null if \(p^{\mu}p_{\mu} = 0\).

Under a Lorentz transformation the components of \(p = p^{\mu}e_{\mu}\) are mapped to the new basis and the vector components transform as \(p^{\mu} \to p^{\mu}\Lambda^{\nu}_{\mu} \equiv P^{\mu}\). Using the raising and lowering properties we have \(P^{\mu}P_{\mu} = P^{\mu}P^{\nu} \eta_{\mu\nu}\). We insert the definition of \(P^{\mu}\), \(P^{\mu}P^{\nu} \eta_{\mu\nu} = (p^{\sigma}\Lambda^{\mu}_{\sigma})(p^{\rho}\Lambda^{\nu}_{\rho}) \eta_{\mu\nu} = (p^{\sigma}p^{\rho})(\Lambda^{\mu}_{\sigma}\Lambda^{\nu}_{\rho} \eta_{\mu\nu} )\). By definition we have that \(\eta_{\sigma\rho} = (\Lambda^{\mu}_{\sigma}\Lambda^{\nu}_{\rho} \eta_{\mu\nu} )\) and therefore \((p^{\sigma}p^{\rho})(\Lambda^{\mu}_{\sigma}\Lambda^{\nu}_{\rho} \eta_{\mu\nu} ) = (p^{\sigma}p^{\rho})\eta_{\sigma\rho} = p^{\sigma}p_{\sigma}\). Therefore \(P^{\mu}P_{\mu} = p^{\nu}p_{\nu}\). Lorentz transformations lead the norm of a vector unchanged.

The light cone is defined by points which are on null trajectories from the apex of the cone. The light cone whose apex is (t,x) = (0,0) is defined by points which satisfy \(\sum_{i}x_{i}^{2} = t^{2}\). This is the equation of a sphere centred on the origin whose radius is proportional to time. It can be rearranged to \(-t^{2}+\sum_{i}x_{i}^{2} = 0\). In terms of vector components we have \(x^{\mu}x_{\mu} = 0\) and in terms of the metric \(x^{\mu}x^{\nu}\eta_{\mu\nu} = 0\). By the above result this is Lorentz invariant, \(y^{\mu}y_{\mu} = 0\) and therefore the light cone maps to the light cone.

We consider this explicitly for a Lorentz boost in the \(x_{1}\) direction by speed v. The explicit form of the Lorentz transformation is determined by the coordinate relationships.

\(\tau = \gamma (t - vx_{1})\), \(y_{1} = \gamma ( x_{1} - vt)\), \(y_{i} = x_{i}\) for \(i \in \{2,\ldots,N\}\)
Inverting these transformations
\(t = \gamma (\tau + vy_{1})\), \(x_{1} = \gamma ( y_{1} + v\tau)\), \(x_{i} = y_{i}\) for \(i \in \{2,\ldots,N\}\)

The definition of the light sphere is \(\sum_{m=1}^{N} x_{m}^{2} = (ct)^{2}\). We rearrange this to collect the non-trivially transformed coordinates, \(\sum_{m=2}^{N}x_{m}^{2} + \Big( -t^{2} + x_{1}^{2} \Big) = 0\). The summation trivially changes to \(\sum_{m=2}^{N}y_{m}^{2}\). The bracketed second term is of the form seen above and thus Lorentz invariant in structure, \(-t^{2} + x_{1}^{2} = -\tau^{2} + y_{1}^{2}\). We shall demonstrate this explicitly; \(-t^{2} + x_{1}^{2} = -(\gamma (\tau + vy_{1}))^{2} + (\gamma ( y_{1} + v\tau))^{2} = -\gamma^{2}(1-v^{2})\tau^{2} + \gamma^{2}(1-v^{2})y_{1}^{2} = -\tau^{2}+y_{1}^{2}.\)

Therefore the light sphere \(\sum_{m=1}^{N} x_{m}^{2} = (ct)^{2}\) Lorentz transforms into the light sphere \(\sum_{m=1}^{N} y_{m}^{2} = (c\tau)^{2}\). We have also demonstrated that a single Lorentz transformation on an N-sphere reduces to the case of a 1-sphere as all other directions are transverse to the boost and thus unchanged.

Summary : We have constructed the component action of Lorentz transformation in a matrix representation from the definition of Lorentz transformations. We have demonstrated that such a transformation does not alter the norm of vectors in TM. We have demonstrated that the light cone in a given set of coordinates (aka frame 1) is mapped to the light cone in a second set of coordinates (aka frame 2) if the change of basis transformation is a Lorentz transformation.

Definition : Given a pseudo-Riemannian space with n space-like directions and N-n time-like directions normal coordinates \(x^{\mu}\) are such that \(\eta_{\mu\nu} = \eta(e_{\mu},e_{\nu})\) is diagonal with N-n entries of -1 and n entries of +1.

If \(\delta(k)\) is the k by k Kronecker delta then \(\eta = (-\delta(N-n)) \oplus (\delta(n))\). We take \(x^{\mu} \to (x^{a},x^{A})\) under this decomposition.

Definition : The Weyl rotation trick is the Euclideanisation of M by the coordinate rotation \(x^{a} \to -ix^{a} \equiv y^{a}\) and \(y^{A} = x^{A}\) and thus \(e_{a} \to -if_{a}\) and \(e_{A} \to f_{A}\) such that \(\eta(f_{\mu},f_{\nu}) \sim \delta(N)\).

Definition : Given a fibre bundle E over base M, \(E \to^{\pi} M\) whose fibres \(\pi^{-1}(p)\) for \(p\in M\) are morphic to some vector space V then there is a principle G-bundle such that a representation of G is the symmetry group of V. Or more clearly given a vector space you can examine it either by examining its vector structure or its symmetry structure.

Definition : The symmetry group associated to a space with n space-like directions and N-n time-like directions is SO(N-n,n). Euclidean space \(\mathbb{R}^{n}\) has symmetry group SO(n). Minkowski space-time previously defined has symmetry group SO(1,N).

Under the Weyl rotation trick Minkowski space-time is Euclideanised and has symmetry group SO(N+1), which is that of N+1 dimensional space in Newtonian physics. Hence the mathematical structure of Minkowski space-time, and by corollary special relativity and electromagnetism, is morphic to that of Euclidean space. Hence an inconsistency in one feeds through into the other. Furthermore by the equivalence of vector spaces and their symmetry groups if the mathematical structure of geometry in Minkowski space-time is flawed then it feeds through into all group theory and all Lie theory.

Hence a flaw in special relativity wouldn't be tucked away in some convoluted physical set up, it would be endemic to pretty much all maths and physics for the last 400+ years. Given various areas have been proven to be mathematical consistent this would contradict the notion the geometry of Minkowski space-time is inconsistent.

Feel free to ask any questions on any algebra or logic I use Jack, I have nothing to hide. I'll provide references to specific results if you don't believe me or you want to find out more. Most of it is stuff I'd expect any competent first year to know and be able to work through themselves. Other than the fibre bundles any one whose done more than 2nd year vector calc would be able to grasp it. Its covered in all relativity textbooks too and is assumed knowledge in any graduate course. The fact null vectors map to null vectors proves your claim about LTs mapping points on a light cone to points off a light cone false. If you claim otherwise explain where my algebra is wrong and provide your alternative working.

Failure to respond will mean I simply post a link to this post every time you whine about wanting someone to 'show you the math' or provide mathematical justification or to 'put up or shut up'. If you can't convince a 'primitive' theoretical physics PhD then you've got no chance with professors who'd run rings around me. Try not to fall at the first hurdle.

 

To be fair I think it's pretty easy to say that he's got people thinking. If you guys thought he was an absolute crank, rather than just stubborn, arrogant, and wrong, then you would've left the thread long ago.

His paper has plenty of math, btw. Maybe he has already submitted to a journal and was rejected..?

 

James R.

Post one is still sitting in this thread without any refutation.

Reading your views of me would indicate you can refute the twins contradiction.

Since, you clearly are more intelligent than I am, ad I am stupid from your view, I will expect you will refute it quickly.

How long will this take?

 

No, I am measuring the intellect I will run up against.

If you have noticed, I have presented more pictures and explanation as time goes on.


I continue to present this simple stuff, but I have no bites.

http://www.proofofabsolutemotion.com/srproblems.html

They will not understand my recursive algorithms and the like without first understanding the basics.