Alternative Twins Paradox

When in the time coords of the O frame does O' see light a distance r in all directions.

That is what this discussion is all about.

 

Log in or Sign up to hide all adverts.

OK, here is a little exercise. Light hits target T for two frames with a light emission at both orgins.

Once light hits T, tell me where the two light emission points are.

You will note, each frame uses its origin for its own light emission point for the light path to T.

You will also note, these two light emission points are not at the same place.

Sure, it is a simple proof.

If SR is complete and it claims as one of it fundamentals that light is spherical in the moving frame, then it should predict a time in the stationary frame when light is r from O'.

SR cannot answer this simple question without my help. Then, when I help SR, I run it into a contradiction.

You can perhaps save SR by indicating that unique time.

Oh, sorry, this is certainly unclear. I retract it because it is not necessary.

Nope, it does not answer a simple question. When is light a distance r from O'.

 

Log in or Sign up to hide all adverts.

OK, I do not want to talk anymore.

 

Log in or Sign up to hide all adverts.

No problem, You haven't said anything coherent for some time anyway.

Wrong, Jack. SR predicts the light is spherical around O' in the moving frame, not in the stationary frame.
In the stationary frame, the light flash is never spherical around O' (except for the trivial case of r=0).
So, SR should not predict a single time in the stationary frame when light is r from O'.

This has been explained to you over and over again.

Now you're done talking, perhaps you'll start listening... but I seriously doubt it.

 

Part of the issue here is that everyone is working on applying Lorentz transformations to coordinates. Lorentz transformations are not defined on coordinates but on vectors, as the have the schematic behaviour of \(TM \to TM\) on the tangent bundle. Jack doesn't like that at a time after light emission there's a disagreement on the centre of the light sphere formed by time slicing the light cone when you apply Lorentz transformations. But we're not talking about applying Lorentz transformations on cooerdinates at some time after emission, we're talking about applying the transformation to vectors at the moment of emission. Two frames are related by a Lorentz transformation defined at a point in space-time. That point is the light emission point and thus two different frames by construction agree on the point of emission, the apex of the light cone.

From this you can work out the coordinate version of Lorentz transforms but you have to make a particular assumption, namely that the space is flat and empty. This is because you obtain the coordinate version by solving the geodesic equation and in SR all geodesics are just straight lines, there's no curving due to gravity.

More formally a vector \(X = X^{a}\partial_{a}\) transforms under a LT to \(X = (X')^{a}\partial_{a'}\) where \(\partial_{a} \to \partial_{a'} = \Lambda^{b}_{a}\partial_{b}\) for \(\Lambda\) a LT. The geodesic equation is 2nd order and you need 2 initial conditions which can be taken as the starting position and the vector to provide the tangent of the geodesic at the starting position. All frames agree on the first initial condition, which is the apex of the light cone, a LT doesn't alter this. But since LTs alter vectors different geodesics will be obtained by solving the geodesic equation in different frames as they have different initial directions. A vector which is null, ie \(X^{a}X^{b} \eta_{ab} = 0\) will map to another null vectoor \((X')^{a}(X')^{b} \eta_{ab} = 0\) so causal structure is preserved. In a given frame with coordinates (t,x,y,z), which induces a coordinate basis on TM of \((\partial_{t},\partial_{x},\partial_{y},\partial_{z})\), the vector \(X = \partial_{t}\) will result in a geodesic which that frame sees as always being in the centre of the light cone. But if you boost to a different frame then \(X = \partial_{t} \to (X')^{a}\partial_{a'}\) and generally this will not be \(X = \partial_{t'}\) and so the new frame won't see that geodesic's location at any given time after emission as in the centre of the light sphere. Conversely the geodesic obtained in the second frame from \(X = \partial_{t'}\) will not generally map to \(X = \partial_{t}\) and thus the first frame won't see it in the centre of its light sphere.

That's the procedure you should use if you want to be rigorous, rather than transforming the locations of all objects at some time after emission from one frame to another you should consider the Lorentz transform acting on the vectors which define the motion of light and emitters at the emission point and then obtain the physical implications by solving the relevant geodesic equations for emitters and photons. This makes it manifest that the different frames give equivalent physics and that they agree on the light cone. Jack claims there's a problem with multiple points of emission but that's because he's muddled himself up by considering coordinates, not vectors. By considering the vectors its clear there's no "multiple emission points" or any contradiction.

Consider a set of emitters all at the same place initially but with different initial velocities and one of them emits a flash of light so you also have a set of photons starting at the same place and with different velocities (but obviously the same speed). In a given frame you can write down the velocities of the emitters and photons in specific coordinates and then solve the geodesics. To find out what a different frame says you go back to your initial setup, transform all velocities via a LT and then solve the geodesic equations again. Its obvious that there's no issue of 'multiple emission points' here since we're only considering the instant of emission and every oobject is in the same place and all frames agree on that location at time of emission. An object's velocity in the first frame is mapped to one and only one velocity in the second frame. Hence there's no issue of multiple results, since the LT map the vectors bijectively, each vector in frame 1 is related to one and only one vector in the second.

So this gives us a method to check physical predictions between frames to see if the use of Lorentz transformations on the coordinates and not vectors is valid. First you work out the physical results in frame 1 and then apply a LT to get the results of frame 2. Then you woork backwards. If you know the position and motion of everything in frame 1 at some given time after emission you can solve the geodesic equations and wind time back to the t=0 moment of emission. You can then LT from frame 1 to frame 2 and then use the geodesic equations to wind time forwards. This gives the same result as you're working with a global Lorentz symmetry, space-time is flat. This second method completely removes the qualatitive issue someone (ie Jack) might have with the photon spheres formed from different time slices of the light cone since the uniqueness of the geodesics (ie 2nd order vector ODE with 2 initial conditions has unique solution) and the invertibility of all Lorentz transformations means there's a bijective relationship between all physical objects in the different frames.

This line of reasoning and examinations of symmetries of systems comes up a lot in physics but not something many people outside the physics/maths community might be familiar with. By cutting corners and transforming coordinates rather than vectors and then computing geodesics you don't see the more formal consistency of LT and SR so readily and you run the risk of having the problems Jack has in understanding. Furthermore the application of LT to coordinates is only valid in SR. In GR the geodesics are not generally straight so you can't jump from vectors to coordinates, you must solve the geodesic equations and LT symmetries exist only point by point.

If Jack isn't lying about his education then this methodology should be familiar to him. Furthermore he should know where to look to find the mathematics relevant to this reasoning, since he should have already come across it in the past. Of course if he's unwilling or unable to do I'm happy to go into more details but he should definitately already know about the nature of geodesics in SR and GR and now LT are only universally applicable on the fibres of the tangent bundle, not the base space itself.

And Jack, if you're going to reply to this don't just quote it all in one go and then ignore it entirely. If you disagree with anything I've said or you want clarification quote the specific bit of this post and make it clear what you're saying. You keep demanding people address your claims and when I or anyone else does you don't address the points we raise or answer to questions we ask you. Please practice what you preach for once.

 

Thanks Aplha.

I readily affirm that I have no education in relativity, and that the limit of my amateur toolkit is simple coordinate transformations in flat space. I flatter myself that I know the limitations I'm working under, ie I know when my simple tools are not useful... but I acknowledge that they're kind of clunky, and definitely not pretty.

The prettiest I can get is spacetime diagrams. I tried those with Jack in the [post=2478774]burn mark thread[/post], but Jack seemed incapable of comprehending them.

 

No, that's fine, I have no issue with you and infact I think you're a very good explainer of this stuff to Jack. Its not required to be rigorous and use vectors rather than computing geodesics to give coordinate expressions in order to explain the basics. But Jack claims there's some underlying contradiction so the distinction between vectors and coordinates makes explaining his mistake clearer. Its only necessary to be pedantic when people complain the fundamentals are wrong when they don't even know the fundamentals.

No, you're right. If he did understand them he'd immediately see what I've just said.

Given two frames related by a LT at the origin, ie (t,x) = (0,0) and (t',x') = 0, they share the same light cone. The choice of frame is equivalent to the choice of which vector is equivalent to the t axis, ie vertical. Jack thinks because there's different choices but they all give the same light cone then its inconsistent yet its obvious from a space-time diagram all you're doing is just squishing or stretching the layout of the system in a manner which preserves causality. I've even shown him the relevant diagram before :

Please Register or Log in to view the hidden image!

The way the dots move about as the frame is constantly boosted and rotated by the motion of the object is such that there's no violation of causality and all the physics is the same. But Jack can't understand this.

 

One thing's for sure... we've both invested way to much time in Jack! At the expense of doing more important things, in my case at least. Sciforums is great for procrastination.

You've written some good stuff that should be put to good use. Perhaps I should read try to learn some vector calculus. I've always balked at the notation before...

It's all Greek to me! I can sympathize a little with Jack. My eyes glaze over and slip down the post looking for something understandable. Is there a 1st year level textbook you'd recommend?

 

Part of the issue here is that everyone is working on applying Lorentz transformations to coordinates. Lorentz transformations are not defined on coordinates but on vectors, as the have the schematic behaviour of on the tangent bundle. Jack doesn't like that at a time after light emission there's a disagreement on the centre of the light sphere formed by time slicing the light cone when you apply Lorentz transformations. But we're not talking about applying Lorentz transformations on cooerdinates at some time after emission, we're talking about applying the transformation to vectors at the moment of emission. Two frames are related by a Lorentz transformation defined at a point in space-time. That point is the light emission point and thus two different frames by construction agree on the point of emission, the apex of the light cone.

Your problem with your vectors is that they do not include the dynamics of the whole system.

Sure, if the vectors were in a stationary frame, they would work just fine.

But, your vectors are decided before the events themselves take place.

In particular, your vectors operate from a common light emission point.

You forget that the beginning of your vectors do not remain in one position as they would in Newtonian mechanics and so they are not vectors.

This is your flaw. If you had any sense, I am showing a major design flaw in SR.

 

Jack, your ignorance is very clear. Even I know enough vector maths to see that you have absolutely no idea what you're talking about.

Perhaps you should just stick to time and distance coordinates until you can learn a little about spacetime. Hint: we're not dealing with (x,y,z) vectors, but (t,x,y,z).

 

Well done on demonstrating you never did any vector calculus. If you had you'd not have the lack of understanding about the formal mathematics behind vectors. Unfortunately that's the kind of understanding you need to realise you're mistaken about your claims so you're still spouting ignorance. I'd suggest you learn about tangent spaces and tangent bundles but obviously even basic vectors is beyond you.

You've not posted for a week or so. Did you spend that time writing up your work to submit to a journal? If not then you demonstrate you're dishonest and have no intention of putting your physics where your mouth is.

You're all mouth with nothing to say Jack.

 

Ooop North we say "All mouth and no trousers". No, I haven't a clue what it means either...

Jack_: I strongly urge to listen carefully to what AlphaNumeric has to say: he is often right about physics, and occasionally right about mathematics (*wink*).

Specifically

is completely nonsensical; what is the "beginning of a vector" ? Maybe you think it has an "end" too? And even assuming that vectors had beginnings and ends, why should the "transport" of a vector to other beginning and end points NOT be a vector?

Alpha suggests you read up on tangent spaces and their bundles; I don't. First you need to know what a vector space IS, then to know what a manifold IS, which clearly you don't, and then so much more before you continue with your arrogant crap

 

He is making the mistake I commented on previously (I think its in this thread), that all too often people do not realise the important discussion between coordinate positions, vectors and lines. I commented on how the discussion of LT is often phrased in terms of positions when infact they are only defined on vectors at a given point and that the coordinate expressions come from solving geodesic equations where the point and a vector at that point serve as initial conditions. Only through the fact the space-time of SR is flat and thus geodesics straight lines do we turn the vector expressions to coordinate expressions. By thinking in terms of positions and not vectors Jack is not approaching the scenario properly and this is the source of much, if not all, of his confusion and mistakes.

The fact he seems unaware of the requirement that to compare two vectors they must be defined at the same point in your space testifies to him having a less than complete understanding of vector calculus. Of course it'd be silly of me to claim he's making an uncommon mistake, few people ever really have to consider the difficulties of comparing vectors defined at different points in a non-trivial manifold. Unfortunately for Jack I happen to be one and so does anyone else who has done research into general relativistic models. His comment about how I might wish to compare vectors at different points implies he thinks this is a concept unfamiliar to me. If he'd read any of the reading material I've suggested to him or done a course in the formal construction of GR or even just differential geometry he'd know its of great interest and relevance to the GR community. Things like 'parallel transport' and 'Lie derivative' are things relevant to the issue of comparing vectors by moving them through space. I can think of several books I've got which devote considerable discussion to this and its relevance to the physics the book is about. Heck, the word 'holonomy' is in the title of one book and is of such fundamental relevance to string theory it is the guiding principle which leads to the interest in Calabi-Yaus.

Of course Jack doesn't know that these concepts are not obscure or difficult or poorly understood in the mainstream community since he has, by his own admission, no wish to read material which might provide him with insight into these areas of mathematics and physics, as doing so would not reinforce his preconceptions. After all, being wrong about something because you're deliberately ignorant is better than knowingly lying, right Jack?

Jack, you aren't doing anything advanced here, your topics of discussion aren't stretching anyone's knowledge and certainly not stretching the limits of the mainstream community. You are naive about just how much is known and understood when it comes to vectors, symmetries, transportation of vectors, curved space-time, differential geometry and anything else relevant. You keep wording your posts as if you think you're teaching me something, that the notion of transporting vectors etc is new to me. You have no clue how much I'm dumbing down my posts, lest my response is entirely over your head (as opposed to mostly over your head, as it is at the moment). The application of differential geometry to relativistic models is home turf for me. I did try to discuss with you tangent bundles, an area entirely relevant to the issue of vectors defined at different points. You claimed it was not relevant, yet it is you who has now taken the discussion to where not only are tangent bundles relevant they are fundamental to understanding the mathematical formalism of special relativity we're talking about.

If you were intellectually honest enough to read what the mainstream knows about this stuff you'd realise how low your level of discourse is. The fact you won't tells everyone you're not as certain as you want us to believe, else you'd not be afraid to find out. After all finding out what the mainstream says would make it easier to demolish it so its not like you'd be wasting your time even if you're right.

 

Yes, that helped make the whole "multiple emission points" thing a lot clearer. Trying to explain it in terms of coordinates only got murky.
I also think that Jack is still unable to deal with spacetime, that he doesn't (can't, won't?) consider time as a dimension that can be plotted on a chart or modeled with vectors.

 

Oh, looks like you have it.

My bad.

Oh, question. Let a light emit from two co-located frames at a common origin.

This is defined by vectors.

Where will the light vector be for the moving frame when the light hits some target T?

Thanks.

 

see post 295

you may answer the question also.

 

The fact he seems unaware of the requirement that to compare two vectors they must be defined at the same point in your space testifies to him having a less than complete understanding of vector calculus.

Exactly, but they must also stay put no?

The moving light beam is a vector that moves with the moving frame.

Normally, when dealing with vectors, the origins remain the same when the operation is complete.

Under SR, this is false.

So, I would like to see vector calculus where the origins will not be the same once the target is hit.

Hunt: LT does not handle this.

Further, this origin thing is a major problem for SR.

If the light sphere emerges at the origin of the moving frame, then using vector calculus, when in the time of the stationary frame will this occur?

If all the vectors remained origined, this answer would be easy.

But, since all the light beams/vectors move, what is the answer?

 

The multiple emission point thing is not solved by vectors.

The oirigination points of the vectors are co-located at origination but not at termination.

Show me the vector calculus for such operations.

 

You are confusing vectors and curves, as well as the difference between staying still in space and staying still in space-time. A vector is defined at a single point in space-time. It moves in neither space nor time. If you wish to consider a set of vectors which are all defined at the same point in space for a period of time then you're considering a curve whose spacial coordinates do not change and whose set of tangent vectors define the vectors all defined at the same point in space.

You're continuing to fail to grasp the fundamentals and no matte how much you profess otherwise its a demonstrable fact. Remember, if you're so confident in your claims being right you should have no problem reading a few bits of literature on the matter of vector spaces because surely they'll agree with you? I've got no fear, I'm willing to engage you in discussion and to aid you in putting your work to journal review. It would seem you're unable to stand by your own rhetoric.

No, the light instantaneously, in any given time slice, has a velocity vector and this defines a null path through space-time. A frame is a choice of time-slice, unlike the tangent vector to the curve it is not uniquely defined by the curve. The velocity of the object defines the curve it sweeps out through space-time. A frame is a choice in how you label the points in space-time in terms of coordinates. Its a fundamental property of manifolds that many different frames are equally good as descriptions of the dynamics of things in the manifold. Once again, you'd do well to read an introductory book on the matter.

'Normally'? And who are you to make claims about what happens more often than anything else in physics or mathematics, when you have no understanding or knowledge or experience of either? Why do you persist in making claims about things you know full well you have little or no information on and which you know others here do? Are you compelled to lie no matter how transparent the lie?

Besides, you're still confusing coordinates and vectors. I've explained many times that any change of basis acts on the set of basis vectors (obviously, but this is something you're not understanding). The basis before and the basis afterwards are both defined at the same point in the manifold, as you can only compare vectors which belong to the same fibre in a tangent bundle (look this up because its vital to any grasp of vectors and yet you don't know it!). Given a new set of vectors in order to work out the motion of objects within that manifold you have to solve different equations of motion or geodesic equations but ultimately you end up with the same set of curves. The essential property of a manifold is that your results are independent of how you describe them. Tensors are, by definition, independent of your choice of coordinates and basis. Look it up.

Since a change of basis acts on a given tangent fibre it must preserve the base point, ie \(\pi(T_{p}M) = p\) must remain unchanged. Since this is an initial condition in any curve moving through that point when you compute said curves or geodesics you end up with a point being left unchanged.

Notice that I haven't actually said 'Lorentz transformation' because this is not restricted to Lorentz transformations, it's true of ALL changes of basis because all of them have the action on a given vector space V of \(V \to V\) and so they belong to End(V), the space of endomorphisms on V, where \(V \sim T_{p}M \sim \pi^{-1}(p)\). A change of basis, be it a dilation, rotation, boost, shear, whatever can only act as \(T_{p}M \to T_{p}M\), not \(T_{p}M \to T_{q}M\) for \(p \neq q\).

So your singling out of SR is simply wrong. And you continue to fail to understand the difference between a point in space-time and the set of points defined by a straight line geodesic through space-time such that any time-like singles out one and only one point in the line. The former is akin to the apex of the light cone and the latter is akin to a point in space of a given frame at some point in time.

Seriously, your continued inability to grasp the difference between a point in space-time and a point in space is severly hindering your understanding.

No, it is simply your failure of understanding. Nothing you have said is in the slighest bit a problem for SR. Your unwillingness to read what SR or mathematics says about Lorentz transformations proves you know you're not right, else you'd have no fear in reading a bit. The moment you say "I'm not going to look at anything else" you admit to being intellectually dishonest, wilfully ignorant and aware that there are refutations of your argument and that you're scared to see them.

If I'm wrong in this then submit your work to a journal. I've been offering to help for months now, you clearly have the time to write it up. You are simply afraid.