Mordred

do you agree the mathematics are required to calculate the age in order for the twin paradox to have any meaning Yes or no.

this is the SR Lorentz transformations \[\acute{x}=(x-vt)\] \[\acute{t}=\gamma(t-\frac{vx}{c^2})\] they are not the same as in that link now calculate the age using the link you just provided. You need the math to calculate age so obviously the math is a huge part of the solution to the twin paradox. Yet you choose to ignore the mathematics while claiming Einstein theory is incorrect. Yet in the same breath claim they are the same

no they are not the transformation rules differ between LET and Lorentz as per SR. Show you can actually calculate the age difference

No the mathematics used in LET in his transformations are not the same as those used in SR. Provide the solution using Lorentz ether transformation rules

I asked you to provide his mathematical solution not an interpretation

great provide his solution to the twin paradox in his mathematical form using LET. I for one have never seen his solution using Lorentz ether for the twin paradox so provide it for reference with the mathematics

Do you honestly not realize that every paper Lorentz published included equations in his physical explanations ? why do you ignore those equations but attempt to falsify Einstein relativity for using equations when Lorentz himself also used equations. It was through his equations that others could point out numerous flaws which led to Lorentz modifying his equations numerous times in various papers. for example one of his papers had an energy momentum violation which he later corrected. you know what lets jump back to the actual topic. provide the mathematics for the twin solution using Lorentz ether theory give us a demonstration that you actually understand his theory

You know you really do mix up a lot of physics without actually understanding any of it. The very term invariant literally means to all observers regardless of model. I have copies of Lorentz ether theories they are readily available in reprint archives. You obviously do not understand its mathematics nor its transformations you have made far too many mistakes this thread alone that has been pointed out by others. Those others include 2 physicists Dr Swansont and myself my primary field is cosmology but I also hold a Bachelors degree in particle physics. This has a couple others extremely well versed in Relativity such as Markus pointing out your mistakes. So instead of just reading the words in your various links why don't you sit down and spend time understanding what the math states. Then perhaps you can better defend LET.

Just to add the relativistic correction to decay rates \[L_o=\beta\gamma c\tau\] Fermi's Golden Rule \[\Gamma=\frac{2\pi}{\hbar}|V_{fi}|^2\frac{dN}{DE_f}\] density of states \[\langle x|\psi\rangle\propto exp(ik\cdot x)\] with periodic boundary condition as "a"\[k_x=2\pi n/a\] number of momentum states \[dN=\frac{d^3p}{(2\pi)^2}V\] decay rate \[\Gamma\] Hamilton coupling matrix element between initial and final state \[V_{fi}\] density of final state \[\frac{dN}{dE_f}\] number of particles remaining at time t (decay law) \[\frac{dN}{dt}=-\Gamma N\] average proper lifetime probability \[p(t)\delta t=-\frac{1}{N}\frac{dN}{dt}\delta t=\Gamma\exp-(\Gamma t)\delta t\] mean lifetime \[\tau=<t>=\frac{\int_0^\infty tp (t) dt}{\int_0^\infty p (t) dt}=\frac{1}{\Gamma}\] relativistic decay rate set \[L_o=\beta\gamma c\tau\] average number after some distance x \[N=N_0\exp(-x/l_0)\] There are several examples in that thread for different cross section and relativistic velocities

In SR neither time nor length in the direction of travel remain constant. You have time dilation as well as length contraction. So both the distance as well as time will also vary depending on the observers . That's where you apply the v and c relations here is the time dilation and contractions \[\acute{x}=(x-vt)\] \[\acute{t}=\gamma(t-\frac{vx}{c^2})\] \[\gamma=v/c\]

I believe part of the confusion may have to do with the fact that there is numerous ways to express proper velocity. Further adding to the mix different hyperbolic functions are used different between coordinate speed and proper speed https://en.m.wikipedia.org/wiki/Proper_velocity This link provides several different methods. Rapidity being one of them. The graph in the link shows the distinctions between each. It also describes rapidity in more detail "Proper speed divided by lightspeed c is the hyperbolic sine of rapidity η, just as the Lorentz factor γ is rapidity's hyperbolic cosine, and coordinate speed v over lightspeed is rapidity's hyperbolic tangent" so you can see that the hyperbolic sine of rapidity is the proper speed whereas the coordinate speed uses the hyperbolic tangent... the link provides details as other key and useful proper velocity methods and relations. This is also one of the reasons I always present other references when concepts such as this being discussed. Sometimes it gets rather tricky finding a mathematical method that a given person can relate to. A good example is components of a vector such as where and when you use the inner/outer or cross product of those components they each have their own purpose and unfortunately v being a vector is subjective to those components in velocity addition rules. I already provided a link above showing the vector addition rules for rapidity. The link included a good listing of other related velocity additions. The other problem is that although rapidity describes the proper velocity it also describes coordinate time depending on the trig function used. Lmao it also doesn't help that in particle physics we use rapidity in reference to a bar. Short hand for a probability current that's describes the path integrals with a number density of state's describing an ensemble for a specific particle. For particle resonance it's often used in the lab frame via Breit Weigner. (Used for cross sections). I mentioned as if one isn't careful and was trying to self study rapidity he could very well google up a reference particular to particle treatment as opposed to how it's used with Lorentz/Minkiwskii. Typically those papers will mention in reference to a bar. I think maybe an example of rapidity in the twin paradox may help. Later on I will type in the Lewis Ryder treatment Mainly because he further extrapolated an interesting function with regards to rapidity. As well as other useful relations.

sigh you do realize that rapidity is a hyperbolic function. it isn't the velocity itself . I gave you those functions above. However here it is again. \[\varsigma=\tanh{-1}\beta\] from which \(cosh\varsigma=\gamma\) and \(\sinh\varsigma=\beta\gamma\) if your not familiar with hyperbolic functions. https://en.wikipedia.org/wiki/Hyperbolic_functions the hyperbolic function describes the region a/2 in that link. the hyperbolic tangent is on that link tanH now look at the image in this link. https://en.wikipedia.org/wiki/Rapidity the area of the hyperbolic sector is the yellow in the second graph. I also provided the hyperbolic rotations for observer a to observer b in the above the underscript A is Alice. under b for Bobs whose x,y inertial frame changes according to the rotation group above. as per the second link for proper acceleration. "Proper acceleration (the acceleration 'felt' by the object being accelerated) is the rate of change of rapidity with respect to proper time (time as measured by the object undergoing acceleration itself). Therefore, the rapidity of an object in a given frame can be viewed simply as the velocity of that object as would be calculated non-relativistically by an inertial guidance system on board the object itself if it accelerated from rest in that frame to its given speed. The product of β and γ appears frequently, and is from the above arguments" quoting the second link...note rate of change of rapidity I provided you the transformation matrix involved in both case of acceleration. \[\begin{pmatrix}x_a\\y_a\end{pmatrix}=\begin{pmatrix}\cos\varphi&-sin\varphi\\sin\varphi&cos\varphi\end{pmatrix}\begin{pmatrix}x_b\\v_b\end{pmatrix}\] \[\frac{dy}{dx}=tan(\theta)\] this expression describes that change in direction How is that not showing acceleration? the first matrix on the RHS is the rotation matrix.... The second matrix on the right hand side is Bob's 2d reference plane \[\begin{pmatrix}t_a\\y_a\end{pmatrix}=\begin{pmatrix}\cosh\varsigma&-sinh\varsigma\\sinh\varsigma&cosh\varsigma\end{pmatrix}\begin{pmatrix}x_b\\v_b\end{pmatrix}\] \[\frac{dx}{cdt}=tanh(\varsigma)\] this expression is your transformations due to change in velocity via a boost of the rapidity which is a also a change in velocity Do you understand this much ? Its obvious proper acceleration uses rapidity in the quoted section of link 2. it clearly shows you in that link. these expressions are the hyperbolic tangents The acceleration is there for both cases. Can you agree on that with the links provided ? I also gave the curves in each case by the way. https://www.wolframalpha.com/input?i=plot+tan(y) there's your tanH(x) plot as per the second link.

I'm actually going to state that I agree the manner its often presented requires a more detailed presentation. I know I'm one of those parties as velocity can also be treated via rapidity its actually more accurate to state acceleration is boosting or rotating the rapidity. As we have two forms of acceleration change in direction and change in velocity. lets take a problem set. Lets have a constant acceleration for however many years. A couple of key notes there is more than one type of boost. the boost for velocity depends on \(\beta\) the velocity parameter. Now I'm sure you agree using velocity addition for a constant accelerating observer can get clunky. So this is where rapidity steps in. this describes the Lorentz boost equations ( for other readers I know you know these details) \[\acute{x}=(x-vt)\] \[\acute{t}=\gamma(t-\frac{vx}{c^2})\] \[\gamma=\frac{1}{\sqrt{1-b^2}}\] \[\gamma=v/c\] so the constant velocity observer will have the above Lorentz boost.. however constant acceleration its more useful to use the boost parameter not the speed parameter. This is is the rapidity given by the tanh function. \[\varsigma=\tanh{-1}\beta\] from which \(cosh\varsigma=\gamma\) and \(\sinh\varsigma=\beta\gamma\). Now that's the Lorentz boosts in terms of rapidity. So rapidity can be used instead of velocity. however we need acceleration so lets have the x,y,t planes. for simplicity. now a rotation in the x,y plane describes the change in angle \[\begin{pmatrix}x_a\\y_a\end{pmatrix}=\begin{pmatrix}\cos\varphi&-sin\varphi\\sin\varphi&cos\varphi\end{pmatrix}\begin{pmatrix}x_b\\v_b\end{pmatrix}\] \[\frac{dy}{dx}=tan(\theta)\] for changes in {x,t} we are boosting the velocity or alternately the rapidity . As rapidity can also describe velocity. \[\begin{pmatrix}t_a\\y_a\end{pmatrix}=\begin{pmatrix}\cosh\varsigma&-sinh\varsigma\\sinh\varsigma&cosh\varsigma\end{pmatrix}\begin{pmatrix}x_b\\v_b\end{pmatrix}\] \[\frac{dx}{cdt}=tanh(\varsigma)\] so yes we need to be more clear I agree or rather I need to be more clear. you certainly use rapidity for both types types of acceleration but you can also use rapidity for velocity. As mentioned I should state acceleration via change in velocity is a boost in the rapidity, while a change in direction is a rotation of rapidity. Does acceleration require rapidity no and you and I have agreed on this in the past if you recall. However as shown it certainly does apply to acceleration. yes of course I agree on that but it also applies to acceleration as shown you can also boost the rapidity or rotate it. If you understood the relation of rapidity to relativistic velocity I really have a hard time understanding why you would think it wouldn't apply to change in velocity at relativistic speeds or changes in direction at relativistic velocity ?

That's the real crux a total lack of evidence.

No I'm familiar with the research that went into Lorentz ether theroy I also know it's transformation rules including many of the other variations . The thing is the physicists performing those Lorentz invariant tests are also well aware of neo-lorentz. So they also conducted tests for that in that article. Here is the detail many miss. In Lorentz time the only known particles were the photon the electron and the proton. That was at that time the entire standard model. The neutron wasn't even discovered until the mid 30's. So it was quite natural to think there was am ether. Modern physics has gone beyond that including particles that are so weakly interactive they could pass through a chunck of lead one light year in length without a single interaction. (Neutrinos). However it's also well known every particle species contributes to the blackbody temperature in particular the CMB including those neutrinos. So why do we not detect any temperature contribution from the Lorentz ether ? Why does it have no influence on universe expansion ? Every other particle does. The way is if you have a static 100 percent non interacting field but then it wouldn't even interact with gravity let alone photons. Or any other particle. We can certainly gather indirect or direct evidence of every other particle in the standard model. Why not the Lorentz ether ? Then why would you claim otherwise and argue that c isn't invariant ? Sounds like you don't even understand Lorentz ether theory.... Then why would you claim otherwise and argue that c isn't invariant ? Sounds like you don't even understand Lorentz ether theory.... in point of detail. Had you actually studied its mathematics. It was a valiant effort to meet observational evidence and keep c invariant to all observers. That is actually harder than one realizes when you have light travellings through a medium.