studiot

I still don't know what you want ? There are many 'definitions' of sinx and cos x. Obviously they all define the same things. So do you want me to tell you what sin and cos are? Or do you want a single formula that will calculate the value of sin or cos for any x. Remembering this value will only be 'correct' to a specific number of digits? Or do want ways of obtaining exact numeric values for any angle.? Remembering there is no single way to do this for every angle. Here are some single formulae in the form of continued products which converge at or before infinity. [math]\sin x = \left( {1 + \frac{x}{\pi }} \right)\left( {1 - \frac{x}{\pi }} \right)\left( {1 + \frac{x}{{2\pi }}} \right)\left( {1 - \frac{x}{{2\pi }}} \right)\left( {1 + \frac{x}{{3\pi }}} \right)\left( {1 - \frac{x}{{3\pi }}} \right)...[/math] [math]\cos x = \left( {1 + \frac{{2x}}{\pi }} \right)\left( {1 - \frac{{2x}}{\pi }} \right)\left( {1 + \frac{{2x}}{{3\pi }}} \right)\left( {1 - \frac{{2x}}{{3\pi }}} \right)\left( {1 + \frac{{2x}}{{5\pi }}} \right)\left( {1 - \frac{{2x}}{{5\pi }}} \right)...[/math]

Your Google translated definitions seem very oddly phrased and over complicated. Can you explain simply what you are trying to do? All trigonometric functions are already very well defined.

Substances don't 'absorb force'. It is possible to build a mechanism that sort of answers your requirement using a dome shaped piece of spring steel foil set against a ring foundation. But the foundation would have to be set against something to ultimately resist the applied force.

Try the handy answer book series, some of them are rather good and all are modern. The Handy Anatomy Answer Book The Handy Diabetes Answer Book The Handy Biology Answer Book The Handy Chemistry Answer Book etc https://www.visibleinkpress.com/s17/The-Handy-Answer-Book-Series

Indeed +1

In the low speed situations you describe Newton always applies. If course it depends exactly what you mean Mechanical Advantage = Load / Effort and an ME of greater than 1 is certainly achievable with suitable arrangements. Of course you need to go a long way further back than Newton for some of these, right back to the ancient Greeks. "Give me a long enough lever and a fulcrum and I will lift the world." https://en.wikipedia.org/wiki/Mechanical_advantage

Glad to hear you are feeling calmer today. Perhaps you can now review your perspective? What is Free Will and why do you think it is all or nothing situation? Are there not degrees or levels of free will? And do not (have not always) different individuals possessed different levels of it? How much free will does a drug addict have (about drug taking) ? How does that compare with when I have a second, third , fourth.... chocolate I know I shouldn't eat?

Having a metric is not an essential requirement for topological spaces. If a topological space has a metric it is a metric topological space. This is important because there is no requirement to measure the 'length' of the sticks in a topological network of connected sticks. The connectivity is all important in determining precedence or causality. So I maintain it would just be different, although topologically equivalent. I do agree that if you restrict the use of 'spacetime' to Minkowski (who coined the word after all) then it would not necessarily be spacetime. Note also that in the first millenium and a half before coordinate systems were invented Geometry functioned perfectly well. In fact the introduction of coordinate systems, principally by Descartes, introduced extra information into Geometry which was not present before. This extra information is that everything now has an orientation. Before an equilateral triangle was the same whichever way up it presented. Whereas the same triangle standing on a vertex or a base are considered to be different different. The issue then becomes is this redundant or required information ? There is a move in modern Geometry to return to the pre Descartes era.

How is this possible without an underlying coordinate system ? I can't agree with this since the sticks (intervals) have a clearly defined measure. And clearly there exists a stick between each pair of events in the set. Even if the set includes every number in [math]\Re \otimes \Re \otimes \Re \otimes \Re [/math]

Despite the middle paragraph of the Wiki quote? here is the full reference. https://en.wikipedia.org/wiki/ADM_formalism I also don't see how using a t axis like that is compatible with the "Principle of Relativity"

So have the boffins got this to work yet, with or without "auxiliary fields" ?

1) Yes it applies to both. The quote was actually from Eddington's "The Mathematical Theory of Relativity" - and it does get very mathematical, which covers both SR and GR. The language is strangely more arcane than Einstein's, but he was a very clear thinker and likes to explain what he is doing and why. He also wrote simpler book "Space, Time and Gravitation" with lots more words and rather less formal maths. 2) A very emphatic no I'm afraid. Dropping the coordinate idea of contours or isolines (t = a constant) is the most important idea both Marcus and Eddington stress. The idea of t = a constant is dangerously close to leading towards an absolute coordinate system - an anathema to relativity. 3) Yes you can separate space and time but then you have immediately the same problem for both as in (2). You can regard the relations or links as like a building or fairground framework or better, the 'ball and stick' models of molecules in Chemistry. The events are the balls and the sticks are the realtions which are the relativistic invariants. Like the molecule, the configuration of the sticks alone, regardless of which way up they are, is fixed or the same. To take the analogy one step further. At the moleculer scale you can separate space and time and just consider the spatial configuration, without bringing in relativistic 'corrections'. But as soon as you get to the astronomic scale (solar systems, galaxies, etc) you have to either include the corrections or use a 4 D spacetime. Swansont's work is intermediate between these and leads to small corrections most people don't know about or bother with.

Wouldn't it just be different?

Yes welcome back. +1 This is (almost) Eddington's presentation. However his answer to these follow up questions places the 'network' at the forefront ie the most important part of 'reality'. What do you think

Maya whispered in my ear that it was really 168421 +1 apiece.

What is your list supposed to represent ? Number 1 is inadequate and untenable. Are you suggesting space ceases to exist when it is 'occupied' and suddenly spring back into existence when the occupying object moves away? What do you mean by continuous and why does it have to be continuous? What do you mean by occupied? Does this include say, light? What happens when a ray of light passes from a vacuum into a glass block? Hey I like this 11111 posts

these equations are very concerning since they imply that space is neither isotropic nor homogeneous. The situation is even worse when we extend it to 3D since in your model when we are aligning the x and x' axes and considering V directed along these, we have y= ct z=ct

Thank you for a response deserving serious consideration. It will certainly receive that. Meanwhile a couple of quick first impressions. This implies a discontinuity at x' = 0. unless we add these equations (equivalent to taking the average) 2x' = 2ct x' = ct This is, of course identical to the equation for observer1 viz x = ct You seem to have finally correctly identified the necessary second postulate of SRT. If this is your replacement postulate it doe not provide the means to numerically evaluate the apparent change of position. Where do your formulae come from?

This is in no way consistent either with the very clear and simple scenario I outlined, Nor is is a possible equation unless the light source was modulated by some sinusoidal signal, something I most definitely did not say. Furthermore I definitely specified the position of the second observer. I did not say that there was any separation between the observers. and everything that follows is completely and utter rubbish and thoroughly discouraging to any sensible discussion. Good night. Are you just being plain akward or what?

So you are saying that your postulate is that if the light source and the observer are one and the same, there is an apparent change in the zero difference in position between source and observer? This must lead to my equation for what observer 1 sees to be incorrect. So what is your version of the 'correct' equation and how may it be tested. I ask for tests because, to my knowledge, no one has ever observed this effect. I am still waiting for a mathematical description of the light front surface for observer 2. Why are you so hung up on wave motion or the ether ? Einstein did not use any particular form of motion in SRT. Have you read the paper? In it he specifically refutes the ether. In fact a contemporaneous paper (The photoelectric effect) he resurrected Newton's corpuscular theory by proposing the photon. Have your read this paper? In fact in that same paragraph he introuces his postulate you have taken issue with, But it again does not say what to attribute to Einstein. Actually this shows you have not read the SRT paper. because the phenomena you mention (and others) are purely theoretical deductions from Einsteins actual postulates (not your misstatements of them) It does not make the postulate that the speed of light is constant for all observers The presentation in the 1905 paper actually deduces from Einstein's own postulate, given on page 1. The idea constant for all observers idea came later and has been used since to develop SRT as it leads to easier mathematics. I will say it again. There is no requirement in SRT for light to be a form of wave motion.

Are you planning to answer my question, now asked several times ?

Now that we have reached this point let us review your beginning. Consider two observers in relative motion (which might be zero) v between them. Observer 1 is located at the origin O of the coordinate system x,y,z,t. Observer 2 is located at the origin O' of a second coordinate system x',y',z',t'. As the second system origin O' passes through and is coincident with the first system origin O, it happens that observer 1 emits a pulse of light. Observer 1 sees a spherical light front travelling out from his source in all directions and obeying the equation x2 + y2 + z2 = c2t2 Where c is the velocity of propagation of light as measured by Fizeau et al. Now the questions arise What does observer 2 see? And what is the resulting equation for his observation? And how is this connected to what observer 1 sees along with his equation. This is where any hypothesis must introduce a postulate and you say the postulate introduced by Einstein is incorrect. So my question to you is What is your alternative hypothesis and what conclusion (equation) does it come to? Note that so far c is just a symbol for a constant in the equation distance = speed x time It is nothing more than this and at this point has no special significance nor can it descibe the mechanism of travel.

Thank you for your response to my question. You very clearly mean quite different things in your use of the words group velocity and phase velocity, and even perhaps velocity. Do you offer any sort of base definitions? I have disentangled your statements of your own theory from you statements of misunderstanding of conventional Physics Conventionally we do not derive the wave nature of light from the wave equation, nor have we ever done this. The attribution of a wavelike nature comes from two directions Historically the first clues were experimental, diffraction effects such as appear the umbra and penumbra of shadows and many others. Then along came this guy Maxwell and his famous four equations concerning electric and magnetic fields (none of which are wave equations) and he further discovered that if he played around mathematically with them as simultaneous equations and applied a bit of calculus he could combine them into one differential equation which had the same mathematical form as what we now call 'The Wave Equation' That is he discovered that combined electric and magnetic fields have the capacity to produce and support the coupled oscillations of wave motion. He further noted that such motion would have the same characteristic speed as recent (to him) measurements indicated light travelled at. He therefore proposed in the mid 1800s that his oscillations and light were one and the same. We now know that that is not enough, but it was pretty good for his time. Half a century later, Einstein carefully considered the relationship between light, regardless of how it works or is propagated, and both Maxwell's laws and the laws of Mechanics and came up with his first theory of relativity which included a velocity which was also had the same value as that measured (by then more accurately) and predicted by Maxwell. We now call that velocity c. Since you have not fully answered my question about the difference between phase velocity and group velocity in a vacuum (have you now dropped this requirement, you didn't mention it lately). Conventionally according to Maxwell they are both the same and both equal c in a vacuum. It doesn't actually matter in Relativity since the mechanism of propagation is irrelevant to relativistic considerations.

Me too, which is why I posed it as a question (more than once) rather than stating the result. The stated aim is to offer an alternative to relativity that results in the same mathematical consequences, but for different reasons. Therefore we must work from the initial postulate through the derivations to see if this is the case . This has prooved rather slippery and elusive. The OP appears to want to jump from postulate to final conclusion without the intermediate working, offering instead just assurances.

Don't you think QM in any form is irrelevent here? The OP has not introduced it. The OP's description is about purely classical and continuum matters, viz the velocity of light and (special?) relativity. He has also failed to see that the mechanism of propagation of light is actually irrelevent to relativity. (Although I am not saying that relativity has no consequences for those properties of light)