In reference to a certain person's theories about space and time, it's interesting to review the issue of symmetries of space-time. Euclid is justly famous for his attempt to give mathematics an axiomatic foundation. His axioms included his fifth or parallel postulate, something that is far from obvious. However, he flubbed a chance to recognize space symmetry. In some of his constructions, he moved geometrical figures while assuming that they would stay undistorted. Over the last two centuries, however, mathematicians have come up with better axioms for Euclid-style geometry, and one of them is the "SAS postulate", after the side-angle-side construction of trigonometry. It states that one gets the same values for the rest of the triangle no matter where one does the construction and in what orientation. It is equivalent to full space symmetry. Flat-space symmetry can be expressed as symmetry under these transforms: x' = R.x + D where R is a rotation/reflection matrix and D is an offset. R is orthogonal; it satisfies R[sup]T[/sup].R = I where T means transpose. This operation can be expressed as a matrix: {x',1} = {{R,D},{0,1}}.{x,1} . It is easy to show that doing (transform 1) . ((transform 2) . x) = (transform 3) . x where #3 is from composition of #1 and #2. Transforms satisfy the properties of abstract-algebra groups, thus we speak of "symmetry groups". Needless to say, there are numerous subgroups of the spatial symmetry group for some number of dimensions, sometimes called the Euclidean group, E(n) or Euc(n). Restricting the offsets to a lattice gives crystal-symmetry groups, and having no offset gives point groups. They have a structure that gets complicated very quickly with increasing numbers of dimensions. For finite point groups: 1D: 2 of them: identity and reflection 2D: 2 infinite families: pure rotation and rotation + reflection -- rosette groups 3D: 7 infinite families and 7 additional ones -- axial / prismatic groups and quasi-spherical groups: tetrahedral, octahedral, icosahedral The continuous ones are called O(n) for n-D rotations and reflections and SO(n) for rotations only. "Orthogonal" and "special". ... For discrete-lattice space groups: 1D space, 1D lattice -- 2 line groups 2D space, 1D lattice -- 7 frieze groups 2D space, 2D lattice -- 17 wallpaper groups 3D space, 3D lattice -- 219 or 230 crystal groups, depending on whether one counts mirror images separately ... As with space-symmetry recognition, I don't know for sure how far back time-symmetry recognition goes back. So there we have it. Space has symmetry group Euc(3) and time has symmetry group Euc(1). As to space and time being related, the first hint was Galileo's thought experiment of being inside a ship and not being able to tell how it is moving. Many of you people have likely had similar experiences with various vehicles. An important feature of Newtonian mechanics is symmetry under what may be called Galilean boosts: x' = x + v*t, y' = y, z' = z, t' = t for x-direction velocity v. I don't know when that was explicitly recognized, however. But the issue likely came up in the late 19th cy., as physicists worked out that Newtonian mechanics and Maxwellian electrodynamics do not coexist very well, and that some proposed coexistence solutions were not supported by experiment, like the Michelson-Morley experiment. Various physicists proposed various solutions, with the most successful one being that of Henrik Antoon Lorentz's one. He proposed that motion relative to a medium called the electromagnetic ether produces distortions in space and time that we may call a Lorentz boost: x' = γ*(x - v*t), y' = y, z' = z, t' = γ*(t - v*x/c[sup]2[/sup]), γ = (1 - v[sup]2[/sup]/c[sup]2[/sup])[sup]-1/2[/sup] This electromagnetic ether was the most common reconciliation, and the Michelson-Morley experiment and various other ones disproved various simple forms of it. Einstein recognized that this solution made the ether physically meaningless, and he worked out how to modify momentum and energy to make them Lorentz-boost-friendly. Hermann Minkowski went a step further, noting that if one modifies the spatial distance to include time contributing in the opposite direction, the result is invariant not only under rotations, but also under Lorentz boosts. Mathematically, the distance invariant is S = x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] - c[sup]2[/sup]*t[sup]2[/sup] This makes Lorentz boots very rotation-like, and it also puts time and space on a roughly equal footing. Hermann Minkowski - Wikipedia notes that he himself had recognized that: In fact, space-time symmetries may be described with the same sort of equation that I presented early in this post: x' = R.x + D where x is a 4-vector, a vector that includes both space and time. However, here, R satisfies R[sup]T[/sup].g.R = g where g is a matrix called the metric tensor. The distance invariant for a 4-vector x is x.g.x . Note about the distance invariant that if it is negative, it becomes a time invariant, and if it is zero, it's an invariant for lightlike motion. The group of space-time symmetries is called the Poincaré group, and it may be denoted Euc(3,1), because space and time contribute in different directions to the distance/time invariant. Without translations, it becomes the Lorentz group, which may be denoted O(3,1). General relativity goes even farther. Space-time can have much less symmetry in GR than in SR, and even no symmetry at all, and this means that it may be hard to find coordinates with clear meanings as space or time coordinates.
If you're referring to me, I'm not some my-theory guy. Typically I refer to Einstein, and you typically tell everybody to ignore Einstein. Einstein reintroduced ether for General Relativity, See Wikipedia and note the quote by Robert B. Laughlin: "It is ironic that Einstein's most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed [..] The word 'ether' has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum..." Also see Einstein's Leyden Address where he said according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether.
If you advocate something, then it's effectively your theory. Treating him as an inspired prophet of revealed truth. (rest of book-thumping snipped) A very different sort of ether from the luminiferous or electromagnetic ether of the late 19th cy.
This post is a great review of Minkowski space time concepts, including invariance of 4D intervals. "S = x2 + y2 + z2 - c2*t2" Yes, these once appeared to me also to be "rotation-like", and this math captures that relation nicely, wrongheaded though the whole concept actually is. For reasons related to discussions I've more recently had with one of my colleagues, I no longer think of Minkowski nor rotation as any sort of solution to the dilemma of space-time. Instead, we propose the concept is backward. Space is subordinate to time: "timespace" is closer to reality than Minkowski's 'spacetime'. Why do we believe this? Time has no meaning without motion, and the most fundamental and unstoppable motion that exists is in the vacuum, not in matter (which derives both its inertial mass and all of its binding energy from the vacuum). In essence, we are saying that the geometry lesson (Euclid through Minkowsky) needs to stop. Time does not progress at the same rate at any two points in the universe. If there actually were a space with no virtual energy or fields whatsoever contained within it, that area would be without any meaning related to the concept of time. But because of the continuous operation of the Higgs mechanism, time is very real for us in our corner of space. Time has won. This universe consists only of time and energy. Matter and space are emergent. All we can say for certain so far is that the speed of light is constant, and that energy itself has a property that is very much like "inertia". Newton and the developers of the mathematics known as calculus did us all a disservice by describing only the laws of motion related to matter (stationary, moving with constant v, moving with constant a, moving with changing a). Why were there no equivalent expressions about generalized motion and acceleration of energy? When it is accelerated, energy can either 1) change direction, or 2) change frequency. Energy is always conserved, even in the motion of space we call gravitation. Energy, whether it is stored in matter, or free EM energy is always an indeterminate quantity. Its value always depends on the state of motion of the observer compared to the state of motion of the source of the energy. Hence, ALL energy is virtual, not just the energy in the vacuum. Give us a calculus that describes the mathematical form for energy in motion the way we currently provide only for things made of the bound energy known as matter. Thanks for not including light cones. They are by far the most useless feature of Minkowsky space-time. Might as well say something like: only a single event at the same location can be simultaneous with itself, the reason every particle in the universe has its own light cone, will never know about events that happened to be obscured from them by something as transparent as a distance of 1 angstrom, because the photon propagated in the opposite direction. Useful for what?
That's unsupportable. It does not explain what produces space, and why space and time are so closely related. As to light cones, they are an important part of the geometry of space-time, and Hermann Minkowski himself had made a light-cone diagram in his paper "Space and Time". Every nonzero vector belongs into one of these five categories: Forward timelike Backward timelike Forward null Backward null Spacelike In a Euclidean space, like familiar 3-space, there is only one category. Here is how it happens. A vector v has a time component, vt, and a spatial part, vs, with length vsl = |vs| = sqrt(vs.vs). Using a timelike metric signature, the length invariant V is given by V = v.g.v = vt[sup]2[/sup] - vsl[sup]2[/sup] V > 0: timelike V = 0: null V < 0: spacelike Consider the time component of a timelike vector. Given the spatial part of that vector, the time component has two possible values: Forward: vt = + sqrt(V + vsl[sup]2[/sup]) Backward: vt = - sqrt(V + vsl[sup]2[/sup]) There is a gap between those two solutions, a gap between -sqrt(V) and +sqrt(V). That is why forward and backward timelike vectors are separate categories. Turning to null vectors, Forward: vt = + vsl Backward: vt = - vsl There is a gap between these two solutions also, a gap at 0, the zero vector. Thus, forward and backward null vectors are also separate categories. But for spacelike vectors, one finds that they can have vt = 0 with vsl nonzero. Thus, a spacelike vector can be either forward or backward, depending on what coordinates one chooses, and spacelike vectors do not split, except if there is only one space dimension.
You have forgotten the fact that as velocities approach c, you get the UNSUPPORTABLE result that space either 1) reduces from 3 dimensions to only 2, or 2) rotates a space-like vector into a timeline one, and vice-versa. This might be OK in terms of the mathematics you have been taught (and so was I), but by the time you are my august age, you will figure out when someone has been scamming you. That someone was Minkowski. The greatest thing anyone has ever written in such math is E=mc^2, but this is only a beginning.
Returning to the transform expression x' = R.x + D, let's consider the continuity of R and D in the most general case. D is always continuous, so let us turn to R. First, the Euclidean case, where the signature of g is all the same sign, and we can take g = I. The R's split up into two continuous parts, rotations and reflections. Rotations: continuous with the identity value of R. Reflections: take a rotation and add a flip along one coordinate direction. Flipping along two coordinate directions gives a rotation, as is easy to show. So an even number of flips gives a rotation, while an odd number gives a reflection. These parts can be distinguished by taking the determinant of R: Rotations: det(R) = 1 Reflections: det(R) = -1 The group of n-D R's is O(n), and group of rotations SO(n) ("special orthogonal"). Adding the D's gives E(n) or Euc(n) in general, and SE(n) or SEuc(n) for pure-rotation R's ("Special Euclidean"). - For signatures of g that contain both +'s and -'s, the R's break down into four continuous parts. Rotations Reflections +: flip along a metric + coordinate direction Reflections -: flip along a metric - coordinate direction Reflections +-: flip along one of each of those coordinate directions Taking the determinant does not completely classify them: the first and fourth have det(R) = +1, and the second and third have det(R) = -1. The full group of R's is called O(n1,n2) for n1 +'s and n1 -'s in the metric; it is equivalent to O(n2,n1). For the rotation part only, I've seen both SO(n1,n2) and SO[sup]+[/sup](n1,n2), with the latter one being for SO(n1,n2) being the det(R) = 1 part. Adding the D's, the full group is E(n1,n2) or Euc(n1,n2), and the R-only-rotation group is SE(n1,n2) or SEuc(n1,n2). Turning to the Lorentz group, the group of rotations, boosts, and reflections, it is O(3,1), with the rotations and boosts making up the "restricted Lorentz group" or SO(3,1). The full group has these four continuous parts: Rotations and boosts Space reflections Time reflections Space and time reflections together
What do you mean physically by "approach c" ? By the velocity composition law, "c" isn't approached by any finite number of sub-c Lorentz boosts. Thus c isn't approached in a sense, made concrete experimentally by measuring the speed of light in vacuum, and illustrated by the use of rapidity instead of velocity, that even after accelerating 10 years with a proper acceleration of 10 g, that you are not "closer" to c, just like 1 million is not "closer" to infinity than zero. If \(\Lambda(v) = { \begin{pmatrix} \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} & \quad & \frac{v}{c^2 \sqrt{1-\frac{v^2}{c^2}}} \\ \frac{v}{\sqrt{1-\frac{v^2}{c^2}}} & & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \end{pmatrix} } = { \begin{pmatrix} \cosh \, \tanh^{-1} \, \frac{v}{c} & \quad & c^{-1} \, \sinh \, \tanh^{-1} \, \frac{v}{c} \\ c \, \sinh \, \tanh^{-1} \, \frac{v}{c} &\quad & \cosh \, \tanh^{-1} \, \frac{v}{c} \end{pmatrix} } = { \begin{pmatrix} c^{-1} & \quad & 0 \\ 0 & & 1 \end{pmatrix} \quad e^{\tanh^{-1} \, \frac{v}{c} \begin{pmatrix} 0 & \quad & 1 \\ 1 & & 0 \end{pmatrix} } \quad \begin{pmatrix} c & \quad & 0 \\ 0 & & 1 \end{pmatrix} } = \begin{pmatrix} c^{-1} & \quad & 0 \\ 0 & & 1 \end{pmatrix} \begin{pmatrix} -\frac{1}{\sqrt{2}} & \quad & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & & \frac{1}{\sqrt{2}} \end{pmatrix} \begin{pmatrix} e^{-\tanh^{-1} \, \frac{v}{c}} & \quad & 0 \\ 0 & & e^{\tanh^{-1} \, \frac{v}{c}} \end{pmatrix} \begin{pmatrix} -\frac{1}{\sqrt{2}} & \quad & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & & \frac{1}{\sqrt{2}} \end{pmatrix} \quad \begin{pmatrix} c & \quad & 0 \\ 0 & & 1 \end{pmatrix} \) then since \(\tanh^{-1} \frac{u}{c} \; + \; \tanh^{-1} \frac{v}{c} \; = \; \tanh^{-1} \frac{u + v}{1 + \frac{ u \, v }{c^2}\) it follows that \(\Lambda(u) \Lambda(v) = \Lambda \left( \frac{u + v}{1 + \frac{ u \, v }{c^2} \right)\) and \(\left( \Lambda(u) \right)^n = \Lambda \left( c \, \tanh \left( n \, \tanh^{-1} \, \frac{u}{c} \right) \right) = \Lambda \left( c \frac{(c+u)^n - (c-u)^n }{ (c+u)^n + (c-u)^n } \right) \). So it's pointless to worry about \(\Lambda(c)\) until empirical physics allows for an instantaneous boost to light speed. Because no finite number of lesser boosts will get you to light speed. This "rapidity multiplication law" is not limited to positive integer multiples, n, of the original boost velocity, u: \(n = 0 \quad \Rightarrow \quad c \frac{(c+u)^n - (c-u)^n }{ (c+u)^n + (c-u)^n } = 0 n = -1 \quad \Rightarrow \quad c \frac{(c+u)^n - (c-u)^n }{ (c+u)^n + (c-u)^n } = - u n = \frac{1}{2} \quad \Rightarrow \quad c \frac{(c+u)^n - (c-u)^n }{ (c+u)^n + (c-u)^n } = \frac{c^2 - c \sqrt{c^2 - u^2}}{u} ; \quad \quad \quad u \neq 0\)
With the discovery of the Higgs boson, confirming both the Higgs field and the Higgs mechanism, we now understand that the speed of light in a vacuum is much more than just a universal constant value measurable from any inertial reference frame. It is the speed of light which provides both the basis for multidimensional time and the means by which the illusion of space is created from it in the universe of matter. Without a hard limit on the speed of light in a "vacuum", space and matter could not exist. Time and spatial dimensions neither rotate nor contract the way Minkowsky proposed. The manner in which space manifests, if it even manifests at all, is relative to the observer, and it works that way because only time and energy actually exist. We observe only "virtual" space. In this universe, and we cannot even observe every aspect of the space we appear to be suspended in. Minkowsky's light cones suggested this, but few really understood them. The energy of a photon or even of moving particles of matter is indeterminate unless you know something about the reference frame which created it. Minkowsky was Einstein's calculus teacher, so it was only natural that everyone would ask him what his take was on the whole relativity idea his student Einstein came up with. They must have been impressed, as evidenced by all of the math that has been devoted to that idea (including, most recently, the "boost" matrices). But it's a fact: this particular corner of relativity has long been the weakest link. It has made no predictions along the lines of E=mc^2, although with some effort it could easily be made to reproduce that result. So the best that can be said about it is, that it is only a redundant part of the theory his student came up with. I suppose if the math needs checking, it served a purpose. But we know better now. E=mc^2 was headed in the direction of the realization that atomic structure itself is very much dependent on the speed of light. It most certainly is, and more influential physicists than I have said so. The spatial component of that structure likewise is dependent only on time, energy, and the speed of light being constant. Give it up. Only time and energy are fundamental, and Minkowski was a good calculus teacher, not much of a physicist.
Citation needed. According to Barry Parker in Einstein's Brainchild: Relativity Made Relatively Easy!, “Einstein took a laarge number of math courses from Minkowski, but he skipped classes so often Minkowski rarely saw him. ... [Einstein] quickly signed up for [Minkowski's class "Applied Mathematical Physics"]” but no details that would allow someone to demonstrate Minkowski taught Calculus are supplied here or in other tertiary sources I consulted. You have condemned yourself by your own words to be regarded as nothing more than a dilettante. You can't have physics without mathematics. So says Newton (an inventor of calculus and the inventor physics taught over the past 325 years), Maxwell and Einstein. Prior to 1912, it is reported that Einstein regarded the introduction of tensors into relativity as "überflüssige Gelehrsamkeit" (superfluous learnedness). In a letter to Sommerfeld on October 29, 1912, Einstein himself wrote about the development of his general theory of relativity: ( See The Collected Papers of Albert Einstein (CPAE, Vol. 5): The Swiss Years: Correspondence, 1902–1914, Princeton University Press, 1993, Doc. 421.) Since Minkowski died in 1909, we really don't know if student and teacher could have reconciled and teamed up; we only know that Einstein went on to build on Minkowsi's introduction of four-dimensional space, tensors and similar geometric thinking. See also notes compiled here: http://arxiv.org/abs/1210.6929
We might not know exactly what it was Minkowsky was supposed to teach Einstein. There was an emphasis on calculus in university higher mathematics, as there should have been. There's this: https://answers.yahoo.com/question/index?qid=20090328061958AAEus1i "(Einstein) was labelled a 'lazy dog' by his mathematics professor - Hermann Minkowski. In August 1900, Einstein only scored 4.9 out of 6 to graduate with his diploma, while his later wife and fellow student Mileva Maric scored only 4.0 and failed the course. " Although I have the greatest respect for mathematicians and their work, I have recounted in these forums only one example of why it is what they do is not always trustworthy to the work of physics. You can see it in String or M Theory. They don't seem to care any more how many dimensions they need to include to get the answer(s) they want. Ever since the incredible success of complex numbers as applied to the expression of kinetic, potential energy math, they seem to feel they have freedom to posit extra ones whenever and wherever they think they perceive a symmetry. One cannot argue with their continued success, but if we impose no limits to our collective credibility of such nonsense, we eventually get exactly what Minkowsky gifted us: math that is light on the bindings that maintain a semblance of a touch with whatever they are modeling. One of the most biting criticisms leveled against the mathematics involved in String or M Theory, for example, is that it literally can be tweaked to provide almost any answer you want. This is something that contrasts with getting the "right" answer, or more particularly, getting the right answer the FIRST TIME, for most people. Einstein did not always get the right answer the first time either. Between the time Eddington's expedition to the Crimea to observe the perihelion of Mercury and the release of the General Theory, a mistake was found and corrected that would have meant the calculation was off by a factor of 2. On coffee cups and T-shirts emblazoned with the Lagranginan commemorating the discovery of the Higgs boson, an additional Hermitian conjugate term was in evidence that would have meant the calculation was off by a factor of 2. This mistake was not found and corrected until after John Ellison sported such a T shirt in a lecture about the discovery. It's one reason that when I gave my own lecture on the same topic, I instead chose the T-shirt emblazoned with the Sheldon Cooper charades game "Higs bowzone pear-tickle". The lack of a g in Higgs was nothing short of genius. Also, when it is pronounced in this manner, it almost sounds like Stephen Hawking himself said it. Hawking lost a bet against the Higgs ever being found. The modern physics scene is littered with mathematically supported fiascos from Edward Witten's misadventures to the Bogdan brothers to the endless honing and polishing of mirrors at the LIGO replay of the Michaelson-Moreley experiment. Skepticism is usually a fine thing in science, but gullibility is decidedly not.
Can you provide an image or link to the Higs T shirt, and explain what "bowzone" and "pear-tickle" mean as substitutes for boson and particle? I'm sure there is some humor in the alternative meaning?
They have nothing to do with c. Nothing. In fact, that constant is nowadays a units factor for relating length and time, and in theoretical work, it's typically to set it to 1. Multidimensional time? How do the extra time dimensions get turned into space dimensions? Why don't you work out mathemtically how it is supposed to happen? There is a certain problem with getting from a 4D Euclidean space to a 4D Minkowskian space. The distance measure becomes singular. What gives you that idea? Can you demonstrate that your notions of space-time are consistent with local Lorentz/Poincaré invariance?
Nothing to do with c? Nothing except that without the Higgs mechanism, electrons, quarks, W and Z bosons ALL LOSE INERTIA, ATTAIN ENERGIES EQUAL TO c AND FLY OFF INTO … WHATEVER THERE IS WITHOUT SPACE. This isn't just a nit. This is the foundation of the Standard Model we are talking about. Or are you still betting your friends that it was all just a bad mathematical dream? There sure is (a problem with getting from a 4D Euclidean space to a 4D Minkowskian space). Why do you suppose that is? No. The Lorentz/Poincare invariance of which you speak is the 4D distance (interval) between events in "space-time". This quantity is of no interest to us, because it is fiction derived of mathematics divorced from actual physics, but fiction nonetheless. I thought I made that clear. I'm serving notice that all of my mathematics professors (exceptional ones they were) are dead, and all save one of my former theoretical physics profs too, so you won't be able to ask any of them if I am doing my math correctly. Maybe that will help. We don't do "spacetime" any more. The correct term is "timespace". As for the correct model of the math, we are working on it. It really doesn't pay to publish such things until and unless you are certain it is right. PS: If someone else beats us to the right formulation, that will be fine. There will be no messy copyright issues. Please, by all means, go for it. It's only about 100 years overdue.
Sure. Please give us a reference for someone doing physics with "timespace". I'm also still waiting for your space-free physics.
It is time for me to ask ... If I speak of a volume of space geometrically in three dimensions, and if that space is filled with the Higgs field, how do you refer to that space. Is it space, or not? Is it time-space? Can you say how you differentiate it from space that contains vacuum energy in the current model?
