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Everything posted by DrRocket

  1. http://en.wikipedia.org/wiki/Einstein%E2%80%93de_Haas_effect
  2. InigoMontoya is right, this is not any easy problem. Good penetration mechanics analysis requires sophisticated hydrocodes and very high rate naterials properties all implemented on high-end workstations. It is not some direct correlation between kinetic energy and dent depth or penetration. That said. 2.54 mm (0.10 in) is not a very thick metal plate, and it should be easily penetrated by a bullet, enen a .22 at the modest velocities that you quote. The bullet weights and velocities that you quote suggest a .22 long rifle. If what you are seeing is only a dent, then I would venture to guess that the bullet has lost a lot of energy and likely been deformed by the fabric, probably an aramid fiber like Kevlar, before it encounters the plate. Dent depth will be highly dependent of the the specific alloy and heat-treatment used which affect hardness, ductility and toughness. As a practical matter, if it were me and I was getting that sort of response with a .22 rimfire, I would get a better vest -- or better yet avoid being shot. A bigger gun would probably be a serious threat.
  3. These "researchers" have obviously not met my mother-in-law.
  4. Put a high-grequency choke filter on the power line and shield the box. If the box is metal it is probably pretty well shielded already. This applies to high altitude generated EMP (HEMP), which is what most people mean by EMP. HEMP is noy nearly as threatening as it is usually portrayed in the popular press. Protection for many devices is quite easy. I once carried a very cheap calculator (free) in a pocket and accidently subjected it to a full threat-level EMP pulse, with no adverse effect at all. If you are talking about source-region EMP or system-generated EMP (SGEMP) then it is much harder and you are probably out of luck.
  5. This is the gist of it. What we call "length" and "time" are bound up in the "deformed 3D+1D measurements" which are different for the alien, astronaut and technician. Those measurements are "real" and they are different, but they are observer-dependent. The world lines of the ends of the bar are invariant and are the same for the alien, astronaut and the technician. But those world lines translate differently into "time" and "length/space" for each of them. That is why I prefer, to avoid confusion, to think about things in terms of invariants and translate back to coordinate-dependent "time" and "space" only at the last minute. But our senses and measurement apparatus only allow us to directly perceive the 3D+1D picture, so a translation to that picture is necessary at some point. You need to understand the difference between "coordinate time" and propertime. Proper time is what clocks, all clocks, measure. This is actually more clear in the context of general relativity than in the context of special relativity where the two often coincide. http://www.scienceforums.net/topic/54990-proper-time/
  6. Thanks. I had not visited that page in a long time. Now that I know more of your background it occurs to me that the book Quantum Mechanics, A Functional Integral Point of View by Glimm and Jaffe might be of interest.
  7. What would you do with a point in some other connected component of the "universe" ? That idea would pretty well make a hash of general relativity. Once you allow a non-connected manifold in the model you can get just about anything. But the connected component in which we find ourselves is, given the apparent accelerating expansion of "space", already large enough to contain regions that are and will remain causally disconnected from us on or near Earth and the galaxy, and hence consideration of anything larger is rather more philosophical than scientific.
  8. The contraction only applies between two reference frames in relative motion. But the traveling twin really will be younger even after the trip is over and the twins are re-united. The reason for the younger twin is quite simple using general relativity with its Lorentz/Minkowski metric. I'll show you how: For simplicity let's start and stop the trip at the South Pole, which along with the Earth is in orbit, free fall. Twin A stays there. Twin B travels to some star and returns. Twin A, in free fall, has a world line that is a spacetime geodesic. Twin B, accelerates and decelerates several times and therefore has a world line that is not a geodesic. Both world lines start at the South pole at the same spacetime point and end there later at another common spacetime point. Now use the metric to calculate the arc length of the two world lines. The geodesic of twin A will be of maximum length, while the non-geodesic of twin B will be shorter. ( On a lorentzian manifold, unlike a purely Riemannian manifold with a positive-definite metric, geodesics maximize rather than minimize length). That "length" is proper time. Ergo twin B isyounger.
  9. It is a matter of taste, personal perspective and inclination. I tend toward your viewpoint and personally much prefer Naber's treatment. But it sometimes helps to see the perspective of "the other guy" as well. Rindler can turn on the mathematics when he wants to, and is a far more important figure in relativity than is Naber. Rindler's two volumes, with Penrose, Spinors ans space-time ought to be mathematical enough for either of us.
  10. Damn. I had not heard that he died. The latest edition of that book is available from the American Mathematical Society. Any of Walter Rudin's books are superb(in order of difficulty and sophistication): Principles of Mathematical Analysis Real and Complex Analysis Functional Analysis Fourier Analysis on Groups Also Analysis, Manifolds and Physics (including Part II, a second volume) by Yvonne Choquet-Bruhat and Cecile DeWitt-Morette
  11. Hint #2 : If A is an SPD think abou the simplest form in which it can be expressed by a suitable choice of basis.
  12. http://www.ams.org/home/page professional information and publications for mathematicians, discounts for members http://www.claymath.org/ info on mathematics and home of the Millenium Problems -- solve a famous problem and win a million $ (or read and understand the problem statements written by world-class mathematicians) http://genealogy.math.ndsu.nodak.edu/ follow the academic geneology of mathematicians (I tracked mine back to Isaac Newton -- that gets me a cup of coffee at Starbucks for about $5)
  13. Design & Analysis of Experiments -- Douglas C. Montgomery Statistics for Experimenters -- Box, Hunter and Hunter Experimental Designs -- Cochran and Cox Statistical Methods -- Snedecor and Cochran Mathematical Statistics -- van der Waerden An Introduction to Error Analysis -- Taylor
  14. Not at all. Suppose that we start with two identical such bars. Leave one in Pavillon de Breteuil with a technician and give its twin to the Alien. Now let the Alien come cruising past the technician at a large fraction of c. The technician will compare his bar to the Alien's bar as they pass. And the Alien will do likewise (on the fly). Both will measure the other's bar as being the shorter of the two. The bars have not "changed", and it is not a matter of measurement technique. It is a matter of the very nature of "space" and "time". That is the nature of relativity. What changes is the very decomposition of spacetime into space and time. It is indeed Reality with a big "R", but it is not the reality of everyday experience. This theory has a mountain of experimental data supporting it. That solid bar not only can but does "contract a lot for the alien, a little for the (slower) astronaut, and not at all for the visitor at rest". And it most certainly is neither "like stating that 1=2=3" nor is "simply wrong'. It may not match your intuition, but your intuition is what is "simply wrong".
  15. I suspect that the answer is "yes", but am not certain what "Reality with a big R exists" means beyond the obvious. I have no desire for some endless philosophical debate on ontology.
  16. This was intended as a post in the thread bon spacetime in special relativity. However the forum software (or me) resulted in the piece being superimposed over the initial post, much editing and extreme confusion. So here it is in a separate thread. It is self-contained and therefore a bit redundant to the spacetime thread. The purpose is to show why "proper time" defined in terms of the "spacetime interval" has anything to do with what clocks measure. This explanation is based on a geometric treatment of special relativity. It is largely based on the treatment given in the book The Geometry of Minkowski Space time by Gregory L. Naber. You can refer to that book for more detailed mathematics and to Essential Relativity, Special General and Cosmological by Wolfgang Rindler for a less mathematical and more physical treatment. Some of Rindler's perspective is also a part of this note. This argument carries over unchanged to general relativity simply by a localization observation -- the metric of GR is locally the Minkowski metric.   What is Minkowski spacetime ? Minkowski spacetime is the setting for special relativity. It is by definition ordinary 4-space with a non-degenerate quadratic form of signature (+,-,-,-). Equivalently one can use a quadratic form of signature (-,+,+,+) and this is the convention used by Naber, but we will use the other convention here. The quadratic form defines an inner product on Minlkowski space. It is analogous to the dot product of ordinary Euclidean space, except that it is not positive-definite. This means that It is possible for the inner product (squared length) of a vector with itself to be negative , and it is possible for the inner product of a vector with itself to be zero even if the vector is not the zero vector. So, in Minkowski space a nonn-zero vector can be perpendicular to itself. That fact requires you to readjust your intuition with regard to some geometric ideas, so don't get blindsided by some of this weirdness. Now just as with the ordinary inner product, there is no a priori need to define a basis so as to express it as a "dot product". To take the ordinary product of two vectors in 2-space just form the product of their lengths and the cosine of the angle between them – no basis needed to do this geometrically. So think of the Minkowski inner product as a geometric idea, and we'll talk about the relationship to a basis set. It takes a little more work than in the usual case of a positive-definite inner product, but one can show that given a non-degenerate inner product one can find an orthonormal basis for the underlying vector space. In this more general case an orthonormal basis is a basis in which distinct elements are orthogonal (have inner product 0) and in which the inner product of any basis element with itself is 1 or -1. One can prove and any two orthonormal basis sets always have the same number of elements with inner product with themselves equal to -1, and that defines the "signature" of the quadratic form. In the case of Minkowski space the signature is (+,-,-,-). The inner product of a vector with itself is called the "squared norm". A vector with a negative squared norm is called "space-like" and one with a positive squared norm is called "time-like". We will denote the inner product, using the Minkowski quadratic form, of 4-vectors X and Y by <X,Y> and then then length of a vector X is the norm of X, [math] |X|=\sqrt{<X,X>}[/math] Transformations that preserve the inner product are (inhomogenous) Lorentz transforms, sometimes called Poincare transforms. One generally restricts attention to a subset of the full set of Lorentz transforms for physical reasons, but that is a subject for another time. Lorentz transforms correspond to individual observers and serve to relate coordinate measurement for one observer to another observer. Objects that are preserved by Lorrentz transforms are called invariants of special relativity. For the purposes herein we will work in units in which the speed of light is 1. That makes the usual formula for gamma simply [math] \gamma=\frac {1}{\sqrt {1-v^2}}[/math] Length in Minkowski space The length of a vector X is just |X|. The length of a curve is given in the usual way. A parameterized curve in Minkowski space is just a function from the real line, or a line segment taking as its values 4-vectors in Minkowski space. Let [math] \phi [/math] be such as curve, defined on [0,1]. Then the length of [math] \phi [/math] (arc length)is just [math] \int_0^1 | \frac {d \phi (\tau}{d \tau}| d \tau[/math] as in the case of ordinary Eudlidean space with a positive definite inner product, one can define an arc length parameter for φ, call it s by (http://www.math.hmc....es_and_arc.html) [math] s(t) =\int_0^t |\frac {d \phi}{d \tau}| d \tau[/math] One can parameterize a curve using arc length, and one finds then that the "speed" along the curve is simply 1.   Proper Time in Minkowski Space The proper time separating the end points of a curve in Minkowski space is simply the length of the curve, and the proper time parameter, τ, is simply arc length. This is a definition. The obvious question is what the definition of "proper time" , τ , has to do with "time", t, since t Is what is measured by the clock of an observer and τ on the surface is nothing but distance associated with an unconventional notion of "length". So far we have worked purely in terms of mathematics and the geometry of Minkowski space. To address this new question requires physical reasoning. \ Consider a curve in Minkowski space that consists of short displacements in space at constant speed. Any smooth curve is approximable by such a curve. This curve represents the trajectory of a particle in Minkowski space, and in the reference frame of that particle we select an orthonormal basis x,y,z,t. Now consider one increment of displacement, from[math] (x_0, y_0,z_0,t_0 )=X_0[/math] to [math](x_1,y_1,z_1,t_1 )=X_1[/math] where the displacement is timelike the length of the displacement τ is just [math] T=|X_1-X_0 |=\sqrt {t_1-t_0)^2-(x_1-x_0)^2-(y_1-y_0 )^2-(z_1-z_0)^2}[/math] =[math]\sqrt{\frac{1-((x_1-x_0)^2+(y_1-y_0 )^2+(z_1-z_0)^2)}{(t_1-t_0)^2 }} (t_1-t_0) [/math] =[math] \sqrt {1-v^2 } \Delta t [/math] =[math] \frac {1}{\gamma} \Delta t[/math] Or [math]\Delta t=\gamma T[/math] This shows that τ is the time sensed by a clock that is co-moving with the particle on this small segment. Since any smooth curve is approximated by a series of such segments it follows that the arc length along the curve is identifiable with the time experienced by a particle moving along the cure. So the parameter τ is deserving of the term "proper time". Note that proper time is defined geometrically, and since it is preserved by Lorentz transforms, it is an invariant of the theory
  17. There have been some questions surrounding special relativity, particularly "time dilation" that might benefit from a global and somewhat abstract perspective on spacetime. This is a bit long, and I thought might therefore be appropriate in a new thread. Yes, keeping various reference frames straight can be confusing, and generates tons of confusion plus the odd "paradox". But there is a way around that mess, and you can thank Minkowski for it. After Einstein built on the work of Lorentz and Poincare to produce special relativity, Minkowski re-cast special relativity in terms of the geometry of a 4-dimensional space with a metric of signature +,-,-,- (or equivalently -,+,+,+). Einstein took that idea and from it constructed general relativity. So, what follows is essentially "special relativity from a general relativity perspective". For the moment, forget about "time". Also forget about "space". You live in a 4-dimensional world called spacetime. It is an affine space, but for clarity pick an arbitrary point and call it the origin. Spacetime is now a 4-dimensional real vector space. Spacetime comes equipped with a non-degenerate inner product, <.,.> We need some theorems that I will state as "Facts". Fact: Given a non-degenerate inner product on 4-space there are vectors [math]x_1, x_2, x_3, x_4[/math] such that [math]<x_i,x_j> = 0 , i \ne j [/math] and [math] <x_i, x_j> = \pm 1[/math]. These vectors form a basis for the vector space and are called an orthonormal basis. A vector [math]x[/math] for which [math]<x,x>=0[/math] is called a null vector. A vector [math]x[/math] for which [math]<x,x> > 0[/math]is called timelike, and a vector vector [math]x[/math] for which [math]<x,x> < 0[/math]is vcalled spacelike. Fact: The number of vectors in an orthonormal basis for which [math] <x_i, x_i> = 1[/math] is the same for any orthonormal basis. If an orthonormal basis has just one such vector the signature of the inner product is said to be +,-,-,-. Minkowski spacetime is 4-space with a metric of signature signature +,-,-,- Special relativity is really just the study of the geometry of Minkowski spacetime. A linear transformation, [math]\Lambda[/math] that leaves the Minkowski inner product invariant, ie. [math] < \Lambda x, \Lambda x>= <x,x>[/math] is called a Lorentz transformation. If in addition, given an orthonormal basis [math] e_1,e_2,e_3,e_4[/math] the sign of [math]<x,e_j>[/math] and [math]< \Lambda x, e_j>[/math] are the same for every null vector [math]x[/math] and each[math] e_j[/math] then [math]\Lambda[/math] is called orthochronous (preserves the direction of time). Hence the term "Lorentz transformation" will mean "orthochronous Lorentz transformation". These are the Lorentz transformations of special relativity. Given an observer and a second one in uniform motion relative to the first one, the relative velocities uniquely determine a Lorentz transformation that give an orthonormal basis for the second observer in terms of an orthonormal basis for the first observer – spacetime is transformed to spacetime but the space/time distinction is lost in the process Consider a smooth timelike curve in spacetime, i.e. a map [math] \phi : [0,1] \rightarrow \mathbb R^4[/math]. Then given the Minkowski metric one can consider the arc length, [math] \int_0^1 |\frac {d \phi (\tau}{d \tau}| d \tau[/math] where [math]|x| = \sqrt {<x,x>}[/math]. A timelike curve is called a world line, and the arc length is called the proper time of the world line. Fact: Given a standard clock which follows a world line in spacetime the time recorded on the clock is the proper time of the world line. The only time measured by ANY clock is the proper time of the world line of the clock. So, just what is "time" ? Consider some observer sitting in his own reference frame at the origin. His wrist watch keeps track of proper time along a world line with coordinates [math](\tau, 0,0,0)[/math]. It then follows immediately that [math]\tau[/math] is proper time. It is also the usual time coordinate. Consider some clock in the same reference at a different fixed "spatial" location [math](\tau, x,y,z)[/math] The proper time for that clock is also [math]\tau], which is still the coordinate time for that reference frame. So, proper time and coordinate time, referred to any single reference frame are the same thing What does a different observer see ? He sees the same proper time for any world line, because the Lorentz transformation preserves the Minkowski inner product. But be has a different orthonormal basis so his mix of space and time is different. This is the source of "time dilation" and "length contraction". Confusion arises from the natural human tendancy to think in terms of space and time as distinct. They are not distinct. They are not invariant. They are coordinate-dependent.
  18. You are confused because it is confusing. Let's stick to Newtonian gravity and consider the problem from the perspective of a distant inertial observer. A gravitational field is conservative. So if you consider a planet in isolation a spacecraft appraching it speeds up and as it goes away from the planet it slows down. The speeding up and slowing exactly compensate and the spacecraft gains no net energy. There is no "slingshot" effect. But a planet is in reality not isolated. It is in orbit around the sun and moving very quickly. So, a spacecraft approaching a planet "from behind" is being dragged along by the planet via a gravitational tether. The planet has a lot of mass and momentum and slows down by a miniscule amount. The spacecraft is much less massive and speeds up appreciably. That is the source of the "slingshot" effect. The energy comes, not from gravity per se, but rather from the kinetic energy of the planet. Gravity merely acts as a "rope" to couple the spacecraft to the planet. Both momentum and energy are conserved.
  19. http://front.math.ucdavis.edu/1009.4107
  20. I think that there are two completely different ideas here being treated as one. Hugh Everett's "many worlds" interpretation of quantum mechanics, the subject of his dissertation under John Archibald Wheeler, is one. Everett's interpretation of QM is not my favorite, but it is simply an interpretation and nothing more. It produces precisely the same predictions as does QM under the Copenhagen interpretation, and is therefore experimentally indistinguishable from it. It is perfectly valid, just as valid as the usual textbook QM. On the other hand the "landscape" of Susskind is an entirely different kettle of fish. I have read Susskind's book, but not the more recent book by Brian Greene (though a copy has now been ordered). My impression is that Greene's multiverse is, if not identical to, at least very similar to Susskind's "megaverse" which realizes his "cosmic landscape". I have to agree with your assessment of the untestable being presented to the lay public as "true" as being, at the very least, a disservice. It is in fact a self-serving disservice, IMO bordering on fraud. The idea behind the "landscape" is roughly as follows: 1. String theory has failed to meet its its initial goal of producing a unique mathematically consistent theory of the fundamental forces. Despite the inability to rigorously define any string theory, it appears to some string proponents that there are something like 10^500 consistent string theories. (Witten seems notably absent from this group.) 2. These string theories are so compelling in their beauty that they must ALL be true and there is a, take your pick -- pocket universe, baby universe, etc -- in a larger, again take your pick -- multiverse, megaverse, etc -- in which the laws of each of these string theories govern physics. 3. In fact, by misapplying probability theory it is concluded that infinitely many copies of each pocket universe must exist. This of course ignores the lack of any probability space in which to apply probability theory -- noted very kindly as "the measure problem", by the more honest but ignored by at least Susskind. 4. With the megaverse in hand proponents then apply "anthropic reasoning" to explain why we find ourselves in a pocket universe in which the laws of physics are "fine-tuned" for life as we know it. They then thumb their noses at deists having concocted a "scientific" explanation for the laws of our little piece of the multiverse. 5. In all of this advocates conveniently ignore their ability to produce any single concrete theory that actually matches the known physics of our pocket. With 10^500 choices available, the right one must be in there somewhere ! So far as I can tell this story is the result of three things: 1) militant atheism converted to its own dogmatic religion, 2) embarrassment that years of string theory and other research have produced no clear concrete candidate for a "theory of everything", and 3) the recognition by the writers that they are mortal and have a very limited amount of time left. The proponents are safe, in that the theory being advocated is not testable, even in principle, so no one can prove them wrong. On the other hand, and for the same reason, no one should care. What would be useful, but much harder and not as profitable as sensational books written for the naive and gullible, would be to find a no-kidding predictive theory that describes the physics that we can actually, at least in principle, observe.
  21. I mean that and a bit more. Invariance of the spacetime interval is just a re-phrasing of the statement that Lorentz transformations preserve the Minkowski inner product (analogous to orthogonal matrices on a Euclidean space). When you formulate special relativity as a theory on Minkowski space 4-vectors are invariant and you can write the equations of dynamics in a completely coordinate-free way, without the need to explicitly identify a reference frame. Only when you need to use measurements from some specific reference frame need the frame be identified. The physical events are the same in all reference frames -- the physics is invariant. You do the same thing in Newtonian mechanics, often without thinking about it. F=ma is not dependent on the specific inertial frame in which F or a are measured. You can pick any convenient inertial frame in which to describe a dynamical system -- the physics is invariant. For instance, an elastic collision in one inertial reference frame is an elastic collision in all reference frames. If you step in front of a bus, the kinetic energy of the bus and the kinetic energy of your body are different if you choose the rest frame of the bus versus your own rest frame, but the kinetic energy lost in the inelastic collision is invariant, and you are dead in either frame. Coordinate-dependent quantities are useful, but the fundamental physics is embodied in the invariants. You can become very confused with coordinate-dependent quantities -- see any of the "paradoxes" of special relativity. But ultimately these things are just components of some 4-vector that is invariant. This is often easier in concept than in concrete realization since translating a common problem statement into 4-vector language takes some thought. The typical "paradox" involves some problem statement that subtly invokes a notion of "simultaneity" which is not invariant, but which our psyche is prepared to accept as invariant. The fundamental problem is that neither time nor space are invariants They are very much dependent on a coordinate choice (aka observer's reference frame). Any timelike vector in spacetime (a vector of positive length if one takes the signature of the metric as +,-,-,-) is coordinate "time" for some observer. Time should really be "proper time". Take two points in spacetime that are joined by a timelike curve (a curve with tangent vectors that are everywhere timelike). The arc length of that curve, called a "world line", using the Minkowski metric, is the proper time of the world line. It is the time that would be measured by a clock having that world line (this is not obvious, but it is true). Proper time is invariant under Lorentz transformations, because the Minkowski norm is invariant. Coordinate time is proper time only in the rest frame of the observer. A lot of this is perspective, analogous to abstract linear algebra as opposed to matrix theory.. One starts with an abstract vector space and only at the last minute chooses an explicit basis or coordinate system. A vector or a linear transformation is an object with certain properties, and only after a choice of a basis is it an n-tuple or matrix. The fundamental properties of vector spaces and linear transformations are made clear and the confusion that comes with an orgy of superscripts and subscripts for matrix elements is avoided.
  22. 4-dimensional spacetime has nothing to do with relativity. It is perfectly natural to cast ordinary Newtonian mechanics in a 4-dimensional spacetime. The difference between Newtonian mechanics and special relativity lies in the geometry of spacetime, not the dimension. The geometry of Minkowski spacetime is determined by bthe Minkowski inner product, which determines the Minkowski norm, or what you call the "spacetime interval. The Significance of Lorentz transformations is that they preserve the Minkowski inner product (and preserve the "direction" of time). The Galilean group leaves invariant the Euclidean metrics on each summand of [math] \mathbb R \oplus \mathbb R^3[/math]. It defines the kinematics of Newtonian mechanics.
  23. My point is not philosophical. I am talking rigorous special relativity. If you let the debate become philosophical then I can pretty much guarantee that it will be fruitless and endless. I have no idea what it takes to change your mind. Try reading a book on relativity, perhaps An Introduction to Special relativity by Wolfgang Rindler. Rulers and clocks are associated with a particular choice of spacetime coordinates. The very fact that different observers see different coordinate times and different coordinate lengths shows that their clocks and rulers are different (record different values) That length contraction and time dilation are real are demonstrated by the decay times of relativistic particles. One of the early pieces of evidence for the validity of special relativity is the deep penetration into the Earth's atmosphere by muons formed by cosmic ray collisions high in the atmosphere. I think that meets the criteria for a "real" effect. A physical event in rigorous relativity is a point in spacetime. Alternately and more loosely a physical event could be a collision, falling into a hole, etc -- something that "happens" at a point in spacetime..
  24. You have to be careful or you will get into an endless and fruitless philosophical debate over what is meant by "real". Physics is in general invariant, even when described by coordinate quantities that are not invariant, Take your bus collision, and look at the Newtonian physics. Yes, the absolute and momentum are coordinate-dependent. But momentum will be conserved in any inertial frame. And the net loss of kinetic energy in the collision is independent of the inertial reference frame (do the calculation, which requires conservation of momentum and see for yourself). In particular, if the collision is elastic it is elastic in all inertial frames. In any frame you will be hit by the bus. In any frame you will be dead. Similarly in relativity, it is the invariants that describe physical events. The real lesson of relativity is that one deals with spacetime, not space and time. What is invariant is the Minkowski norm of a 4-vector. Conventional "length" and "time" are coordinate dependent, not invariant. They are "real", but they are not fundamental. Most of the confusion and so-called "paradoxes" of relativity come from thinking about coordinate dependent quantities as though they were invariant. The Lorentz transformations are important precisely because they preserve the Minkowski inner product and hence the geometry of flat spacetime. Length "contraction" is real, but it is also nothing more and nothing less than a relationship between two separate coordinate systems. Two distinct observers simply have different rulers, and different clocks. But they see the same physical events. Physical events are invariant.
  25. This argument would be more convincing if at least one of the statements in the premise were true. Such is not the case.
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