Bernardhlavenda.com
https://bernardhlavenda.com/rss.xml
enQuantum Non-Demolition Experiments are Impossible
https://bernardhlavenda.com/node/205
<span class="field field--name-title field--type-string field--label-hidden">Quantum Non-Demolition Experiments are Impossible</span>
<span class="field field--name-uid field--type-entity-reference field--label-hidden"><span lang="" about="https://bernardhlavenda.com/user/1" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">adminlavenda</span></span>
<span class="field field--name-created field--type-created field--label-hidden">Fri, 02/15/2019 - 11:48</span>
<div class="clearfix text-formatted field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="tex2jax_process"><p>To salvage the application of the Michelson interferometer for gravitational wave measurements, the LIGO team has resorted to all sorts of ways to avoid the Heisenberg Uncertainty Principle (HUB). Things like Standard Quantum Limit (SQL), Quantum Non-Demolition (QND) experiments, and "back-action" are completely meaningless. All this is the result of confusion between the wave-particle duality of quanta. To talk of the reflection of a photon from a mobile mirror, and to calculate its change in frequency as Braginsky et al do in \it{Quantum Measurement} is the wrong picture to use. Rather, it is the wave picture that is appropriate in which the wave recoils from its target, and to speak about the uncertainty in the wave number that results. Anywhere the wave function in momentum space is appreciable $\phi(k)$, it is impossible to control the momentum of a body to within $\Delta k$, where $k$ is the wave number, while anywhere the wave function $\psi(x)$ is appreciable, there will be an uncertainty in its position $\Delta x$. One cannot consider that "kicks" the position of a particle due to momentum fluctuations, and consider there are no "kicks" in the momenta due to fluctuations in the position of a particle by saying there is no "back action". In fact, back action or back reaction or whatever you want to call it is completely foreign to quantum mechanics. If force cannot be defined (Newton's second law), then surely equal action and reaction (Newton's third law) cannot. Since uncertainty in conjugate variables are determined by the minimum action, $\hbar$, amplitude and phase of an oscillator or an electromagnetic wave are not conjugate variables. The claim that two conjugate variables, $E$, (instead of amplitude) and phase $\varphi$ are related by $$\Delta E\cdot\Delta\varphi=\hbar\omega/2$$ is definitely incorrect since $\Delta\varphi=\omega\Delta t$ so that $$\Delta E\Delta t=\hbar/2$$ is the correct uncertainty relation. The claim that $\Delta E=\hbar L/2\tau\Delta x$ is "not constrained by SQL since by taking long enough time for the measurement to be made, one can obtain any desired sensibility" (Braginsky et. al.) shows a complete insensibility for the meaning of what is a quantum measurement. It is tantamount to considering the amplitude of radiation pressure, given by $\Delta A\propto 1/\omega^2$ to be an error that is not constrained by SQL since by taking a high enough angular frequency, $\omega$, it is possible to reach any desired sensibility. QND measurements [cannot] be made on: 1. observables that are conserved during the free motion of the body; 2. observables in which there is no perturbation by its conjugate variable; 3, observables whose conjugate variables are perturbed according to the uncertainty principle. There are clear distortions and misunderstanding of the nature of quantum measurements. The whole idea of converting a Michelson interferometer making position measurements into a "speed meter" because velocity is a QND variable is ludicrous. Moreover, if amplitude and phase are non-conjugate variables that do not satisfy HUP, what is the meaning of "squeezing" one of them and "stretching" the other? Finally, the introduction of correlations between conjugate HUP variables converts an equality into an inequality so that the proportionality between the uncertainty in one and the uncertainty in inverse of the other is lost.</p>
</div></div>
Fri, 15 Feb 2019 10:48:07 +0000adminlavenda205 at https://bernardhlavenda.comHow LIGO Redefines the Meaning of Quantum Uncertainty
https://bernardhlavenda.com/node/204
<span class="field field--name-title field--type-string field--label-hidden">How LIGO Redefines the Meaning of Quantum Uncertainty</span>
<span class="field field--name-uid field--type-entity-reference field--label-hidden"><span lang="" about="https://bernardhlavenda.com/user/1" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">adminlavenda</span></span>
<span class="field field--name-created field--type-created field--label-hidden">Wed, 02/13/2019 - 11:45</span>
<div class="clearfix text-formatted field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="tex2jax_process"><p>To avoid the negative consequences of Heisenberg's Uncertainty Relations (HUP), LIGO borrows from Braginsky et. al. to introduce what is supposed to be novel developments like the Standard Quantum Limit (SQL) and Quantum Non-Demolition (QND) experiments that supposedly beat the HUP. Yet, we know that HUP stands in the "defense" of quantum mechanics, to use a colorful expression borrowed from Feynman. When we get down to the quantum limit, the measuring device interacts with what is being measured to create unavoidable disturbances to transform what was the system prior to measurement into an entirely new system after measurement. Shining light on an electron imparts a velocity to it so that the more precisely we measure its position that less we know about its speed. This is summarized by HUP: $$S_xS_v=\hbar/m$$ where the spectral density for position measurements is $S_x=\sqrt{\overline{\Delta x)^2}/\Delta t}=\sqrt{\hbar}$, and velocity measurements $S_v=\sqrt{\overline{\Delta v^2}\Delta t}=\sqrt{\hbar}/m$. To "beat" the quantum limit, we must live in a world where the unit of action, $\hbar$ is small or the mass we are measuring is large enough so that it will be impervious to the impulses that photons subject it to. HUP says that measurements in position and speed are perfectly, negatively, correlated: the increase in the variance of one leads to a corresponding decrease in the other. It is therefore inaccurate to claim that a "fundamental premise of SQL is that the optical noise sources--radiation pressure nose and photon shot noise--together enforce SQL \it{only if they are uncorrelated}." (T Corbitt \& N Mavalvala, "Quantum noise in gravitational-wave interferometers" LIGO-P030067-00-R) This mistaken premise allows "the SQL [to] be overcome by creating correlations between radiation pressure and shot noise.,,,QND interferometers are achieved by creating correlations between radiation pressure and shot noise." If anything, such correlations will transform the above equality into an inequality so that there will be no direct relationship between the variances in the radiation pressure and shot noise. If fluctuations in phase are associated with those in position, while fluctuations in amplitude are associated with those in velocity, then such an analogy leads to two orthogonal quadratures of the radiation field where reducing one leads to a corresponding increase in the other. This is supposedly the origin of "squeezed" light. But such correlations will destroy the equality in HUP leading to an inequality so that the direct relations ship between the two uncertainties is lost. Moreover, it is contended that the spectral density of shot noise if flat, where $S_x\propto I_0$, the initial intensity of the beam, while radiation pressure has a spectral density $S_v\propto 1/I_0\omega^2$ "showing that radiation pressure falls off as $1/\omega^2$", where $\omega$ id the angular frequency at which the measurement is made. Consequently, their product will also fall off as $1/\omega^2$, and will not be a universal constant as in HUP. This says the variance of the radiation pressure can be reduced to zero merely by increasing the frequency of the laser beam---something that contradicts all logic. In statistics, HUP go under the name of Cramer-Rao. Non-perfect correlations convert the equality into an inequality which is none other than the Schwarz inequality between correlations in conjugate variables and their standard deviations. In thermodynamics the go under the name of thermodynamic uncertainty relations (TUR) where the inequality is a measure of irreversibility (B H Lavenda, \it{Statistical Physics: A Probabilistic Approach} (Dover, 2016)). Irreversibility is seen to decrease the "information" between conjugate thermodynamic variables, like energy and inverse temperature. They cannot beat the standard thermodynamic limit (STL) in which they are perfectly negative correlated. Finally, any modification of the nature of the laser beam will have unavoidable consequences on the measurement of the displacement of the masses. Here, again, there is an \it{ad hoc} mixture of quantum (photon) and classical measurements (displacement of the 40 kg mirrors). LIGO has first to overcome what ails it before it can say that it has beaten the HUP limit. And it is starting from a disadvantage because a Michelson interferometer do not mix the macroscopic and the microscopic. Moreover, to attempt to convert position into speed measurements, via a Michelson interferometer, because the latter at not acted upon by "kicks" the produce "back action" is to change entirely what the basic principles of a Michelson interferometer are. In fact, "back action", which is sort of a reaction to an action is completely foreign to quantum mechanics which knows no definition of a force (cf. D Bohm, \it{Quantum Theory} (Dover, 1989)).</p>
</div></div>
Wed, 13 Feb 2019 10:45:36 +0000adminlavenda204 at https://bernardhlavenda.comLast Minute Update!
https://bernardhlavenda.com/node/202
<span class="field field--name-title field--type-string field--label-hidden">Last Minute Update!</span>
<span class="field field--name-uid field--type-entity-reference field--label-hidden"><span lang="" about="https://bernardhlavenda.com/user/1" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">adminlavenda</span></span>
<span class="field field--name-created field--type-created field--label-hidden">Tue, 02/12/2019 - 21:41</span>
<div class="clearfix text-formatted field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="tex2jax_process"><p>My new book,</p>
<p><em>Seeing Gravity</em></p>
<p>will be in all the bookstalls early next year, being published by World Scientific Publishing Company.</p>
</div></div>
Tue, 12 Feb 2019 20:41:10 +0000adminlavenda202 at https://bernardhlavenda.comGravitational Waves Masquerading as Schumann Resonances
https://bernardhlavenda.com/node/203
<span class="field field--name-title field--type-string field--label-hidden">Gravitational Waves Masquerading as Schumann Resonances</span>
<span class="field field--name-uid field--type-entity-reference field--label-hidden"><span lang="" about="https://bernardhlavenda.com/user/1" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">adminlavenda</span></span>
<span class="field field--name-created field--type-created field--label-hidden">Tue, 02/12/2019 - 11:43</span>
<div class="clearfix text-formatted field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="tex2jax_process"><p>Metallic conductors like the 4 km long steel tubes of the LIGO interferometer are naturals for the attraction of electromagnetic disturbances. Back in 1893 FitzGerald noticed that the upper atmosphere was a good conductor of electricity. With a depth of 100 km, the cavity formed from what is now known as the ionosphere and the earth facilitates the establishment of very long wavelength electrodynamic oscillations having a period of roughly 0.1 sec. The electromagnetic waves are excited by lightning discharges and have a (discrete) spectrum ranging from 3 to 60 Hz--right smack in the frequency range of the detected initial frequencies of gravitational waves, Part of the electromagnetic disturbance exponentially penetrates through the metal walls, generating currents. The magnetic field is circular around the tube, and Poynting's vector shows that the energy flow is directed inward appearing as Joule heat. This is what Poynting argued. However, Heavside showed that there is a flux of energy directed along the tube, and this could generate a detectable phase shift in the mirrors. Since the mirrors weigh in at some 40kg, it would take some time to get them moving which could be attributed to a rise in frequency. Once the electromagnetic disturbance has passed, the mirrors return to a state of rest thereby mimicking the three phases of inspiralling, chirping, and ringdown that has been attributed to gravitational waves. And because the cavity consists of the earth and the ionosphere, it would explain the near coincidence of the two interferometers in Washington and Louisiana responding to the same electromagnetic perpeturbation. Moreover, it predicts that the "initial" frequencies of the gravitational waves should be at the extremely low frequencies of 14.3, 20.8, 27.3 and 33.8 Hz. If such electromagnetic disturbances are not gravitational waves, how has LIGO excluded such possibilities?</p>
</div></div>
Tue, 12 Feb 2019 10:43:26 +0000adminlavenda203 at https://bernardhlavenda.comWhere Lies the Optical Properties of Matter in General Relativity?
https://bernardhlavenda.com/node/201
<span class="field field--name-title field--type-string field--label-hidden">Where Lies the Optical Properties of Matter in General Relativity?</span>
<span class="field field--name-uid field--type-entity-reference field--label-hidden"><span lang="" about="https://bernardhlavenda.com/user/1" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">adminlavenda</span></span>
<span class="field field--name-created field--type-created field--label-hidden">Tue, 02/12/2019 - 11:39</span>
<div class="clearfix text-formatted field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="tex2jax_process"><p>In the December 12 1936 issue of Nature, Ludwik Silberstein publishes a letter entitled "Minimal lines and geodesics within matter: A fundamental difficulty of Einstein's theory." The issue Silberstein address was that Einstein equates geometry $G$ with a material tensor $T$ so that one should know what the other is talking about. Otherwise, you would be equating apples and oranges. Now according to Einstein's generalization of the Minkowski indefinite metric, $$ds^2=g_{\nu\mu}dx^{\nu}dx^{\mu}$$ its vanishing should still represent "light-lines". Its banal to show that in vacuo the light-lines $ds=0$ represent light trajectories that propagate at speed $c$, while in matter that is characterized by a material tensor $T_{\nu\mu}$, "the minimal lines manifestly cannot represent light propagation, even to a rough approximation." In such an isotropic medium, the light velocity is not $c$ but $c/n$, where $n$ is the index of refraction in the material medium. He uses the case of a Schwarzschild incompressible liquid sphere where the components of the mixed matter tensor are $$T_1^1=T_2^2=T_3^3=-p/c^2, \hspace{30pt} T_4^4=\varrho,$$ where $p$ is the hydrostatic pressure and $\varrho$ is the constant density of the liquid, Since the $T$'s determine the $g$'s via Einstein equation, if there is no trace of $n$, the index of refraction, in $T$, there can be no trace of it in the $g$'s. Wrong! The Schwarzschild metric is determined by Einstein's condition of emptiness, the vanishing of all the components of the Ricci tensor, $R_{\n\mu}=0.$ Due to the fact that the $g$'s depend on $r$ since the medium is isotropic and static, there will be a nonunity index of refraction. But, if we want to consider the Schwarzschild metric immersed in an incompressible liquid sphere who tells $T$ how to react since the index of refraction lies within the $g$'s and not in $T$ which cannot account for gravitational energy at all. Therefore, you are, in effect, equating apples and oranges. Moreover, the radius of curvature of the sphere is $c\surd(3/8\pi G\varrho)$, which is the product of $c$ and the Newtonian free-fall time. So says $T$, but what does $G$ say? It says that the radius of curvature is $c\surd(r^3/GM)$. So which is it? Are we working with a model of constant density, $\varrho$, or one with constant mass, $M$? According to Dirac, the "emptiness" of the system is unaffected by the gravitational field whereas other fields do. But, then there can be no mass, $M$ and the gravitational field lies in the $g$'s. For if there is mass, then there is a density of matter defined as $M/V$, where $V$ is the volume of the incompressible fluid sphere. It is therefore a logical inconsistency to consider the gravitational field to be buried in the metric and not reflected in the matter tensor which is equated to it via Einstein's equations. But if the gravitational field is in $G$ how can it simultaneously be in $T$? The linearized Einstein equations supposedly describe gravitational waves whose source is a pseudo-tensor density. Now the right-hand side of the equation accounts for the source of the gravitational field. So which is it: Is gravity geometry or gravity a force that can do work and should be included in the matter tensor? You can't have your cake and eat it too! Search</p>
</div></div>
Tue, 12 Feb 2019 10:39:56 +0000adminlavenda201 at https://bernardhlavenda.comA Black Hole without Gravity
https://bernardhlavenda.com/node/186
<span class="field field--name-title field--type-string field--label-hidden">A Black Hole without Gravity</span>
<span class="field field--name-uid field--type-entity-reference field--label-hidden"><span lang="" about="https://bernardhlavenda.com/user/1" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">adminlavenda</span></span>
<span class="field field--name-created field--type-created field--label-hidden">Fri, 02/08/2019 - 09:57</span>
<div class="clearfix text-formatted field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="tex2jax_process"><p>A "black" hole by definition is black to anyone observing it from the outside. But, is it "black" inside? The same gravitational potential exists whether "inside" or "outside". Then what distinguishes "inside" from "outside"? It was shown by Luneburg [Luneburg, The Mathematical Theory of Optics, p. 172.] that light rays acted upon by a Coulomb potential cannot penetrate into regions greater than a fourth of the inverse of the absolute value of the total energy. We claim that this should be revised to one-half the inverse of the total energy, which is the difference between the minimum and maximum values of radial coordinate of a ray. Here we develop the basic approach, and, in the next blog, we apply it to the Schwarzschild metric in a static gravitational potential. A variant of the phenomenon will give advance of the perihelion of Mercury, but with a completely different explanation. This will illustrate the complete lack of correspondence between the physics that is used to explain phenomena and what is actually happening. In Landau and Lifshitz, Classical Theory of Fields, $\S 101$, the ultra-relativistic Hamilton-Jacobi equation (HJ), $$g^{ij}\frac{\partial S}{\partial x^i}\frac{\partial S}{\partial x^j}-(mc)^2=0,$$ for the action $S$ is used to derive the advance of the perihelion of Mercury. In the HJ equation they introduce the relativistic energy and rest momentum, $mc$, only to claim later on that they use "the non-relativistic energy (without the rest energy)." Certainly, this should have raised eyebrows, but it didn't. It appears that anything goes as long as you come out with the correct numerical result, which, keeps changing as the technology advances. Even Kevin Brown, in his Reflections on Relativity uses numerical equivalence to discredit Paul Gerber's model of the deflection of light by a massive body even though it gives the correct numerical result for the advance of the perihelion of Mercury. This is absurd, and so too, the physics employed to derive the results. Why use the HJ equation, either ultra-relativistic, or non-relativistic, when Einstein's equations can do the job? Note that the HJ equation contains two masses, the central, $M$, and peripheral, $m$ mass---something that general relativity can't do. So what is the role of the peripheral mass when it enters the HJ equation through the rest momentum, $mc$? It serves only to introduce the Newtonian potential, and disappears when discussing the bending of light by a massive body---as it should. But, the physics is completely wrong, and has absolutely nothing to do with the phenomena of the perihelion advance and the bending of light. Even the generalized line element is completely absurd. We are dealing with static phenomena so that there is no distinction between proper and coordinate time. Are we to believe that proper time belongs to Mercury orbiting the sun and coordinate time to us on earth? The influence of a gravitational field on the motion of a massive body or light is liked to a medium with a varying index of refraction. Although the analogy between an inhomogeneous optical medium and the general relativistic formulation has long been known (beginning with Eddington in his Mathematical Theory of Relativity, it nevertheless leads to incorrect interpretations. An example is the red-shift due to the presence of a static potential. Here, one forgets the space part and transforms the proper and coordinate times to frequencies of emission and reception. The metric implies that this be described by a second-order Doppler shift, even though the geometry of observer and emitter has not been specified! When we throw a ball up in the air, the kinetic energy of the ball is being converted into potential energy so that it finally returns to the ground. Light has no mass so its velocity cannot change. What does change, however, is its wavelength; it becomes longer. And since the medium is static, there is no change in the frequency. So we come to the conclusions that gravity cannot cause red-shifts, and the phenomenon involved has nothing whatsoever to do with a Doppler shift. Frequency changes require a source of energy light a moving mobile; yet, here, there is none. The model we are searching for can be found in Luneburg's book, precisely equation (27.31), where he writes the square of the index of refraction as $$ n^2=C+\frac{1}{r}.$$ According to Luneburg, "[t]he light rays in this medium are identical with the paths of particles which move in a Coulomb field of potential $\phi=1/2r$, and with energy $C/2$." Luneburg gives the equation of the trajectory of light rays as $$\varphi-\varphi_0=\int_{r_0}^{r}\frac{Ldr}{r\sqrt{n^2r^2-L^2}},$$ where $L$ is the angular momentum, a constant of the motion. This can easily be derived from Fermat's principle of least time treating the angle variable, $\varphi$ as a cyclic, or ignorable, or kinosthenic, variable. Evaluating the integral he finds $$r=\frac{2L^2}{1+\sqrt{1+4CL^2}\cos\varphi},$$ which are conic sections. For $C>0$, or positive energy, the curves are hyperbolas, for $C=0$, parabolas, and, finally, for $C<0$, ellipses. We are interested in the latter. Extreme values of $r$ are determined by the vanishing of the denominator in the integral of the trajectory, i.e., $(nr)^2=L^2$, or specifically as $$r=\frac{1\pm\sqrt{1-4|C|L^2}}{2C}.$$ When this is equated with the maximum value of $r$ from the equation of the ellipse, $2L^2$, only the negative root is seen to be acceptable. Hence, the conclusion that no light ray can penetrate a region $r>1/2|C|$. This is, effectively, the inverse of a black hole, which we have obtained for a Coulomb potential. A Newtonian potential will work just as well. The equation for the orbit can be derived from the more general equation $$A\dot{r}^2+r^2\varphi^2=n^2,$$ where the index of refraction is $$n^2=B\dot{t}^2+C.$$ The coefficients $A$ and $B$ can be functions of $r$, which make them automatically, functions of $t$. In the case $A=B=1$, the energy integral has been defined as $\dot{t}$, where $t$ is an affine parameter, having absolutely nothing to do with proper time. This is another inaccuracy committed in making the analogy with the mechanical HJ equation. In a previous blog we have defined the Legendre transform with respect to the radial velocity and showed that $B=1/A$. Moreover, $t$ is a cyclic, ignorable, or kinosthenic, variable so that a first integral, $\dot{t}/A=c$, a constant. If $A\neq 1$, the first integral indicates a conservation in contraction with the above analogy with the total energy of the HJ equation. The proof lies in the fact that the radial equation is a geodesic equation and for $n=c/\surd r$, it coincides with the vanishing of the Weber force! Here, $A=1/r$, and the index of refraction given above becomes $$n^2=\frac{c^2}{r}+C,$$ where the constant $C$ still introduces the Coulomb field, $\phi=1/2r$ with energy $C/2$. There is, therefore, no need to transform, say, the Schwarzschild metric into a conformally related metric, $$d\tau^2=Bd\tau^{\prime\;2}$$ $$c^2 d\tau^{\prime\;2}=c^2 dt^2-A^2dr^2-Ad\Omega^2,$$ where $d\Omega$ is an element of a $2$-sphere. Now, since $g_{00}=1$, it is claimed that the energy integral, $\mathcal{E}=dt/ds'$ exists, and $t$, the coordinate time, is an affine parameter. However, this is not the Schwarzschild metric. Obviously not, it is the Beltrami metric for $B=1-(8\pi/3)G\varrho r^2/c^2$, where $1/\surd(G\varrho)$ is the so-called Newtonian free-fall time. Luneburg's choice $A=1/r$ will not give to a systematic (secular) change in the perihelion of the orbit, but Schwarzschild's choice $A=(1-\mathcal{R}/r)^{-1}$, where $\mathcal{R}$ is the Schwarzschild radius, will. It will also give the deflection of light about a massive body, In the former case, the index of refraction become imaginary $n^2=C$, while in the latter case, the constant $C$ disappears, and with it, the gravitational potential. The equation of the trajectory that correctly describes the advance of the perihelion is $$\frac{dr}{d\varphi}=\pm\frac{r^2}{L}\sqrt{-|C|^2+\frac{1}{r}|C|\mathcal{R}-\frac{L^2}{r^2}(1-\mathcal{R}/r)}.$$ The coefficients of the first two terms under the square root have no particularly interesting effect on the orbit (see above). It is "the change in the coefficient of ^1/r^2$", according to Landau and Lifshitz, that "leads to a more fundamental effect---to a systematic (secular) shift in the perihelion of the orbit." Albeit, that "orbit" only applies to Mercury. However, in the derivation of their expression, they have gone from an ultra-relativistic HJ equation to a negative, non-relativistic energy "without the rest energy", although the rest momentum was fundamental in introducing the Newtonian potential! Consistency appears to be lacking. In the bending of light we don't need the constant, $C$, and with an index of refraction given by $$n^2=\frac{c^2}{1-\mathcal{R}/r},$$ the equation of the orbit becomes $$\frac{dr}{d\varphi}=r^2\sqrt{\frac{1}{\Delta^2}-\frac{1}{r^2}(1-\mathcal{R}/r)},$$ where $\Delta=L/c$, the impact parameter. Neglect of the last term under the square root gives the equation $r=\Delta/\cos\varphi$ of a straight line passing at a distance $\Delta$ from the origin. It is the last term that is responsible for the bending of light due to a "fictitious" mass buried in $\mathcal{R}$. It is "fictitious" insofar as the expression for the coefficient $A$ has been derived under Einstein's condition of emptiness. However, that condition results from the assumption that $A$ is time independent. But, how can it be time independent when it depends on the radial coordinate, $r$? As I trust it is clear from the above analysis, there is no need for any of these shenanigans. Maxwell found no need to introduce a Ricci tensor when he derived the index of refraction for his "fish eye" . It turned out that the index of refraction led to a line element of a sphere that allows a mapping of points on the sphere to points in the plane via a stereographic projection. The light rays in such a medium are curves that a stereographic images of geodesics on the unit sphere, i.e. great circles. A great circle intersects the equator at two opposite points. Hence light rays will intersect the unit circle (the image of the equator) at antipodal points. Hence, the optical image, the antipode of the curve, has the same optical image as the original since it has the same path length. This is Maxwell's solution of a perfect optical instrument which images stymatically in $3D$. Maxwell had no idea he was trespassing into the world of non-euclidean elliptic geometry!</p>
</div></div>
Fri, 08 Feb 2019 08:57:21 +0000adminlavenda186 at https://bernardhlavenda.comThe Relativity of Kepler's Laws
https://bernardhlavenda.com/node/197
<span class="field field--name-title field--type-string field--label-hidden">The Relativity of Kepler's Laws</span>
<span class="field field--name-uid field--type-entity-reference field--label-hidden"><span lang="" about="https://bernardhlavenda.com/user/1" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">adminlavenda</span></span>
<span class="field field--name-created field--type-created field--label-hidden">Fri, 02/08/2019 - 07:25</span>
<div class="clearfix text-formatted field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="tex2jax_process"><p>Everyone thinks of Kepler's laws as being static with the period of orbital motion enters into his third and not time increments. That is, it is the period that is proportional to the area to the power $3/2$, and not the time increments themselves. Yet, time does enter into his law, and there are two different time depending on which focus is used as a reference. We will show that Kepler's laws are fully compatible with the laws of geometrical optics. It will also help us to think of the eccentricity of an elliptical orbit, and its sign, as a relative velocity for all the laws related to Kepler's theory obey the laws of refraction and aberration. The energy integral is $$\mu\left(\frac{2}{r}-\frac{1}{a}\right)r^2-r^2\dot{r}^2=\mu a(1-\varepsilon^2),$$ where $\mu=GM$, $a$ the semi-major axis and $\epsilon<1$, the eccentricity. By considering, $$n_K=\sqrt{\frac{2a}{r}-1},$$ as the index of refraction, the above equation can be recast as the equation for a light ray, where $r\dot{r}=Kz$, where $K$ is the conserved angular momentum and $z=\frac{1}{r}dr/d\theta$ is the slope of the curve. The analogy with geometrical optics will turn out to be more than a mere analogy. The eccentric anomaly, $E$ is defined by $$r=a(1\pm\varepsilon\cos E),$$ depending on which focus, $f^{\prime}$ or $f$ in the figure below, we are considering, $$1\mp\varepsilon\cos E=\frac{a(1-\varepsilon^2)}{1\pm\varepsilon\cos\theta},$$ which relates the eccentric anomaly to the true anomaly, $\theta$. Kepler's equation Solving for cosine and sine of the true anomaly, we find $$\cos\theta_{+/-}=\frac{1\pm\varepsilon}{1\pm\varepsilon\cos E},$$ and $$\sin\theta_{+/-}=\frac{\sqrt{1-\varepsilon^2}\sin E}{1\pm\varepsilon\cos E}.$$ These relations bear an extraordinary similarity to aberration when we regard the eccentricity as a constant relative velocity. For the far focus, $f^{\prime}$, the area of the relevant triangle is added to the area of the sector, $(1/2)Ea^2$ instead of subtracting it as we would due for the near focus $f$. As a preliminary remark, consider the lineal ratio-property $$\sin\theta=\frac{b}{r}\sin E,$$ where $b=a\sqrt{1-\varepsilon^2}.$ Employing the above aberration formula, it follows that $d\theta/dE=b/r$, so that $$\frac{d\theta}{\sin\theta}=\frac{dE}{\sin E},$$ can be integrated to obtain $$\ln\tan\theta/2=\ln\tan E/2+\mbox{const}.$$ Again, by the use of the above formula it can be shown that the constant is $$\frac{1}{2}\ln\left(\frac{1+\varepsilon}{1-\varepsilon}\right)=\tanh^{-1}\varepsilon=\bar{\varepsilon},$$ the hyperbolic measure of eccentricity, so that the solution is $$\tan\frac{\theta}{2}=\sqrt{\frac{1+\varepsilon}{1-\varepsilon}}\tan\frac{E}{2}.$$ As an aside, if we set $\theta=\pi/2$, the side of the triangle corresponds to the semi-latus rectum, shown below, Angle of parallelism the above expression becomes the definition of the angle of parallelism, $$\tan\frac{E}{2}=\exp(-\overline{\varepsilon}),$$ where the hyperbolic eccentricity $\overline{\varepsilon}=\tanh^{-1}\varepsilon$. As $\varepsilon$ varies between $0$ and $1$, its hyperbolic counterpart varies from $0$ to $\infty$. The eccentric anomaly give above is the largest angle that $CP$ touches $Pf$. For all larger angles there are two parallels through the given point $C$, and, moreover, the angle of parallelism is a sole function of the eccentricity, $\varepsilon$, independent of whatever radius $a$ we may choose. Returning to the relation $$\sin\theta=\frac{b}{r}\sin E,$$ we may write it as Snell's law by defining the index of refraction to be $$n=\frac{b}{r}.$$ Since $nr=b$, a constant, the ray equation $$\left(\frac{dr}{d\theta}\right)^2=\frac{r^2}{K^2}\left(n^2r^2-K^2\right),$$ vanishes This determines where $r(\theta)$ reaches an extreme value. However, returning to the above aberrational expressions, we use $$\sin\theta_{-}=\frac{\sqrt{1-\varepsilon^2}\sin E}{1+\varepsilon\cos E},$$ to eliminate $\sin E$ from $$\sin_{+}=\frac{\sqrt{1-\varepsilon^2}\sin E}{1-\varepsilon\cos E},$$ to obtain $$\sin\theta_{+}=\left(\frac{1+\varepsilon\cos E}{1-\varepsilon\cos E}\right)\sin\theta_{-},$$ which is the square of the Keplerian index of refraction, $n_K$, given above. The intermediate angle, $E$, gives a "double dose" to Snell's law. We can "compound" Snell's law, like we do Doppler shifts, when light pases through more than two media of differing index of refraction. We also have $$\frac{dt_{+}}{dt_{-}}=n_K^2, $$ since the two mean anomalies, depending on our reference point, are $$t_{\pm}=\frac{2\pi}{T}\left(1\pm\varepsilon\cos E\right).$$ The two mean anomalies are related to the gravitational field created by the central mass $M$, which gives rise to a non-unitary index of refraction, $n_K$. However, unlike the usual form of the index of refraction, $n_K$ is proportional to the angular velocity rather than be proportional to the inverse of the translational velocity. Finally, if we divide the left side of Snell's law by $1+\cos\theta_{+}$ and the right side by $1+\cos\theta_{-}$ while multiplying it by $n_K^{-2}$ so as to leave the identity intact, we get $$\tan\frac{\theta_{+}}{2}=\left(\frac{1+\varepsilon}{1-\varepsilon}\right)\tan\frac{\theta_{-}}{2},$$ This is precisely the square of what we found in the above expression leading to the angle of parallelism. If we consider $\varepsilon$ as a relative velocity, the above relation is what we would get for a Doppler shift upon reflection by a mirror. Because of reflection, the light ray gets a double dose of a Doppler shift. Parenthetically, I want to correct a typo in eqn (10.1.5b) in my book, A New Perspective on Relativity: there should not be an exponent of $1/2$ in that expression. Now, to go from the square root to its square, we define a new variable, $$\gamma\equiv\frac{2\varepsilon}{1+\varepsilon^2},$$ and write $$\frac{1+\gamma}{1-\gamma}=\left(\frac{1+\varepsilon}{1-\varepsilon}\right)^2.$$ The same transformation is used to go from the Poincare' disc model (conformal} to the Klein disc model (non-conformal) of hyperbolic geometry. Here, we see that it is related to the conformal transformation, $$w=\frac{1}{2}\left(z+\frac{1}{z}\right),$$ known as the Joukowski transform, where $w=1/\gamma$ and $z=1/\varepsilon$ that transforms a one parameter family of ellipses onto a doubly covered family of ellipses. Its square, $$z^2+\frac{1}{z^2}+2,$$ is still an ellipse, but displaced by two units. For hyperbola, $\varepsilon>1$, we would be dealing with the direct quantities, and not their inverses. In terms of the conformal transforms we used in earlier blogs, the $z^2$ term gives the dual laws of Hooke and Newton, while the $1/z^2$ gives the new, and physically unknown, laws of the inverse fourth and seventh force laws. We know that $|z\pm f|=\mbox{const}$ are ellipses, since their sum (arithmetic mean) $$|z+f|+|z-f|=\mbox{const},$$ while their square, $|z^2\pm f^2|$ (geometric mean) $$|z+f|\cdot|z-f|=R^2=\mbox{const},$$ lead to Cassinian ovals. For $R>|f|$, we get one closed oval, while for $R<|f|$, the Cassinian splits into two separate ovals. At the boundary $R=|f|$ there results an ordinary leminiscate. In our next blog we we relate these to force laws. Here we have given the motivation for writing down a specific conformal transformation.</p>
</div></div>
Fri, 08 Feb 2019 06:25:19 +0000adminlavenda197 at https://bernardhlavenda.comThe Luminosity Attributed to Gravitational Waves is Incompatible with a Black-Body Spectrum
https://bernardhlavenda.com/node/199
<span class="field field--name-title field--type-string field--label-hidden">The Luminosity Attributed to Gravitational Waves is Incompatible with a Black-Body Spectrum</span>
<span class="field field--name-uid field--type-entity-reference field--label-hidden"><span lang="" about="https://bernardhlavenda.com/user/1" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">adminlavenda</span></span>
<span class="field field--name-created field--type-created field--label-hidden">Wed, 02/06/2019 - 18:36</span>
<div class="clearfix text-formatted field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="tex2jax_process"><p>In our last blog we derived the speed of gravity from the observational period and variation in the period of the binary pulsar PSR 1913+16 that was discovered by Taylor and Hulse. The luminosity spectrum they used went back to a 1963 paper of Peter and Mathews who took the rate of energy emission of Einstein's rotating dumbbell, $$L=\frac{G}{5c^5}\left(\frac{d^3Q_{ij}}{dt^3}\frac{d^3Q_{ij}}{dt^3}-\frac{1}{3}\frac{d^3Q_{ii}}{dt^3}\frac{d^3Q_{jj}}{dt^3}\right),$$ and evaluated the quadrupole moments using a Keplerian ellipse $$d=\frac{a(1-e^2)}{1+e\cos\psi},$$ and the angular "velocity" $$\dot{\psi}=\frac{G(m_1+m_2)a(1-e^2)]^{1/2}}{d^2}.$$ However, $\dot{\psi}$ will only be the angular speed, which implies that $$\dot{\psi}^2a^3=\frac{GM(1+e\cos\psi)^4}{(1-e^2)^3},$$ which should be independent of both the eccentricity $e$ and the true anomaly, $\psi$, for then it would reduce to Kepler's III. Be that as it may, they find a luminosity given by $$L=\frac{2}{5}\frac{GM^2}{c^5}a^4\omega^6,$$ where $\omega$ is the angular speed of a binary system of two equal masses, each with mass $M/2$ in a circular orbit of radius $a/2$ about the system's center of mass. The LIGO team took this as the "gravitational wave luminosity". How electromagnetic radiation became gravitational radiation is anyone's guess. However, their expression is incompatible with the luminosity of a star considered as a perfect black body. Omitting numerical factors, the Luminosity is $$L\sim\omega E\sim \omega\frac{(kT)^4a^3}{(\hbar c)^3},$$ where $E$ is the total radiated energy. In comparison their proposed luminosity can be written as $$L\sim\omega\frac{GM}{a}\left(\frac{a\omega}{c}\right)^3.$$ In order that this be associated with gravitational waves, we must assume with Poincare' that gravitational waves travel at the speed of light. Over a decade later, Einstein made the same assumption but forgetting to reference the source. Equating the two expressions, we come to a blatant contradiction: the absolute temperature cannot depend on the speed of light, or for that matter, any other speed. We have point out in a previous blog, "An error in the LIGO calculation of the 'chirp' mass" that the luminosity must be given by $$L\sim\omega\frac{GM}{a}\left(\frac{a\omega}{c}\right)^3,$$ for in this case the temperature will be given by $$kT=\left[\frac{GM}{a}(\hbar\omega)^3\right]^{1/4},$$ independent of $c$. We recall from yesterday's blog that the cubic term in the expression for the luminosity is a third-order aberration effect. Einstein would raise it by a power of two. For then the temperature would be a function of $c$, and temperature measurements would allow the determination of the speed of light. Gravitational waves are not the same as electromagnetic waves confined to a black-body cavity! And even if it were, it would not be given by the Peters-Mathew expression. We may venture to derive the expression for the luminosity of gravitational waves that should (and must) propagate at a velocity $v_G$, which we have no reason to suppose is the same as $c$. The luminosity would be given by $$L\sim\omega\frac{GM}{a}\frac{a\omega}{v_G}.$$ The corresponding thermal luminosity would be $$L\sim\omega\frac{(kT)^2}{\hbar v_G}a.$$ This would correspond to a one dimensional system of dimension $a$. The temperature of such a system would be $$kT=\left(\frac{GM}{a}\hbar\omega\right)^{1/2}.$$ We would also venture so far as the say that the dynamics of a rotating dumbbell is incompatible with Keplerian dynamics, as it is with black-body radiation.</p>
</div></div>
Wed, 06 Feb 2019 17:36:18 +0000adminlavenda199 at https://bernardhlavenda.comFrom Newtonian Gravitation to Maxwell's Fish-Eye to Cassini Ovals
https://bernardhlavenda.com/node/196
<span class="field field--name-title field--type-string field--label-hidden">From Newtonian Gravitation to Maxwell's Fish-Eye to Cassini Ovals</span>
<span class="field field--name-uid field--type-entity-reference field--label-hidden"><span lang="" about="https://bernardhlavenda.com/user/1" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">adminlavenda</span></span>
<span class="field field--name-created field--type-created field--label-hidden">Tue, 02/05/2019 - 11:20</span>
<div class="clearfix text-formatted field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="tex2jax_process"><p>It is known [Needham, Am. Math. Monthly 100 (1993) 119-137] that there are only three pairs of integer exponents for dual force laws $$[1,-2]\hspace{25pt}[-5,5]\hspace{25pt}[-4,-7].$$ $[\tilde{a},a]$ stands for pair of exponents in the force laws $$\frac{d^2z}{dt^2}=-z|z|^{\tilde{a}-1},$$ and $$\frac{d^2w}{d\tau^2}=-w|w|^{a-1},$$ where $z$ and $w$ are related by the conformal transform, $$w=z^{\alpha},$$ where $\alpha=(\tilde{a}+3)/2$, and $(\tilde{a}+3)(a+3)=4$. The two times are related by $$\frac{d\tau}{dt}=|z|^{\tilde{a}+1}.$$ The first pair stands for the dual Hooke-Newton's law, the second, as we have shown, to the inverse fifth power law in points of inversion in Maxwell's fish-eye, while the last one to Cassin ovals. You cut a cone by a plane and get conic sections; you cut a doughnut (torus) you get Cassini ovals. For an ellipse with eccentricity less than one, the sum of the distances from the two foci to any point is a constant. Cassini ovals replace the arithmetic mean by the geometric mean: the product of the distances between any point and the two foci is a constant. Therefore, the Cassini ovals are the next simplest closed curves after ellipses, and this probably drove Cassini in 1680. This is the 'next' and last generalization since it constitutes the third pair of exponents in the above list of dual laws. A little history. Cassini ovals are known as spiric sections (spiric=torus). In a sense they are generalizations of conic sections that were first constructed by Menaechmus in 150 BC. Two centuries later, Perseus considered slicing a torus with a plane, and obtained the spiric sections. Now most of the planets describe ellipses with low eccentricity. For such ellipses, the angular sectors created by rays from a focus are approximately equal to the corresponding angle at the other focus. Using this property Ward attempted to find the true anomaly of a planet given the mean anomaly. The ratio between the infinitely small sector and the corresponding angle is like a rectangle of two lines passing through the focus, and for an ellipse of low eccentricity this rectangle remains constant. Cassini reasoned that such a rectangle will be constant in an oval, and this gave birth to the cassinoid. Let us begin with the first pair. A conic section is given in polar coordinates by $$\cos\theta=\frac{p\epsilon-r}{r}=n^2<1,$$ where $p$ is the focal parameter, and $\epsilon>0$ is the eccentricity. The second equality defines the index of refraction, $n$. Expressing Kepler's area law as $$\sin\alpha=\frac{K}{nr}=\frac{K}{\sqrt{r(p\epsilon-r)}},$$ it is easy to show that the radius of curvature, $\rho$, is given by $$\rho\sin^3\alpha=K,$$ the angular momentum. This leads immediately to Newton's inverse square law of force. That is, introducing $-n^2r/n^{\prime}K$ for the curvature, where the prime stands for differentiation with respect to $r$ and $\sin\alpha=K/nr$, there results $$n^2=\frac{2K}{r}+2C,$$ upon integration, where $C$ is an arbitrary constant of integration. For $C=-K/2a$, we have an elliptic orbit with semi-major axis $a$, $0$ for a parabolic one, and $C=K/2a$ for a hyperbolic orbit. The parabolic orbit equates the index of refraction with the escape velocity where $K=GM$. Finally, for circular orbits, $n^2=K/r$, which is none other than Kepler's third law, $rn^2=K$ There is little symmetry between the two angle expressions. This changes when we go to the sphere, where the angles are given by $$\cos^{-1}\left[\frac{K}{\sqrt{1-4K^2}}\left(r-\frac{1}{r}\right)\right]=\theta,$$ and $$\sin^{-1}\left[K\left(r+\frac{1}{r}\right)\right]=\alpha.$$ The arguments are a Jukowski transform which shifts the center of the ellipse to the left and right foci when squared. A Hookean ellipse is thus transformed into a Newtonian ellipse simply by squaring. The index of refraction is that of Maxwell's fish-eye $$n=\frac{1}{1+r^2},$$ and the force directed to the center of its source, $$F=-\frac{1}{2}\frac{d}{dr}n^2=\frac{r}{(r^2+1)^3},$$ is an inverse fifth order law for both points of inversion. Cassini ovals appear when we square the Jukowski transforms $$\cos 2\theta=\frac{1}{2a^2}\left(r^2+\frac{a^4-b^4}{r^2}\right).$$ and $$\sin2\alpha=\frac{1}{2(a\epsilon)^2}\left(r^2-\frac{a^4-b^4}{r^2}\right),$$ where $a$ is the distance to either focus, and $rr'=b^2$ is the bipolar equation, proportional to the geometric average of two distances to either fixed point (foci). The first equation gives the polar equation, $$r^4-2a^2r^2\cos 2\theta+a^4=b^4,$$ with eccentricity $\epsilon=b/a$. The constant $K=1/(a\epsilon)^2$, and $$n=\frac{2r}{r^4-(a^4-b^4)}.$$ This gives the (known) radius of curvature, $$\rho=\frac{2b^2r^3}{3r^4+(a^4-b^4)},$$ with inflection points located on the lemniscate $r^2=2a^2\cos 2\theta$, for which the force law will become a central force law, as we now show. The force directed to its source is $$F_C=2rb^2\frac{3r^4+(a^4-b^4)}{(r^4-(a^4-b^4))^3}.$$ This force governs trajectories of the motion of a central field of force whose strength is inversely proportion to the 7th power of the distance to the source. For $a=b$, or $\epsilon=1$, the Cassini oval transforms into a lemniscate of Bernoulli, $r^2=2a^2\cos 2\theta$, with a central law of force $$F_C=\frac{12a^4}{r^7}.$$ This is the last pair of integer exponents in the trio given above. There are no other integer exponents for the dual laws. It remains, among other things to analyze the dual force law $F\sim 1/r^4$. It has been fantasized [Tapia, Int J Mod Phys D 2 (1993) 413] that spiric sections may lead to a gravitational theory with a fourth-rank metric tensor. Such is the influence of general relativity. Equally fanciful is the association of Bernoulli's lemniscate with an oriented matroid theory, which is a combinatorial structure that has been proposed as the underlying structure of $M$-theory [Nieto, hep-th/0506106]. On thing is sure: Nature is frugal and doesn't waste its integer exponents on nothing. But that nothing is still to be discovered.</p>
</div></div>
Tue, 05 Feb 2019 10:20:52 +0000adminlavenda196 at https://bernardhlavenda.com'Naked' Singularities Exist: the Schwarzschild Metric Doesn't Apply
https://bernardhlavenda.com/node/192
<span class="field field--name-title field--type-string field--label-hidden">'Naked' Singularities Exist: the Schwarzschild Metric Doesn't Apply</span>
<span class="field field--name-uid field--type-entity-reference field--label-hidden"><span lang="" about="https://bernardhlavenda.com/user/1" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">adminlavenda</span></span>
<span class="field field--name-created field--type-created field--label-hidden">Wed, 01/23/2019 - 10:08</span>
<div class="clearfix text-formatted field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="tex2jax_process"><p>Newton was well-aware of the fact that an inverse cubic orbit implies a bound orbit reaches the origin "by an infinite number of spiral revolutions..." This he first mentioned to Hooke in a letter addressed to him on the 13th of December 1679. General relativity, basing itself on the outer Schwarzschild solution, claims that "naked" singularities, like the inverse cubic law, don't exist. But, has anyone ever asked to what central force law the Schwarzschild metric corresponds? It certainly will not due to claim asymptotic flatness, and in the weak field limit, claim that a Newtonian potential exists. Moller, in his Theory of Relativity dissects Einstein's expression for the deflection of light by a massive body into the sum of a light ray passing through a medium with a nonuniform index of refraction, $$n_S^2=\frac{c^2}{1-\mathcal{R}/r},$$ and a second contribution coming from the non-Euclidean nature of the metric, $$\frac{\dot{r}^2}{1-\mathcal{R}/r}+r^2\dot{\varphi}^2=c^2.$$ A general property of uniform deflection is that the medium possess singularities. The index of refraction should not diverge more strongly than $1/r$ as $r\rightarrow 0$. This is clearly not case in Schwarzschild's case. So in order to rationalize the result, Penrose came up with the assumption that Nature does not like "naked" singularities. Apart from the fact that Schwarzschild's inner solution has been swept under the carpet, an index of refraction does not meet the requirements of uniform deflection. In fact, optics clearly shows that "naked" singularities exist. Hence, there must be something wrong with Schwarzschild's outer metric which is bounded by the Schwarzschild radius, $\mathcal{R}$. In fact, we will show that it violates the conservation of energy. It has long been known (1911, known as Bohlin's principle) that there is a certain duality between Hooke\s law, described by the differential equation $$\frac{d^2z}{dt^2}+z=0,$$ and Newton's law $$\frac{d^2w}{d\tau^2}+\frac{4Ew}{|w|^3}=0,$$ where $2E$ is the total energy of the oscillator $$\left|\frac{dz}{dt}\right|^2+z^2=2E.$$ We also require that Kepler's law of areas be obeyed: the planets sweep out equal areas in equal times, with the sun at one of the foci of the ellipse. This means for $z$ that $$|z^2|\frac{d\varphi}{dt}=L,$$ while for $w$, it implies $$|w|^2\frac{d\varphi}{d\tau}=L.$$ This means that the two times, $t$, and $\tau$, must be related by $$\frac{d\tau}{dt}=\frac{|w|^2}{|z|^2}.$$ For the conformal transformation, $$w=z^2,$$ $\tau$-time will increase more rapidly that $t$-time by $$\frac{d\tau}{dt}=|z|^2.$$ The duality can be expressed by the Maupertuis-Jacobi (MJ) principle: $$E-U(z)=\left|\frac{dw}{dz}\right|^2\left(E^{\prime}-U^{\prime}(w)\right),$$ which bears a striking similarity to Fermat's optical principe when we define the index of refraction as $$n^2=E-U.$$ Let $w=z^{\alpha}$ be any conformal mapping. Then, such a mapping transforms the field with potential $U(z)=\left|\frac{dw}{dz}\right|^2$ into trajectories of motion of a field with potential $U^{\prime}(w)=\left|\frac{dz}{dw}\right|^2$. That is, the duality principle is expressed by the conditions $$U(z)U^{\prime}(w)=EE^{\prime}=-1.$$ For $\alpha=2$, the conformal mapping transforms trajectories with potential $U(z)=4|z|^2$ into those with potential $-1/4|w|$. The former is the potential of an harmonic oscillator, while the latter is Newton's gravitational potential. Thus, the Maupertuis-Jacobi principle is satisfied identically $$1-4|z|^2=4|z|^2\left(-1+\frac{1}{4|w|}\right).$$ Now, the same principle is also satisfied by a linear repulsion law: $$\frac{d^2z}{dt^2}-z=0.$$ The same mapping now transforms trajectories with potential $U(z)=-4|z|^2$ into those with potential $U^{\prime}=+1/4|w|$. In other words, repulsion appears as negative gravitation. Again, the MJ principle, $$-1+4|z|^2=4|z|^2\left(1-\frac{1}{4|w|}\right),$$ is satisfied identically. The attractive case contains turning points, such as those that exist in scattering theory. Denoting $4|z|$ by $r$, the left-hand side of the MJ principle: $$n_L^2=1-r^2,$$ while, the right-hand side gives an index of refraction $$n_E^2=\frac{1}{r}-1,$$ where $r$ denotes $4|w|$. The former is the Luneburg index of refraction while the latter is known as Easton's index of refraction. It satisfies the condition for uniform deflection, diverging no greater than $1/r$ as $r\rightarrow 0$.Eaton's index of refraction profile is shown below for $r<1$. Rays come from infinity and return to infinity in the same direction, cat's eye it a perfect cat's eye. Unlike the Schwarzschild index of refraction, $n_S$, $n_E$ has a naked singularity at $r=0$. $n_E^2$ also happens to be Newton's index of refraction for an inverse square central force law. It will account for any conic orbit, regardless whether the origin is at the center or a focus since the two are related by the squaring the former. It is commonly believed that optical orbits are the zero-energy limit of mechanical ones. The barrier is completely illusory as Maxwell's fish-eye has demonstrated: Newtonian elliptic orbits with an index of refraction $n_E^2$ are stereographic projections of curves on the surface of a sphere representing Maxwell's fish-eye. In the case of repulsion, we define the indices of refraction as the negative inverses of attraction. On the left-hand side we have an index of refraction. $$n^2=\frac{1}{r^2-1},$$ while on the right-hand side of the MJ principle the index of refraction is $$n^2_S=\frac{1}{1-1/r}.$$ This would precisely the Schwarzschild index of refraction given above. What was gravitational attraction now has become gravitational repulsion. In comparison with the Eaton lens, where the index of refraction tends to infinity as $r\rightarrow 0$ as $r^{-1/2}$, Schwarzschild's index of refraction tends to $\infty$ as $r\rightarrow 1$! The barrier, or horizon, which has been put into place has nothing to do with a scattering process, regardless of whether it be electromagnetic or gravitational. It is as if gravity has become repulsive, in the sense discussed above. General relativity was so impressed with the (outer) Schwarzschild solution that they invented a "Cosmic Censorship" principle to hide 'naked' singularities. Optics has no such qualms, and the Eaton lens manifest the general feature of a uniformly deflecting lens as we have discussed above. Without singularities there can be no uniform deflection. Then what is the significance of using the (outer) Schwarzschild metric as a prototype metric for discussing the deflection of light rays by a massive body? For small $r$, the so-called turning parameter $\rho\equiv nr$ must tend to zero, implying that $n$ must diverge more slowly than $1/r$. This is certainly not satisfied by the Schwarzschild index of reflection. In fact, there is absolutely no justification for using the Schwarzschild metric in the calculation of the angle of deflection of light rays, other than it was the only one on the market. And instead of attracting light rays, the massive body repulses them according to the Schwarzschild metric. This is manifested by the fact that the scattering angle which the general relativists calculate is positive, meaning that light rays bend outward in violation of Snell's law which says that when light rays traverse a medium with an index of refraction greater than $1$ they must bend inward. Search</p>
</div></div>
Wed, 23 Jan 2019 09:08:19 +0000adminlavenda192 at https://bernardhlavenda.com