tag:blogger.com,1999:blog-1500584444083499721Tue, 26 Jul 2016 17:11:47 +0000theory of flightgreenhouse effectclimate sciencemathematics educationmyth of backradiationquantum mechanicsclimate politicsRoyal Swedish Academy of Sciencesblack body radiationmatematik-ITAIAAKTH-gateDLRclimate simulationclimate sensitivityQuantum ContradictionsKTH-gate3theory of relativityphysicsscienceengineering education reformsimulation technologyBodyandSoulKammarrättenmathematicsPlanck's Lawthermodynamics of global climateStefan-Boltzmann's LawIPCCLennart Bengtssonphysical quantum mechanicsSchrödinger's equationHögsta FörvaltningsdomstolenKTH-gate2CO2Einsteingravitationradiative heat transfer2nd law of thermodynamicscosmologyemissivityclay problemNew View on gravitationSky DragonKVAinterviewsMattelyftetPlanck's constantfluid mechanicsclimategatefinite precision computationStandard Caculusmysticism of modern physicsFalse-SBOLRmany-minds relativityradiating atomCFDFundamental Theorem of CalculusKTHOffentlighetsprincipenbig bluffsvensk klimatpolitikinfrared thermometerpyrgeometermodtranIPCC Trickphotonsphysics illusionsroy spencerturbulencePrandtlJudy CurryStefan Lövenfred singerradiative forcinguncertainty principleclimate modelsdark energyfinite element quantum mechanicsresonanceNADAPrandtl Medalbolometerlapse ratemany-minds physicsCERESKVAs klimatuttalandeMuir Russell InquiryNavier-Stokesarrow of timedark matterlindzensecret of separationteknikdelegationentheory of sailingIR cameraMSTclimate skepticdAlembertgatephotoelectric effectpiano acousticsDr FaustusFEniCSFakultetsnämnden KTHLaplace demonRoyal SocietySVTSvenska Mekanikdagareconomical crisisfree willrelativityBert Bolin Centre for Climate ResearchCMBEquivalence PrincipleGlenn Research CenterKirchhoff's lawSvenska Matematikersamfundetazarcensorshipcosmological principledirection of timeextended Newtonian gravitationibookkollegialt lärandephlogistonthermal imaging3rd Nobel SymposiumAcademic Rights WatchAnders ÅngströmBergsonEuler equationsFreedom FestHubble's LawKlimatupplysningenLIGOLorentz transformationOckham's razorP = NP?Penguin logicRiksbankenSchwarzschildZeno's arrow paradoxboundary layerconstructive physicscrisis in physicsdefinition vs physical factiducationperiodic tableprinciple of least actionrepo rateshorttime vs longtime accuracystring instrumentsBig PhysicsChandrasekharFinal SolutionHans RoslingHelmholtz ReciprocityHiggs mechanismMargot WallströmMichelson-MorleyPopperQEDSMHISULFdigitaliseringskommissionenentropyfaint sun paradoxfinite elementinstant action at distancelord moncktonnobel prizenumerical analysisstellar aberrationwave-particle duality2nd lawBig BangCJ70DNDeleuzeDiracDiscussion ForumEinstein's equationFeynmanGoogle BooksHamstenHawkingHeisenbergKTH-studenterKarolinskaKnut ÅngströmKodcentrumKutta-ZhukovskyMOOCMacchiariniMagnus effectMaxwell's equationsN=NP? 2nd LawNewton's 2nd lawP=NP? 2nd lawSRSagnac effectTheoryOfEverythingUlf DanielssonUnicornWien's displacement lawabsolute simultaneityaeroacousticsaniconismatmosphere spectrumbigbrunobutterfly effectconduction vs radiationcopernicuscosmological modelcritical thinkingdark age of sciencedefinition as factdynamical systememissvitygalileohäggströminteractive computationlogical fallacymattekomissionenmillikanpeer reviewpokerprypyrgeomterscientific methodseminarssimultaneityspiral galaxystability-wellposednessstring theorysvarta håltim balltyndallvon NeumannwellposednessÖppna GöteborgClaes Johnson on Mathematics and Sciencetowards understanding by critical constructive inquiryhttp://claesjohnson.blogspot.com/noreply@blogger.com (Claes Johnson)Blogger1542125tag:blogger.com,1999:blog-1500584444083499721.post-5116196218121688760Mon, 25 Jul 2016 14:32:00 +00002016-07-25T16:34:43.464+02:00New Quantum Mechanics 6: H2 MoleculeComputing with the model of the previous post in polar coordinates with origin at the center of an H2 molecule assuming rotational symmetry around the axis connecting the two kernels, gives the following results (in atomic units) for the ground state using a $50\times 40$ uniform mesh:<br /><ul><li>total energy = -1.167 (kernel potential: -4.28, electron potential: 0.587 and kinetic: 1.147)</li><li>kernel distance = 1.44</li></ul><div>in close correspondence to table data (-1.744 and 1.40). Here is a plot of output:</div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-HlwlKFeRGsE/V5YiNwQojFI/AAAAAAAA8ZA/FPRhlneKyy8opgOAvxFO6KjEk_hxmXQawCLcB/s1600/IMG_0005.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://1.bp.blogspot.com/-HlwlKFeRGsE/V5YiNwQojFI/AAAAAAAA8ZA/FPRhlneKyy8opgOAvxFO6KjEk_hxmXQawCLcB/s640/IMG_0005.png" width="480" /></a></div><div><br /></div>http://claesjohnson.blogspot.com/2016/07/new-quantum-mechanics-6-h2-molecule.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-5817148563466368048Sun, 24 Jul 2016 15:12:00 +00002016-07-26T06:47:00.025+02:00New Quantum Mechanics 5: Model as Schrödinger + NeumannThis sequence of posts presents an alternative Schrödinger equation for an atom with $N$ electrons starting from a wave function Ansatz of the form<br /><ul><li>$\psi (x,t) = \sum_{j=1}^N\psi_j(x,t)$ (1)</li></ul>as a sum of $N$ electronic complex-valued wave functions $\psi_j(x,t)$, depending on a common 3d space coordinate $x$ and a time coordinate $t$, with non-overlapping spatial supports $\Omega_j(t)$ filling 3d space, satisfying for $j=1,...,N$ and all time:<br /><ul><li>$i\dot\psi_j + H\psi_j = 0$ in $\Omega_j$, (2a)</li><li>$\frac{\partial\psi_j}{\partial n} = 0$ on $\Gamma_j(t)$, (2b)</li></ul>where $\Gamma_j(t)$ is the boundary of $\Omega_j(t)$, $\dot\psi =\frac{\partial\psi}{\partial t}$ and $H=H(x,t)$ is the (normalised) Hamiltonian given by<br /><div><ul><li>$H = -\frac{1}{2}\Delta - \frac{N}{\vert x\vert}+\sum_{k\neq j}V_k(x)$ for $x\in\Omega_j(t)$,</li></ul><div>with $V_k(x)$ the repulsion potential corresponding to electron $k$ defined by </div><ul><li>$V_k(x)=\int\frac{\psi_k^2(y)}{2\vert x-y\vert}dy$, </li></ul><div>and the electron wave functions are normalised to unit charge of each electron:</div><div><ul><li>$\int_{\Omega_j(t)}\psi_j^2(x,t) dx=1$ for $j=1,..,N$ and all time. (2c)</li></ul><div>The differential equation (2a) with homogeneous Neumann boundary condition (2b) is complemented by the following global free boundary condition:<br /><ul><li>$\psi (x,t)$ is continuous across inter-electron boundaries $\Gamma_j(t)$. (2d)</li></ul><br />The ground state is determined as a the real-valued time-independent minimiser $\psi (x)=\sum_j\psi_j(x)$ of the total energy<br /><ul><li>$E(\psi ) = \frac{1}{2}\int\vert\nabla\psi\vert^2\, dx - \int\frac{N\psi^2(x)}{\vert x\vert}dx+\sum_{k\neq j}\int V_k(x)\psi^2(x)\, dx$,</li></ul><div>under the normalisation (2c), the homogeneous Neumann boundary condition (2b) and the free boundary condition (2d).</div><div><br /></div><div>In the next post I will present computational results in the form of energy of ground states for atoms with up to 54 electrons and corresponding time-periodic solutions in spherical symmetry, together with ground state and dissociation energy for H2 and CO2 molecules in rotational symmetry.</div><div><br /></div><div>In summary, the model is formed as a system of one-electron Schrödinger equations, or electron container model, on a partition of 3d space depending of a common spatial variable and time, supplemented by a homogeneous Neumann condition for each electron on the boundary of its domain of support combined with a free boundary condition asking continuity of charge density across inter-element boundaries. </div><div><br /></div><div>We shall see that for atoms with spherically symmetric electron partitions in the form of a sequence of shells centered at the kernel, the homogeneous Neumann condition corresponds to vanishing kinetic energy of each electron normal to the boundary of its support as a condition of separation or interface condition between different electrons meeting with continuous charge density.<br /><br />Here is one example: Argon with 2-8-8 shell structure with NIST Atomic data base ground state energy in first line (526.22), the computed in second line and the total energies in the different shells in three groups with kinetic energy in second row, kernel potential energy in third and repulsive electron energy in the last row. Note that the total energy in the fully filled first (2 electrons) and second shell (8 electrons) are nearly the same, while the partially filled third shell (also 8 electrons out of 18 when fully filled) has lower energy. The color plot shows charge density per unit volume and the black curve charge density per unit radial increment as functions of radius. The green curve is the kernel potential and the cyrano the total electron potential. Note in particular the vanishing derivative of charge density/kinetic energy at shell interfaces.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-MDA16nX6UaM/V5XBvCx12LI/AAAAAAAA8Yw/3R9Yy1fIlVU6Db4dkegQFqz_POP_1sosQCLcB/s1600/IMG_0004.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://3.bp.blogspot.com/-MDA16nX6UaM/V5XBvCx12LI/AAAAAAAA8Yw/3R9Yy1fIlVU6Db4dkegQFqz_POP_1sosQCLcB/s640/IMG_0004.png" width="480" /></a></div> </div><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><br /></div></div></div>http://claesjohnson.blogspot.com/2016/07/new-quantum-mechanics-5-model-as.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-4587804826291458175Sat, 02 Jul 2016 10:37:00 +00002016-07-02T16:12:32.169+02:00New Quantum Mechanics 4: Free Boundary ConditionThis is a continuation of previous posts presenting an atom model in the form of a free boundary problem for a joint continuously differentiable electron charge density, as a sum of individual electron charge densities with disjoint supports, satisfying a classical Schrödinger wave equation in 3 space dimensions.<br /><br />The ground state of minimal total energy is computed by parabolic relaxation with the free boundary separating different electrons determined by a condition of zero gradient of charge density. Computations in spherical symmetry show close correspondence with observation, as illustrated by the case of Oxygen with 2 electrons in an inner shell (blue) and 6 electrons in an outer shell (red) as illustrated below in a radial plot of charge density showing in particular the zero gradient of charge density at the boundary separating the shells at minimum total energy (with -74.81 observed and -74.91 computed energy). The green curve shows truncated kernel potential, the magenta the electron potential and the black curve charge density per radial increment. <br /><br />The new aspect is the free boundary condition as zero gradient of charge density/kinetic energy.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-QzFfLrqGf40/V3eTz_PPhlI/AAAAAAAA8JU/oJi1g8qrLuEk6w5KJiNrSUPQpn-Rg1dTACLcB/s1600/IMG_0001.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://1.bp.blogspot.com/-QzFfLrqGf40/V3eTz_PPhlI/AAAAAAAA8JU/oJi1g8qrLuEk6w5KJiNrSUPQpn-Rg1dTACLcB/s640/IMG_0001.png" width="480" /></a></div><br />http://claesjohnson.blogspot.com/2016/07/new-quantum-mechanics-3-free-boundary.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-7031858645886626024Tue, 21 Jun 2016 19:47:00 +00002016-06-21T21:47:30.013+02:00New Quantum Mechanics 3: Why?Modern physics is being based on (i) relativity theory and (ii) quantum mechanics, both viewed to be correct beyond any conceivable doubt, but nevertheless (unfortunately) being incompatible. The result is a modern physics based on shaky grounds of contradictory theories from which anything can emerge, and so has done in the form of string theory and multiversa beyond thinkable experimental verification. <br /><br />The basic model of quantum mechanics is Schrödinger's equation as a linear equation in a wave function depending on $3N$ spatial dimensions for an atom with $N$ electrons. Schrödinger's equation is an ad hoc model arrived at by a purely formal extension of classical mechanics without direct physical meaning and rationale. Schrödinger's equation is thus viewed as being given by God with the job of physical interpretation being left to humanity in endless quarrels. In this sense quantum mechanics is rather religion than science and the present state of physics maybe a fully logical result.<br /><br />Experimental support for Schrödinger's equation is in incontestable form only available in the case of Hydrogen with $N=1$, since for larger $N$ the multidimensionality prevents both analytical and computational solution. The message of books on quantum mechanics that solutions of Schrödinger's equation always (have to) agree with observations, rather reflect a belief that a God-given equation cannot be wrong, than actual human experience. <br /><br />But if we as scientists do not welcome the idea of an equation given by God beyond human comprehension, then we may find motivation to search for an alternative atomic model which is computable and thus is possible to compare with physical experiment. This is my motivation anyway. <br /><br />And God said:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-zrp6PTfER2A/V2mZtN4xYFI/AAAAAAAA8Ig/zth8N6DfHCADamwsLghD2hGf5QN1UJc7gCLcB/s1600/schrodingerequation1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="70" src="https://4.bp.blogspot.com/-zrp6PTfER2A/V2mZtN4xYFI/AAAAAAAA8Ig/zth8N6DfHCADamwsLghD2hGf5QN1UJc7gCLcB/s320/schrodingerequation1.jpg" width="320" /></a></div><br />And then there were Atoms!http://claesjohnson.blogspot.com/2016/06/new-quantum-mechanics-3-why.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-7862389003444768183Tue, 21 Jun 2016 13:13:00 +00002016-06-21T20:25:26.202+02:00New Quantum Mechanics 2: Computational ResultsI have now tested the atomic model for an atom with $N$ electrons of the previous post formulated as a classical free boundary problem in $N$ single-electron charge densities with non-overlapping supports filling 3d space with joint charge density as a sum of electron densities being continuously differentiable across inter-electron boundaries.<br /><br />I have computed in spherical symmetry on an increasing sequence of radii dividing 3d space into a sequence of shells filled by collections of electrons smeared into spherically symmetric shell charge distribution. The electron-electron repulsive energy is computed with a reduction factor of $\frac{n-1}{n}$ for the electrons in a shell with $n$ electrons to account for lack of self repulsion.<br /><br />Below is a typical result for Xenon with 54 electrons organised in shells with 2, 8, 18, 18 and 8 electrons with ground state energy -7413 to be compared with -7232 measured and with the energy distribution in the 5 shells displayed in the order of total energy, kinetic energy, kernel potential energy and inter-electron energy. Here the blue curve represents electron charge density, green is kernel potential and red is inter-electron potential. The inter-shell boundaries are adaptively computed so as to represent a preset 2-8-18-18-8 configuration in iterative relaxation towards a ground state of minimal energy.<br /><br />In general computed ground state energies agree with measured energies within a few percent for all atoms up to Radon with 86 electrons.<br /><br />The computations indicate that it may well be possible to build an atomic model based on non-overlapping electronic charge densities as a classical continuum mechanical model with electrons keeping individuality by occupying different regions of space, which agrees reasonably well with observations. The model is an $N$-species free boundary problem in three space dimensions and as such is readily computable for any number of $N$ for both ground states, excited states and dynamic transitions between states.<br /><br />We recall the the standard model in the form of Schrödinger's equation for a wave function depending on $3N$ space dimensions, is computationally demanding already for $N=2$ and completely beyond reach for larger $N$. As a result the full $3N$-dimensional Schrödinger equation is always replaced by some radically reduced model such as Hartree-Fock with optimization over a "clever choice" of a few "atomic orbitals", or Thomas-Fermi and Density Functional Theory with different forms of electron densities.<br /><br />The present model is an electron density model, which as a free boundary problem with electric individuality is different from Thomas-Fermi and DFT.<br /><br />We further recall that the standard Schrödinger equation is an ad hoc model with only formal justification as a physical model, in particular concerning the kinetic energy and the time dependence, and as such should perhaps better not be taken as a given ready-made model which is perfect and as such canonical (as is the standard view).<br /><br />Since this standard model is uncomputable, it is impossible to show that the results from the model agree with observations, and thus claims of perfection made in books on quantum mechanics rather represent an ad hoc preconceived idea of unquestionable ultimate perfection than true experience.<br /><br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-23m8o4zx1U0/V2kkQfzdDMI/AAAAAAAA8IM/nL0GecXnFg02pAhGR4IlVhi0kzg5gv-5QCLcB/s1600/IMG_0023.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://3.bp.blogspot.com/-23m8o4zx1U0/V2kkQfzdDMI/AAAAAAAA8IM/nL0GecXnFg02pAhGR4IlVhi0kzg5gv-5QCLcB/s640/IMG_0023.png" width="480" /></a></div>http://claesjohnson.blogspot.com/2016/06/new-quantum-mechanics-computational.htmlnoreply@blogger.com (Claes Johnson)2tag:blogger.com,1999:blog-1500584444083499721.post-3841990563878665531Wed, 01 Jun 2016 08:12:00 +00002016-07-02T15:14:55.040+02:00physical quantum mechanicsNew Quantum Mechanics 1 as Classical Free Boundary Problem Let me (as a continuation of the sequence of posts on<a href="http://claesjohnson.blogspot.se/2015/09/finite-element-quantum-mechanics-5-1d.html"> Finite Element Quantum Mechanics 1-5</a>) present an alternative formulation of the eigenvalue problem for Schrödinger's equation for an atom with $N$ electrons starting from an Ansatz for the wave function<br /><ul><li>$\psi (x) = \sum_{j=1}^N\psi_j(x)$ (1)</li></ul>as a sum of $N$ electronic real-valued wave functions $\psi_j(x)$, depending on a common 3d space coordinate $x\in R^3$ with non-overlapping spatial supports $\Omega_1$,...,$\Omega_N$, filling 3d space, satisfying<br /><ul><li>$H\psi = E\psi $ in $R^3$, (2)</li></ul>where $E$ is an eigenvalue of the (normalised) Hamiltonian $H$ given by<br /><div><ul><li>$H(x) = -\frac{1}{2}\Delta - \frac{N}{\vert x\vert}+\sum_{k\neq j}V_k(x)$ for $x\in\Omega_j$,</li></ul><div>where $V_k(x)$ is the potential corresponding to electron $k$ defined by </div><ul><li>$V_k(x)=\int\frac{\psi_k^2(y)}{2\vert x-y\vert}dy$, for $x\in R^3$,</li></ul><div>and the wave functions are normalised to correspond to unit charge of each electron:</div><div><ul><li>$\int_{\Omega_j}\psi_j^2(x) dx=1$ for $j=1,..,N$.</li></ul><div>One can view (2) as a formulation of the eigenvalue problem for Schrödinger's equation, starting from an Ansatz for the total wave function as a sum of electronic wave function according to (1), as a classical free boundary problem in $R^3$, where the electron configuration is represented by a partition of $R^3$ into non-overlapping domains representing the supports of the electronic wave functions $\psi_j$ and the total wave function $\psi$ is continuously differentiable.</div><div><br />Defining $\rho_j = \psi_j^2$, we have</div><div><ul><li>$\psi\Delta\psi = \frac{1}{2}\Delta\rho-\frac{1}{4\rho}\vert\nabla\rho\vert^2$, </li></ul></div><div>and thus (2) upon multiplication by $\psi$ takes the form<br /><ul><li>$-\frac{1}{4}\Delta\rho+\frac{1}{8\rho}\vert\nabla\rho\vert^2-\frac{N\rho}{\vert x\vert}+V\rho = E\rho$ in $R^3$, (3)</li></ul><div>where</div></div><div><ul><li>$\rho_j\ge 0$, $support(\rho_j)=\Omega_j$ and $\rho_j=0$ else, </li><li>$\int_{\Omega_j}\rho_jdx =1$,</li><li>$\rho =\sum_j\rho_j$,</li><li>$V\rho=\sum_{k\neq j}V_k\rho_j$ in $\Omega_j$,</li><li>$\Delta V_j=2\pi\rho_j$ in $R^3$.</li></ul></div><div>The model (3) (or equivalently (2)) is computable as a system in 3d and will be tested against observations. In particular the ground state of smallest eigenvalue/energy E is computable by parabolic relaxation of (3) in $\rho$. Continuity of $\psi$ then corresponds to continuity of $\rho$.<br /><br />We can view the formulation (3) in the same way as that explored for gravitation, with the potential $V_j$ primordial and the electronic density $\rho_j$ defined by $\rho_j =\frac{1}{2\pi}\Delta V_j$ as a derived quantity, with in particular total electron-electron repulsion energy given by the neat formula<br /><ul><li>$\frac{1}{2\pi}\sum_{k\neq j}\int V_k\Delta V_jdx=-\frac{1}{2\pi}\sum_{k\neq j}\int\nabla V_k\cdot\nabla V_jdx$</li></ul><div>in terms of potentials with an analogous expression for the kernel-electron attraction energy.<br /><br />For the choice of free boundary condition <a href="http://claesjohnson.blogspot.se/2016/07/new-quantum-mechanics-3-free-boundary.html">see later post.</a></div><div><br /><br /></div></div><div><div><br /></div></div></div></div>http://claesjohnson.blogspot.com/2016/06/new-quantum-mechanics-as-classical-free.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-5103214935449622910Wed, 01 Jun 2016 07:25:00 +00002016-06-01T09:25:13.522+02:00dark energydark matterNew View on gravitationMany Big Bangs: Universe Bigger Than You Think<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-ii-vPKd1F4k/V06Nu3PDHuI/AAAAAAAA8HA/gnpKYHhZxAoXnYLjgjH0SiBHqN75_8p2wCLcB/s1600/lordrees.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /><img border="0" height="209" src="https://4.bp.blogspot.com/-ii-vPKd1F4k/V06Nu3PDHuI/AAAAAAAA8HA/gnpKYHhZxAoXnYLjgjH0SiBHqN75_8p2wCLcB/s320/lordrees.jpg" width="320" /></a></div><br /><a href="http://www.telegraph.co.uk/science/2016/05/31/multiple-big-bang-theory-could-bring-copernican-revolution-says/">Astronomer Royal Lord Rees has made a statement</a>:<br /><ul><li><i>There may have been more than one <a href="http://www.telegraph.co.uk/news/science/large-hadron-collider/11489442/Big-Bang-theory-could-be-debunked-by-Large-Hadron-Collider.html">Big Bang</a>, the Astronomer Royal has said and claims the world could be on the brink of a revolution as profound as Copernicus discovering the Earth revolved around the Sun.</i></li><li><i>Many people suspect that our Big Bang was not the only one, but there’s a whole ensemble of Big Bangs, a whole archipelago of Big Bangs.</i></li><li><i>The theory is still highly controversial, but Lord Rees said he would ‘bet his dog’ on the theory being true.</i></li></ul><div>This fits with the view I have presented in posts on a new view on gravitation, dark matter and dark energy, with gravitational potential $\phi$ viewed as primordial from which matter density $\rho$, which may be both positive and negative, is generated by </div><div><ul><li>$\rho = \Delta\phi$, </li></ul><div>through local action in space of the Laplacian $\Delta$. In this model a Big Bang corresponds to a small local fluctuation of $\phi$ around zero, which generates an much bigger fluctuation of matter density by the action of the Laplacian. </div><div><br /></div><div>In this model substantial matter may be generated locally from small fluctuations of gravitational potential opening the possibility of an endless number of Big Bangs seemingly created out of nothing. </div><div><br /></div><div>You can test the model in the app <a href="https://itunes.apple.com/se/app/dark-energy-newmath/id1091609721?l=en&mt=8">Dark Energy on App Store.</a></div></div>http://claesjohnson.blogspot.com/2016/06/many-big-bangs-universe-bigger-than-you.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-6400470718074102074Tue, 31 May 2016 14:15:00 +00002016-06-01T09:02:45.608+02:00Kodcentrummatematik-ITSamtal med Kodcentrum om Matematik-ITJag har idag haft ett konstruktivt och mycket trevligt samtal med <a href="http://www.kodcentrum.se/om-oss/kontakt">Jessica Berglund och Lisa Söderlund på Kodcentrum</a> om eventuellt samarbete vad gäller att sprida programmeringens evangelium till svenska elever och svensk skola.<br /><br />Kodcentrum har hittills satsat på Scratch som introduktion till programmering och verkar ha behov av att kunna leverera vidareutbildning, och kanske Matematik-IT därvid kan vara ett alternativ. Vi får se om Kodcentrum vill utnyttja denna möjlighet under nästa läsår. Chansen finns...<br /><br />Vad gäller programmeringsplattform har jag alltså använt Codea (programmering på iPad för iPad), men det finns många alternativ, tex Corona för PC/Mac som använder samma språk som Codea (Lua). Codea kostar några kronor, medan Corona är gratis.<br /><br />Sen finns det ju många andra möjligheter som Xcode, Swift, Python, JavaScript, Perl....Till slut måste man välja något specifikt språk/plattform, om man vill säga/göra något konkret med någon mening... på samma sätt som med kärleken, som är ju evig och det bara är föremålen som växlar...<br /><br /><b>PS</b> Det finns en inställning som verkar ha många företrädare, att möta Regeringens uppdrag till Skolverket att införa programmering i skolan, inte med att helt sonika följa uppdragets direktiv och göra så, utan istället ersätta konkret programmering med betydligt mindre konkreta slagord som "digital kompetens" och "datalogiskt tänkande". Tanken är alltså att gå runt den heta gröten och inte ta del av den och istället konsumera ev urvattnade derivat av den goda och stärkande gröten.<br /><br />Inte att lära sig programmera (det behöver man inte eftersom det finns så många programmerare), utan istället lära sig att det finns något som kallas programmering. Inte att de facto få lära sig multiplikationstabellen och att använda den, utan istället få lära sig att det räcker att veta att den finns och att några kan den (den behövs ju egentligen inte eftersom det finns så många mini-räknare).<br /><br />Tanken borde istället vara att om man äter programmeringsgröten och smälter den, så får man bättre förmåga att utveckla både "digital kompetens" och "datalogiskt tänkande", om det nu är huvudsaken, än om man bara går runt gröten.<br /><br />Det är bättre att kunna multiplikationstabellen och kunna använda den, än att inte kunna den och inte veta att använda den, även om det finns miniräknare. Varför? Därför att människan är en tänkande varelse och tänkande bygger på förståelse.<br /><br />http://claesjohnson.blogspot.com/2016/05/samtal-med-kodcentrum-om-matematik-it.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-4513438436630176042Mon, 30 May 2016 08:44:00 +00002016-05-30T13:30:51.124+02:00Kutta-ZhukovskyPrandtltheory of flightNew Theory of Flight: Time Line<div class="separator" style="clear: both; text-align: center;"></div><div style="margin-left: 1em; margin-right: 1em;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Gfl4ijd7gDw/V0v95sHdOQI/AAAAAAAA8Gc/NtuuRP7biPMRo_UqS0hJoVaGCCEpS7rNwCLcB/s1600/pot-vel.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://2.bp.blogspot.com/-Gfl4ijd7gDw/V0v95sHdOQI/AAAAAAAA8Gc/NtuuRP7biPMRo_UqS0hJoVaGCCEpS7rNwCLcB/s200/pot-vel.jpg" width="200" /></a><a href="https://1.bp.blogspot.com/-eka0119h37g/V0wCUc2RR2I/AAAAAAAA8Go/NI8z-0jmKxAj9I1YTbpSGca3w8wcsWSmwCLcB/s1600/cylindersep.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://1.bp.blogspot.com/-eka0119h37g/V0wCUc2RR2I/AAAAAAAA8Go/NI8z-0jmKxAj9I1YTbpSGca3w8wcsWSmwCLcB/s1600/cylindersep.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1500584444083499721" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><img border="0" height="123" src="https://1.bp.blogspot.com/-eka0119h37g/V0wCUc2RR2I/AAAAAAAA8Go/NI8z-0jmKxAj9I1YTbpSGca3w8wcsWSmwCLcB/s200/cylindersep.jpg" width="200" /></div><br /><br /><br /><br /><br /><i><br /></i><i>Potential flow around circular cylinder with zero drag and lift (left). </i><br /><i>Real flow with non-stationary turbulent 3d rotational slip separation and non-zero drag (right). </i><br /><br />The <a href="http://link.springer.com/article/10.1007/s00021-015-0220-y?wt_mc=internal.event.1.SEM.ArticleAuthorOnlineFirst">New Theory of Flight published in J Mathematical Fluid Mechanics</a>, can put be into the following time line:<br /><br /><b>1750 formulation by Euler</b> of the <b>Euler equations</b> describing<b> incompressible flow with vanishing viscosity</b> expressing Newton's 2nd law and incompressibility in Euler coordinates of a fixed Euclidean coordinate system.<br /><br /><b><a href="https://www.youtube.com/watch?v=QPTjuS5G3dY">1752 d'Alembert's Paradox</a></b> as zero drag and lift of<b> potential flow</b> around a wing defined as stationary flow which is<br /><ol><li>incompressible</li><li>irrotational</li><li>of vanishing viscosity</li><li>satisfies slip boundary condition </li></ol>as exact solution of the Euler equations.<br /><b><br /></b><b>1904 resolution of d'Alembert's para</b><b>dox of zero drag</b> by Prandtl stating that potential flow is unphysical, because 4. violates a requirement that real flow must satisfy<br /><div><ul><li>no slip boundary condition.</li></ul><div><b>1904 resolution of d'Alembert's paradox of zero lift</b> by Kutta-Zhukovsky stating that potential flow is unphysical, because 2. violates that a sharp trailing edge in real flow creates </div></div><div><ul><li>rotational flow.</li></ul><div><b><a href="http://link.springer.com/article/10.1007%2Fs00021-008-0290-1#/page-1">2008 resolution of d'Alembert's paradox of zero drag and lift</a></b> by Hoffman-Johnson stating that potential flow is unphysical, because </div></div><div><ul><li>potential flow it is<b> unstable at separation</b> and develops into <b>non-stationary turbulent</b> <b>3d rotational slip separation</b> as a viscosity solution of the Euler equations with substantial drag and lift. </li></ul><div>Recall that d'Alembert's paradox had to be resolved, in one way or the other, to save theoretical fluid mechanics from complete collapse, when the Wright brothers managed to get their Flyer off ground into sustained flight in 1903 with a 10 hp engine. </div><div><br /></div><div>Prandtl, named the Father of Modern Fluid Mechanics, discriminated the potential solution by an ad hoc postulate that 4. was unphysical (without touching 2.) and obtained drag without lift.</div><div><br /></div><div>Kutta-Zhukovsky, named Fathers of Modern Aero Dynamics, discriminated the potential solution by an ad hoc postulate that 2. was unphysical (without touching 4.) and obtained lift without drag. </div><div><br /></div><div>Hoffman-Johnson showed without ad hoc postulate that the potential solution is unstable at separation and develops into <b>non-stationary turbulent</b> <b>3d rotational slip separation</b> causing drag and lift. </div><div><a href="https://www.blogger.com/blogger.g?blogID=1500584444083499721" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><br /></div><div>The length of the time-line 1750-1752-1904-2008 is remarkable from scientific point of view. Little happened between 1752 and 1904 and between 1904 and 2008, and what happened in 1904 was not in touch with reality. For detailed information, see <a href="https://secretofflight.wordpress.com/">The Secret of Flight.</a><br /><br />1946 Nobel Laureate Hinshelwood made the following devastating analysis:<br /><ul><li><i>D’Alembert’s paradox separated fluid mechanics from its start into theoretical fluid mechanics <b>explaining phenomena which cannot be observed </b>and practical fluid mechanics or hydraulics <b>observing phenomena which cannot be explained.</b></i></li></ul><div>The only glimpse in the darkness was offered by the mathematician<a href="http://onlinelibrary.wiley.com/doi/10.1002/zamm.19610411025/abstract"> Garret Birkhoff in his 1950 book Hydrodynamics</a>, by asking if any potential flow is stable, a glimpse of light that was directly blown out by a <a href="https://projecteuclid.org/euclid.bams/1183516308">devastating critique of the book from fluid dynamics community,</a> which made Birkhoff remove his question in the 2nd edition of the book and to never return to hydrodynamics.<br /><br />The 2008 resolution of d'Alembert's Paradox leading into the New Theory of Flight by Hoffman-Johnson, has been met with complete silence/full oppression by the fluid mechanics community still operating under the paradigm of Hinshelwoods analysis. </div><br /></div><div></div></div>http://claesjohnson.blogspot.com/2016/05/new-theory-of-flight-short-version.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-3679328782448277546Sun, 29 May 2016 08:13:00 +00002016-05-29T18:42:37.636+02:00physical quantum mechanicsQuantum Contradictionsquantum mechanicsRestart of Quantum Mechanics: From Observable/Measurable to Computable<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-3JMm4Y_ZNEE/V0qmUL699cI/AAAAAAAA8F8/7g9y-KB5YnwI0uwgSrJKRgM_SUHHcbOggCLcB/s1600/pw1412085.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /><img border="0" height="238" src="https://4.bp.blogspot.com/-3JMm4Y_ZNEE/V0qmUL699cI/AAAAAAAA8F8/7g9y-KB5YnwI0uwgSrJKRgM_SUHHcbOggCLcB/s320/pw1412085.gif" width="320" /></a></div> <i> Schrödinger and Heisenberg receiving the Nobel Prize in Physics in 1933/32..</i><br /><br />If modern physics was to start today instead of as it did 100 years ago with the development of quantum mechanics as atomistic mechanics by Bohr-Heisenberg and Schrödinger, what would be the difference?<br /><br />Bohr-Heisenberg were obsessed with the question:<br /><div><ul><li><i>What can be observed?</i></li></ul><div>motivated by <i>Bohr's Law</i>:</div></div><div><ul><li><i>We are allowed to speak only about what can be observed.</i></li></ul><div>Today, with the computer to the service of atom physics, a better question may be:</div></div><div><ul><li><i>What can be computed?</i></li></ul><div>possibly based on an idea that</div><div><ul><li><i>It may be meaningful to speak about what can be computed. </i></li></ul><div>Schrödinger as the inventor of the Schrödinger equation as the basic mathematical model of quantum mechanics, never accepted the Bohr-Heisenberg Copenhagen Interpretation of quantum mechanics with the Schrödinger wave function as solution of the Schrödinger equation interpreted as a probability of particle configuration, with collapse of the wave function into actual particle configuration under observation/measurement. </div><div><br /></div><div>Schrödinger sought an interpretation of the wave function as a physical wave in a classical continuum mechanical meaning, but had to give in to Bohr-Heisenberg, because the multi-dimensionality of the Schrödinger equation did not allow a direct physical interpretation, only a probabilistic particle interpretation. Thus the Schrödinger equation to Schrödinger became a monster out of control, as expressed in the following famous quote: </div><div><div class="page" title="Page 1"><div class="layoutArea"><div class="column"><ul><li><i>If we have to go on with these damned quantum jumps, then I'm sorry that I ever got involved.</i></li></ul>And Schrödinger's equation is a monster also from computational point of view, because solution work scales severely exponentially with the number of electrons and thus is beyond reach even for small $N$.<br /><br />But the Schrödinger equation is an ad hoc model with only weak formal unphysical rationale, including the basic ingredients of (i) linearity and (ii) multi-dimensionality. <br /><br />Copenhagen quantum mechanics is thus based on a Schrödinger equation, which is an ad hoc model and which cannot be solved with any assessment of accuracy because of its multi-dimensionality and thus cannot really deliver predictions which can be tested vs observations, except in very simple cases. <br /><br />The Copenhagen dogma is then that predictions of the standard Schrödinger equation always are in perfect agreement with observation, but a dogma which cannot be challenged because predictions cannot be computed ab initio.<br /><br />In this situation it is natural to ask, in the spirit of Schrödinger, for a new Schrödinger equation which has a direct physical meaning and to which solutions can be computed ab initio, and this is what I have been exploring in many blog posts and in the book (draft) <a href="http://www.csc.kth.se/~cgjoh/ambsquantum.pdf">Many-Minds Quantum Mechanics.</a> <br /><br />The basic idea is to replace the linear multi-d standard Schrödinger equation with a computable non-linear system in 3d as a basis of a new form of physical quantum mechanics. I will return with more evidence of the functionality of this approach, which is very promising...<br /><br />Note that a wonderful thing with computation is that it can be viewed as a form of non-destructive testing, where the evolution of a physical system can be followed in full minute detail without any form of interference from an observer, thus making Bohr's Law into a meaningless limitation of scientific thinking and work from a pre-computer era preventing progress today.<br /><br /><b>PS</b> It is maybe wise to be a little skeptical to assessments of agreement between theory and experiments to an extremely high precision. It may be that things are arranged or rigged so as to give exact agreement, by changing computation/theory or experiment.<br /><br /></div></div></div></div></div></div>http://claesjohnson.blogspot.com/2016/05/restart-of-quantum-mechanics-from.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-7861357375498130449Sat, 28 May 2016 15:43:00 +00002016-05-29T18:31:56.312+02:00logical fallacyphysicsAristotle's Logical Fallacy of Affirming the Consequent in Physics<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-JK3nXTT_x64/V0m-IX73rSI/AAAAAAAA8E4/j2n-SVUGzR88XXDL4kznDhUwCDYncUM9QCLcB/s1600/logic.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://4.bp.blogspot.com/-JK3nXTT_x64/V0m-IX73rSI/AAAAAAAA8E4/j2n-SVUGzR88XXDL4kznDhUwCDYncUM9QCLcB/s400/logic.jpg" width="400" /></a></div><br />One can find many examples in physics, both classical and modern, of <a href="http://changingminds.org/disciplines/argument/fallacies/affirming_consequent.htm">Aristotle's logical fallacy of Affirming the Consequent</a> (confirming an assumption by observing a consequence of the assumption):<br /><ol><li>Assume the Earth rests on 4 turtles, which keeps the Earth from "falling down". Observe that the Earth does not "fall down". Conclude that the Earth rests on 4 turtles.</li><li>Observe a photoelectric effect in accordance with a simple (in Einstein's terminology "heuristic") argument assuming light can be thought of as a stream of particles named "photons" . Conclude that light <b>is</b> a stream of particles named photons. </li><li>Assume light is affected by gravitation according the general theory of relativity as described by Einstein's equations. Observe apparent slight bending of light as it passes near the Sun in accordance with an extremely simplified use of Einstein's equations. Conclude universal validity of Einstein's equations.</li><li>Observe lift of a wing profile in accordance with a prediction from potential flow modified by large scale circulation around the wing. Conclude that there is large scale circulation around the wing. </li><li>Assume that predictions from solving Schrödinger's equation always are in perfect agreement with observation. Observe good agreement in some special cases for which the Schrödinger equation happens to be solvable, like in the case of Hydrogen with one electron. Conclude universal validity of Schrödinger's equation, in particular for atoms with many electrons for which solutions cannot be computed with assessment of accuracy.</li><li>Assume there was a Big Bang and observe a distribution of galaxy positions/velocities, which is very very roughly in accordance with the assumption of a Big Bang. Conclude that there was a Big Bang.</li><li>Assume that doubled CO2 in the atmosphere from burning of fossil fuel will cause catastrophic global warming of 2.5 - 6 C. Observe global warming of 1 C since 1870. Conclude that doubled CO2 in the atmosphere from burning of fossil fuel will cause catastrophic global warming of 4 - 8 C.</li><li><a href="http://claesjohnson.blogspot.se/2016/02/absurdity-of-modern-physics-ligo.html">Assume that two massive black holes merged about 1.3 billion years ago</a> and thereby sent a shudder through the universe as ripples in the fabric of space and time called gravitational waves and five months ago washed past Earth and stretched space making the entire Earth expand and contract by 1/100,000 of a nanometer, about the width of an atomic nucleus. Observe a wiggle of an atom in an instrument and conclude that two massive black holes merged about 1.3 billion years ago which sent a shudder through the universe as ripples in the fabric of space and time called gravitational waves...</li><li>Observe experimental agreement of the <i>anomalous magnetic dipole moment of the electron</i> within 10 decimals to a prediction by Quantum Electro Dynamics (QED). Conclude that QED is universally valid for any number of electrons as the most accurate theory of physics. Note that the extremely high accuracy for the specific case of the <i>anomalous magnetic dipole moment of the electron,</i> compensates for the impossibility of testing in more general cases, because the equations of QED are even more impossible to solve with assessment of accuracy than Schrödinger's equation.</li></ol><div>The logic fallacy is so widely practiced that for many it may be difficult to see the arguments as fallacies. Test yourself!<br /><br /><b>PS1.</b> Observe that if a theoretical prediction agrees with observation to a very high precision, as is the case concerning the Equivalence Principle stating equality of inertial and gravitational (heavy) mass, then it is possible that what you are testing experimentally in fact is the validity of a definition, like testing experimentally if there are 100 centimeters on a meter (which would be absurd).<br /><br /><b>PS2</b> Books on quantum mechanics usually claim the there is no experiment showing any discrepancy whatsoever with solutions of the Schrödinger equation (in the proper setting), which is strong evidence that the Schrödinger equation gives an exact description of all of atom physics (in a proper setting). The credibility of this argument is weakened by the fact that solutions can be computed only in very simple cases. </div>http://claesjohnson.blogspot.com/2016/05/logical-fallacies-in-physics-of.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-865235780641534846Fri, 27 May 2016 10:34:00 +00002016-05-27T16:17:03.482+02:00Emergence by Smart Integration of Physical Law as Differential Equation<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-ZNuIgt1dKEQ/V0g2oAO1nqI/AAAAAAAA8DM/3ZaZhau5A6kkg234ZCOy5xQjbTOmtJwPQCLcB/s1600/European_Parliament_Strasbourg_Hemicycle_-_Diliff.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="225" src="https://4.bp.blogspot.com/-ZNuIgt1dKEQ/V0g2oAO1nqI/AAAAAAAA8DM/3ZaZhau5A6kkg234ZCOy5xQjbTOmtJwPQCLcB/s400/European_Parliament_Strasbourg_Hemicycle_-_Diliff.jpg" width="400" /></a></div><i>Perfect Harmony of European Parliament: Level curves of political potential generated by an empty spot in the middle.</i><br /><br />This is a continuation of previous posts on a new view of Newton's law of gravitation. We here connect to the <i>Fundamental Theorem of Calculus </i>of the previous post<i>,</i> allowing a bypass to compute an integral by tedious laborious summation, using a primitive function of the integrand:<br /><ul><li>$\int_0^t f(s)ds = F(t) - F(0)$ if $\frac{dF}{dt} = f$.</li></ul>This magical trick of Calculus of computing an integral as a sum without doing the summation, is commonly viewed to have triggered the scientific revolution shaping the modern world.<br /><br />The magic is here computing an integral $\int_0^t f(s)ds$ in a smart way, rather than computing a derivative $\frac{dF}{dt}$ in a standard way.<br /><br />The need of computing integrals comes from the fact that physical laws are usually expressed in terms of derivatives, for example as an<i> initial value problem</i>: Given a function $f(t)$, determine a function $F(t)$ such that<br /><ul><li>$DF(t) = f(t)$ for $t\ge 0$ and $F(0) = 0$,</li></ul><div>where $DF =\frac{dF}{dt}$ is the derivative of $F$. In other words, given a function $f(t)$, determine a primitive function $F(t)$ to $f(t)$ with $F(0)=0$, that is, determine/compute the integral<br />by the formula</div><div><ul><li>$\int_0^t f(s)ds = F(t)$ for $t\ge 0$. </li></ul><div>Using the Fundamental Theorem to compute the integral would then correspond to solving the initial value problem by simply picking a primitive function $F(t)$ satisfying $DF = f$ and $F(0)=0$ from a catalog of primitive functions, allowing to in one leap jump from $t=0$ to any later time $t$. Not very magical perhaps, but certainly smart!</div><div><br /></div><div>The basic initial value problem of mechanics is expressed in Newton's 2nd Law $f=ma$ where $f$ is force, $m$ mass and $a(t)=\frac{dv}{dt}=\frac{d^2x}{dt^2}$ is acceleration, $v(t)=\frac{dx}{dt}$ velocity and $x(t)$ position, that is,</div></div><div><ul><li>$f(t) = m \frac{d^2x}{dt^2}$. (1)</li></ul><div>Note that in the formulation of the 2nd Law, it is natural to view position $x(t)$ with acceleration $\frac{d^2x}{dt^2}$ as given, from which force $f(t)$ is derived by (1) . Why? Because position $x(t)$ and acceleration $\frac{d^2x}{dt^2}$ can be observed, from which the presence of force $f(t)$ can be inferred or derived or concluded, while direct observation of force may not really be possible. In this setting the 2nd Law acts simply to define force in terms of mass and acceleration, rather than to make a connection with some other definition of force.</div><div><br /></div><div>Writing Newton's 2nd law in the form $f=ma$, thus defining force in terms of mass and acceleration, is the same as writing Newton's Law of Gravitation:</div><div><ul><li>$\rho = \Delta\phi$, (2)</li></ul><div>thereby defining mass density $\rho (x)$ in terms of gravitational potential $\phi (x)$ by a differential equation. </div><div><br /></div><div>With this view, Newton's both laws (1) and (2) would have the same form as differential equation, and the solutions $x(t)$ and $\phi (x)$ would result from solving differential equations by integration or summation as a form of emergence. </div></div><div><br /></div><div>In particular, this reasoning gives support to an idea of viewing the physics of Newton's Law of Gravitation to express that mass density somehow is "produced from" gravitational potential by the differential equation $\rho =\Delta\phi$. </div><div><br /></div><div>To solve the differential equation $\Delta\phi =\rho$ by direct integration or summation in the form<br /><ul><li>$\phi (x) = \frac{1}{4\pi}\int\frac{\rho (y)}{\vert x-y\vert}dy$,</li></ul>would then in physical terms require instant action at distance, which is difficult to explain. </div><div><br /></div><div>On the other hand, if there was a "smart" way of doing the integration by using some form of Fundamental Theorem of Calculus as above, for example by having a catalog of potentials from which to choose a potential satisfying $\Delta\phi =\rho$ for any given $\rho$, then maybe the requirement of instant action at distance could be avoided.<br /><br />A smart way of solving $\Delta\phi =\rho$ would be to use the knowledge of the solution $\phi (x)$ in the case of a unit point mass at $x=0$ as<br /><ul><li>$\phi (x)=\frac{1}{4\pi}\frac{1}{\vert x\vert}$ </li></ul>which gives Newton's inverse square law for the force $\nabla\phi$, which is smart in case $\rho$ is a sum of not too many point masses. But the physics would still seem to involve instant action at distance.<br /><br /></div><div>In any case, from the analogy with the 2nd Law we have gathered an argument supporting an idea to view the physics of gravitation as being expressed by the differential equation $\rho =\Delta\phi$ with mass density $\rho$ derived from gravitational potential $\phi$. Rather than the opposite standard view with the potential $\phi$ resulting from mass density $\rho$ by integration or summation corresponding to instant action at distance.</div><div><br />The differential equation $\Delta\phi =\rho$ would thus be valid by an interplay "in perfect harmony" in the spirit of Leibniz, where on the one hand "gravitational potential tells matter where to be how to move" and "matter tells gravitational potential what to be".<br /><br />This would be like a Perfect Parliamentary System where the "Parliament tells People where to be and what to do" and "People tells Parliament what to be".<br /><br /><b>PS</b> There is a fundamental difference between (1) and (2): (1) is an initial value problem in time while (2) is a formally a static problem in space. It is natural to solve an initial value problem by time stepping which represents integration by summation. A static problem like (2) can be solved iteratively by some form of (pseudo) time stepping towards a stationary solution, which in physical terms could correspond to successive propagation of effects with finite speed of propagation.</div><div><br /></div><div><br /></div></div>http://claesjohnson.blogspot.com/2016/05/emergence-by-smart-integration-of.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-1515367244982155082Thu, 26 May 2016 12:07:00 +00002016-05-27T10:49:26.793+02:00extended Newtonian gravitationNew View on gravitationFatal Attraction of Fundamental Theorem of Calculus?Calculus books proudly present the Fundamental Theorem of Calculus as the trick of computing an integral<br /><ul><li>I=$\int_a^b f(x)dx$,</li></ul>not by tedious summation of little pieces as a Riemann sum<br /><ul><li>$\sum_i f(x_i)h_i$</li></ul><div>on a partition $\{x_i\}$ of the interval $(a,b)$ with step size $h_i = x_{i+1} - x_i$, but by the formula<x_2 as="" b="" but="" by="" div="" formula="" h_i="x_{i+1}-x_i," nbsp="" simply="" size="" step="" the="" with="" x_2="" x_="" x_n=""></x_2></div><div><ul><li>$I = F(b) - F(a)$, </li></ul></div>where $F(x)$ is a primitive function to $f(x)$ satisfying $\frac{dF}{dx} = f$, <br /><br />The trick is thus to compute an integral, which by construction is a sum of very many terms, not by doing the summation following the construction, but instead taking just one big leap using a primitive function. <br /><br />On the other hand, to compute a derivative no trick is needed according to the book; you just compute the derivative using simple rules and a catalog of already computed derivatives.<br /><br />In a world of analytical mathematics, computing integrals is thus valued higher than computing derivatives, and this is therefore what fills Calculus books.<br /><br />In a world of computational mathematics, the roles are switched. To compute an integral as a sum can be viewed to be computationally trivial, while computing a derivative $\frac{dF}{dx}$ is a bit more tricky because it involves dividing increments $dF$ by small increments $dx$.<br /><br />This connects to Poisson's equation $\Delta\phi =\rho$ of Newton's theory of gravitation discussed in recent posts. What is here to be viewed as given and what is derived? The standard view is that the mass density $\rho$ is given and the gravitational potential $\phi$ is derived from $\rho$ as an integral<br /><ul><li>$\phi (x) = \frac{1}{4\pi}\int\frac{\rho (y)}{\vert x-y\vert}dy$,</li></ul><div>seemingly by instant action at distance. </div><div><br /></div><div>In alternative Newtonian gravitation, as discussed in recent posts, we view instead $\phi$ as primordial and $\rho =\Delta\phi$ as being derived by differentiation, with the advantage of requiring only local action.</div><div><br /></div><div>We thus have two opposing views:</div><div><ul><li>putting together = integration requiring (instant) action at distance with dull tool.</li><li>splitting apart = differentiation involving local action with sharp tool. </li></ul></div><div>It is not clear what to prefer?</div><br />http://claesjohnson.blogspot.com/2016/05/fatal-attraction-of-fundamental-theorem.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-8835288413708310922Thu, 26 May 2016 07:58:00 +00002016-05-26T09:59:09.676+02:00Einstein's equationextended Newtonian gravitationNew View on gravitationConnection between Neo-Newtonian and Einsteinian Gravitational Theory <div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-1tUyjh3-S2w/V0ar0nj-BoI/AAAAAAAA8C0/fRTJEAQDXbc-ghj_RjjFhnqKbmu152WZQCLcB/s1600/Hen-eggs_1370969c.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://1.bp.blogspot.com/-1tUyjh3-S2w/V0ar0nj-BoI/AAAAAAAA8C0/fRTJEAQDXbc-ghj_RjjFhnqKbmu152WZQCLcB/s320/Hen-eggs_1370969c.jpg" width="320" /></a></div> <i> Hen laying eggs by local action. </i><br /><br />If you are a strong supporter of Einstein's general theory of relativity, like almost all modern physicists, then maybe you would be open to see the following connection with the Neo-Newtonian<br />theory I have been exploring in recent posts, with the gravitational potential $\phi$ viewed as primordial and matter density $\rho =\Delta\phi$ as derived by local action in space of the Laplacian $\Delta$ and with the gravitational potential playing the same role as the "space-time curvature" of Einstein:<br /><ul><li>space-time curvature tells matter to move along geodesics</li><li>gravitational potential tells matter to move according to Newton's 2nd Law</li></ul><div>with </div><div><ul><li>space-time curvature connected to matter by Einstein's equation</li><li>gravitational potential connected to matter by Poisson's/Newton's equation.</li></ul><div>This connects to the iconic summary of general relativity by John Archibald Wheeler:</div><ul><li><i>Spacetime tells matter how to move; matter tells spacetime how to curve,</i></li></ul><div>where the "telling" goes both ways. </div><div><br /></div><div>But maybe it is enough that the "telling" only goes one way, maybe it suffices that the gravitational potential tells where matter will be and how it is to move. After all, it is only the equality $\rho =\Delta\phi$ that counts and thus has to be established is some way, and then possibly through one-way communication from $\phi$ to $\rho$ in some form of local action. </div></div><div><br /></div><div>Maybe it is enough to understand/explain how a hen can lay an egg by local action in a poultry yard, and leave out the much more difficult problem of how a hen can come out of an egg by global action outside the poultry yard.</div>http://claesjohnson.blogspot.com/2016/05/connection-between-neo-newtonian-and.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-7416413569058577431Wed, 25 May 2016 17:14:00 +00002016-05-30T06:57:24.529+02:00New View on gravitationNewton's Genius and New View on Gravitation<div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-WV3okUHwj8I/V0XdQKBcO8I/AAAAAAAA8CU/SHXJrZ-7aDwYDcENR78wwZNS1Khnw7KswCLcB/s1600/gravity.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-WV3okUHwj8I/V0XdQKBcO8I/AAAAAAAA8CU/SHXJrZ-7aDwYDcENR78wwZNS1Khnw7KswCLcB/s400/gravity.gif" width="362" /></a></div><div><br /></div><div>Newton computed the gravitational attraction of a planet as a spherically symmetric distribution of matter, to be equal to that of a point mass of the same total mass at the center of the planet, away from the planet. </div><div><br /></div><div>This made it possible for Newton to model gravitational interaction of planets as gravitational interaction of point masses, as a much simpler problem from computational point of view. </div><div><br /></div><div>Newton thus could simplify the computationally impossible problem of instant gravitational interaction at distance of all the individual atoms of one planet with all the individual atoms of another planet, to the interaction between two point masses of the same total mass. Genial and absolutely necessary to make the theory useful and thereby credible.</div><div><br /></div><div>But let us reflect a bit about the physics of instant individual interaction at distance of each atom of one planet with each atom of another planet, which we agreed is computationally impossible. We now ask if it is physically possible?</div><div><br /></div><div>Is it thinkable that each atom can instantly at distance exchange details about position and mass with all others atoms by using some form of world wide web? Think about it! </div><div><br /></div><div>The answer can only be NO. Atoms cannot have access to such technology. It is unthinkable.</div><div><br /></div><div>The result is that we have to view gravitation in a different way, not as individual instant attraction between small pieces of matter at distance, and then why not in the other way as suggested in recent posts: </div><div><br /></div><div>What is primordial is then a gravitational potential $\phi$ with associated gravitational force $\nabla\phi$, to which matter density $\rho$ is connected by $\rho =\Delta\phi$ through the local operation in space of the Laplacian $\Delta$.</div><div><br /></div><div>With this view there is no instant action at distance between atoms to explain, but instead local production of matter without demand of atomistic resolution into pieces, which at least is thinkable.</div><div><br /></div><div>It would be interesting to listen to Newton's reaction to this idea. <br /><br /><b>PS</b> <a href="http://www.businessinsider.com/earth-core-younger-than-crust-relativity-2016-5?utm_source=feedburner&utm_medium=referral%3fr=US&IR=T&IR=T">Business Insider reports that</a>:<br /><ul><li><i>Earth's core is 2.5 years younger than its crust due to some eerie physics</i></li></ul><div>the eerie physics being Einstein's general theory of relativity claiming that clocks slow down with increasing gravitation. Yes, maybe your feet are a bit younger than your head...</div><div> </div></div>http://claesjohnson.blogspot.com/2016/05/newtons-genius-and-new-view-on.htmlnoreply@blogger.com (Claes Johnson)7tag:blogger.com,1999:blog-1500584444083499721.post-78281747550288960Wed, 25 May 2016 13:55:00 +00002016-05-25T17:51:52.996+02:00climate sciencetim ballThe Value of Compulsory (Climate) Science?<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-_qCCJAk4PZw/V0W0uls8gmI/AAAAAAAA8CE/9D2vNRT1nqsqMRtgTa97KPu6Epi0yp-sQCLcB/s1600/fossilfree.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="183" src="https://2.bp.blogspot.com/-_qCCJAk4PZw/V0W0uls8gmI/AAAAAAAA8CE/9D2vNRT1nqsqMRtgTa97KPu6Epi0yp-sQCLcB/s320/fossilfree.jpg" width="320" /></a></div> <i>Sunset over fossil free state without CO2 polluting people and welfare.</i><br /><br />Tim Ball asks for <a href="http://drtimball.com/2016/compulsory-courses-for-any-curriculum-the-science-dilemma/">Compulsory Courses for Any Curriculum; The Science Dilemma</a>:<br /><div><ul><li><i>Science is pervasive directly and indirectly in every phase of modern life. </i></li><li><i>This knowledge must be a fundamental part of any school curriculum.</i><i> </i></li></ul><div>Tim suggests that compulsory science could save people from a meaningless ban on fossil fuel: <i></i></div><ul><li><i>Climate skeptics struggle with getting the majority of people to understand the problems with the UN Intergovernmental Panel on Climate Change’s (IPCC) anthropogenic global warming (AGW) story. The problem is much wider because it relates to the lack of scientific abilities among a majority of the population.</i></li></ul><div>Tim is not happy with the present situation:</div><ul><li><i>I was involved in many curricula fights, few of them ever resolved much</i><i>.</i></li><li><i> Ever subject area and discipline considered theirs essential to an education. They failed in achieving curricula useful to the student and society. </i></li><li><i>This was because they were controlled by people ensuring what interested them or what ensured their job, rather than what the student needed to become an effective informed citizen.</i></li><li><i>Students are not given the tools to avoid being exploited. Indeed, sometimes I think the system keeps them ignorant so it can exploit them as adults. </i></li><li><i>Peoples of the Rainforest teach their children what they need to survive in the real and dangerous world in which they live.</i></li><li><i>We don’t do this at any level. For most North American university or college students the experience is a socially acceptable and ridiculously expensive form of unemployment. Most of them learn more about life and themselves in part-time and summer jobs. </i></li></ul><div>I agree with Tim about the incitaments behind curricula, but I am not sure compulsory science would be beneficial. The trouble with anthropogenic global warming is that it is massively backed by scientists and academic institutions, and compulsory science could just mean more backing of fake science.</div></div><div><br /></div><div>The real Science Dilemma is maybe rather to distinguish real science from fake science.<br /><br /><b>PS</b> Our new social democratic Minister of Climate and Vice Prime Minister Isabella Lövin <a href="http://www.regeringen.se/regeringens-politik/regeringens-prioriteringar/sverige-som-foregangsland-for-minskade-klimatutslapp/">today proudly announces that</a><br /><ul><li><i>Sweden will be the first<a href="http://www.regeringen.se/regeringens-politik/regeringens-prioriteringar/sverige-som-foregangsland-for-minskade-klimatutslapp/fossilfritt-sverige/"> fossil free welfare state</a> in the Milky Way Galaxy.</i></li></ul><div><a href="https://en.wikipedia.org/wiki/Isabella_L%C3%B6vin">Isabella Lövin of course has no climate science education,</a> but would that have changed anything?</div></div>http://claesjohnson.blogspot.com/2016/05/the-value-of-compulsory-climate-science.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-5689961349059088491Tue, 24 May 2016 12:01:00 +00002016-05-25T14:57:42.940+02:00Margot WallströmWallström: Förmån (utan Tjänstesamband) Ingen Muta<a href="http://www.svd.se/flashinget-atal-mot-wallstrom">Chefsåklagare Alf Johansson vid riksenheten mot korruption meddelar till TT:</a><br /><ul><li><i>Det begicks inget mutbrott när utrikesminister Margot Wallström fick en lägenhet av Kommunal. Åklagaren lägger ner förundersökningen.</i></li><li><i>Det är också uppenbart att Wallström har mottagit en förmån, anser Johansson, men det kan alltså inte bevisas att hon har fått den just i egenskap av minister.</i></li><li><i>Sammanfattningsvis har jag inte kunnat finna att det föreligger ett så kallat tjänstesamband, det vill säga en förmån i form av ett hyreskontrakt som har lämnats med anledning av statsrådets uppdrag.</i></li></ul><a href="http://www.lawline.se/blogg/2016-01-20">Lawline gjorde an analys</a> när lägenhetsaffären uppdagades i januari med titeln MARGOT WALLSTRÖM UPPFYLLER REKVISITEN FÖR TAGANDE AV MUTA, med bl följande motivering:<br /><div><ul><li><i>Av kontraktet parterna emellan framgår att hyresförhållandet gäller så länge Margot Wallström har sin nuvarande anställning.</i></li></ul><div>Chefåklagaren anser alltså att Wallström tagit emot förmån, men anser, i synbar strid med kontraktet, att denna förmån inte har haft samband med Wallströms anställning som minister. </div><div><br /></div><div>Eller kanske har tjänsten haft visst samband, men jämfört med det rena vänskapssamband som Johansson vet måste ha funnits (ja, vad annat skulle det ha kunna vara?) mellan Wallström och Kommunal (dock utan att ha hört Wallström), måste väl tjänstesambandet anses som försumbart?<br />Så klart att Wallström säger "Jag älskar Kommunal" och så klart att kärleken är besvarad!</div><div><br /></div><div>Under alla förhållanden är Sverige det land i världen som har minst korruption, om man inte räknar vänskapskorruption förstås, och det är ju inte detsamma som den riktiga rejäla korruption som florerar i alla andra länder...<br /><br />Det rimliga är väl nu att Wallström åter flyttar in i Kommunals lägenhet, eftersom som Wallström säger "inget fel är begånget". Det är ju trots allt en ganska trevlig lägenhet, stor, möblerad, central, låg hyra...<br /><br />Se också <a href="http://claesjohnson.blogspot.se/2016/01/margot-wallstrom-och-utomrattsliga.html">tidigare post.</a><br /><br />Jämför också med <a href="http://sverigesradio.se/sida/artikel.aspx?programid=83&artikel=6347527">SVT: Wallströms lägenhet direkt kopplade till hennes politiska uppdrag. </a></div></div>http://claesjohnson.blogspot.com/2016/05/wallstrom-forman-utan-tjanstesamband.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-8553784011296480032Tue, 24 May 2016 11:31:00 +00002016-05-25T14:59:24.824+02:00absolute simultaneityEinsteintheory of relativityThe Stupid Demand of Absolute Simultaneity which Destroyed Rational Physics <div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-J3z3QYFUaVc/V0Q6-PyrUYI/AAAAAAAA8Bs/ujiYUzXRyOkLb4-JdAcmOI14fVpanf7XgCLcB/s1600/Alice-In-Wonderland-Quote-Tea-Time-Clock-Mad-Hatter-Vinyl-Wall-Art-Sticker-Decal-Room-Kitchen.jpg_640x640.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="301" src="https://3.bp.blogspot.com/-J3z3QYFUaVc/V0Q6-PyrUYI/AAAAAAAA8Bs/ujiYUzXRyOkLb4-JdAcmOI14fVpanf7XgCLcB/s320/Alice-In-Wonderland-Quote-Tea-Time-Clock-Mad-Hatter-Vinyl-Wall-Art-Sticker-Decal-Room-Kitchen.jpg_640x640.jpg" width="320" /></a></div><div> Absolute simultaneity of Tea Time</div><div><br /></div><div>The book <i><a href="http://www.amazon.com/Einstein-Relativity-Absolute-Simultaneity-William/dp/041559166X">Einstein, Relativity and Absolute Simultaneity edited by William Lane Craig and Quentin Smith</a>, </i>starts out</div><div class="page" title="Page 4"><div class="layoutArea"><div class="column"></div></div></div><ul><li><i>2005 marked the centenary of one of the most remarkable publications in the history of science, Albert Einstein’s ‘‘On the Electrodynamics of Moving Bodies,’’ in which he presented a theory that later came to be known as the Special Theory of Relativity (STR). </i></li><li><i>This 1905 paper is <b>widely regarded as having destroyed the classical conceptions of absolute time and space,</b> along with absolute simultaneity and absolute length, which had reigned in physics from the times of Galileo and Newton to the dawn of the twentieth century. </i></li></ul><div>Einstein is thus commonly viewed to have destroyed classical Newtonian physics, and to judge if this is to applaud or not, it is necessary to take a look at Einstein's reason for the destruction as presented in the 1905 article. And that is a perceived impossibility of synchronising clocks with different positions and velocities, a perceived impossibility of fulfilling an absolute need of absolute simultaneity which Einstein attributed to classical Newtonian mechanics. </div><div><br /></div><div>But what says that the world as classical mechanics requires absolute simultaneity to go around? Yes, it is needed for navigation by the Sun or GPS by humans, but birds navigate without synchronised clocks. And wasn't the world going around pretty well before there were any Poincare or Einstein worrying about clock synchronisation and absolute simultaneity? </div><div><br /></div><div>So is there no need of absolute simultaneity in classical Newtonian mechanics? Yes, the standard idea is that the gravitation from the Sun is pulling the Earth around by instant action at distance, and that seems to require (i) synchronisation of Sun time and Earth time and (ii) a mechanism for instant action at distance. </div><div><br /></div><div>Since no progress has been made concerning (i) and (ii) over the centuries since Newton, I have in recent posts tested a way to circumvent these difficulties or impossibilities, and that is to view the gravitational potential $\phi$ as primordial from which matter density $\rho =\Delta\phi$ is derived by the local action in space of the Laplacian $\Delta$. </div><div><br /></div><div>With this view, which is Newtonian mechanics with just a little twist on what comes first, matter or gravitational potential, there is no need for absolute simultaneity and thus no longer any need to destroy a most beautiful and functional Newtonian mechanics. </div><div><br /></div><div>Einstein thus attributes an unreasonable requirement of absolute simultaneity to Newtonian mechanics, and then proceeds to kill Newton. Of course this can be seen as an example of the all too well known tactics of attributing some evil quality (true or false) to your enemy, and then killing him. </div><div><br /></div><div>And the book also ventilates such criticism:</div><ul><li><i>Unfortunately for Einstein’s Special Theory, however, its epistemological and ontological assumptions are now seen to be questionable, unjustified, false, perhaps even illogical. </i></li><li><i>The precise philosophical arguments for the illogicality, falsity, or unjustifiably of the epis- temological, semantic, and ontological presuppositions of the Special Theory remain, with a few exceptions, unknown among physicists.</i></li></ul><div>Pretty tough words, but how to cope with lack of knowledge and ignorance?</div><br />http://claesjohnson.blogspot.com/2016/05/the-stupid-demand-of-absolute.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-6320771592162239995Mon, 23 May 2016 07:34:00 +00002016-05-24T11:48:06.641+02:00cosmological modelextended Newtonian gravitationNew View on gravitationNeo-Newtonian Cosmology: Progress!<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-AtxyV2FeaYo/V0KxEAMJ1tI/AAAAAAAA8BU/S1YRfqn44joOr4DrVrQJs4rqbpWvGNrEwCLcB/s1600/spacetime_strip.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="172" src="https://3.bp.blogspot.com/-AtxyV2FeaYo/V0KxEAMJ1tI/AAAAAAAA8BU/S1YRfqn44joOr4DrVrQJs4rqbpWvGNrEwCLcB/s400/spacetime_strip.jpg" width="400" /></a></div><br />We consider a <i>Neo-Newtonian cosmological model</i> in the form of <b><i>Euler's equations</i></b> for a <b style="font-style: italic;">compressible gas </b>subject to<b style="font-style: italic;"> Newtonian gravitation</b>: Find $(\phi ,m, e,p)$ depending on a Euclidean space coordinate $x$ and time $t$, such that for all $(x,t)$:<br /><ul><li>$\Delta\dot\phi + \nabla\cdot m =0$ (1)</li><li>$\dot m +\nabla\cdot (mu) +\nabla p + \rho\nabla\phi =0$ (2)</li><li>$\dot e +\nabla\cdot (eu) +p\nabla\cdot u +\rho\nabla\cdot m=0$, (3)</li></ul>where $\phi$ is <i>gravitational potential,</i> $\rho =\Delta\phi$ is <i>mass density</i>, $m$ is momentum, $u=\frac{m}{\rho}$ is <i>matter velocity</i>, $p$ is <i>pressure</i>, $e$ is <i>internal energy </i>as the sum of <i>heat energy $\rho T$ </i>with $T$ temperature and <i>gravitational energy </i>$\rho\phi$<i>, </i>and<i> </i>the dot indicates time differentiation, see <a href="http://books.google.se/books?id=dTjpb329P-4C&lpg=PP1&dq=many+minds+relativity&pg=PP1&redir_esc=y#v=onepage&q=many%20minds%20relativity&f=false">Many-Minds Relativity 20.3</a> and <a href="http://www.csc.kth.se/~cgjoh/ambsthermo.pdf">Computational Thermodynamics Chap 32</a>. Here $x$ is space coordinate in a fixed Euclidean coordinate system, and $t$ is a local time coordinate which is not globally synchronized.<br /><br />The primary variables in this model are the gravitational potential $\phi$ and the momentum $m$ connected through (2) expressing conservation of momentum or Newton's 2nd law. We view matter density $\rho =\Delta\phi$ as being derived by local action of the differential operator $\Delta$. The model is complemented by a constitutive equation for the pressure.<br /><br />The essential components of this model are:<br /><ol><li>Newton's law of gravitation $\rho =\Delta\phi$ connecting mass to gravitational potential</li><li>$\nabla\phi$ as gravitational force </li><li>Newton's 2nd law (2) connecting motion to force,</li><li>(1) expressing conservation of mass and (3) conservation of energy,</li></ol><div>with the following features:</div><ul><li>no action at distance with $\phi$ primordial and $\rho =\Delta\phi$ derived quantity</li><li>global clock synchronisation not needed because all action is local</li><li>equivalence of inertial and gravitational mass by (2)</li><li>$\Delta\phi$ of variable sign opens to positive and negative matter</li><li>no limit on matter speed</li><li>no electro-magnetics or nuclear physics so far included in the model.</li></ul><div>It may well by that a model of this form is sufficient to describe the mechanics of the universe we can observe, a universe resulting from an interplay of gravitational force and motion of matter. You can test the model in the app<a href="https://itunes.apple.com/se/app/dark-energy-newmath/id1091609721?l=en&mt=8"> Dark Energy at App Store.</a> Try it!<br /><br />Some form of starting values are needed for simulations using the model, but like in weather prediction initial values at a given global time are not known, but have to be constructed from observations over time possibly involving synchronisation of nearby clocks. </div><div><br /></div><div>The primordial quantity in this Newtonian model is the gravitational potential and gravitational force. It is the opposite of Einstein's model, where gravitational force is eliminated and replaced by "space-time" curvature. It is no wonder that Einstein expressed "Forgive me Newton!!" when taking this big extreme step.</div><div><br /></div><div>A fundamental problem with modern physics is the incompatibility of Einstein's theory of gravitation in "curved space-time" and quantum mechanics in Euclidean space. This big obstacle would disappear if Einstein's gravitation was given up, and Newton's gravitation was resurrected in suitable form. </div><div><br /></div><div>What is the reason to not take this step and open for progress?<br /><br />Recall that nobody understands what "curved space-time" is, while everybody can understand what a Euclidean coordinate system is and how to measure local time. If we follow Einstein's device of always seeking to "make things as simple as possible, but not simpler", then Newton would have to be preferred before Einstein, or what do you think?<br /><br />The basic force of cosmology is gravitation, and thus it may appear from rationality point of view to be irrational to seek to eliminate gravitational force from the discussion altogether, which is what Einstein did and which maybe paradoxically gave him fame bigger than that of Newton. <br /><br /><b>PS1</b> What drove Einstein into his extremism? Well, the reception of the special theory of relativity Einstein presented in a short sketchy note in 1905, did not draw any attention the first years and when it did, the reaction was negative. The only thing left for Einstein before getting called and kicked out of academics, was to increase the bet by generalising the special theory, which did not cover gravitation, into a general theory of relativity including gravitation. The only thing Einstein had in his scientific toolbox was the Lorentz transformation between non-accelerating inertial systems and the only way to bring that in contact with gravitation was to introduce coordinate systems in free fall, which in the presence of gravitation required strange transformations of space and time coordinates.<br /><br />Einstein's "happiest thought" was when he realised that sitting in a freely falling elevator cannot be distinguished from sitting in an elevator at rest assuming no gravitation... until the freely falling elevator hits ground....It was this idea of free fall seemingly without gravitation, which allowed him to keep the Lorentz transformation with all its wonderful effects of the special theory without gravitation, when generalising to include gravitation...but the price was high...and the free fall is going on...Compare with <a href="http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_pathway/">Einstein's Pathway to General Relativity.</a><br /><br /><b>PS2 </b>Another fact not to suppress is that the special theory of relativity was focussed on propagation of light with the same speed in all inertial coordinate systems if connected by the Lorentz transformation, which gave strange effects for the mechanics of matter (without gravitation) including dilation in time and contraction in space. But the Lorentz transformation was shaped for light propagation and not for mechanics of matter and so it was no wonder that strange effects came out. Since the Lorentz transformation also underlies the general theory of relativity, it is even less wonder that strange effects come out when adding gravitation to the picture.<br /><br />The lack of scientific logic is clear: If you apply a theory designed to describe a a certain phenomenon (light propagation) to a different type of phenomenon (mechanics of matter), then you must be prepared to get in trouble, even if your name is Einstein... <b> </b></div><br /><br />http://claesjohnson.blogspot.com/2016/05/neo-newtonian-cosmology.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-2522938420836255892Sun, 22 May 2016 10:22:00 +00002016-05-22T12:23:16.893+02:00Equivalence Principle Equivalence Principle as Definition: ExperimentHere is little experiment you can do yourself on the kitchen table supporting the idea that intertial mass is made equal to gravitational mass by definition as a result of a definition of force in terms of gravitational force:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Dd6C7m5y6MA/V0GEu3AIC0I/AAAAAAAA8BA/Oh2DyW-oMFIzGwhGX6zDmYBC09-9oWOmQCLcB/s1600/equivalenceprinciple.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://2.bp.blogspot.com/-Dd6C7m5y6MA/V0GEu3AIC0I/AAAAAAAA8BA/Oh2DyW-oMFIzGwhGX6zDmYBC09-9oWOmQCLcB/s320/equivalenceprinciple.png" width="320" /></a></div>Take two identical pieces of material, put one of the pieces on a horisontal table with frictionless surface and connect the other by a weightless rope as indicated in the picture and let go from rest.<br /><br />Record the acceleration of the system. Observe that it is half of the gravitational acceleration of one of the pieces in free fall. Conclude that inertial mass = gravitational mass and that force ultimately is defined in terms of gravitational force as expressed by the green arrows.<br /><br />Understand that what you test with the experiment is if mass = force/acceleration is independent of orientation of Euclidean coordinate system.http://claesjohnson.blogspot.com/2016/05/equivalence-principle-as-definition.htmlnoreply@blogger.com (Claes Johnson)3tag:blogger.com,1999:blog-1500584444083499721.post-3938145401801413067Fri, 20 May 2016 15:03:00 +00002016-05-22T11:01:27.258+02:00Equivalence Principlemysticism of modern physics Gravitational Mass = Inertial Mass by Definition: Hard Thinking<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-FzFZt_isZ5U/Vz8mypK3YOI/AAAAAAAA8Ak/U_3Idix5WAoZRRFvMJXXtbaFWcutRY8GgCLcB/s1600/equivalence_princ.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="167" src="https://2.bp.blogspot.com/-FzFZt_isZ5U/Vz8mypK3YOI/AAAAAAAA8Ak/U_3Idix5WAoZRRFvMJXXtbaFWcutRY8GgCLcB/s320/equivalence_princ.gif" width="320" /></a></div> <i>Typical illustration of equivalence principle. Get the point?</i><br /><br />In Newtonian mechanics as already observed and understood by Galieo, inertial and gravitational (heavy) mass are equal, because there is only one form of mass and that is inertial mass as a measure of acceleration vs force per unit of volume. Since Newtonian gravitation is a force per unit of volume, gravitational mass is equal to inertial mass, by definition, as expressed by the fact that the dimension of gravitation is $m/s^2$. See also <a href="http://www.csc.kth.se/~cgjoh/ambsrelativity.pdf">Chap 18 of Many-Minds Relativity</a>.<br /><br />Let us compare this insight with what modern physics says as told by Nigel Calder in Magic Universe:<br /><ul><li><i>A succession of experiments to check the equivalence principle—the crucial proposition that everything falls at the same rate—began with Lorand Eötvös in Budapest in 1889. After a century of further effort, physicists had improved on his accuracy by a factor of 10,000. The advent of spaceflight held out the possibility of a further improvement by a factor of a million.</i></li><li><i>If another theory of gravity is to replace Einstein’s, the equivalence principle cannot be exactly correct. Even though it’s casually implicit for every high-school student in Newton’s mathematics, Einstein himself thought the equivalence principle deeply mysterious. ‘Mass,’ he wrote, ‘is defined by the resistance that a body opposes to its acceleration (inert mass). It is also measured by the weight of the body (heavy mass). That these two radically different definitions lead to the same value for the mass of a body is, in itself, an astonishing fact.’</i></li><li><i>Francis Everitt of Stanford put it more forcibly. ‘In truth, the equivalence principle is the weirdest apparent fact in all of physics,’ he said. ‘Have you noticed that when a physicist calls something a principle, he means something he believes with total conviction but doesn’t in the slightest degree understand.’</i></li><li><i>Together with Paul Worden of Stanford and Tim Sumner of Imperial College London, Everitt spent decades prodding space agencies to do something about it. Eventually they got the go-ahead for a satellite called STEP to go into orbit around the Earth in 2007. As a joint US–European project, the Satellite Test of the Equivalence Principle (to unpack the acronym) creates, in effect, a tower of Pisa as big as the Earth. Supersensitive equipment will look for very slight differences in the behaviour of eight test masses made of different materials— niobium, platinum-iridium and beryllium—as they repeatedly fall from one side of the Earth to the other, aboard the satellite.</i></li><li><i>‘The intriguing thing,’ Everitt said, ‘is that this advance brings us into new theoretical territory where there are solid reasons for expecting a breakdown of equivalence. A violation would mean the discovery of a new force of Nature. Alternatively, if equivalence still holds at a part in a billion billion, the theorists who are trying to get beyond Einstein will have some more hard thinking to do.’ </i></li></ul><div>So Einstein thought to be deeply mysterious, what every high school student directly understands, and was able to imprint his idea into the brains of all modern physicists, who now have some hard thinking to do...<br /><br />Einstein skillfully jumped between definition as a tautology true by construction and physical principle/law, which may be valid/true or not, thereby creating a total confusion. Another aspect is the constancy of the speed of light, which today is used as definition with the meter defined by distance traveled by light in certain time, yet physicists go around and believe that this works because the speed of light <b>is</b> constant. If you cannot distinguish between a definition without content and statement with content, then you may find yourself in trouble and mislead others...<br /><br /><b>PS</b> This previous post may be consulted: <a href="http://claesjohnson.blogspot.se/2014/05/the-principal-difference-between-laws.html">The Principal Difference between Principles and Laws in Physics.</a> Note in particular the distinction that a law is typically expressed as a formula, while a principle is expressed in words e.g. as equality of inertial and gravitational mass.</div><br /><br />http://claesjohnson.blogspot.com/2016/05/inertial-mass-gravitational-mass-by.htmlnoreply@blogger.com (Claes Johnson)14tag:blogger.com,1999:blog-1500584444083499721.post-3761393952070007951Fri, 20 May 2016 12:20:00 +00002016-05-23T13:13:35.113+02:00New Theory of Flight Presented at KTH<div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/Dic5mCopbXs/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/Dic5mCopbXs?feature=player_embedded" width="320"></iframe></div><br />The New Theory of Flight developed together with Johan Hoffmann and Johan Jansson will be presented at a KTHX seminar on May 26, see<br /><ul><li><a href="http://www.kth.se/foreningar/kthx/">KTHX</a></li><li><a href="http://www.kth.se/foreningar/kthx/fluid.pdf">Flow Theory Beyond Prandtl: Change of Paradigm?</a></li><li><a href="https://secretofflight.wordpress.com/">The Secret of Flight</a> (website)</li><li><a href="http://www.csc.kth.se/~cgjoh/ambsflying.pdf">The Secret of Flight</a> (book).</li></ul><div>Note that the starting point is the incompressible Euler equation formulated by Euler around 1750 and presented as follows:</div><ul><li><i>Everything that the theory of fluids contains is embodied in the two equations I have formulated. It is not the laws of mechanics that we lack in order to pursue this research, only the analysis which has not been sufficiently developed for this purpose.</i></li></ul><div>What we do is to compute turbulent solutions of the Euler equations after having realised for the first time the development Euler is asking for, and we then discover as predicted by Euler "everything that the theory of fluids contains" or at least lots of it. </div><div><br /></div><div>Sometimes it takes a long time for a correct idea to bear fruit.</div><br />http://claesjohnson.blogspot.com/2016/05/new-theory-of-flight-presented-at-kth.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-2674112103469264980Thu, 19 May 2016 15:05:00 +00002016-05-21T08:17:52.307+02:00Einsteinmysticism of modern physicsSchrödinger's equationUnsolvable Incompatible Equations of Modern Physics = Complete Success!<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-XL9c-7lxj6M/Vz3VACkFCQI/AAAAAAAA8AE/btBU41fzfRkzIL4tNU2kO9oHvGLjoo0swCLcB/s1600/quantumrelativity.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://3.bp.blogspot.com/-XL9c-7lxj6M/Vz3VACkFCQI/AAAAAAAA8AE/btBU41fzfRkzIL4tNU2kO9oHvGLjoo0swCLcB/s400/quantumrelativity.jpg" width="400" /></a></div><br />All books on modern physics start out with a praise to Schrödinger's equation of quantum mechanics and Einstein's equation of general relativity as the highest achievement of science. Here is what Stephen Hawking says in The Grand Design (while signaling that something is weird):<br /><ul><li><i>The quantum model of nature encompasses principles that contradict not only our everyday experience but our intuitive concept of reality. Those who find those principles weird or difficult to believe are in good company, the company of great physicists such as Einstein and even Feynman, whose description of quantum theory we will soon present. In fact, Feynman once wrote, “I think I can safely say that nobody understands quantum mechanics.” But quantum physics agrees with observation. It has never failed a test, and it has been tested more than any other theory in science.</i></li></ul>Then comes a little caveat saying that unfortunately the equations are incompatible, and so one of them must be wrong, but in any case both equations are certainly valid/true and since both are written in stone none of them can be wrong, after all. Here is what Wikipedia says about the situation and the prospects:<br /><ul><li><i>Through years of research, physicists have experimentally confirmed with tremendous accuracy virtually every prediction made by these two theories when in their appropriate domains of applicability. In accordance with their findings, scientists also learned that GR and QFT, as they are currently formulated, are mutually incompatible – they cannot both be right.</i></li><li><i>Some theoretical physicists currently believe that a quantum theory of general relativity may require frameworks other than field theory itself, such as <a href="https://en.m.wikipedia.org/wiki/String_theory">string theory</a> or <a href="https://en.m.wikipedia.org/wiki/Loop_quantum_gravity">loop quantum gravity</a>. Some models in string theory that are promising by way of realizing our familiar <a href="https://en.m.wikipedia.org/wiki/Standard_model">standard model</a> are the <a href="https://en.m.wikipedia.org/wiki/Perturbative">perturbative</a> <a href="https://en.m.wikipedia.org/wiki/Heterotic_string">heterotic string</a> models, 11-dimensional <a href="https://en.m.wikipedia.org/wiki/M-theory">M-theory</a>, Singular geometries (e.g. <a href="https://en.m.wikipedia.org/wiki/Orbifold">orbifold</a> and <a href="https://en.m.wikipedia.org/wiki/Orientifold">orientifold</a>), <a href="https://en.m.wikipedia.org/wiki/D-branes">D-branes</a> and other <a href="https://en.m.wikipedia.org/wiki/Branes">branes</a>, <a href="https://en.m.wikipedia.org/wiki/Flux_compactification">flux compactification</a> and <a href="https://en.m.wikipedia.org/wiki/Warped_geometry">warped geometry</a>, and <a href="https://en.m.wikipedia.org/wiki/Non-perturbative">non-perturbative</a> <a href="https://en.m.wikipedia.org/wiki/Type_IIB_superstring">type IIB superstring</a> solutions (<a href="https://en.m.wikipedia.org/wiki/F-theory">F-theory</a>).</i></li></ul>So string theory is the only hope, but the string hype seems to be fading and so the prospects seem pretty dim. But there is another even more cumbersome problem with these equations: They cannot be solved!<br /><br />Schrödinger's equation for an atom with $N$ electrons is formulated in terms of a wave function which depends on $3N$ space coordinates, which makes numerical solution impossible for N > 10<br /><a href="http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/kohn-lecture.pdf">according to Nobel Laureate Walter Kuhn</a>, and already the case N=2 of Helium is filled with difficulty not to speak about the case of Oxygen with N=8. And analytical solution is known only in the case of Hydrogen with N=1.<br /><br />Einstein's equation is even more difficult to solve and only a few analytical solutions in extreme simplicity are known (e.g. vacuum solution of a spherically symmetric gravitational field for a static mass), and numerical solution is not really an issue because data such as initial and boundary values and forcing are completely up in the air and the choice of coordinate system is unclear. We quote from Baumgarte Numerical Relativity: Solving Einstein's equations on the Computer:<br /><div class="page" title="Page 17"><div class="layoutArea"><div class="column"><ul><li><span style="font-family: "timesnewromanps"; font-size: 11pt;"><i>Chapters 12 and 13 focus on the inspiral and coalescence of binary black holes, one of the most important applications of numerical relativity and a promising source of detectable gravitational radiation. These chapters treat the <b>two-body problem</b> in classical general relativity theory, and its solution represents one of the major triumphs of numerical relativity. </i></span></li></ul>We understand that modern physics is based on two equations which cannot be solved, except in very simplistic cases. Yet the equations are claimed to be true in the sense that solutions of the equations always agree with observations, in all cases including complex cases. But wait, how can you know that solutions of the equations always agree with observation if you cannot solve the equations and produce solutions to compare with observations?</div></div></div><br />Of course you cannot know that. But this troublesome fact for modern physicists, is twisted into: Since solutions cannot be computed, it is impossible to find any discrepancy between predictions according to the equations and observations! There are no predictions from solving the equations and thus there is no discrepancy!<br /><br />More precisely it works this way: Suppose you have computed a what you view as an approximate solution of Schrödinger's equation for some atom with many electrons, by an ingenious choice of "atomic orbitals" combined with some optimisation to choose a best combination of orbitals, and that the predicted energy is in perfect agreement with observation. Then you congratulate yourself and say that you have produced yet another piece of evidence that solutions of Schrödinger's equation always agree exactly with observations. On the other hand, if your approximate solution does not agree exactly with observation, then you blame the approximation and take that as evidence that without approximation the agreement certainly would be complete, and then you try some other orbitals...until complete agreement...possibly by twisting the observation under the firm conviction that solutions to Schrödinger's equation give an exact description of atomic physics.<br /><br />The net result is that the unsolvable equations of modern physics, which unfortunately are incompatible, anyway both must be valid to an unprecedented precision, since there are no examples of slightest discrepancy between prediction based on solving the equations and observation.<br /><br />In other words the equations serve like <a href="https://en.wikipedia.org/wiki/Oracle">oracles</a> who know exact answers to important questions, but are not willing to reveal the full truth. Not very helpful.<br /><br />Do you buy this? Or is there something fishy about solutions to unsolvable mathematical equations, which always give results in perfect agreement with observations? Doesn't perfect agreement sound little bit too good?<br /><br />Some quotes, among many similar:<br /><ul><li><i>In fact, it is often stated that of all the theories proposed in this century, the silliest is quantum theory. Some say that the only thing that quantum theory has going for it, in fact, is that it is unquestionably correct.</i> (<a href="http://todayinsci.com/K/Kaku_Michio/KakuMichio-Quotations.htm">Michio Kaku</a>)</li><li><i>When thinking about the new relativity and quantum theories I have felt a homesickness for the paths of physical science where there are ore or less discernible handrails to keep us from the worst morasses of foolishness.</i> (<a href="http://todayinsci.com/E/Eddington_Arthur/EddingtonArthur-Quotations.htm">Sir Arthur Stanley Eddington</a>)</li><li><i>Einstein, my upset stomach hates your theory [of General Relativity]—it almost hates you yourself! How am I to' provide for my students? What am I to answer to the philosophers?!!</i>(<a href="http://todayinsci.com/E/Ehrenfest_Paul/EhrenfestPaul-Quotations.htm">Paul Ehrenfest</a>)</li><li><i>I count Maxwell and Einstein, Eddington and Dirac, among “real” mathematicians. The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are at present at any rate, almost as “useless” as the theory of numbers. (</i><a href="http://todayinsci.com/H/Hardy_Godfrey/HardyGodfrey-Quotations.htm">G. H. Hardy</a>)</li><li><i>Quantum field theory, which was born just fifty years ago from the marriage of quantum mechanics with relativity, is a beautiful but not very robust child. (</i><a href="http://todayinsci.com/W/Weinberg_Steven/WeinbergSteven-Quotations.htm">Steven Weinberg</a>)</li><li><i>Niels Bohr brainwashed a whole generation of theorists into thinking that the job of interpreting quantum theory was done 50 years ago.</i> (1969 Nobel Laureate Murray Gell-Mann) </li><li><i>Let me say at the outset, that in this discourse, I am opposing not a few special statements of quantum physics held today (1950s), I am opposing as it were the whole of it, I am opposing its basic views that have been shaped 25 years ago, when Max Born put forward his probability interpretation, which was accepted by almost everybody. I don’t like it, and I’m sorry I ever had anything to do with it.</i> (Erwin Schrodinger talking about Quantum Physics) </li><li><i>One might very well be left with the impression that the theory (of general relativity) itself is rather hollow.: What are the postulates of the theory? What are the demonstrations that else follows from these postulates? Where is the theory proven? On what grounds, if any, should one believe the theory? ....One’s mental picture of the theory is this nebolous mass taken as a whole.....One makes no attempt to derive the rest of the theory from the postulates. (What, indeed, could it mean to “derive” somtheing about the physical world?). One makes no attempt to “prove” the theory, or any part of it. </i>(Robert Geroch in General Relativity from A to B)</li></ul><br />http://claesjohnson.blogspot.com/2016/05/unsolvable-incompatible-equations-of.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-1721450848532975445Thu, 19 May 2016 12:10:00 +00002016-05-19T14:19:03.995+02:00dark matterEuler equationsextended Newtonian gravitationNew View on gravitationspiral galaxySpiral Galaxy Formation in Extended Newtonian Gravitation<h4></h4><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Kt_oihOrcWc/Vz2s45rPV9I/AAAAAAAA7_0/BxKFL4_CjUoqjtY4E9WQtS3URARJmEVjACLcB/s1600/spiral%2Bgalaxy.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="248" src="https://3.bp.blogspot.com/-Kt_oihOrcWc/Vz2s45rPV9I/AAAAAAAA7_0/BxKFL4_CjUoqjtY4E9WQtS3URARJmEVjACLcB/s400/spiral%2Bgalaxy.png" width="400" /></a></div><h4>1. Cosmological Model </h4>This is a continuation of previous posts on dark matter and <a href="http://claesjohnson.blogspot.se/2014/02/the-universe-weakly-compressible-gas.html">The Universe as Weakly Compressible Gas subject to Pressure and Gravitational Forces</a>, which post we recall:<br /><br />We consider a cosmological model in the form of <b><i>Euler's equations</i></b> for a <b style="font-style: italic;">compressible gas </b>subject to<b style="font-style: italic;"> Newtonian gravitation</b>: Find $(\rho ,m, e ,\phi ,p)$ depending on a Euclidean space coordinate $x$ and time $t$, such that for all $(x,t)$:<br /><ul><li>$\dot\rho + \nabla\cdot (\rho u ) =0$ (or $\frac{D\rho}{Dt} = -\rho\nabla\cdot u$)</li><li>$\dot m +\nabla\cdot (mu) +\nabla p + \rho\nabla\phi =0$</li><li>$\dot e +\nabla\cdot (eu) +p\nabla\cdot u +\rho\nabla\cdot m=0$,</li></ul>where $\rho$ is <i>mass density</i>, $u=\frac{m}{\rho}$ is <i>matter velocity</i>, $p$ is <i>pressure</i>, $\phi$ is <i>gravitational potential</i>, and $e$ is <i>internal energy </i>as the sum of <i>heat energy $\rho T$ </i>with $T$ temperature and <i>gravitational energy </i>$\rho\phi$<i>, </i>and<i> </i>the dot indicates time differentiation and<br /><ul><li>$\frac{D\rho}{Dt}=\dot\rho +u\cdot\nabla\rho$</li></ul>is the convective time derivative of $\rho$, see <a href="http://books.google.se/books?id=dTjpb329P-4C&lpg=PP1&dq=many+minds+relativity&pg=PP1&redir_esc=y#v=onepage&q=many%20minds%20relativity&f=false">Many-Minds Relativity 20.3</a> and <a href="http://www.csc.kth.se/~cgjoh/ambsthermo.pdf">Computational Thermodynamics Chap 32</a>.<br /><br />These equations express <i>conservation of mass</i> $\rho$, <i>conservation of momentum</i> $m$ with $\nabla p$ <i>pressure force</i> and $-\nabla\phi$ <i>gravitational force</i>, and <i>conservation of internal energy</i> $e$. These laws of conservation are complemented with <i>constitutive laws</i> connection $p$ and $\phi$ to density, of the following form:<br /><br />A1:<b> Weakly compressible gas </b>($\delta$ small positive constant):<br /><ul><li>$\Delta p =\frac{\nabla\cdot u}{\delta}= - \frac{1}{\delta\rho}\frac{D\rho}{Dt}$</li></ul>or<br /><br />A2:<b> Compressible perfect gas </b>($0 < \gamma < 1 $):<br /><ul><li>$p=\gamma \rho T$.</li></ul>B:<b> Newton's law of gravitation:</b><br /><ul><li>$\Delta\phi =\rho$ with $\phi =0$ at infinity. </li></ul><div>We observe</div><div><ol><li>Similarity of $\nabla p$ and $\nabla\phi$ in momentum equation. </li><li>Similarity between A1 and B connecting $\Delta p$ to $-\frac{D\rho}{Dt}$ (or $-\rho$) and $\Delta\phi$ to $\rho$.</li><li>$p \ge 0$ and $\phi \le 0$.</li></ol><div>Here 1. can be seen as the Equivalence Principle (equality of heavy and inertial mass) expressing that there is no difference between gravitational and other forces (pressure) in Newton's 2nd law expressing conservation of momentum.</div></div><div><br /></div><div>Further, 2. expresses that the constitutive laws A1 and B both can be viewed as action at distance if $\rho$ is viewed as the cause, but represent local action of differentiation if $\rho$ is viewed as the effect. </div><div><br /></div><div>For a weakly compressible gas described by A1, there is no need per se to identify a cause-effect relation between $p$ and $\rho$; it is enough to say that $p$ and $\rho$ are connected in a certain way expressing a form of "perfect harmony". </div><div><br /></div><div>In the same way, there is no need per se to identify a cause-effect relation between $\phi$ and $\rho$; it is enough to say that $\phi$ and $\rho$ are connected in certain way expressing a form of "perfect harmony" in the spirit of Leibniz.<br /><br />The relation $\Delta\phi =\rho$ is explored in <a href="http://www.csc.kth.se/~cgjoh/cosmology.pdf">Newtonian Matter and Antimatter</a> with $\Delta\phi > 0$ identifying matter and $\Delta\phi < 0$ antimatter, with dark matter where $\Delta\phi$ is smooth and visible matter where $\Delta\phi$ is singular, typically as a sum of multiples of delta functions representing matter in point form. We refer to such a model as <i>Extended Newtonian Gravitation. </i><br /><br /><h4>2. Galaxy Formation</h4></div><div>We start from a spherical distribution of matter of low density of dark matter (a halo) with $\Delta\phi$ a smooth function, which we assume to be in static equilibrium with the the gravitational force balanced by a weak pressure force with $\nabla p = - \rho\nabla\phi$. </div><div><br /></div><div>Starting from this halo of low density dark matter, we assume that some visible matter (stars) is formed by concentration of dark matter by gravitational attraction into point masses with $\rho$ becoming large locally with the result that the gravitational force $\rho\nabla\phi$ can no longer be balanced by a weak pressure force $-\nabla p$. This is an effect of the different action of pressure and gravitational force, with pressure scaling with surface and gravitational force with volume.</div><div><br /></div><div>The combined effect of the presence of a halo of dark matter and gravitational collapse of visible matter as a system of point masses, may then create a spiral galaxy of visible matter surrounded by a halo of dark matter, which is the standard view of the nature of a spiral galaxy, with in particular a characteristic distribution of velocity of visible matter as roughly independent of the distance to the galaxy center as an effect of the dark matter halo. </div><div><br /></div><div>It thus appears that an extended Newtonian model with $\Delta\phi$ of variable sign and concentration may be sufficient to explain essential aspects of galaxy formation, for which Einstein's equation equation is useless. </div><br />http://claesjohnson.blogspot.com/2016/05/spiral-galaxy-formation-in-extended.htmlnoreply@blogger.com (Claes Johnson)0tag:blogger.com,1999:blog-1500584444083499721.post-7353604908355126507Tue, 17 May 2016 10:34:00 +00002016-05-18T21:58:06.486+02:00EinsteinEquivalence PrincipleLorentz transformationrelativitytheory of relativityEinsteins "Scientific Method": Magic Physics from Definition<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-PJBiv4Npn0o/VzsA-jTQ4kI/AAAAAAAA7_Q/wNNQAJ3KbUgd53mNOPCfQh_ycOXvaxYOwCLcB/s1600/einsteinmagician.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-PJBiv4Npn0o/VzsA-jTQ4kI/AAAAAAAA7_Q/wNNQAJ3KbUgd53mNOPCfQh_ycOXvaxYOwCLcB/s320/einsteinmagician.jpg" width="240" /></a></div><br /><br />Einstein "scientific method", which brought him immense fame as the greatest physicist of all times, consists of:<br /><ul><li><i>Start from a definition, convention or stipulation/law without physical content, and then draw far-reaching consequences about the physics of the world.</i></li></ul><div>It is not hard to understand that such a "method" cannot work: You cannot draw meaningful conclusions about the world simply from a definition empty of physics content. You cannot develop a meaningful scientific theory from a definition that there are 100 centimeters on a meter. </div><div><br /></div><div>Einstein cleverly covered up by naming his definitions or conventions or stipulations/laws, "principles":</div><div><ol><li><i>Equivalence Principle: Gravitational mass<b> is </b>equal to inertial mass.</i></li><li><i>Relativity Principle: Observations in inertial coordinate systems moving with constant velocity with respect to each other, <b>are to</b> be connected by the Lorentz transformation.</i></li><li><i>Covariance Principle: Physical laws <b>are to </b>have the same form independent of the choice of coordinate system.</i></li></ol></div><div>Here 1. is an empty definition, because there is only one mass, and that is inertial mass, which measures acceleration vs force and gravitational force is a force. Gravitational mass is equal to inertial mass by definition. Attempts to "prove/verify" this experimentally, which are constantly being made with ever increasing precision and always with the same result of equality, are as meaningful as experiments attempting to verify that there are 100 centimeters on a meter, which could very well be the next grand challenge for LHC, in the spirit of Einstein.<br /><br />2. stipulates that different physical phenomena are to be viewed to be the same. This is because the Lorentz transformation is<b> <a href="http://claesjohnson.blogspot.se/2016/05/bergson-with-history-vs-einstein.html">not invariant</a></b><a href="http://claesjohnson.blogspot.se/2016/05/bergson-with-history-vs-einstein.html"> with respect to initial conditions,</a> and thus Einstein stipulates that two waves satisfying the same form of wave equation, but having different initial conditions, shall be viewed to be the same. No wonder that with this play with identities, all sort of strange effects of time dilation and space contraction can be drawn out of a magicians hat.<br /><br />It is clear that physical laws in general take different forms in different coordinate systems, and thus 3. is an absurd stipulation. Alternatively, it is trivial and just says that a physical law will have to transform when expressed in different coordinates so that the law has the same physical content. So 3. is either absurd or trivial, in both cases devoid of physics.<br /><br />It is depressing that none of this can be understood by leading modern physicists. Nada. Even more depressing is that the discussion is closed since 100 years. <br /><br /><br /><br /></div>http://claesjohnson.blogspot.com/2016/05/einsteins-scientific-method.htmlnoreply@blogger.com (Claes Johnson)3