tag:blogger.com,1999:blog-15005844440834997212016-05-26T11:45:35.474+02:00Claes Johnson on Mathematics and Sciencetowards understanding by critical constructive inquiryClaes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.comBlogger1529125tag:blogger.com,1999:blog-1500584444083499721.post-88352884137083109222016-05-26T09:58:00.002+02:002016-05-26T09:59:09.676+02:00Connection 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>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-74164135690585774312016-05-25T19:14:00.000+02:002016-05-25T21:44:33.204+02:00Newton'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 thus 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>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com2tag:blogger.com,1999:blog-1500584444083499721.post-782817475502889602016-05-25T15:55:00.000+02:002016-05-25T17:51:52.996+02:00The 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>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-56899613490590884912016-05-24T14:01:00.002+02:002016-05-25T14:57:42.940+02:00Wallström: 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>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-85537840112964800322016-05-24T13:31:00.000+02:002016-05-25T14:59:24.824+02:00The 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 />Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-63207715921622399952016-05-23T09:34:00.003+02:002016-05-24T11:48:06.641+02:00Neo-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 />Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-25229384208362558922016-05-22T12:22:00.000+02:002016-05-22T12:23:16.893+02:00 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.Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com4tag:blogger.com,1999:blog-1500584444083499721.post-39381454018014130672016-05-20T17:03:00.000+02:002016-05-22T11:01:27.258+02:00 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 />Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com14tag:blogger.com,1999:blog-1500584444083499721.post-37613939520700079512016-05-20T14:20:00.001+02:002016-05-23T13:13:35.113+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 />Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-26741121034692649802016-05-19T17:05:00.000+02:002016-05-21T08:17:52.307+02:00Unsolvable 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 />Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-17214508485329754452016-05-19T14:10:00.000+02:002016-05-19T14:19:03.995+02:00Spiral 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 />Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-73536049083551265072016-05-17T12:34:00.000+02:002016-05-18T21:58:06.486+02:00Einsteins "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>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com3tag:blogger.com,1999:blog-1500584444083499721.post-344944966501068182016-05-16T12:29:00.002+02:002016-05-17T08:10:37.192+02:00The Blind Space Traveler with Gravitational Potential Meter<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Y32TanHvlAk/Vzq1eqP5avI/AAAAAAAA7-8/EIwvsoHlH3wIeFAw2i0p0txj-RE4gmAjACLcB/s1600/Hawking-zero-G.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="266" src="https://3.bp.blogspot.com/-Y32TanHvlAk/Vzq1eqP5avI/AAAAAAAA7-8/EIwvsoHlH3wIeFAw2i0p0txj-RE4gmAjACLcB/s400/Hawking-zero-G.jpg" width="400" /></a></div> <i>Hawking inside a space ship without windows with a Gravitational Potential Meter</i><br /><br />Imagine you are a space traveler locked into a space ship without windows, or traveling through a region of invisible dark matter. Imagine that in this difficult situation, you have access to an instrument capable of recording the gravitational potential around the space ship from near to far away, an instrument or sense which we may call a <i>Gravitational Potential Meter. </i>Below I discuss how such an instrument might be designed.<br /><span id="goog_1467420581"></span><br />Would that allow you to create a normal picture of the distribution of celestial objects/matter around you including your own position, which would be the picture you could see if there were windows or dark matter somehow was made visible, a standard picture/map making it possible to navigate?<br /><br />Yes, it would because the mass distribution $\rho (x)$ depending on a Euclidean space coordinate $x$ at any instant of time, is related to the gravitational potential $\phi (x)$ by Poisson's equation (in normalised form):<br /><ul><li>$\rho = \Delta\phi$, (*)</li></ul><div>where $\Delta$ is the Laplacian with respect to $x$. In this setting you would naturally view the gravitational potential $\phi (x)$ as primordial, because this is what you can record/sense, and you would view the mass distribution $\rho (x)$ as a derived quantity, because this is what you can compute knowing $\phi (x)$ by applying the Laplace operator, which is a differential operator acting locally in space. </div><div><br /></div><div>In this new setting you would not, as in the classical setting of viewing $\rho (x)$ as primordial and $\phi = \Delta^{-1}\rho$ as derived by the inverse of the Laplacian as a non-local operator, have to explain <i>instant action at distance</i>, only the <i>local action</i> of (*), and you would thus have eliminated the question of the physics of instant action at distance, which does not seem to have an answer, and as such may be the wrong question. </div><div><br /></div><div>We conclude that depending on what we can see through instruments or senses, we are led to questions, which may have answers or not. It is natural to think that questions, which may have answers, are better questions than questions which do not have answers.<br /><br />As to the design of a Gravitational Potential Meter or Gravitational Force Meter, imagine a system of little satellites in free fall distributed over the space of interest and connected to a GPS system allowing tracing of the satellites, thus giving information about the Gravitational Force and from that the Gravitational Potential. It is not unthinkable that such a system could cover any space accessible for space travel and beyond. </div><div><br /></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-75495102651357564662016-05-16T09:16:00.002+02:002016-05-16T12:33:45.953+02:00Simultaneity as Non-Physical Convention along with Special RelativityThe book <a href="https://jhupbooks.press.jhu.edu/content/concepts-simultaneity">Concepts of Simultaneity: From Antiquity to Einstein and Beyond</a> is presented by:<br /><ul><li><i>Max Jammer's Concepts of Simultaneity presents a comprehensive, accessible account of the historical development of an important and controversial concept—which played a critical role in initiating modern theoretical physics—from the days of Egyptian hieroglyphs through to Einstein's work in 1905, and beyond. </i></li><li><i>Beginning with the use of the concept of simultaneity in ancient Egypt and in the Bible, the study discusses its role in Greek and medieval philosophy as well as its significance in Newtonian physics and in the ideas of Leibniz, Kant, and other classical philosophers. </i></li><li><i>The central theme of Jammer's presentation is a critical analysis of the use of this concept by philosophers of science, like Poincaré, and its significant role in inaugurating modern theoretical physics in Einstein's special theory of relativity. </i></li><li><i>Particular attention is paid to the philosophical problem of whether the notion of distant simultaneity presents a factual reality or <b>only a hypothetical convention.</b> The study concludes with an analysis of simultaneity's importance in general relativity and quantum mechanics.</i></li></ul><div>In earlier post on I have argued that <i>simultaneity in time at distant points in space</i> is a man-made convention, which is useful to humanity in many ways including GPS, but as convention has no role in describing the physics of material bodies without GPS receivers. Jammer presents much evidence supporting this view without closing the door to simultaneity as some form of factual reality.</div><div><br /></div><div>Einstein's special relativity came out from an a simple thought experiment showing that agreement on distant simultaneity defined by a certain conventional form of clock synchronization set up by Einstein, cannot be established for different observers moving with speeds comparable to the speed of light with respect to each other. </div><div><br /></div><div>Einstein thus started from a certain ad hoc man-made convention and from the impossibility of making the convention work for moving observers Einstein jumped to the conclusion that our concepts of the physics of space and time will have to be fundamentally changed. And the world jumped along. But is it possible to change physics by man-made convention? Can we change physics by changing our man-made conventions to measure time and space, by changing from yard to meter? I think not. </div><div><br /></div><div>Why believe that special relativity is real physics, when special relativity is based on an impossibility to make a certain man-made convention work?</div><div><br /></div><div>I have stressed that the notion of distant simultaneity is present in the standard form of Newton's law of gravitation as Poisson's equation $\Delta\phi =\rho$, seemingly creating a gravitational potential $\phi (x)$ depending on a Euclidean space coordinate $x$ from instant action at distance by a primordial matter distribution $\rho (y)$ with $y$ different from $x$, represented as $\phi =\Delta^{-1}\rho$ with the inverse $\Delta^{-1}$ a non-local (integral) operator.<br /><br /></div><div></div><div>On the other hand, viewing the gravitational potential $\phi$ as primordial and $\rho =\Delta\phi$ as derived by local differentiation, there is no need to explain the physics of instant action at distance, which Newton left open under the criticism of Leibniz and which has resisted all attempts after Newton.</div><div><br /></div><div>We conventionally view matter $\rho$ as primordial, since we can see matter at distance if it is sending out light, while we cannot see the gravitational potential $\phi$, only feel that it is there. </div><div><br /></div><div>But with a different eyes we may be able to see the gravitational potential $\phi$, but not $\rho$, and we would then naturally view $\phi$ to be primordial. With such eyes we might be able to see a gravitational potential of dark matter and dark energy, which we now cannot see, only feel that it is there. </div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-64747593398918959902016-05-15T14:36:00.002+02:002016-05-15T14:47:54.323+02:00The Quest for the Ultimate Theory of Time: Physical Stability or Empty Probability?<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-N5VLrfGxvqE/VzhvFxlStjI/AAAAAAAA7-c/pYYa_fztVP0pBvJVzLKvJUB5YWpANU1OwCLcB/s1600/eternity_main_1867100f.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-N5VLrfGxvqE/VzhvFxlStjI/AAAAAAAA7-c/pYYa_fztVP0pBvJVzLKvJUB5YWpANU1OwCLcB/s1600/eternity_main_1867100f.jpg" /></a></div><br /><br />The question of the direction of time, or the arrow of time, is still haunting physicists with the physicist and cosmologist Sean Carrol expressing state of art in e.g. the book <a href="http://www.amazon.com/Eternity-Here-Quest-Ultimate-Theory/dp/0452296544">From Eternity to Here: The Quest for the Ultimate Theory of Time</a>, which is basically to say following old Boltzmann: There is a quantity named<i> entropy,</i> which cannot decrease with time and when strictly increasing sets a direction of time motivated by Carroll as follows in an introduction:<br /><ul><li><i>The reason why entropy wants to increase is deceptively simple:</i></li><li><i>There are more ways to be disorderly than orderly, so an orderly arrangement will naturally tend toward increasing disorder.</i></li></ul><div>But Carroll is not very happy with this his explanation:</div><div><ul><li><i>If everything in the universe evolves toward increasing disorder, it must have started out in an exquisitely ordered arrangeement...a state of very low entropy.</i></li><li><i>Why were conditions in the early universe set up in a very particular way? That is the question this book sets out to address.</i></li><li><i>Unfortunately, no one yet knows the right answer.</i></li></ul><div>And then follows the rest of the book, without answer. The only attempt to give reason to the tendency of entropy to increase, is to argue following Boltzmann, that things naturally evolve from less probable/low entropy states to more probable/higher entropy states. But of course this is circular: To say that more probable is more probable than less probable is a tautology without actual content.</div></div><div><br /></div><div>In the book <a href="https://books.google.se/books?id=W0xp9JMnhFwC&lpg=PP1&pg=PP1&redir_esc=y&hl=en#v=onepage&q&f=false">The Clock and the Arrow: A Brief Theory of Time</a> I argue that there is another way of explaining the arrow of time and that is with reference to the physics of stability instead of the non-physics of probability of Boltzmann. The key point is:</div><div><ul><li>A system cannot remain in an unstable state because the inevitable effect of small fluctuations will have a major effect and thus transform the system to either a more stable state of more or less rest or to another unstable state of non-rest. </li><li>The transition from unstable to stable rest is irreversible since the reverse process from stable rest to unstable is impossible without major exterior forcing. </li><li>The transition from unstable is sensitive to small perturbations along with the formally reversed process, and thus cannot be reversed under any form of finite precision physics. </li></ul></div>Here is a summary of my view and that of Boltzmann/Carroll:<br /><ol><li>An arrow of time is given by physical stability properties of certain systems making them irreversible, without asking any specific order of an early universe.</li><li>An arrow of time is motivated by an empty tautology stating that systems evolve from less probable to more probable states, asking for a highly improbable highly ordered early universe. </li></ol><div>You may decide yourself between 1. and 2. Which is more probable?</div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-24094479291738485542016-05-15T12:43:00.001+02:002016-05-16T12:35:15.730+02:00Instant Action at Distance and Simultaneity not Needed in New Theory of Gravitation including Dark Energy <div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-j52DtxV0tDo/VzhRjQ3RH0I/AAAAAAAA7-M/HNNKVQcllyQWuM-9-9GubiSN90MxTDrpACLcB/s1600/relay.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="https://1.bp.blogspot.com/-j52DtxV0tDo/VzhRjQ3RH0I/AAAAAAAA7-M/HNNKVQcllyQWuM-9-9GubiSN90MxTDrpACLcB/s320/relay.jpg" width="320" /></a></div> Einstein won the game. But what was the game about? Simultaneity?<br /><br />Einstein's theory of relativity grew out from a question of <i>simultaneity</i> in time of events at different locations in space, which Einstein could not answer in a non-ambiguous way and then jumped to the conclusion that a fundamental revision of our concepts of space and time was necessary. Einstein took so on the responsibility in the service of science and humanity to make the revision and thereby open the door to a modern physics of "curved space-time" with all its wondrous new effects of time dilation and space contraction, albeit too small to be detected.<br /><br />It is clear that simultaneity plays an important role in our society, to set schedules and allow people to meet at the same place and for these purposes we all have clocks synchronized to a reference clock. And to decide which scientist first submitted an article reporting a certain new scientific break-through and to navigate...<br /><br />But what role does simultaneity play in physics? In what sense do distant physical objects care about simultaneity? Do they all have synchronised clocks? Of course not. What they do is to react to local forces acting locally in time, and no simultaneity with the action of distant objects is involved.<br /><br />Or is it? What about gravitation, isn't it supposed to act instantly over distance and thus require a form of exact simultaneity? Yes, it so seems because in Newtonian gravitation the Earth is instantly acted upon by a gravitational force from the Sun directed towards the present position of the Sun, and not towards the position where we see the Sun because of the 8 minute time delay of the light from the Sun.<br /><br />The standard view on gravitation, is thus that the presence of matter instantly generates a gravitational potential/force (Newton) or "curvature of space" (Einstein) at distance. This view comes with the following questions:<br /><ol><li>What is the physics of the instant action at distance? Gravitons?</li><li>What is the physics of the simultaneity associated with instant action? </li></ol>Since no progress towards any form of answer has been made over all the centuries since Newton, it is natural to shift and instead view the gravitational potential $\phi$ as primordial from which matter density $\rho$ is obtained by the differential equation acting locally in space and time:<br /><ul><li>$\Delta\phi =\rho$. (*) </li></ul><div>With this view there is no instant action at distance to explain and no associated simultaneity, since the action of Laplacian $\Delta$ as differential operator is local is space and time. </div><div><br /></div><div>It may thus be that the questions 1. and 2. are not the right questions, and then also that Einstein's relativity originating from a question about simultaneity, is not the right answer to the right question.<br /><br />More precisely, simultaneity does not appear to be a matter of the physics of the world, since atoms are not equipped with a man-made system of synchronised clocks, and so it is not reasonable to make a complete revision of Newtonian mechanics starting from an ad hoc idea of probably little significance. </div><div><br /></div><div>The equation (*) further suggests that with $\phi$ primordial there is no reason to insist that $\rho$ as a derived quantity must be non-negative, thus (*) opens to the possible existence of matter density $\rho$ of both signs, that is to both positive and negative matter. </div><div><br /></div><div>This idea is explored 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> with in particular a simulation of a universe resulting from a fluctuation of the gravitational potential with associated positive and negative matter, with the negative matter forcing a positive matter world into accelerating expansion, which may be the missing dark energy you are looking for. Try it!</div><div></div><div><br /></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-78402377145041760242016-05-11T18:44:00.001+02:002016-05-14T14:18:00.715+02:00Bergson with History vs Einstein without History: Tragedy of Modern Physics<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-_a0KvxGPWqc/VzRG_mnhAFI/AAAAAAAA79M/F9bjbhwL0M0nzH3T0QfX2vfl6rm7qfn0QCLcB/s1600/albert_einstein_quote.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://2.bp.blogspot.com/-_a0KvxGPWqc/VzRG_mnhAFI/AAAAAAAA79M/F9bjbhwL0M0nzH3T0QfX2vfl6rm7qfn0QCLcB/s400/albert_einstein_quote.jpg" width="400" /></a></div><br /><a href="http://claesjohnson.blogspot.it/2016/05/time-again-for-bergson-vs-einstein.html">The clash between Bergson and Einstein in 1922 about the physics of special relativity</a> can be described as the clash between the physics of Herakleitos as change and Parmenides as no change.<br /><br />Let us recall Einstein's position of no change with motionless space-time trajectories without beginning and end or "world lines" frozen into a block of space-time, expressed with the typical Einsteinian ambiguity:<br /><ul><li><b style="-webkit-text-size-adjust: auto; font-family: Garamond, 'Book Antiqua', 'Footlight MT Light', 'Times New Roman'; text-indent: 19.200000762939453px;">..<i>.</i></b><i>for us physicists believe the separation between past, present, and future is only an illusion, although a convincing one.</i></li><li><i>Since there exists in this four dimensional structure [space-time] no longer any sections which represent "now" objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence.</i></li></ul>Einstein's special theory of relativity is defined by the following linear transformation between two space-time coordinate systems $(x,y,z,t)$ and $(x^\prime ,y^\prime ,z^\prime ,t^\prime )$ denoted by $S$ and $S^\prime$, named the Lorentz transformation:<br /><ul><li>$x^\prime =\gamma (x - vt)$,</li><li>$y^\prime =y^\prime$</li><li>$z^\prime =z^\prime$</li><li>$t^\prime =\gamma (t - vx)$, </li></ul>where $\gamma = \frac{1}{\sqrt{1-v^2}}$ assuming the speed of light is 1 and $0 < v < 1$. Here $(x,y,z)$ and $(x^\prime ,y^\prime ,z^\prime)$ are supposed to represent orthogonal space coordinates and the origin $x^\prime = 0$ in $S^\prime$ can be seen to move with velocity $(v,0,0)$ in $S$. Einstein's strike of genius is to claim that the Lorentz transformation represents the coordinate transformation between two orthogonal coordinate systems "moving with velocity $(v,0,0)$ with respect to each other" both describing the same physics of light propagation at speed = 1 according to one and the same wave equation taking the same form (being invariant) in both systems.<br /><br />In the physics of change of Bergson the wave equation in $S$ is combined with an intial condition in the form of position $u(x)$ and velocity $\dot u(x)$ of a wave with extension at a given time instant say $t=0$, which forms the history for subsequent evolution for $t > 0$ of the wave as described in $S$. And the same for a wave described in $S^\prime$.<br /><br />But initial conditions are not invariant under the Lorentz transformation, because $t=0$ translates to $x^\prime = \gamma x$ and $t^\prime =-\gamma vx$, and not $t^\prime =0$ as in a Galilean coordinate transformation. Two waves connected by the Lorentz transformation satisfying the same wave equation will satisfy different initial conditions and therefore represent different physical phenomena. No wonder that different waves can exhibit what is referred to as time dilation and space contraction if the different waves are identified!<br /><br />Bergson's physics of change describes phenomena with different histories/initial values as different phenomena even if they happen to satisfy the same wave equation in subsequent time, which is completely rational.<br /><br />In Einstein's physics of no change there are no intial conditions for extended waves, which allows Einstein to claim that there is no way to tell that representations connected by the Lorentz transformation do not describe the same physical phenomenon. This is used by Einstein as negative evidence that indeed the phenomena are the same, which leads to all the strange effects of special relativity in the form of time dilation and space contraction. By covering up history Einstein thus can insist that two different waves with different histories are the same wave, and from this violation of logic strike the world with wonder. But of course Einstein's insistence to cover up initial values, is fully irrational. <br /><br />Einstein circumvents the question of initial value/history by only speaking about <i>space-time</i> <i>events</i> without extension in space recorded by space-time point coordinates $(x,y,z,t)$. By focussing on points in space-time without extension in space, Einstein can cover up the crucial role of initial value/history for a phenomenon with extension in space. But physical objects have extension in space and so Einstein's physics of points is not real physics. Einstein's physics is about "events" as isolated points in space-time, but real physics is not about such "events" but about the position in space and time of physical objects with extension both in space and time.<br /><br />What has existence for Einstein as extended objects are "world lines" as trajectories extended in time of spatial points without extension frozen into a block of space-time, not objects extended in space changing over time. This is so weird and irrational that rational arguments fall short and the tragic result is modern physics without rationality, where only what is weird has a place.<br /><br />In other words, a picture consisting of just one dot carries no history, just presence. A picture with many dots can carry history. It is not rational to identify two different persons arguing that they are the same person because they were born at the same place at the same time and live under the same conditions, while forgetting that they have different ancestors and histories. Or the other way around, if you identify such people, then you obtain a strange new form of parapsychology of shifting personalities and if you believe this is science then you are fooling yourself.<br /><br />Einstein's special theory of relativity is about measurement of "space-time events" using "measuring rods" and "clocks", without ever telling what instruments these are and without caring about the underlying physics. It is thus a like an ad hoc tax system imposed by the government without caring about the underlying economy. <br /><br />It is now up to you to decide if you think that the point physics of no change/without history of Einstein, is more useful for humanity than the real physics of change/with history of Bergson, or the other way around.<br /><div><br /></div><div>Maybe you will then come to the conclusion that it is a tragedy that modern physicists have been seduced by Einstein to believe in point physics without change and history, and even more tragical that no discussion of this tragedy has been allowed after 1922, by a dictate of leading physicists.<br /><br />You can read more about the contradictions of special relativity in <a href="http://www.csc.kth.se/~cgjoh/ambsrelativity.pdf">Many-Minds Relativity</a>, with the non-invariance of initial conditions under Lorentz transformation observed in section 5.9.<br /><br /></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com1tag:blogger.com,1999:blog-1500584444083499721.post-23914737680904536082016-05-10T13:01:00.000+02:002016-05-14T15:44:12.436+02:00Bergson and Deleuze on Duration of Time (and Irreversibility)<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-aN9rVrZen7o/VzSvAssZ8MI/AAAAAAAA790/BKrzgSbljssgZ4xvbkYvvZCJXN0GEDlDQCLcB/s1600/Bergson_homenagem_1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="144" src="https://2.bp.blogspot.com/-aN9rVrZen7o/VzSvAssZ8MI/AAAAAAAA790/BKrzgSbljssgZ4xvbkYvvZCJXN0GEDlDQCLcB/s320/Bergson_homenagem_1.jpg" width="320" /></a></div> Duration of thinking.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-k6Awew7IUmg/VzSvF4w1_DI/AAAAAAAA794/cSCSBagQxWUya92rJjY-7p-wMDcQFw1xACLcB/s1600/deleuze.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://1.bp.blogspot.com/-k6Awew7IUmg/VzSvF4w1_DI/AAAAAAAA794/cSCSBagQxWUya92rJjY-7p-wMDcQFw1xACLcB/s320/deleuze.jpg" width="320" /></a></div> Duration of smoking.<br /><br />Both Bergson and Delueze say something essential about time by stressing the concept of duration or "thickness of the present" as a transformer changing past into future. To Bergson and Deleuze change is thus a fundamental aspect of time, and maybe time can simply be identified with change.<br /><br />Deleuze makes a distinction between a more simple elementary concept of time represented by the Greek god Aion as the simple tick of a clock in linear progression with an infinitely thin present tick as an infinitely thin cut between past and future, and a more complete concept of time represented by Chronos as the duration transforming past to present. One of Bergson's masterpieces is <a href="https://archive.org/stream/DurationAndSimultaneityHenriBergson/Duration%20and%20Simultaneity_Henri%20Bergson_djvu.txt">Duration and Simultaneity.</a><br /><br />Chronos thus represents the continuity of a phenomenon or being under change, where the past is transformed into present into future in a continuous change which loads the present with history, and which then gives time a direction. Aion represents discontinuous time without history and direction with every tick the same.<br /><br />We understand that Chronos gives a fuller picture of time than Aion, which is the view of Einstein with his space-time events without history which has created so much confusion and misunderstanding.<br /><br />Chronos concept of time is present in the generic mathematical model of a time-dependent problem as an intial value problem of the form: Find a function $u(t)$ of a real variable $t$ named time, such that<br /><ul><li>$\frac{du}{dt} = f(u)$ for $t > 0$ </li><li>$u(0) = 0$ </li></ul><div>where $f(v)$ is a given function of a vector-valued real variable $v$. Here the differential equation in time-discretized form expresses the transformation of the state $u(t)$ from one time instant $t$ to the next $t + dt$ with the length of the Chronos duration equal to the time step $dt > 0$, carrying along (some of) the history of previous states, in an update of the form $u(t+dt) = u(t) + f(u(t))dt$ with $u(t)$ carrying the history and $f(u(t))dt$ the change.<br /><br />With the duration of the present equal to the time step, we understand that the length of the duration of the present is not given once and for all, but like the time step can be smaller or bigger depending on the precision of time resolution of an underlying continuous evolution in time we may choose, and like the time step is never zero.<br /><br />The irreversibility of certain processes as time evolution of $u(t)$, then is expressed in the initial value problem as stability in forward/increasing time and instability in reverse backward/decreasing time. The reversibility or irreversibility of certain processes is thus determined by stability aspects with actual physical processes being represented by intial value problems which are stable in forward time, and irreversible physical procesess being represented by intial value problems which are unstable in backward time and thus not realizable as physical processes.<br /><br />The 2nd law of Thermodynamics as the law of irreversibility of certain physical processes thus can be based on stability which is a physical property, instead of probability which is unphysical and has ruined modern physics.<br /><br />It is more illuminating to give evidence of irreversibility as physicsl break-down or blow-up of unstable processes, than to say following Boltzmann that natural processes have a tendency to move from improbable to probable states (rather than the opposite), which is an empty tautological statement.<br /><br /><b>PS1</b> Concerning simultaneity, which is so important to Einstein, one may say that physics does not care about this concept, because in physics things happen locally and if two particles collide they do it at the same time. It would be ridiculous to report as a curious fact of a new physics that you met a person in a certain street corner at 1.00 pm once during a day according to your clock, while the person you met insisted that he/she met you once at 2.00 pm according to his/her clock the same day at the same street corner.<br /><br />This would not be accepted as evidence of new physics, only as evidence that at least one of the clocks was off time. Right? And yes, Einstein's physics is at most epistemology but not real physics, according to Lorentz.<br /><br /><b>PS2</b> Concerning time as change, one may identify time with motion as change of position, of the Earth, of the arms of a clock or with propagation of light. The basic question is then to answer how motion is possible as a solution of Zeno's Paradox of the arrow which at each moment is still and yet moves or changes position. A resolution is presented in <a href="http://claesjohnson.blogspot.it/search/label/Zeno%27s%20arrow%20paradox">posts on Zeno's paradox.</a><br /><br /><b>PS3 </b><a href="https://en.wikipedia.org/wiki/Fashionable_Nonsense">Bricmont and Sokal as Fashionable Nonsense</a> accuses Deleuze and Guattari of using mathematical language in imprecise way in philosophical texts, which may be more or less meaningful, but forgets that Einstein in a scientific physics text is even more imprecise which is not meaningful. </div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-27704159340417995592016-05-10T12:23:00.003+02:002016-05-13T14:39:01.068+02:00Time Again for Bergson vs Einstein<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-490LK3pTjaQ/VzRfonComhI/AAAAAAAA79c/zhXKHKTDvt8X-nX8LzjmAIjoue-uaTQcQCLcB/s1600/einstein-bergson-getty_custom-72e0b13f6e25c2fcbde960dadddad055e2bc45d9-s900-c85.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="266" src="https://4.bp.blogspot.com/-490LK3pTjaQ/VzRfonComhI/AAAAAAAA79c/zhXKHKTDvt8X-nX8LzjmAIjoue-uaTQcQCLcB/s400/einstein-bergson-getty_custom-72e0b13f6e25c2fcbde960dadddad055e2bc45d9-s900-c85.jpg" width="400" /></a></div> Who was more clever? Who was a better physicist? Who was a better philosopher?<br /><br />The debate at Societe Francaise de Philosophie in Paris on April 6 in 1922 between the German physicist Albert Einstein and the French philosopher Henri Bergson represents a critical moment deciding much of 20th century physics and philosophy. Bergson articulated strong criticism of Einstein's theory of relativity, in particular its new concepts of time dilation and time inseparable from space during a long presentation, and was met by only a short statement by Einstein that Bergson's philosophy time was not Einstein's physics time and so all Bergson's arguments could be dismissed without consideration.<br /><br />Einstein's tactics was necesserary because Bergson was very clever, much more clever than Einstein, and even better, it worked. Bergson was defeated, although his criticism was instrumental for the decision by the Nobel Committee to award the 1921 Nobel Prize to Einstein <b>explicitly not</b> for relativity theory but instead for "the discovery of the law" of photoelectricity and then <b>explicitly not</b> for Einstein's derivation of the law from an idea of light as a flow of particles. Evidently the Committee had a hard time finding a reason to avoid <b>not giving</b> Einstein the prize.<br /><br />So Einstein won the game to physics with tactics quickly adopted by the physics community which have been dominating the debate into our time: As physicist you have the priority whatever criticism is expressed by anyone outside the inner group of physicists, to simply dismiss the criticism as being based on misunderstanding of a basic question which was settled long ago.<br /><br />This is what happened when the Bergson-Einstein debate resurfaced in the 1990s in the form of an assault by Bricmont representing physics on postmodern philosophers including Deleuze again claiming exclusive priority to questions about space and time to physicists.<br /><br />The string physics Lubos Motl expresses this attitude on his blog in ultimate defense of quantum mechanics as settled once and for all and thus beyond any form of criticism by anybody except Lubos himself.<br /><br />All of this is very unfortunate, because Bergson was very intelligent and knowledgable and so was Deleuze and many other people outside the inner group of physicists, and to kill debate as Einstein did, always stops progress of science.<br /><br />The result today of lacking constructive debate is an extreme form of modern physics, which paradoxically is beyond the most speculative philosophy and parapsychology as multiversa and string theory way beyond any thinkable observation.<br /><br />It is clear that it is necessary for Bergson to restart the debate and thereby open for physics of the 21st Century which is connected to both realities and to human perception and understanding.<br /><br />The between Bergson and Einstein is described in detail in the recent book by <a href="http://press.princeton.edu/titles/10445.html">Jimena Canales The Physicist and the Philopsopher.</a><br /><br />Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-81694539103849153432016-05-08T17:32:00.000+02:002016-05-09T17:54:00.086+02:00Making Sense of Quantum Mechanics??Jean Bricmont starts his new book <a href="http://www.springer.com/gp/book/9783319258874">Making Sense of Quantum Mechanics</a> with:<br /><ul><li><i>This book is both apparently ambitious and modest in its aims. Ambitious, as it attempts to achieve something that has been declared impossible by some of the greatest physicists since the 1920s: making sense of what quantum mechanics really means. </i></li><li><i>But modest, because that goal was actually already attained many years ago in the work of Louis de Broglie, David Bohm, and John Bell. I will simply try to explain what they achieved.</i></li><li><i>It would seem that, given all the claims to the effect that such a theory is impossible, its mere existence should be a subject of considerable interest, but this is not the case. Although interest in the de Broglie–Bohm theory is probably increasing, it is still widely ignored or misrepresented, even by experts on foundations of quantum mechanics.</i></li><li><i>This book is written especially for all those students who feel that they have not understood the subject of quantum mechanics, not because they fail to master the mathematics or because they cannot do the exercises, but because they do not see what the theory means.</i></li></ul><div>The message is that still 100 years after its conception quantum mechanics is not understood, neither by the greatest physicists nor by students. Any theory about physics with these qualities should have been dismissed long ago, but this is not the case and Bricmont gives us the reason: </div><div><ul><li><i>Since its beginnings in 1900, the quantum theory has led to the most spectacularly well confirmed predictions ever made in science (some experimental results agree with the theoretical predictions up to one part in a billion), and it underpins all modern electronics and telecommunications. </i></li><li><i>It explains the stability of atoms and of stars, and lies at the foundation of the whole of particle physics, but also solid state physics, chemistry, and thus, in principle, biology. </i></li><li><i>It is truly our most fundamental theory of the world. Yet, to quote the famous American physicist Richard Feynman that “nobody understands quantum mechanics”.</i></li></ul><div>We read that no human being understands quantum theory, but nevertheless it is the most successful theory ever giving predictions which fit incredibly well with experiments.</div><div><br /></div><div>We understand that this can only mean that quantum theory somehow has been given to humanity as ready-made, not to understand but to use to make predictions for human needs, like a clock of a construction which cannot be understood given to us from Heaven, but yet always giving the exact time for our needs. The difference between religion and science is supposed to be that science can be understood by all educated or at least by some expert scientists, whereas religion is hidden to understanding for all people. The conclusion can only be that the quantum theory which Bricmont speaks about, is not science.</div><div><br /></div><div>To get out of this hopeless mess from scientific point of view, because science means to understand, it is necessary to go to the root of the trouble, which is to insist that quantum mechanics as atom physics must be based on a (i) linear (ii) multi-dimensional wave equation named Schrödinger's equation. This is an equation which by (i) allows unphysical superposition of states and which by (ii) cannot be solved for many electrons/atoms and thus can make real predictions only in very simple cases. </div><div> </div><div>But there may be a way out of the hopeless mess, and that is to start from a different form of Schrödinger's equation without (i) and (ii). This is explored in <a href="http://www.csc.kth.se/~cgjoh/ambsquantum.pdf">Many-Minds Quantum Mechanics.</a></div><div>Why not take a look, if you like Feynman and everybody else, do not understand quantum mechanics.<br /><br />The key step is to replace an uncomputable linear multi-dimensional unphysical form of Schrödinger's equation with a computable system in 3d physical space, and in this way eliminate the unfortunate unphysical aspects which has driven modern physics into meaningless scholastics of mystery and fantasy. </div><div><br /></div><div> </div></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-57673066434461401862016-04-27T09:57:00.000+02:002016-05-04T18:16:27.044+02:00Reformulation of Clay Navier-Stokes Problem Needed 5<a href="http://www.claymath.org/sites/default/files/navierstokes.pdf">Fefferman concludes the official formulation of the Clay Navier-Stokes Problem</a> with:<br /><ul><li><i>Let me end with a few words about the significance of the problems posed here. </i></li><li><i>Fluids are important and hard to understand....our understanding is at a very primitive level.</i></li><li><i>Standard methods from PDE appear inadequate to settle the problem. </i></li><li><i>Instead, we probably need some deep, new ideas.</i></li></ul><div>Yes, fluids are hard to understand for a pure mathematician and the understanding appears to be on a very primitive level, and this has led to an unfortunate formulation of the problem leading into a fruitless search for either (i) blowup into infinite fluid velocities in finite time as non-smoothness, or (ii) not blowup as smoothness of solutions.</div><div><br /></div><div>But for a fluid the distinction between smoothness and non-smoothness concerns the size of velocity gradients. It is a well-known fact since long that compressible flow may exhibit non-smoothness in the form of shocks with large velocity gradients but without large velocities.</div><div><br /></div><div>The Clay Problem concerns incompressible flow (at unbounded Reynolds numbers), which does not form shocks but instead becomes turbulent for large Reynolds number with again large velocity gradients as expression of non-smoothness, and (most likely) without large velocities. </div><div><br /></div><div>The clue to solve the Clay Problem offered in the official formulation by searching for infinite velocities in fluid flow, which Tao has picked up in recent attempts to solve the problem, thus appears to be misleading and as such is not helpful to mathematics as science. </div><div><br /></div><div>Fefferman is asking for some new ideas, but closes the door to any form of communication with computational turbulence as a new idea towards understanding and resolution of the problem. </div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-78847917109785076462016-04-26T11:17:00.002+02:002016-04-26T19:01:08.647+02:00Reformulation of the Clay Navier-Stokes Problem 4<a href="http://www.claymath.org/sites/default/files/navierstokes.pdf">The official formulation of the Clay Navier-Stokes Problem by Fefferman</a> includes the following statements (with (A) and (B) global existence+regularity and (1)-(3) Navier-Stokes equations):<br /><ul><li>For initial data $u^0(x)$ not assumed to be small, it is known that (A) and (B) hold (also for $\nu = 0$) if the time interval $[0,∞)$ is replaced by a small time interval $[0,T)$, with $T$ depending on the initial data. </li><li>For a given initial $u^0(x)$, the maximum allowable $T$ is called the “blowup time.” Either (A) and (B) hold, or else there is a smooth, divergence-free $u^0(x)$ for which (1), (2), (3) have a solution with a finite blowup time. </li><li>For the Navier–Stokes equations ($ν > 0$), if there is a solution with a finite blowup time $T$, then the velocity $u_i(x,t)),1≤i≤3$ becomes unbounded near the blowup time.</li></ul><div>We read that Fefferman claims that the distinction between (i) YES or (ii) NO to the question of existence+regularity for the Navier-Stokes equations, is between (i) bounded flow velocity for all time and (ii) unbounded velocity for some "blowup time" $T$.</div><div><br /></div><div>Fefferman here uses the same distinction as in the classical theory of ordinary differential equations (odes) based on a (correct) mathematical analysis showing that the only way a solution trajectory can cease to exist, is to tend to infinity in finite time. </div><div><br /></div><div>But this argument cannot be generalised to partial differential equations (pdes), because a smooth solution to a pde can cease to exist as a smooth solution because of unbounded derivatives of the solution, without the solution itself becoming infinite (as required in the ode case). </div><div><br /></div><div>The basic distinction for Navier-Stokes is instead between (i) laminar/smooth flow and (ii) turbulent/non-smooth for all time without blowup to infinity of the velocity, where non-smooth means large velocity gradients.</div><div><br /></div><div>The official formulation of the problem is unfortunate by (incorrectly) claiming that the question can be reduced to a question of infinite velocities at finite blowup time. The Clay problem thus needs to be reformulated, since an incorrectly formulated problem can only lead in a wrong direction.<br /><br /><a href="http://claymath.msri.org/navierstokes.mov">In a lecture about the problem, Cafarelli</a> falls in the trap of Fefferman. </div><div><br /></div><br /><div class="page" title="Page 3"><div class="layoutArea"><div class="column"></div></div></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-21635512127285160152016-04-25T18:47:00.001+02:002016-04-26T21:05:33.601+02:00Reformulation of Clay Navier-Stokes Problem Needed 3The official formulation of the Clay Navier-Stokes Problem does not include any reference to the Reynolds number $Re =\frac{UL}{\nu}$ with $U$ a typical flow speed, $L$ length scale, and $\nu$ viscosity scale, and thereby makes no distinction between laminar/smooth flow at small Reynolds numbers and turbulent/non-smooth flow at large Reynolds numbers.<br /><br />Since no bound on the Reynolds number is given, it can only mean that the Reynolds number can have any size, in particular be arbitrarily large. By normalizing $U$ and $L$ to unity, the viscosity thus can be arbitrarily small (or normalizing viscosity and length scale to unity and letting $U$ become large), which means that the Clay problem includes the incompressible Euler equations as the incompressible Navier-Stokes equations with vanishingly small viscosity.<br /><br />This is precisely what the book <a href="http://www.csc.kth.se/~jhoffman/pub/v4.pdf">Computational Turbulent Incompressible Flow</a> is about! As a basic example from the book, let us consider flow around a sphere under vanishing viscosity depicted in the following pictures:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-jlUJq1k5Uds/Vx47DxA669I/AAAAAAAA77o/zfVkrUyAqyAxDpTN4YXZIh4Ne4Q-QEctACLcB/s1600/drag-crisis-31.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="148" src="https://4.bp.blogspot.com/-jlUJq1k5Uds/Vx47DxA669I/AAAAAAAA77o/zfVkrUyAqyAxDpTN4YXZIh4Ne4Q-QEctACLcB/s200/drag-crisis-31.jpg" width="200" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-a-Gfb3zj4wk/Vx47GNhdY1I/AAAAAAAA77s/YpIOOYFppJoCPVpwHKWa7QrxCSeaSVWNgCLcB/s1600/drag-crisis-11.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="120" src="https://4.bp.blogspot.com/-a-Gfb3zj4wk/Vx47GNhdY1I/AAAAAAAA77s/YpIOOYFppJoCPVpwHKWa7QrxCSeaSVWNgCLcB/s200/drag-crisis-11.jpg" width="200" /></a></div>We see a distinct large-scale separation pattern developing consisting of 4 tubes of counter-rotating flow attaching to the rear of of the sphere, which are dissolved into turbulent flow further down-stream. We see that the length of the tubes increases with decreasing viscosity, which is consistent with Kolmogorov's conjecture that the total amount of turbulent dissipation stays roughly constant under vanishing viscosity (along with total drag), requiring the surfaces of intense dissipation of the 4 tube pattern to extend further downstream.<br /><br />We thus discover a vanishing viscosity solution to the incompressible Euler equations, which is fundamentally different from the formal exact solution in the form of potential flow, which is symmetric in the flow direction with symmetric attachment and separation without the 4tube gross pattern, a formal exact solution which is unstable at separation and thus not a limit of Navier-Stokes solutions.<br /><br />The official problem formulation does not fit well with this situation, since stability aspects are left out and the presence of vanishing viscosity solutions is hidden, and the Clay Navier-Stokes problem thus asks for a reformulation away from the deadlock of the present (meaningless) formulation.<br /><br /><br /><br />Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-63006669635102981112016-04-25T10:36:00.002+02:002016-04-25T12:28:39.678+02:00Reformulation of Clay Navier-Stokes Problem Needed 2The official formulation of the Clay Navier-Stokes Problem is unfortunate by not mentioning the Reynolds number $Re =\frac{UL}{\nu}$ with $U$ a typical flow speed, $L$ length scale, and $\nu$ viscosity scale, and thereby making no distinction between laminar/smooth flow at small Reynolds numbers and turbulent/non-smooth flow at large Reynolds numbers.<br /><br />Instead the problem is formulated so as lead people to seek a solution in the form of blow-up (or not blow-up) to infinity of flow speeds in finite time, following a simple methodology borrowed from ordinary differential equations saying that as long as a solution trajectory is finite it can be continued for some time. But this is not the real issue for a partial differential equation like Navier-Stokes, where the essential distinction is instead between laminar/smooth and turbulent/non-turbulent flow.<br /><br />The result is a problem formulation which is meaningless because it is both unphysical and unmathematical, and as such cannot be given a meaningful answer.<br /><br />The problem is owned by a small group of pure mathematicians including Fefferman and Tao, who refuse to participate in any form of discussion about the problem and its formulation. This is not in the interest of anybody outside this group and thus not in the general interest of mathematics as science, which must be the interest of mr Clay in particular...<br /><br /> Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-23271053843893754842016-04-21T20:05:00.000+02:002016-04-24T14:29:53.790+02:00Velocity Blow-up to Infinity for Incompressible Euler?In an effort to solve the Clay Navier-Stokes problem as formulated by Fefferman, <a href="https://terrytao.wordpress.com/2016/02/01/finite-time-blowup-for-an-euler-type-equation-in-vorticity-stream-form/">Terence Tao in recent work</a> seeks to construct a solution to the incompressible Euer equations with velocities becoming infinite in finite time, but does "not quite achieve" the goal.<br /><br />Let me present some evidence indicating that the goal cannot be achieved. To this end we compare the incompressible Euler equations:<br /><ul><li>$\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p =0$</li><li>$\nabla\cdot u=0$</li></ul><div>with (i) vector-Burgers as a model of very compressible flow:</div><div><ul><li>$\frac{\partial u}{\partial t}+u\cdot\nabla u=0$</li></ul><div>and (ii):</div></div><ul><li>$\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p =0$, </li><li>$\delta\Delta p=\nabla\cdot u$</li></ul><div>with $\delta >0$ a small constant, as a model of slightly compressible flow.<br /><br />For Burgers equation and so for (i), velocities may become discontinuous corresponding to the development of shocks over time, but velocities do not tend to infinity.<br /><br />In case (ii) solving for the pressure p gives the following equation along a streamline $x(t)$:<br /><ul><li>$\frac{du(x(t))}{dt} + \frac{1}{\delta}\nabla\Delta^{-1}\nabla\cdot u(x(t),t)=0$ </li></ul><div>which formally gives a bound on the possible growth of velocity in terms of $\frac{1}{\delta}$ preventing blow-up to infinity. </div><div><br /></div><div>We conclude that neither very compressible nor slightly compressible flow appears to accommodate blow-up to infinite velocity. Is it then the incompressibility which will squeeze the flow to infinite pressure driving flow velocity to infinity? Far-fetched in my view.<br /><br />On the other hand, we have strong evidence that Euler solutions become turbulent with substantial turbulent dissipation from large velocity gradients, while velocity does not spike to infinity. Again, the formulation of the Clay Navier-Stokes problem without reference to turbulence, appearently leads mathematicians into meaningless dead ends.</div><br /></div><div><ul></ul></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0