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But… the GR curvature tensor is fully calculable from the metric tensor, so does this mean that a metrical connection reduces the independent components of the 4 index curvature tensor to just 10?
So the curvature tensor has 20 independent elements whilst a solution of Einstein’s field equations determines the 10 independent elements of the Ricci tensor (a contraction of the 4 index curvature tensor first&last indices). This means that given a solution of the field equations the full curvature tensor is not uniquely deternined (pairs of Rijkl independent elements can be related to single or combinations of Rjk elements), does this mean anything physically?
One last question for you Dominic, then I’ll let the topic rest. Referring to the quote of yours highlighted by Paul above, don’t the Einstein A coefficients contain all the information regarding the relative strengths of lines?
Hi Paul, I’m glad the discussion revived your grey cells! If you are interested in understanding my work in more detail then you’re very welcome to contact me at my email address above. I suppose I should try to get it reviewed by submitting a paper to a journal, the trouble is that it falls between two stools i.e. to basic for professional journals and a bit maths heavy for amateur journals. I would have thought though that the approach would be good for teaching the basics of stellar spectroscopy at undergraduate level in a similar way to being taught the Bohr model of an atom.
Regards
KenHi Dominic, Amazing! I transferred to Alan Knapp’s group in 2002 (or thereabouts) in the twilight of my career with Philips, avoiding redundancy in the process. I eventually took redundancy and early retirement when what was left of the lab moved to Cambridge in 2008. A sad end to what was part of a prestigeous institution that was up there (almost) with Bell labs. I had many interesting projects and conference/business trips over the years since I joined, in 1973, what was then the Mullard Research Laboratory. My happiest time was through the 1980’s to early 90’s modelling silicon power devices, computing power was increasing according to Moore’s law and the models could therefore become more and more detailed so I’m very familiar with solving thousands of sparse profiled matrix equations. I still have the software to do this (if you’re interested). It can solve (in principle) via LU decomposition any number of equations in any number of dimensions to any level of “fill in” all the way up to a direct solver using various iteration methods developed in the 1980’s (conjugate gradients, bi-conjugate gradients etc). I posted some of my work on the “legacy page” of my website http://www.thewhightstuff.co.uk.
Back to spectroscopy: I think you are referring to the principle of detailed balance in your last reply and that is what I used to determine the Einstein B coefficients but they vary too strongly between the Balmer series lines to reproduce my measurements. They are also temperature independent whereas my measurements, over a number of stars, definitley show a temperature dependance hence the reasoning leading to my equation A.4.12. Another pleasing feature of my model is that all calculated internal parameters have believeable values, the impact parameter for pressure calculation is approximately 8 Bohr radii and photon cross-sections are of the order of 10 Bohr areas.
I suppose I am looking for someone who would look at my model and say “yes this is how a ball of gas in thermal equilibrium would look spectroscopically” or “no it isn’t because….”, whether it’s a good model of any particular star is a separate question though the Sun looks to be well modelled in it’s gross features. I realise this is a big imposition on anyone so if it is of interest to you please continue this discussion via my email address ken.whight@btinternet.com and if you still live in the South East or are attending Astrofest it would be great to meet up. if it’s not of interest then thank you very much for taking the trouble to comment.
Regards
Ken WhightHi Dominic, Thanks for your interesting reply. In my career within Philips research I learned that modelling silicon devices was difficult enough even with the luxury of having them on the bench infront of you! However assuming thermal equilibrium got you a long way. We did go to a “hydrodynamic” model where the silicon lattice, electrons and holes could be at different temperatures but I couldn’t claim that that was a major advance in terms of designing better devices. I also know that “Monte Carlo” methods were necessary for modelling GaAs devices but I never had to get into that thankfully!
Yes stellar photospheres are not in a global thermal equilibrium but I was hoping that estimates of photosphere pressure and thickness would be “reasonable” for a fair number of stellar types (my estimates for the Sun seemed OK – see the attached file and others under my “Spectral Line Modelling topic). Having developed a global equilibrium model it could fairly easily be extended to a local equilibrium model (but not by me!) to cover more (main sequence?) stars.
However, the particular “insight” that I wanted to discuss was that whilst detailed balance under thermal equilibrium allows you to calculate the Einstein B coefficients in terms of a cross-section area (if you express the Planck function as a photon flux). These cross-sections infuriatingly only apply in the monochromatic case. In the polychromatic case they proved useless as each atom now has a choice of photon to capture. I think this is what necessitated the additional calculation of “oscillator strengths”. However if you use the properties of thermal equilibrium you can relate the polychromatic capture cross-sections simply to the Einstein A coefficients of any elemental spectral series as demonstrated in my attached paper Europa.pdf (equation A.4.12) and I’m hoping to test this out on the Sodium principle series for the Sun if I can get the necessary data, sadly my experimental setup is somewhat moribund (offers of data anyone?).Sorry if my post was a little short of information. My request is directly linked to my other topic raised in this spectroscopy section i.e. Spectral Line Modelling.
When I got interested in astronomical spectroscopy I decided to start a project to see what information I could extract from the spectra I was collecting. I was expecting to find, in the literature, a ready made simple global thermodynamic equilibrium analysis but couldn’t find one and so started to develop one myself.
It was a much more difficult task than I anticipated (I am more of a solid state physicist/mathematician/numerical modeller) and my project took about 10 (often frustrating!) years to complete but in the end I was quite successful (gaining interesting insights on the way) at determining the pressure and thickness of the Sun’s photosphere and made predictions for other stars.
Trouble is my model uses thermal equilibrium photon capture cross-sections, calculated from thermal equilibrium functions (Boltzmann & Planck) and Einstein A coefficients rather than conventional “oscillator strengths” which seem to require an additional somewhat mysterious quantum mechanical calculation.
I was hoping to meet someone who was familiar with the conventional approach to discuss how the two approaches fit together.
My sources were:-
“Astronomical Spectroscopy for Amateurs” Ken M. Harrison, Springer
“Spectroscopy: The Key to the Stars” Keith Robinson, Springer
“Atomic Astrophysics and Spectroscopy” Anil K. Pradhan and Sultana N. Nahar, Cambridge University PressI know I posted that I thought this topic had most probably gone as far as it could but, I have another point to make to support the idea that the universe might resemble a converging numerical simulation:-
In the Copenhagan interpretation of Quantum Mechanics the act of observation by an external observer causes the wavefunction of the system being observed to collapse to one of the allowed states with a probability that is calculable. The problem is that when applying QM to the universe as a whole there are no extermal “observers” so we are forced to the multiverse where everything that can happen does (but why am I not in that universe where I won the lottery the one time that I actually did buy a ticket?). In the paradigm I am suggesting, the observer and the system evolve together to the tick of the external clock so one universe one result with order evolving out of chaos like a growing snowflake.I realised that R2301, R1302 and R1203 are linked by a Bianchi identity such that we know R2301+R1203=R1302 so the total reduces to 20 independent elements QED.
I have many books including MTW, if only the knowledge contained could be fed directly into the brain as in the Matrix movies! Thanks for the suggestions, I have a few more questions to ask but will start new topics as I think this one has got as far as it can.
Regarding a “superintelligence” I think everyone must make up their own mind but the quote that has most impressed me went something like “If a god created the universe for mankind then it’s an awful waste of space”. So I’m adopting a wait and see approach and hopefully I’ll be waiting for a long time yet!I agree with you Richard, thank you Paul and Paul for your replies, I just have to try and fully understand them now!
I had an enjoyable career working for Philips Research from 1973 to 2008, mainly in silicon device physics (to 1993) during the time when computing power was rapidly expanding. However, I have always had an interest in the more exotic fields of Cosmology and Quantum/Particle Physics and over the years have tried to
improve my knowledge in these areas. So now that I have completed my stellar spectroscopy project (BAA Forum topic “Spectral Line Modelling – please would somebody review the work) I think my winter project will be to try once again to improve my cosmological knowledge.In the past I have tried to work my way through quite a few books on GR (getting a little further each time) more recently I have worked through the “Theoretical Minimum” books by Lenoard Susskind and got to chapter 10 (before running out of steam!) of “Quantum Field Theory for the gifted amateur” by Tom Lancaster & Stephen j. Blundell, I have also bought “Covariant Physics” by Moataz H. Emam. It’s difficult working alone so would anybody like to join me in studying these books? Are there better books?
One last comment on this topic and I appologise if it sounds a bit mystical, but I don’t think we will get a “theory of everything” until we can fit “free will” or at least the illusion of free will into the picture something that I thought might be possible with my suggestion of a variable Planck “Constant”.
Thanks for all the comments. I am aware (but don’t fully understand) that experiments to test Bell’s in-equalities prove that there is no hidden variable classical interpretation of QM but I was hoping that by “widening the stage” some of the existing certainties might be circumvented.
Classical mechanics is deterministic and the whole of history is pre-determined so that our existence is like being in a completed movie and there is no reason that we should be conscious of any particular time.
Quantum Mechanics is probabalistic and history is built by random interaction outcomes as time flows and that history, once written, is fixed.
In the world view I am suggesting our past is still evolving to the tick of the external “computer clock” and our future is also developing to this clock with everything driven ultimately by “boundary conditions”. Our conscious existance is driven through a spacelike 4D world (there may well be more dimensions to account for all the “forces”) and what we call history exists only in our memories.
p.s. the Tessa paper and an second paper applying the large signal solution methos to electronic circuits is available on the legacy page of my website http://www.thewhightstuff.co.uk.
Hi Adam, I thought I had attached a reference to my last post but it seems to have failed as it’s 4.7MB. I’ve attaced zipped version that comes in at just under the limit.
Regards
KenHi David, I wouldn’t call my proposal a Theory as I have no mathematics to directly support it. I was just trying to find a way to understand how QM and GR could be made to fit together given that the current fashion of assuming GR must be quantised seems to be stalling or at least leading to some pretty wild cosmologies e.g. Multiverse. Perhaps the reverse option should be considered i.e. QM arises from our limited view of a wider more classical universe.
I attach a paper in which a physical problem (semiconductor device simulation) was numerically solved in the type of “space” I am imagining, though the cylindrical topology may not be required (ignore the wayward points in figure 4 and beyond, a faulty mobility model implementation was to blame). Currents are flowing at all points in the space and I would guess in this particular case that convergence to a solution of given accuracy would have been fairly chaotic throughout the space (I never looked at the convergence behaviour). If this was going to be a suitable paradigm to model our universe then the physics would need to be such that the convergence was more structured, like a crystal seeding and growing from a melt.
Sorry that’s all I have but I think there are avenues to explore to explain observations like entanglement, dark energy and dark matter as I set out in my previous attachment so the proposal should be falsifiable.
Regards
KenStudy of Vega/Sirius updated. If anyone wants to play with my software it can be downloaded from my website (www.thewhightstuff.co.uk) and I am happy to guide people throght it.
