From: jthorn@mach.thp.univie.ac.at (Jonathan Thornburg) Subject: Re: Global Position System and general relativity Date: 9 Feb 2000 11:39:38 -0600 Newsgroups: sci.physics.research Summary: Sagnac effect = clock synchronization in a rotating reference frame Keywords: general relativity, special relativity, GR, SR, In article <86srm3$4mh$1@mach.thp.univie.ac.at>, I (Jonathan Thornburg ) wrote: = * the Global Positioning System satellites = [If the biggest general relativity effects there were = ignored in the computer models, GPS positions would drift = by on the order of 10 kilometers/day. The fact that you = can actually measure GPS positions to 10s-of-centimeters = accuracy, implies that the general relativity corrections = built into the GPS computer software, must be *very* = accurate.] In article <87ocvc$b81@charity.ucr.edu>, John Baez asked > 10 kilometers per day? That's a lot! What general relativity > effect would cause positions to drift that much - and how? There are numerous important relativistic effects (both special and general) in GPS, of which the largest are probably: (1) The Sagnac effect due to the Earth's rotation (Einstein clock synchronization isn't consistent in a rotating reference frame). Given the Earth's rotation rate, the size of this effect is 207.4 ns (62 meters of light-travel-time) for a single circumnavigation of the Earth. I'm not sure if the relevant time scale for this is once a day, or once every 16-18 hours (GPS-satellite orbital period), or something else. (2) The fact that we're sitting down in the Earth's gravitational potential well contributes a fractional gravitational red/blueshift of about 6.95e-10 relative to infinity (that's Earth-relative infinity, i.e. we're neglecting gravitational potential wells from Sun, galaxy, etc; since it's only the *difference* in those wells' depth between GPS-satellite orbits and the Earth's surface that counts, these effects are tiny). This is 60.1 microseconds/day = 18.0 kilometers/day position drift. (3) The Earth's J_2 and the 2nd order Doppler shift from the Earth's surface rotation velocity are about 3 and 2 orders of magnitude down (respectively) from (2). (4) The GPS clocks are themselves somewhat down in the Earth's gravitational potential well. Moreover, they're also orbiting the Earth at high speed, which contributes a 2nd order Doppler shift. The sum of these two effects contributes a fractional gravitational blue/redshift of 2.50e-10 relative to infinity = 21 microseconds/day = 6.5 kilometers/day position drift. Effects (2) and (4) have opposite signs, their net algebraic sum is a fractional red/blueshift of -4.465e-10 = 38 microseconds/day = 11.6 kilometers/day at the speed of light. Quoting from the 2nd reference given below, The negative sign [[in the net sum of (2) and (4)]] means that the standard clock in orbit is beating too fast, primarily because its frequency is gravitationally blueshifted [when observed from the Earth's surface]. In order for the satellite clock to appear to an observer on the geoid [[= the Earth's surface, roughly speaking]] to beat at the chosen frequency of 10.23 MHz, the satellite clocks are adjusted lower in frequency so that they proper frequency is [1 - 4.465e-10] * 10.23 MHz = 10.229 999 995 43 MHz This adjustment is accomplished on the ground before the clock is placed in orbit. This reference goes on to discuss other interesting (smaller) GR effects in GPS, and some of the controversies about the physical reality of these effects that have erupted between relativists and the less-knowledgable-in-GR personnel in the US Air Force and its contractors who operate GPS. [Despite its many civilian uses, GPS is funded and operated for the primary benefit of the US military. There have been occasional talk of GPS shutdowns over (say) countries with which the US government wasn't on friendly terms (eg Iraq during the 1992 war), but I don't know if these were ever implemented.] So far the relativists (and Einstein) seem to have been right every time... References (both are fascinating reading, I highly recommended them): "General relativity in the global positioning system" Neil Ashby in Matters of Gravity #9, online at http://vishnu.nirvana.phys.psu.edu/mog/mog9/node9.html @incollection { Ashby-in-GR15, author = "Neil Ashby", title = "Relativistic Effects in the Global Positioning System", pages = "231--258", editor = "Naresh Dadhich and Jayant Narlikar", booktitle = "Gravitation and Relativity: At the Turn of the Millennium", booksubtitle = "Proceedings of the GR-15 Conference held at IUCAA, Pune, India, during December 16--21, 1997", publisher = "Inter-University Center for Astronomy and Astrophysics", address = "Post Bag 4, Ganeshkhind, Pune 411 007, India", year = 1998, isbn = "81-900378-3-8", } -- -- Jonathan Thornburg http://www.thp.univie.ac.at/~jthorn/home.html Universitaet Wien (Vienna, Austria) / Institut fuer Theoretische Physik "Stocks are now at what looks like a permanent high plateau" -- noted economist Irving Fisher, 2 weeks before the 1929 stock market crash ============================================================================== From: jthorn@mach.thp.univie.ac.at (Jonathan Thornburg) Subject: Re: State of the art in numerical simulation Date: Thu, 9 Mar 2000 04:20:52 GMT Newsgroups: sci.physics.research Summary: current practice in numerical (general) relativity Keywords: general relativity, numerical relativity, In article <38C379A7.4979008F@esat.kuleuven.ac.be>, Koen Delaere asked KD? Could you tell me what is the state of the art in theoretical and KD? numerical relativity research these days? In article , Chris Hillman replied CH: for numerical solutions, see first the excellent expository article on CH: the Cauchy problem by Friedrich and Rendall: CH: CH: http://xxx.lanl.gov/abs/gr-qc/0002074 CH: CH: and then see this article by Bruegmann: CH: CH: http://xxx.lanl.gov/abs/gr-qc/9912009 I also highly recommend Bruegmann's article as a general review of the state-of-the-art in numerical relativity. CH: After reading these, you'll know that the two big problems in actually CH: carrying out the original "ADM" approach to numerically integrating the CH: EFE come down to: More accurately, "two of the big problems" -- there are several other "big problems" as well, and many of us don't think that these two are in fact the biggest problems. CH: 1. finding a suitable set of initial data CH: CH: (\Sigma, h_(ab), K_(ab), rho, j_a) CH: CH: where (\Sigma, h_(ab) is a Riemannian three-manifold, K_(ab) is CH: a symmetric tensor (will be the extrinsic curvature), rho is the CH: mass-energy density and j_a is the momentum density covector, In my opinion, this problem is fairly well under control: several reasonably-good solutions are known. (Of course, research is continuing on better solutions.) See Bruegmann's review for details. CH: 2. finding a suitable method of "gauge fixing" during the evolution, one CH: which avoids "kinking", extremely inhomegenous advance of "time", and CH: which avoids curvature singularities. Yes, this is a significant problem. A number of methods have been tried, some of which seem to work fairly well, but many people (myself included) are (still) actively researching better methods. [[...]] CH: With respect to 2, there is a large (oldish) literature on gauge fixing in CH: the ADM formalism (see the appendix in the gtr textbook by Wald for a nice CH: explanation of the York formalism), but it turns out that the ADM CH: formalism and its close kin are for various reasons not very suitable for CH: simulations of phenomena involving curvature singularities (i.e., all the CH: interesting stuff!). So, be sure to see the article by Friedrich and CH: Rendall cited above for important newer formalisms which circumvent many CH: of the problems which, in practice, mitigate against the practical utility CH: of the ADM formalism. Many numerical relativists (myself included) would strongly disagree with this (negative) assessment of the ADM approach. In my opinion, the ADM approach has a great deal of "practical utility" for studying (say) dynamic black hole (BH) spacetimes (which of course contain singularities within the black holes). Some of the literature on coordinate choice ("gauge fixing") is "oldish"... but some if it is rather recent, too. Many numerical relativists (myself included) have used the ADM approach successfully to study (among others) dynamic BH spacetimes. Notably, most of the work on simulating the decay/coalescence of BH binaries has used the ADM approach, and there's a lot of current work continuing to use it. I think the "important newer formalisms" Friedrich and Rednall describe are interesting ideas worthy of further investigation, but at this point it remains unproven how well they will work in practice. It's certainly not in any way clear that they will prove preferable to the ADM approach; I think it likely the two approaches will coexist (and coexist with still other approaches, i.e. the various 2+2 null-based approaches) in the field for a long time, i.e. different researchers will prefer different techniques. Koen Delaere also asked KD? Are there a lot of numerical simulations going on KD? about these things? Or is the numerical search impeded by slow computers KD? and the lack of numerical experts? Both statements are true: Yes, there are a lot of numerical simulations being done. Yes, there is a lot of current research on better ways to do these simulations. But yes, this is a small field (counting grad students, there are at most only a few hundred researchers worldwide), so progress could surely be faster if there were more people working, including "numerical experts". As to slow computers: I've been doing numerical relativity for a bit over 15 years now, and I don't remember ever hearing a numerical relativist complain that her/his computers were too fast or had too much memory or disk space. Today's supercomputers are easily adequate for (say) the simulation of the sparaling decal/coalescence of BH binaries (a.k.a. "the 2BH problem"), but not all researchers (especially grad students) have easy access to these, and today's desktop systems aren't really adequate, in any of cpu speed, memory size, or disk space. To give a (very) rough idea of the scale of problems: To test a new code I'm working on, last week I did a 3-spatial-dimensions x time ADM numerical evolution of Kerr-spacetime initial data for a period of t=20m. Using a rather low-resolution grid covering only 1/8 of the neighbourhood of the BH outside the horizon, this took about 145 MB of memory, and ran for a bit over 2 cpu days on a Pentium 333. Of course much faster computer systems exist, and are rather common, but it's often more convenient to use a local "slow system" than a distant "fast system". (Long-haul internet access is often painfully slow, in particularly trying to visualize monster datasets over a long-haul network connection is often impractically slow.) But having said all this, my opinion is that today, slow computers aren't the biggest limiting factor in (say) the 2BH problem. That is, even given much bigger/faster computers, today we still don't know how to fully solve the 2BH problem. Things are getting better, though, and I expect the first published results (asymptotic h_+ and h_x waveforms) for the radiated gravitational radiation in an equal-mass 2BH binary coalescence, in about 3+/-2 years from now. Koen Delaere also asked KD? And on a side note: does there still exist software dedicated to gtr KD? calculations or are all you guys just plugging away in maple (maybe with KD? dedicated toolboxes within the common mathematical software)? All of these are common. Many people use "tensor packages" for Macsyma/Maple/Mathematica. (My impression is that at least in the numerical relativity community, M/M/M are much more common nowdays than specialized-for-relativity systems like Sheep.) Many people also take the result of symbolic manipulations and have have the symbolic algebra system (M/M/M) generate Fortran/C/C++/etc code for the number-crunching. -- -- Jonathan Thornburg http://www.thp.univie.ac.at/~jthorn/home.html Universitaet Wien (Vienna, Austria) / Institut fuer Theoretische Physik "Washing one's hands of the conflict between the powerful and the powerless means to side with the powerful, not to be neutral." - Freire / OXFAM