csmyth3025

Thanks for that additional input SH. It makes sense. My initial confusion originated because I was uncertain about the roles that accuracy (significant digits) and precision (the placement of the last significant digit relative to the decimal point) played in complex operations. I clearly understood that calculations involving only multiplication and division (or powers and roots), the number of significant digits initially presented determine the significant digits contained in the result (regardless of the placement of the decimal point). Conversely, I understood that in operations involving only addition and subtraction the precision of the initial value dictates the precision of the result. The textbook I cited doesn't explicitly state the precedence of accuracy over precision in complex operations involving multiplication, division, addition and subtraction. Naturally, the text clearly states that addition and subtraction of terms are the last operations to be performed. The exercises illustrate this precedence, though. The small difference between my result and the textbook's correct answer in one of these exercises led to my original question. Your suggestion that two or three guard digits should be retained in the interim results of complex operations (entered sequentially) is one that I'll remember. As your illustration shows, following that principle eliminates the discrepancy I originally encountered. Upon reflection, it seems only logical that the accuracy of the initial values (as approximate numbers) in complex calculations must dictate the accuracy of the final result. As you know, if such calculations are entered in a calculator as a continuous formula, the result will very probably be shown to 8 (or 10) significant digits. Chris

Thanks again. I think I'll try to follow the rule as I now understand it - unless I get information to the otherwise. Chris

Thanks. Let's see if I "get it", though. If the original grouped numbers had been different, for instance: (3872/64) - (2.056*309.6/395.2) , my interim result would be 60.5 (exactly) - 1.610672 and my unrounded result would be 58.889328. If I apply the rule that the least precise interim result should dictate the precision of the final result, I would have a rounded answer of 58.9. If, on the other hand, I apply the rule that the final result shouldn't be more accurate than the least accurate initial figure - the result should be rounded to two significant digits: 59 In the original problem, the initial values were all accurate to four significant figures - and the final answer was also taken to four significant digits. As I now understand it, the accuracy of the initial values (when powers, multiplication, division, addition and subtraction are all part of the calculation) dictate the accuracy of the final result. If a calculation involves only addition and subtraction of terms, the least precise initial term dictates the precision of the final result. Is this a fair description of how these rules are applied? Chris

In my dotage I've decided to revisit a textbook I still have from a class I took about 10 years ago. The book is Basic Technical Mathematics with Calculus (Seventh Edition) by Allyn J. Washington (Addison Wesley Longman, publisher). The specific problem is meant to illustrate the correct rounding techniques as they apply to practical math problems. These techniques concern accuracy (significant digits in multiplication and division) and precision (the decimal position of the last significant digit in addition and subtraction). The problem is presented as follows: (3872/503.1) - (2.056*309.6/395.2) I first evaluated each of the groupings using the principle that the individual results should be accurate to four significant digits. My interim result was thus: 7.696 - 1.611 I then performed the subtraction based on the premise that the precision of the result should match the least precise number in my interim result (thousandths): Answer = 6.085 The book provides an answer to this problem (without explanation) of 6.086 If I don't round off any of my results, the final answer I get (to seven significant digits) is 6.085611 Rounding this off to four significant digits will, of course, produce the 6.086 given answer. This leads me to believe that only the final result of this calculation should be rounded off. The complication in my mind is that the final answer is the result of the subtraction of one number from another (both numbers being, themselves, the product of a prior multiplication operation). I notice that all of the original values are given to four significant digits. Is this the criteria by which only the final answer should be rounded off? I realize that this produces a very minor difference in the final answer, but I'd like to be sure that I understand how these rules for rounding off should be applied. Chris

A reading of the below linked Wikipedia article might help to alleviate your headache and allow you to get some sleep: http://en.wikipedia....of_the_universe BTW - There are gases in space - they're just so rarefied that by our normal Earthly standards it seems to laymen like ourselves that there's "nothing there". Chris

Whatever it is, it's not mainstream science. Chris

Thanks for that explanation. I think I actually understand it. This should make it a bit easier for me to understand some of the other concepts in relativity theory that confuse me. Chris

I think you might be confusing the curvature of space-time predicted by General Relativity with the more philosophical question: "Does space exist?" I'm not in any way well educated about Einstein's General Theory of Relativity, so I hope other members here will correct any mis-statements I might make on the subject. General Relativity is usually considered by the general public as theory that explains the phenomenon of gravity as the curvature of spacetime by objects such as planets, stars, galaxies, etc. It is this, and much more. The following is presented as a reference and is not intended to confuse you. It's a simplified way of writing the Einstein Field Equations (EFE): (ref. http://en.wikipedia....thematical_form ) Note that the explanation given above says that what's on the left of the equals sign represents the curvature of spacetime and what's on the right side of the equals sign represents the matter/energy content. The term (8*pi) is a simple factor equal to about 25.13. The expression is the stress–energy–momentum tensor. Most people think that this stress–energy–momentum tensor includes energy and matter, but this isn't necessarily so. Remember that E=mc2, so mass (matter, as most people think of it) doesn't have to be present. This equation is perfectly valid if represents only energy. The result would be a universe in which no matter exists, but which has spacetime curvature that is perfectly described by the EFE. Such a universe was first postulated in 1917 by Willem de Sitter and is appropriately termed a de Sitter universe. I mention all of this because of your OP comment that: "...It seems that GR claims to know that space and time exist. Not only that it claims to know the nature of space with out providing any proof that space exists in the first place..." General Relativity doesn't "claim to know that space and time exist", it just assumes that it does - as we all do in everyday life. Nor does General Relativity claim to know "the nature of space". It predicts how space, time, energy and (by extension through the equivalence formula E=mc2) matter interact. Don't misunderstand the quote from Einstein that IM Egdall provided. "What fills space doesn't have to be matter. It can be - and in some models it is - just energy. The short answer to your question: "...What if metaphysicians come to the conclusion that space doesn't exist without objects -wouldn't that then invalidate GR?" would be "no". Chris

Can you provide a reference for these values? The most convenient source of information available to me is Wikipedia, which gives the following values: (ref. http://en.wikipedia.org/wiki/Earth ) More importantly, your subjecting your initial values to manipulations that seem entirely arbitrary. Regarding the relationship between the given value for the Astronomical Unit and Planck's constant, please explain what you feel is the logical motivation and significance of the mathematical treatment that you've applied to these numbers. On the surface it seems that you've arbitrarily selected formulas that will give you a predetermined result. You seem to feel that there's a deeper meaning to your results than sheer coincidence, so I have to assume there's a deeper meaning to your method of manipulating these numbers than just the desire to make them ultimately match. Chris

I don't think you understand what the thing that we refer to as the observable universe actually is. It is not the 13.7 billion year light travel time since the big bang. You may want to read up on this concept in the Wikipedia article here: http://en.wikipedia....rvable_universe Chris

(bold added by me) -and- These concepts are really hard for me. Do I understand correctly that the velocity of any object (or massless particle) along its spacetime world line always equals 299,792,458 km/s? (as measured by an observer on the object using his own "ruler" and his own clock - i.e., his own proper time) I think I must be missing something fundamental here. Chris Edited to add parenthetical clause to question

This sounds like a variation on the "luminiferous aether" theory originally proposed in the late 17th century and finally put to rest in the early 20th century. (ref. http://en.wikipedia....niferous_aether ) I have difficulty seeing how "stationary" light in a putative fourth dimension would impinge on objects in our three-dimensional world with the same measured velocity (in a vacuum) regardless of the motion of the emitting object or the observer. It was once thought that time was an entirely separate "dimension" from our three spatial dimensions and the the universe "moved" through time at the same rate for all observers. Einstein mixed this all together and concluded that the space and time that an observer measures (our current four dimensional world) can vary provided we hold the speed of light in a vacuum to be a constant value for all observers. Einstein's proviso was rooted in the equations of James Clerk Maxwell and the null results of the Michelson-Morely experiment. As far as I can tell, your proposal doesn't get us any closer to "understanding exactly what light is" and it doesn't explain our observations any more intuitively than does Einstein' relativity theory. You seek to explain the "constant-ness of the speed of light" by invoking an unknowable fourth dimension through which everything in the universe moves. At the very least your proposal begs the question: "Why does everything in the universe move through the fourth dimension at a constant speed (i.e., the speed of light)?" Chris

If my calculation is correct, the Earth travels 2,977,356.8 cm in one second. Are you saying that when comparing your manipulated Planck constant (having units of g*cm2/s) with the Earth's average orbital distance from the Sun (having units of cm), that the "/s" should be replaced with "/2,977,356.8 cm"? Chris Edited to include distance that the Earth travels in one second.

There are a lot of multiverse speculations - as far as I know, none of them rise to the level of a "legitimate theory". A valid theory must be testable in some way. You propose sending a probe "a very long distance and back". What do you consider "a very long distance" and how many millions of years are you willing to wait for it to return? (Andromeda - the closest spiral galaxy to the Milky Way - is about 2.5 million light years away) BTW - The current co-moving distance to the edge of our observable universe is about 46 billion light years. Chris

I certainly don't have the qualifications to critique this paper, but I immediately notice that Robert begins by manipulating Planck's constant [6.62606957(+/-29)x10-27 erg*s] and subsequently comparing the manipulated number to the sum of the distance measure AU plus 1/4 the distance light travels in one second. (ref. http://en.wikipedia...._constant#Value ) I may be showing my ignorance here, but I fail to see the signifcance of taking a natural physial constant with units of one gram centimeter-squared per second-squared times seconds (g·cm2/s2)(s)=(g*cm2/s), manipulating that number, and then comparing the result to the sum of an arbitrarily chosen distance measurement (the current average orbital distance between the Earth and the Sun) plus another arbitrarily chosen distance measurement (1/4 the distance that light travels in one second). I might add that the second, itself, is an arbitrarily chosen unit of time based originally on Egyptian and then Babylonian and finally Persian systems of measurement: (ref. http://en.wikipedia....chanical_clocks ) Other than coincidence and convenient manipulation, is there any significance to the comparison of these two numerical results? Dimensionally, I have no idea how to compare them. If Robert had used the Planck length [1.616252(81)×10−35metres] as a starting point I could have at least followed the the dimensional comparison a bit better

I have absolutely no idea how or why you would choose such a contorted train of "logic" to arrive at a value of pi for the speed of light. If you think that pi has some special signifcance in the working of the cosmos, why don't you just assign that value to the speed of light and be done with it? You would, of course, have to change our SI definition of the meter in order to accomodate your new definition. Instead of the meter being defined as "...the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second..." we could say that the meter is defined as "...the length of the path travelled by light in vacuum during a time interval of 1/314 159 265 of a second...". This would require us to change all of our textbooks, astronomical calculations, speed limit laws (and signs), etc by a factor of ~1.04792251 (the new meter would be slightly longer than the old meter). This seems like a lot of trouble to go to just to make a correction to what everyone already knows is just an arbitrary number we've given to the speed of light based on our previous arbitrary definition of the meter and our current arbitrary definition of the second. If you're curious about the "natural" value of such things you might want to read up on Planck units. The Wikipedia article on this can be found here: http://en.wikipedia....ki/Planck_units Chris

Try putting your paper back into its original pdf format and attaching it to your reply as I indicated in my post above, like so: chapter1.pdf This is a 494 kb document in pdf format that downloads in my computer with the title "Index.pdf". It's a 21 page document dealing mostly with special relativity that I downloaded to my computer many months ago from this site: http://www.eftaylor....ub/chapter1.pdf I'm not very savvy about computers, but I would think that if you can manage to attach your file to an email, you would also be able to attach it to your post here. Chris

At the bottom of your reply box (if you're using the "fast reply" box, use the "use full editor" option) there's a section entitled "attachments". Below that there's a blank box with a "browse" button next to it. Press the "browse" button and you're computer files will come up in a separate window. Double click on the file you want to attach to your reply. The file name will appear in the blank box. Click on the blue "attach this file" button below the box with the file name in it. At the top, over to the right in this same section two selections will appear: "add to post" and "delete". Click on the "add to post" selection and the file will be added to your post - like so: Musings%20-%20The%20Twins%20Paradox.doc The above file downloads to my computer as "Index.doc (99.0 kb)". It's a 12 page 2009 document created by Mark Egdall (not me - I'm not smart enough). It opens on my computer with OpenOffice.org Writer. It would probably be a good idea to include this type of information in your reply. Your computer then gives you the option to save this file (to your desktop, or wherever), or to open it. The program you use to open it will have to be able to recognize the format that it's in. If your file is very long you may have to split it up into several smaller files. I don't know how to do this, but I'm sure someone here can tell you how it's done if you don't already know. Chris Edited to specify use of full editor

I think I understand your type of operation and your motivation. Right now you're dumping a lot of rock down a cliff face from your mining source so that trucks at the base of the cliff can transport the material to your processing facility. The cliff is, essentially, a shortcut to avoid hauling the rock an additional 10 miles overland to reach the same ultimate destination. Your intention is to recycle some of the kinetic energy of the falling rock in order to generate electricity in place of (or to supplement) wind-generated power that might be proposed for your facility as a cost savings and/or source of revenue. I get the feeling that you would like some small assurance that if you propose this idea to your boss or to the company engineer, you won't be laughed out of the office or (at best) told "it won't work". Heavy duty bucket elevators are widely available through industrial machinery suppliers. Your idea is to use them to drive a generator as the rock goes down instead of using an electric motor to raise the rock up to a higher elevation. Gearing the elevator drive sprockets to match a suitable generator is a simple mechanical problem. Again, an industrial machinery supplier would probably have an off-the-shelf gear drive that would serve your purpose. With the proper engineering I believe this can be done and significant usable energy can be generated. The question is (first) whether your employer is willing to consider the idea , and (second) whether he's willing to spend the money to have an engineer look at the proposal. Only an engineer working on your specific installation can provide you with an estimate of the capital costs involved and the possible cost savings (electrical generation capacity) that can be expected. If your idea is shot down don't be discouraged. In many companies the folks in the front office are predisposed to dismiss suggestions from anyone who spends his workday in mud-smeared coveralls. Personally, I think your idea shows original thinking. I like it. I think it deserves a proper engineering analysis. Chris PS - Generally, you don't need to use your cap-lock or bold letters. In forums, typing in capital and bold letters is interpreted as yelling out your reply (the people you're replying to think you're yelling at them). Also, if you plan to use the forum very much it would be a good idea to figure out how to use the "quote" button at the top of your post box to separate what someone else has said from what your reply is. Edited to improve clarity

To expand on StringJunky's comment, the following may be helpful: (ref. http://en.wikipedia....tons_definition ) I could have just provided the link, but I really like the colors they used in the graphic. There's an old song entitled "Dust in the Wind" (Kansas, 1977) with the refrain "...all we are is dust in the wind..." Modern science tells us that we're actually about 1.5% "dust" and 98.5% energy. Kinda blows your mind, don't it? Chris

First, Welcome to Science Forums.net! Second, if you would like us to comment on your equations your best approach would be to present them here in your post. Most members are reluctant to email someone they don't know (or give someone they don't know their email address). Chris

On the subject of the Tolman-Oppenheimer-Volkoff limit there is a 1995 paper by I. Bombaci (Astronomy and Astrophysics, v.305, p.871) entitled "The Maximum Mass of a Neutron Star". This paper is in pdf format here: http://articles.adsa...8;filetype=.pdf Unfortunately, this paper is way over my head. If one of our more knowledgeable members can translate this paper into something that us mere mortal humans can understand, it might provide the mathematical resolution that EWyatt has been unable to find: In the meantime, I would refer EWyatt to the Wikipedia article on super-massive black holes and, particularly, the passage regarding mass densities contained within the event horizon: (ref. http://en.wikipedia....sive_black_hole ) My read on this is that the question isn't whether the density of a neutron star (of any size) is sufficient to form a black hole event horizon. The Wikipedia article indicates that the density of ordinary water is sufficient to do that if enough of it is piled up in one place. The question is how big can a neutron star get and still be a neutron star - which I believe is what the paper by Bombaci addresses. Chris This is a bit off-topic, but an interesting experiment for a highly advanced civilization with plenty of time and money on their hands would be for them to pick some out-of-the-way cosmic location and send cargo ships laden with pig iron - which they would eject upon their arrival and then go back to the galactic foundries to get more. At some point in time (millennia?) this huge molten ball of iron (molten from the kinetic energy of dropping off the pig iron in the course of their fly-by maneuvers) would get so big that it would collapse into a -- ? (white dwarf? or, neutron star?). Would this be a cataclysmic event or could it occur gradually (from the inside -->out)? This would be essentially the same as a core collapse of a star, but without the hydrogen, helium and other lighter elements layered on top of it. Would this collapse release energy? Chris

As mentioned by SH, the Tolman–Oppenheimer–Volkoff limit provides the mathematical resolution you're searching for: (ref. http://en.wikipedia....r-Volkoff_limit ) As mentioned in the quoted passage: "...most astrophysicists assume, in the absence of evidence to the contrary, that a neutron star above the limit collapses directly into a black hole..." As far as I know, there isn't any "...evidence to the contrary..." yet. Chris

I think Realitycheck was referring to the perception of an observer outside of the event horizon. As ajb commented, no one really knows what lies in the middle of a black hole. If it was possible to stand next to a black hole, you wouldn't feel any heat coming from it like you would standing next to a fireplace - no matter how "hot" it is inside the event horizon. Chris

Because of theoretical considerations, the "neutron star thing" has been rebuffed already: (ref. http://en.wikipedia....ki/Neutron_star ) Because of its event horizon, a black hole cannot emit thermal radiation. It is theorized, however, that it may produce Hawking radiation. The amount of radiation thus produced is inversely proportional to the mass of the black hole and, for stellar-mass black holes or larger, the "temperature" of the Hawking radiation is measured in billionths of a degree Kelvin or less: (ref. http://en.wikipedia....iation#Overview ) Chris