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  1. Locally the equation I gave is a reasonable approximation, I used it to show the sheer triviality of the fact that the constancy of "c" is a geometrical hypothesis. The standard form that you have given is correct generally. The point I was making is that "c" is a ratio of two, dimensional extents, ie: meters per second. Null geodesics given from 0 = gab dxadxb = g'ab dx'adx'b are geometrical entities that describe the path of light rays and express the geometrical nature of "c". Why is the speed of light constant? Because the universe is a four dimensional, pseudo Riemannian manifold where, in the general case, 0 = gab dxadxb = g'ab dx'adx'b gives the path taken by light photons. The discussion about whether metric tensors are sewing together local Minkowskian spaces and coordinate systems or not etc. is non-sequitur to the essential point that the constancy of the speed of light is a geometrical phenomenon due to the nature of spacetime.
  2. juanrga: "Baez is right. The speed of light in general relativity is a constant." pmb: "The speed of light is locally c." granpa: "the speed of light is constant when measured locally." etc.. So c = dr/dt dr = cdt 0 = dr^2 - (cdt)^2 0 = dx^2 + dy^2 + dz^2 - (cdt)^2 So the constant speed of light implies that Lorentz Invariance exists (in the places that the posters claim c is constant). This was my original point, that c is due to the geometrical form of the universe. Part of the answer to the question "Why does light travel at the Speed of Light?".
  3. It only makes sense to talk about the "speed of light in a vacuum" and as a ratio of small or infinitessimal intervals (delta X/delta Y). Given this definition of "speed" I have also been saying that the speed of light is always constant in General Relativity. I have also been saying that the speed of light is constant in general relativity and have described how this might be expected from general or local Lorentz Invariance. Well, its been nice talking with you, I do not really have any more to add.
  4. First, the quote from the link addresses both Inertial Frames and spacetime geometry: "the constancy of the speed of light in inertial frames is a tautology from the geometry of spacetime." This was why I introduced the quote, because it supports my statement that Lorentz Invariance underlies the constancy of the speed of light. Lorentz Invariance means that null vectors are seen as null by all observers and hence there is a constant speed 'c'. Lorentz invariance is a feature of four dimensional spacetime with a signature of three positive and one negative dimension (or vice versa). It is a geometrical invariance and not dependent upon the presence of an inertial frame of reference. In other words the geometry of spacetime specifies the constancy of the speed of light, as it says in the quote. As for whether an Inertial Frame is required for the speed of light to be constant, which seems to be your contention, I can only point out that the geometry is present whether or not an inertial frame is under consideration and invariant null vectors are a property of the geometry. Second, the full quote from the link is: "A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Since Einstein talks of velocity (a vector quantity: speed with direction) rather than speed alone, it is not clear that he meant the speed will change, but the reference to special relativity suggests that he did mean so. This interpretation is perfectly valid and makes good physical sense, but a more modern interpretation is that the speed of light is constant in general relativity." We both agree that the speed of light is constant in General Relativity. The quote above is consistent with what I said above, eg: racing cars etc, the speed of light is constant in General Relativity when taken over a short period. Curvature means that the average speed between two points will not be constant over longer intervals because the light will not go in a straight line. We both know this. I cannot see how this quote resolves your contention that Lorentz Invariance does not underlie the constancy of the speed of light. In each short interval events are governed by the tangent Minkowski Space. The net effect is that along a small element of the path of a light ray the speed is always 'c'.
  5. The speed of light is constant in General relativity in the same way as the speed of a racing car can be constant on a circular race course and this is the result of the light following null geodesics (the path at right angles to null vectors in a 4D manifold). To quote from the "speed of light" FAQ on John Baez's website: "If general relativity is correct, then the constancy of the speed of light in inertial frames is a tautology from the geometry of spacetime. The causal structure of the universe is determined by the geometry of "null vectors". Travelling at the speed c means following world-lines tangent to these null vectors. The use of c as a conversion between units of metres and seconds, as in the SI definition of the metre, is fully justified on theoretical grounds as well as practical terms, because c is not merely the speed of light, it is a fundamental feature of the geometry of spacetime." http://math.ucr.edu/...d_of_light.html The velocity of light and the speed of light over an extended interval are not constant in General Relativity but the speed over an infinitessimal interval is constant and found to be so by all observers. The essential feature of General Relativity is that the tangent space to any point in spacetime is a 4 dimensional Minkowski Space ie: locally spacetime is Lorentz Invariant. This can be imagined as there being at any point a local spacetime that is flat and can be described by Special Relativity. As the quote says, the "causal structure of of the universe is determined by the geometry of 'null vectors'" and the null vectors are where, to a very close approximation: 0 = dx^2 + dy^2 + dz^2 - (cdt)^2 As it points out in the chapter on the book on Special Relativity referenced above (http://en.wikibooks....cial_relativity ) Lorentz Invariance means that every observer observes a null vector to be a null vector whatever their own velocity. As a result there is a universal constant 'c' (see the link to the book for the simple maths of the derivation of 'c').'c' is the velocity at which a null vector occurs and this velocity is the same for all observers - a tautology of dynamics and geometry. Whether this constant velocity is "the speed of light in a vacuum" is another problem. Your point that "But 'in the limit as an inertial reference frame (IRF) approaches the velocity of a photon' the frame ceases to be localizable and cannot be a reliable reference frame (you cannot define/measure distances in it)." is the same as my proposal. At very nearly the speed of light an observer might be able to measure a few Planck lengths in a "stationary" frame of reference but at higher velocities there is no meaningful separation. I agree that a photon is not an Inertial Reference Frame but we are discussing the geometry of spacetime, a four dimensional manifold with three dimensions of one sign and one dimension of another sign, this manifold has paths within it, tangents to null vectors, that can be considered as direct connections from one place and time to another. "Direct" in the sense that anything following the path of a photon takes no time along that path and the path has no length. Either this extrapolation of the geometrical properties of spacetime is correct or the photon is "outside" of spacetime in some unknown way.
  6. In General Relativity light travels along null geodesics (See http://en.wikipedia....l_relativity%29 ) which embodies a very closely related concept to the idea of light travelling along paths specified by a zero space-time interval. "Lorentz Invariance does not apply globally within the General Theory of Relativity because such invariance expresses certain relations between global orthogonal rectilinear coordinate systems, and within the General Theory of Relativity (and in most non-Euclidean spaces) there do not exist such coordinate systems (because under General Relativity mass distorts the flatness of space). Local Lorentz Invariance does however apply." Invariances: the structure of the objective world. By Robert Nozick This local Lorentz Invariance occurs at the point where the photon is located and represents the same constraint that occurs in Special Relativity and gives rise to the existence of a particular constant velocity for all local observers no matter how fast they are moving ©. General Relativity does not undermine the insights available from Lorentz Invariance, it builds on them. Your point about there being no such thing as the frame of reference of a photon is well taken and might be replaced by "in the limit as an inertial reference frame (IRF) approaches the velocity of a photon" there is no separation between an emitter and an absorber of electromagnetic radiation in the direction of travel of the IRF in flat space-time. Still a very interesting point given that almost all EM interactions that we observe occur in flat spacetime. Consider also that we can never observe a photon, what we observe is an interaction that can be explained by the concept of the transfer of a quantum of energy. What we consider to be a host of little particles in flight could also be regarded as a resonance between an emitter and an absorber. Imagine the IRF discussed above, the one moving at .9999... c in the direction taken by a photon, it might observe the interaction between the photon emitter and absorber as the direct resonant contact between emitting and absorbing outer shell electrons. (The space between these electrons having been contracted to very, very nearly zero). You have made me amend the overly broad position that I took earlier but the substance of my points still stands: that there is a constant velocity 'c' (amended to a constant velocity for all observers in flat spacetime and for local observers in curved spacetime) and that EM interactions can be regarded, in flat spacetime at least, as direct interactions between emitters and absorbers with no room left for greater or lesser velocities.
  7. There are two issues here. The first is "why is the speed of light a constant?" and the second is "why do photons travel at this constant velocity?". The answer to the first question is given in plenty of detail at: http://en.wikibooks.org/wiki/Special_Relativity/Spacetime#The_modern_approach_to_special_relativity and is the result of the existence of Lorentz Invariance. The answer to the second question is more interesting nowadays. In the frame of reference of a photon there is no separation between its source and its destination and the transit takes no time. For a photon emitters and absorbers are in contact...
  8. Your introduction has two strands, the first is "what is time?" and the second is a series of speculations about supraluminal travel. The relativistic notion of time is covered brilliantly in the Wikibook called "Special Relativity". The bit on the nature of time begins at link: http://en.wikibooks.org/wiki/Special_Relativity/Spacetime#The_modern_approach_to_special_relativity As for time and light being closely interrelated, this relationship comes from the postulate that the space-time interval is invariant between observers - look at the link above to understand this postulate. This book also has a section on travelling faster than the speed of light, the link is at: http://en.wikibooks.org/wiki/Special_Relativity/Faster_than_light_signals,_causality_and_Special_Relativity It points out that if travel at faster than light speed is possible then either Special Relativity is true or Causality is true but not both. Curiously Causality is less well founded in observation and theory than Special Relativity so if we ever did find a signal travelling at more than light speed my money would be on a violation of Causality... So what is time? Relativity tells us that it fulfils the criterion for being a dimension and common sense tells us that it is involved in change and causality. My guess is that the "time of change" is highly correlated with the dimensional time of relativity but is not the same physical phenomenon.
  9. At 0.5c you are within about 15% of Newtonian approximations so I do not see the relevance of the relativistic calculation above. To travel at 0.5c you need to throw out sufficient mass behind the rocket to give your 1 tonne spaceship 1.5x10^8 m/s velocity. The rocket equation does not really apply here because Relativity comes to your rescue. You can increase the momentum of a small amount of matter, your fuel, to almost any value by accelerating it to near the speed of light. Your only problem will be obtaining enough energy to do the trick. Energy is conserved so you will need at least 10^3x10^16 ie: 10 million trillion joules to get up to speed. This would take the total matter-energy conversion of about 100 kilos. Not impossible but tricky .
  10. I have been daft enough to try this. I used a computer program that used a mixture of statistical extrapolation and learning to predict prices on the Derivatives Market for several months (derivatives because you need to bet on falls and rises). It shadowed the real market and made £2 profit over the period having gambled up to £1000 per transaction many times daily. The program dealt entirely with short term changes in prices (one day or less). Long term changes are due to factors outside of the data available from It would have made a very substantial profit except for the commission charged on each transaction which almost exactly equalled the profit. Rule1: The banks who run the derivatives market are already running software that predicts the market and take any possibility of mechanical profit as commission. The market is really and truly being run as a casino. Rule 2: The banks (who run the market) remove any profit from consistent, predictable changes as commission and so all that remains are random fluctuations. So much for the short term. The longer term depends on data sources and the reliability of these sources. Reliability is the prime problem. The bank, HBOS, publicly announced that they had no exposure to sub-prime mortgage lending. Lloyds TSB truly had no exposure. So, when banking shares fell through the floor your software would buy HBOS and Lloyds TSB shares. It turned out that the directors of HBOS were lying. HBOS collapses. The directors of Lloyds TSB, egged on by the government, make a terrible error of judgement and merge with HBOS. The HBOS directors were criminals, they had already committed one fraud (lying to the public) and now committed another (lying to Lloyds). Both HBOS and Lloyds went into long term decline. (The directors of Halifax Bank Of Scotland got off Scot free despite participating a a multibillion pound fraud). Your software loses.
  11. The question asks: "Is it unscientific to believe that theories should be expressed in a language that is consistent with other theories as well as providing solid maths?" The question at the root of this thread seems to have mixed maths up with science. The greatest achievement of "science" was the discovery of germs and the theory that hygiene would greatly reduce disease. Without this discovery the life expectancy in cities would be about 45. Where is the maths in that? Most science is worded in everyday language and goes little further than arithmetic in its analyses. Most scientists have little more knowledge of the Platonic approach to science so beloved of mathematicians and philosophers than the average vicar. See my previous post for an analysis of Platonic and Empirical science. On a separate point: newts: "it is proof that Carroll is a fool". You may disagree with Carroll but refrain from insults. Time travel into the past seemingly involves paradoxes and these are probably evidence that our understanding of the universe is incomplete. Special Relativity places a barrier to time travel into the past but permits travel into other people's futures. Subatomic particles are routinely travelling into your future and can be observed to do this (ie: time dilation and twin paradox). General Relativity (GR) suggests that we can travel into our own past. However, it is well known that GR is to some extent incompatible with quantum theory and hence necessarily incomplete. We can only speculate about what would happen if a traveller were able to navigate along a loop back to their own past - certainly it creates two possible outcomes for the future of the traveller and world at the moment of arrival so would each become a separate quantum reality? Who knows? Nobody at present because this is at the boundary between science fact and scientific hypotheses.
  12. Science is largely empirical and it is maths that is platonic. There is a good discussion of the difference between these at: http://newempiricism.blogspot.co.uk/2009/01/science-empirical-or-platonic.html As an empirical pursuit anyone indulging in science would need to observe something that is directly related to the spiritual to do a "science of religion". Is there anything observable that is spiritual? Take a look at: http://newempiricism.blogspot.co.uk/2009/01/nature-of-soul.html (The Nature of the Soul).
  13. Mars would be habitable if we landed a factory to produce methane (not cfcs because we need ozone) in vast amounts. Global warming would occur. The subsoil ice would melt and oceans would form. Some sort of alga could be used to produce oxygen - there is plenty of CO2 for photosynthesis. The atmosphere would be thin so you might need a compressor to breathe and would benefit from living in glass domes to shield from radiation.
  14. Why not start by reading a simple introduction to Relativity like the Wikibook "Special Relativity" http://upload.wikimedia.org/wikipedia/commons/7/74/Special_Relativity_V2.11.pdf
  15. Time is a hugely complex subject. The Time of Relativity theory is undoubtedly akin to a negative spacial dimension (cf Weyl's analysis), recent experiments on quantum interference through time reinforce this view (see Lindner et al 2005). The time of Change is highly correlated with the dimensional time of Relativity but may not be exactly the same, for instance according to Multiverse theories two differing outcomes may coexist and be derived at the same location in a common spacetime. The time of causation is also problematical, as Reichenbach pointed out, the spherical symmetry of spacetime means that although chains of cause and effect can be pursued into the future it is difficult to pursue them into the past - for instance try to calculate the inverse of a spreading ripple from a needle dipped into a pond so that the disturbance at the edge of the pond ends up as a dip in the water surface at the exact place that the needle entered. Even more complex, try to think of something that exists for no time at all. Can an object have no temporal parts? Lindner, F., Schaetzel, F.G., Walther, H., Baltuska, A., Goulielmakis, E., Krausz, F., Milosevic, D.B., Bauer, D., Becker, W., and Paulus, G.G.. (2005) Attosecond double-slit experiment. Phys.Rev.Lett. 95,040401 (2005)
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