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Natural Science Forum / Physics / Research / July 2008speed/velocity c in GR
I have noticed that Einstein stresses in various places that, whereas the speed/velocity of light is constant in SR, it is not in GR. The reason for my current source of confusion here is that Einstein wrote in German, where there is no linguistic distinction between vector (velocity) and scalar (speed). It is obvious that the vector must vary since light rays bend in gravitational fields. However, the situation for the scalar seems rather less clear to me, at present. Does anyone know of any accelerating (linear or rotating) reference frame experiments which unambiguously resolve this question one way or the other? (pleaseremove spam from address for emailed response) > I have noticed that Einstein stresses in various places that, whereas > the speed/velocity of light is constant in SR, it is not in GR. One may hear this assertion from time to time. However, its truth depends greatly on the definition of "speed of light" that is used. I think it is best (that is, to avoid as much confusion as possible) to consider the speed of light to be constant everywhere. I discussed this issue in detail in an old post: news:1161021765.752413.52950@h48g2000cwc.googlegroups.com http://groups.google.ca/group/sci.physics.research/msg/5b598d3b6e4fbce7 Hope this helps. Igor > I think it is best (that is, to avoid as much confusion as possible) > to consider the speed of light to be constant everywhere. I am inclined to agree. It certainly makes matters simpler. I have often thought that, given all our observations of the constancy of c were made in the accelerating frames we actually inhabit, it seems a bit peculiar to conclude from this that c is constant in inertial frames but perhaps not in accelerating frames. >> I think it is best (that is, to avoid as much confusion as possible) >> to consider the speed of light to be constant everywhere. > > I am inclined to agree. It certainly makes matters simpler. It's not a question of opinion, or of making things "simpler" ("Keep things as simple as possible, BUT NO SIMPLER." -- A. Einstein). It's a question of how this is measured and modeled. In our current best model, General Relativity, there's no question: over a non-local path the vacuum speed of light need not be c (even though the local speed of light is c at each point along the path). This is a basic instance of the curvature of spacetime. To see this there's no need to consider light bending around the sun (though that's an experimental confirmation of the model). One need not even worry about synchronization of distant clocks. In Schwarzschild spacetime and Schw. coordinates, for a clock at A and a mirror at B directly above A, the round-trip vacuum speed of light is not c, where speed means twice the number of standard meter sticks that can be laid between A and B divided by the elapsed time during the light's flight measured by a standard clock at point A. Adding another clock and mirror, the speed for B->A->B differs from A->B->A; adding a mirror C an equal number of meter sticks below A, the speed for A->C->A is different from either of them. Unfortunately, for the earth and sun these differences are too small to measure directly, and interference methods suffer from the impossibility of making a Michelson interferometer rigid enough to rotate in a vertical plane. Hmmm. I wonder if it could be sensitive enough to rotate in a horizontal plane on earth, and measure the effect due to the sun? Probably not, as these interferometers are notoriously unstable.... And, of course, Kennedy and Thorndike (and similar experiments) could have seen at least some fraction of it. > I have often thought that, given all our observations of the > constancy of c were made in the accelerating frames we actually > inhabit, it seems a bit peculiar to conclude from this that c is > constant in inertial frames but perhaps not in accelerating frames. The accelerations involved are so small they are inconsequential. The math of GR is quite clear: the vacuum speed of light is locally c in any locally inertial frame (using standard clocks and rulers). In an accelerated system the speed of light cannot be isotropic, whether measured one-way or round-trip [#]. But for tiny accelerations the difference is smaller than tiny, and in practice here on earth, unmeasurable. The relevant scale is c^2/g, and for earth's gravity that is about a lightyear. [#] One-way measurements have great difficulties with clock synchronization. Tom Roberts > >> I think it is best (that is, to avoid as much confusion as possible) > >> to consider the speed of light to be constant everywhere. [quoted text clipped - 21 lines] > from either of them. Unfortunately, for the earth and sun these > differences are too small to measure directly, I note from http://en.wikipedia.org/wiki/Schwarzschild_coordinates that the Schwartzchild chart does not represent radial distances accurately. Given this, it is difficult to see how the use of such a coordinate system can prove the reality of effects which are beyond the limits of observation. >> In our current best model, >> General Relativity, there's no question: over a non-local path the [quoted text clipped - 5 lines] > that the Schwartzchild chart does not represent radial distances > accurately. Of course not! The only usage I made of Schw. coordinates is to specify that the clock, mirror, and rulers are at rest in them (at least that was my intent, I see my words do not precisely say that). To compute the distance from A to B one must, of course, integrate the metric along the path of the ruler. Etc. > Given this, it is difficult to see how the use of such a > coordinate system can prove the reality of effects which are beyond > the limits of observation. Look again -- I defined "speed" as the ratio of meter sticks laid along the path to the elapsed time on a single clock, and that is completely independent of coordinates (but not independent of how the objects move, so I mentioned Schw. coordinates to specify how they [don't] move). > Consider points a and b in an accelerating > frame accelerating in the direction a to b, and generated by starting > the acceleration of a and b at the same point in time, with the same > separation. > Observers at both a and b can confirm observationally that clocks on b > run faster than clocks on a. Yes. Assuming Minkowski spacetime. > Now, are you saying that for b, c is < 1, for the round photon trip > bab, and for a, c is >1 for the round photon trip aba, so that they > can also agree on separation? Yes, as long as one uses meter sticks undergoing Born rigid motion to measure the distance used in the computation of speed (as in the Schw. example above -- meter sticks at rest in Schw. spacetime naturally undergo Born rigid motion). > And is that difference effective on the > outward photon trip, the return photon trip, or both? That is a question without a definite answer, because to answer it one must synchronize two clocks at different locations. But clocks at a and b will not remain synchronized! The key point about the Schw. example and yours is that they are round-trip so one need not synchronize clocks. One-way speeds are ALWAYS subject to variation depending on how you happen to synchronize the clocks. Also, please avoid "photons" in such discussions, as they inherently introduce quantum issues that are not relevant. Discuss light pulses instead so one can use the geometrical optics (or in some cases the wave optics) approximation. > Now consider an accelerating frame where both a and b start > accelerating from the same point in spacetime (but in different > spatial directions). Both will then find observationally that clocks > on the other are running slower. > Is c then>or<1 for either / both round trips? I don't know -- work it out for yourself. > Finally, and potentially more subversively, since we cannot tell, > experimentally, whether c is constant in inertial or accelerating > frames, This is not true. We CAN tell for round-trip speed. For measurements of round-trip speed, c is constant in locally-inertial frames occupied by labs on earth; this has been measured literally zillions of times. Of course the accuracy and validity of a locally-inertial frame depends on lots of things, but one can use GR to compute how good this approximation is, and for such measurements here on earth it is FAR better than experimental resolutions. In accelerating frames there is no reason to think the speed of light would be constant. AFAIK no measurements have been made for systems with SIGNIFICANT acceleration, except possibly measurements of Shapiro time delay (but that's a PUN on "acceleration"...). Tom Roberts > > Finally, and potentially more subversively, since we cannot tell, > > experimentally, whether c is constant in inertial or accelerating [quoted text clipped - 3 lines] > round-trip speed, c is constant in locally-inertial frames occupied by > labs on earth; this has been measured literally zillions of times. I don't see how you can describe as 'locally inertial', a frame where inertial bodies achieve speeds of > 15 feet/sec in less than one second, if you stop applying force to keep them 'stationary'. Your argument appears to fly in the face of the general postulate (aka principle of equivalence) >> For measurements of >> round-trip speed, c is constant in locally-inertial frames occupied by [quoted text clipped - 3 lines] > inertial bodies achieve speeds of > 15 feet/sec in less than one > second, if you stop applying force to keep them 'stationary'. Remember that physics is a QUANTITATIVE science. The phrase "locally inertial" is always an approximation. But in practice, for labs on earth and for tabletop light-speed measurements (mentioned above), that approximation is FAR better than the experimental resolutions. As the apparatus is supported against gravity, one normally analyzes such experiments in the locally-inertial frame that is initially at rest relative to the apparatus. Exercise for the reader: Consider a lab measurement of the round-trip speed of light over a 10 meter path. To estimate the error induced by ignoring the acceleration of gravity: A) compute the time duration of the round-trip light path B) compute how far an object would fall during that time, given it is initially at rest (hint: L = 0.5 g t^2). This is an overestimate, as there is a better inertial frame to use (at rest when the light is in the middle). C) now compute the difference between the assumed (horizontal) path and the actual path due to that distance fallen (hint: neglect the curvature of the path and add the vertical fall to the horizontal distance; this clearly overestimates the actual path length). D) now relate that error in path length to an error in the measured speed (hint: fractional errors are the same). Compare to a typical experimental resolution of ~0.1 parts per billion. This is only an (over) ESTIMATE. For real experiments you need to look them up and read their error analysis. > Your argument appears to fly in the face of the general postulate (aka > principle of equivalence) No. It merely includes the fact that physics is a quantitative science, and for this specific physical situation the acceleration due to gravity is completely negligible. > Well, I don't trust Born rigid rulers because they are totally > artificial constructs embodying too many logical contradictions. Not at all! For a ruler, Born rigid motion means that in a steady state the atoms of the ruler arrange themselves so that the inter-atomic distances have their usual values. Yes, this only applies to accelerations small enough so that the stain is negligible, but that was included in the original physical situation. > If you start two separate bodies (or two ends of > the same body) accelerating at the same rate, at the same time, it is > obvious that they will both have the same velocity at all subsequent > times, in that original inertial frame. This is not Born rigid motion. It is not "rigid" in any sense. Born rigid motion is such that for every pair of points in the object the proper distance between them remains constant -- that is normally what one means by "rigid". But it should be clear that this can only be applied to "small" objects, where the actual size limit depends on the acceleration applied and the (in)elastic properties of the object. To induce Born rigid motion in a solid object, you must either: a) attach the acceleration to a single point and let any startup transients damp out, making sure the acceleration is small enough so strain is negligible. or: b) couple the acceleration to every individual atom, varying it appropriately for each atom so the proper distances remain constant. For a steel ruler standing up on the surface of the earth, these approximations are good enough for most practical purposes. Tom Roberts > >> For measurements of > >> round-trip speed, c is constant in locally-inertial frames occupied by [quoted text clipped - 39 lines] > and for this specific physical situation the acceleration due to gravity > is completely negligible. My original argument was that we can't tell the difference, and you objected to that. You are now confirming my argument, quantitatively. Can't you see that? > To induce Born rigid motion in a solid object, you must either: > a) attach the acceleration to a single point and let any > startup transients damp out, making sure the acceleration > is small enough so strain is negligible. So, what (preferably physical) mechanism are you proposing to 'damp out' the consequences of such 'startup transients' ? > or: > b) couple the acceleration to every individual atom, varying > it appropriately for each atom so the proper distances > remain constant. Ditto > > Well, I don't trust Born rigid rulers because they are totally > > artificial constructs embodying too many logical contradictions. > > Not at all! For a ruler, Born rigid motion means that in a steady state > the atoms of the ruler arrange themselves so that the inter-atomic > distances have their usual values. Which is un-physical. If the rulers are laid out vertically, the force inducing acceleration transfers through the rulers at the speed of sound in that material. Consequently, rulers DO stretch if hung from the ceiling, and compress if stacked up from the floor, when acceleration is induced. This would even be the case (but on a smaller scale) for an 'ideal' ruler where the speed of sound in the material equals the vacuum speed of light, and the rulers are completely inelastic in every other sense. It is pointless arguing that these differences are negligible when the rulers are short and the g forces small, if you then apply huge numbers of them stacked end on end to 'prove' that c is not constant in gravitational fields. >> For a ruler, Born rigid motion means that in a steady state >> the atoms of the ruler arrange themselves so that the inter-atomic [quoted text clipped - 5 lines] > the ceiling, and compress if stacked up from the floor, when > acceleration is induced. Sure. But read the very next sentence I wrote, and the entire theme of that post -- this APPROXIMATION is useful for many applications. How many carpenters use different rulers for vertical and horizontal measurements, and adjust the former for altitude? Experimental physics is always an approximation, not the abstract discussion you are pursuing. The key is to know which approximations are useful and which are not. Born rigid motion is useful in many applications; but not all. Locally inertial frames on earth are accurate for many physical situations, but not all. Tom Roberts > How > many carpenters use different rulers for vertical and horizontal > measurements, and adjust the former for altitude? None. You have missed the point. The golden rule of carpentry is measure twice, cut once. This applies generally when we are "on the tools". For example, with a spirit level, you turn it round and measure again, both for vertical and horizontal applications. This is because experience shows it often makes a difference. > > Consider points a =A0and b in an accelerating > > frame accelerating in the direction a to b, and generated by starting [quoted text clipped - 11 lines] > Yes, as long as one uses meter sticks undergoing Born rigid motion to > measure the distance used in the computation of speed Well, I don't trust Born rigid rulers because they are totally artificial constructs embodying too many logical contradictions. To be perfectly honest, I can't even understand why they were invented in the first place. If you start two separate bodies (or two ends of the same body) accelerating at the same rate, at the same time, it is obvious that they will both have the same velocity at all subsequent times, in that original inertial frame. Consequently, I can see no reason for postulating that one must accelerate harder than the other, for their separations to remain constant subsequently. Sure, their separation in the original inertial frame may become relativistically shrunk in due course, but that is irrelevant because the lengths of co-moving rulers would shrink by the same amount. > > Finally, and potentially more subversively, since we cannot tell, > > experimentally, whether c is constant in inertial or accelerating [quoted text clipped - 3 lines] > round-trip speed, c is constant in locally-inertial frames occupied by > labs on earth; this has been measured literally zillions of times. If what you say is true (ignoring your oxymoronic frame definition), then, via the general postulate (strong principle of equivalence), that is literally zillions of lab verifications that c is constant in accelerating frames. > If you start two separate bodies (or two ends of > the same body) accelerating at the same rate, at the same time, it is [quoted text clipped - 5 lines] > shrunk in due course, but that is irrelevant because the lengths of > co-moving rulers would shrink by the same amount. On further reflection, this argument of mine is unsound. AFAICT the separation should remain constant in the original inertial frame. Consequently, with relativistic shrinking of rulers in the co-moving frame, the accelerating observer SHOULD see separations increase, if both accelerations are the same. > >> I think it is best (that is, to avoid as much confusion as possible) > >> to consider the speed of light to be constant everywhere. [quoted text clipped - 51 lines] > > Tom Roberts This IS rather confusing. Consider points a and b in an accelerating frame accelerating in the direction a to b, and generated by starting the acceleration of a and b at the same point in time, with the same separation. Observers at both a and b can confirm observationally that clocks on b run faster than clocks on a. Now, are you saying that for b, c is < 1, for the round photon trip bab, and for a, c is >1 for the round photon trip aba, so that they can also agree on separation? And is that difference effective on the outward photon trip, the return photon trip, or both? Now consider an accelerating frame where both a and b start accelerating from the same point in spacetime (but in different spatial directions). Both will then find observationally that clocks on the other are running slower. Is c then>or<1 for either / both round trips? Finally, and potentially more subversively, since we cannot tell, experimentally, whether c is constant in inertial or accelerating frames, is not a decision that one is true in preference to the other, anything more than an irrational (historically predicated) preference for brand A over brand B? > I have noticed that Einstein stresses in various places that, whereas > the speed/velocity of light is constant in SR, it is not in GR. [quoted text clipped - 14 lines] > > (pleaseremove spam from address for emailed response) The speed of light is constant locally... where it matters. Take the Schwarzschild line element for light ds=0 and solve for dr/ dt. It is not c except at r=inf where the local observers are. This also happens in SR. Switch to an accelerating frame and do a similar analysis. The difference is that in SR there is a global inertial frame which gives you an unambiguous way to compare speeds across different locations on the manifold. Whereas in GR there are only local inertial frames. There are alternate theories of gravity that have a variable speed of light (VSL). They are generally hogwash or merely a coordinate transformation and not physical. > > I have noticed that Einstein stresses in various places that, whereas > > the speed/velocity of light is constant in SR, it is not in GR. [quoted text clipped - 33 lines] > > - Show quoted text - So why is the speed of light different is materials? Does it always seems to be slower according to the refractive index of the material its passing through? On Jun 17, 2:33 pm, festusby...@btinternet.com wrote: > > The speed of light is constant locally... where it matters. > So why is the speed of light different is materials? > > Does it always seems to be slower according to the refractive index of > the material its passing through? The speed of light in a material differs from the speed of light in vacuum. This fact is independent of either special or general relativity. Inside a material, light gets scattered. In a sense, microscopically, it no longer travels in a straight line, but bounces from atom to atom. Take two pairs of points that are the same distance apart. If one pair is in vacuum, light can cover the distance between them in a straight line. If the other pair is inside a material, then light will have to bounce around while traveling from one point to the other. Thus, necessarily, it must travel slower in a material than in vacuum. On a coarser, macroscopic level, the bouncing light waves interfere in a way that light appears to propagate in straight lines. The only signature of the microscopic behavior is refraction, a sharp change of direction of a light ray when crossing the boundary between two materials of different optical properties. Hope this helps. Igor > On Jun 17, 2:33 pm, festusby...@btinternet.com wrote: > [quoted text clipped - 25 lines] > > Igor Yes that's very helpful, Thank you. Also when space-time is said to be curved, does this mean that a spin of a body is actually causing the curviture in space-time? If you toss a massive body in flat space-time, will it have a choice not to be become curved, if there is no rotation? In other words is curviture in space-time controlled by the presence of the mass? Or does the angular momentum of a body contribute to this curviture? Thanks On Jun 18, 4:35 am, festusby...@btinternet.com wrote: > Also when space-time is said to be curved, does this mean that a spin > of a body is actually causing the curviture in space-time? Hmm, it's a bit of a jump from light refraction to space-time curvature. :-) So this is a completely different topic now. The short answer is Yes, but I'm not sure why your question is specific to spin. > If you toss a massive body in flat space-time, will it have a choice > not to be become curved, if there is no rotation? In the context of general relativity, the presence of matter (of any kind, including light) induces space-time curvature. This dependence is expressed as the Einstein equations. > In other words is curviture in space-time controlled by the presence > of the mass? Let me make my last point more explicit. The presence of matter implies, for any point in space-time, (i) an energy density (including heat, rest mass, electromagnetic energy, etc.), (ii) a momentum density (including heat flux, and the electromagnetic Pointing vector), (iii) pressure and other internal stresses (including shear stresses). All of these influence space-time curvature. > Or does the angular momentum of a body contribute to this curviture? In a rotating extended object, each unit volume of the body (except possibly directly on the axis of rotation) has a non-zero velocity and hence a non-zero momentum density. This momentum density falls into category (ii) above. Therefore, a rotating object will affect curvature differently than the same object if it is not rotating. Hope this helps. Igor [snip] > In a rotating extended object, each unit volume of the body (except > possibly directly on the axis of rotation) has a non-zero velocity and > hence a non-zero momentum density. This momentum density falls into > category (ii) above. Therefore, a rotating object will affect > curvature differently than the same object if it is not rotating. There have been many attempts to directly observe spin effects upon gravitation from lab to cosmic scales - all of them unsuccessful to date including gyroballs and polarized spin test masses. Gravity Probe B was to observe local exceedingly small gravitomagnetic effects. There are no observed orbital anomalies associated with 2.15 msec rotation, 1.74 solar-mass pulsar PSR J1903+0327 in a 95.17-day 0.44-eccentricity orbit with its 1.05 solar-mass companion. Large divergences of gravitational binding energy (27% vs. 1.4x10^(-4)%), strong and weak field (1.8x10^11 vs. ~30 surface gees), composition (neutrons and exotica vs. protons and electrons); pulsar equatorial spin greater than 0.11 lightspeed, 2x10^8 gauss magnetic field... are inert within observational error. http://arXiv.org/abs/0805.2396 It is a difficult measurement.  SignatureUncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/lajos.htm#a2 In article <be106d44-274f-48a6-bbd4-6aa0a24208d6@e53g2000hsa.googlegroups.com>, festusbyrne@btinternet.com writes: > So why is the speed of light different is materials? This has nothing to do with the issue being discussed, where implicitly the speed of light in vacuum is what matters. > Does it always seems to be slower according to the refractive index of > the material its passing through? Yes. [Moderator's note: See Igor's post for more details. -P.H.] > I have noticed that Einstein stresses in various places that, whereas > the speed/velocity of light is constant in SR, it is not in GR. The underlying issue is what one means by "speed" in GR. This is nontrivial. But for the case of the earth orbiting the sun, the difficulties can be avoided and a definite measurement and analysis can be made, basically due to the fact that the fields are quite small. So one can construct background Minkowski coordinates that are valid throughout the region of interest, and measure speed relative to them. > It is obvious that the vector must vary since light rays bend in > gravitational fields. At the same level of precision in wording, they also slow down. This is known as the "Shapiro delay", and has been measured accurately for light passing close to the sun; for known pulsed sources like pulsars, spacecraft, or round-trip reflections, the delay can usually be measured much more accurately than the bending. > However, the situation for the scalar seems rather less clear to me, > at present. It has been measured, to sufficient accuracy to solidly demonstrate that the prediction of GR applies to the world we inhabit. Of course the effect is quite small, being just a few hundred microseconds for a path earth->mars->earth that passes close to the sun. Tom Roberts > > I have noticed that Einstein stresses in various places that, whereas > > the speed/velocity of light is constant in SR, it is not in GR. [quoted text clipped - 25 lines] > > Tom Roberts Hmmm....I'm not too sure about this one. If light takes a curved path around the Sun, it is obvious that it will take longer to travel that extra distance. However, that doesn't necessarily mean that speed (dx/ dt) has altered. Can anyone clarify? > I have noticed that Einstein stresses in various places that, whereas > the speed/velocity of light is constant in SR, it is not in GR. Its true. When that is said it refers to the coordinate speed of light. It can't refer to the proper speed of light since there is no such thing when it comes to light. > The reason for my current source of confusion here is that Einstein > wrote in German, where there is no linguistic distinction between > vector (velocity) and scalar (speed). A 4-velocity for light cannot be defined. > Does anyone know of any accelerating (linear or rotating) reference > frame experiments which unambiguously resolve this question one way or > the other? Sure. See http://www.geocities.com/physics_world/gr/c_in_gfield.htm Pete > I have noticed that Einstein stresses in various places that, whereas > the speed/velocity of light is constant in SR, it is not in GR. [quoted text clipped - 14 lines] > > (pleaseremove spam from address for emailed response) In GR, along a null geodesic, locally at each point of the geodesic, in the local tangent space, velocity of the light vector is "c". Regarding a distant vector in a curved space, for knowing its magnitude you shoud perform a parallel transport up to your local space and as the result may depend on the path followed, usually one says that it's a nonsense to speak about celerity of a distant vector. One observer may measure the celerity of a speed vector only in his local space! Jacques |
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