According to 1905 Relativity, a single material point must be always at rest
in [Relativity]
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From: PD on 27 May 2010 13:48 On May 26, 3:49 pm, va...(a)icmf.inf.cu wrote: > On 25 mayo, 10:46, PD <thedraperfam...(a)gmail.com> wrote: > > > On May 25, 10:19 am, va...(a)icmf.inf.cu wrote: > > > > On 24 mayo, 07:27, PD <thedraperfam...(a)gmail.com> wrote: > > > > > On May 24, 6:13 am, va...(a)icmf.inf.cu wrote: > > > > > > On 21 mayo, 10:22, PD <thedraperfam...(a)gmail.com> wrote: > > > > > > > On May 21, 6:27 am, va...(a)icmf.inf.cu wrote: > > > > > > > > On 20 mayo, 13:09, PD <thedraperfam...(a)gmail.com> wrote: > > > > > > > > > On May 20, 8:19 am, va...(a)icmf.inf.cu wrote: > > > > > > > > > > In his first Relativity paper (30June1905), after declaring the ether > > > > > > > > > superfluous, Einstein considers a material point at rest. He uses a > > > > > > > > > system of Cartesian coordinates in which the equations of Newtonian > > > > > > > > > mechanics and Euclidean geometry hold good. If the massive body is > > > > > > > > > alone (and then without a Newtonian gravitational force acting on it), > > > > > > > > > it must remains forever at rest in its own centre of mass inertial > > > > > > > > > frame. > > > > > > > > > Yes. So? > > > > > > > > > Every object lives in an infinite multitude of frames. The ones that > > > > > > > > have constant velocity with the frame you just described, plus this > > > > > > > > frame itself, constitutes the set of inertial frames. > > > > > > > > If we apply that to the single material point case, the result is an > > > > > > > infinite number of inertial frames where the single material point can > > > > > > > be moving with any velocity v. > > > > > > > Any velocity of magnitude less than c, yes. So? > > > > > > > > Once the ether is put out by 1905 > > > > > > > Einstein (and with it the Newtonian absolute space and time), can you > > > > > > > explain to me with respect to what a single material point can have > > > > > > > then a velocity different from zero? > > > > > > > A reference frame does not require a material "anchor" object, for > > > > > > which that anchor object is at rest in that frame. > > > > > > What do you say is totally valid in today Special Relativity, but not > > > > > in 1905 Relativity. See the following references to the 30Jun1905 > > > > > text. > > > > > Almost at the end of the Introduction: > > > > > R1. [The theory to be developed is based like all electrodynamics on > > > > > the kinematics of the rigid body, since the assertions of any such > > > > > theory have to do with the relationships between rigid bodies (system > > > > > of co-ordinates), clocks, and electromagnetic processes.] > > > > > At the beginning of paragraph 3: > > > > > R2. [Let us in stationary space take two systems of co-ordinates, > > > > > i.e. two systems, each of three rigid material lines, perpendicular to > > > > > one another, and issuing from a point.] > > > > > In R1 a system of co-ordinates is identified with a rigid body, and in > > > > > R2 (more detailed) with a system of tree rigid material lines. Even a > > > > > single material point must have some mass, imagine tree material > > > > > lines. Do you continue thinking that in 1905 Relativity a reference > > > > > frame does not require something material to establish the rest? > > > > > Yes. Just because Einstein uses a material framework to *explain* his > > > > position does not mean that it is essential to the conceptual > > > > framework. The idea of reference frames without material anchors > > > > preceded Einstein by at least a century. > > > > And what? How many centuries preceded Ptolemaic view the Newtonian > > > one? > > > Taking out the ether with its Newtonian absolute space and absolute > > > time is the essence of the Einsteinian relativistic view. > > > Yes. > > > > In it remain > > > only the material bodies themselves to determine inertial frames. > > > No. Again, you feel it is essential to anchor a frame to *something* > > material for some reason. Thus, if the aether is removed, then to you > > it MUST be material bodies. And if loosed from material bodies, then > > to you this DEMANDS the aether back again. This is simply wrong. > > Reference frames do not need material anchors, period. And they did > > not, for a long time preceding Einstein. Newtonian mechanics easily > > considered motion in frames in which there was no material object, > > including any supposed ether, at rest. > > The fact is that all Newtonian mechanics is based in its absolute > space and time, the unique privileged inertial frame considered real > and true. I disagree. Newton did believe in absolute space and absolute time. However, in terms of the laws of physics, it was in no way a privileged inertial frame. Indeed, he took the same position (and developed it further) that Galileo had: that the laws of physics are identical regardless of inertial reference frame. Calling out one of those frames as being somehow preferred even though the laws of physics are unchanged in it, is a rather empty statement. And in fact, when the ether was supposed, in order to support electrodynamics, the immediate implication was that the laws of electrodynamics had to be different in this frame than in any other frame. > You are right saying that in it is very easy to consider > motion in frames without a material object, because an empty frame > moving with ANY uniform velocity v with respect to the privileged one > is also an inertial frame (even if considered apparent and false). > Then, we have in Newtonian mechanics ALL possible inertial frames, and > ALL of them without the need of a single material body. Yes. And that "ALL of them" *includes* the absolute frame. > The situation is totally different in 1905 Relativity. Once the > privileged frame with absolute space and time is declared superfluous, > ALL the others empty frames moving with respect to it with any uniform > velocity v result also superfluous. Nonsense. All you've just said is that the inertial frames without a single material body owe their existence to being referred by a relative velocity v to a frame with some material body anchor. Thus, it is "anchoring by association" to a frame with a material body at rest. Nothing could be further from the truth. What you said earlier is correct. ALL inertial frames are perfectly well defined without anchoring to any material body at rest. That "ALL" is comprehensively inclusive. > Velocity with respect to nothing > simply cant exist in 1905 Relativity. It's not nothing. A reference frame is not nothing. You have in your head that the only "somethings" that velocity can be in reference to are material objects. This is not the case. > From where do you take a > reference frame without material bodies? It is moving or at rest? If > moving, with respect to what? It is moving with respect to one (set of) inertial reference frame(s), and it is in motion with respect to other inertial reference frames. > And if at rest, with respect to what? > And if you say that it is at rest with respect to itself, I didn't say with respect to itself. It is at rest with a particular reference frame. That reference does not have to be with respect to a material object. > you are > simply repeating Newtons primary absolute frame already put out by > 1905 Einstein. > In any case, in this thread we are considering only 1905 Relativity, > where a system of coordinates is identified with tree rigid material > lines, as the 1905 Einstein text put out of any doubt. No, he did not make that identification. A reference frame CAN BE but NEED NOT BE represented with three rigid material lines. You confuse a representation with the thing itself. > And you have no > right at all to put in 1905 Relativity what you want without the > support of a 1905 text. > Reading things in isolation is dangerous. Einstein was not ignorant of the use of reference frames unlocked from material objects, as had been understood for at least a hundred years prior, nor did he propose to replace that concept with one solely identified with a material object. > > > > > > > You are free to support the old Newtonian view if you want, but that > > > doesnt give you the right to say that in 1905 Relativity a reference > > > frame does not require something material. You cant change what 1905 > > > Einstein writes, even thinking you that it is something wrong. > > > > > > > If that were the case, then for two reference frames to even exist, > > > > > > you'd need two objects. For three, you'd need three objects. > > > > > > Exactly. See in the previous reference R2 a different material object > > > > > for each one of the two different reference systems introduced.> This however, is not the case. A reference frame has a perfectly well- > > > > > > defined meaning with or without a material anchor in it. > > > > > > A valid assertion in today Special Relativity, but a totally invalid > > > > > one in 1905 Relativity. In this thread we are addressing only 1905 > > > > > Relativity (see the title). > > > > > This, in my opinion, is a boondoggle. > > > > I don't see the point of it at all. > > > > Relativity is relativity. It is not a *creation*, it is a discovery.. > > > > When the first sailors navigated to the New World, they drew maps that > > > > represented their view of their discovery. Of course, those maps did > > > > not at all accurately represent the shape of the Americas. It does > > > > absolutely no good to take one of those maps and say, "But this is the > > > > America of 1500." It is not. The shape of the continent has not > > > > substantively changed since 1500. The America of 1500 is the same as > > > > the America of 2010. It's just that how people have described it is a > > > > little different from 1500 to 2010. > > > > If I am interpreting well, you are suggesting that 1905 Relativity > > > contains errors (first sailors making bad America maps) that were > > > already corrected in today Special Relativity. And surely you think > > > that the need to have material bodies to determine an inertial frame > > > is one of these errors. > > > Not errors per se. Just not completely worked out, and so there is an > > incomplete presentation of the model. > > > A map of the eastern seaboard of the Americas is not inaccurate, but > > it is not a good map of the Americas, either, because it is not > > completely articulated. > > That you consider 1905 Relativity incomplete dont give you the right > to complete it in the way you want. You (or anyone) can modify it in > any way, but not with the right to continue naming it 1905R. We cant > change history, and I have powerful reasons to consider 1905R as it > is. You will know about them in my next comment in this same post. > Again, I want to point out that historical representations are important when you are talking about a *creation*, but relativity is not a *created* thing, it is a DISCOVERED thing. When fossils of dinosaurs were first found, paleontologists inappropriately connected the leg bone fossils, giving the assemblies the postures of lizards, from which they concluded that the animals were cold-blooded and covered in scales. This, of course, has little to do with what dinosaurs REALLY were. So the initial *representation* of dinosaurs may have been one thing, but the intent was to represent something actual, and that representation was not complete. PD
From: PD on 27 May 2010 13:55 On May 26, 3:49 pm, va...(a)icmf.inf.cu wrote: > > > Suppose now that somebody finds the way to explain GR effects (like > > > the ones necessary to adjust GPS clocks in the Earths gravitational > > > field), but using only 1905 Relativity (Newtonian mechanics, Euclidean > > > geometry). Would you consider that a boondoggle? (English is not my > > > mother language, and I dont know the meaning of that word). > > > By my guest. Would love to see it. > > Let be two material points M and m (one with a great mass M, and the > other with a small mass m<<M). We can consider then M practically the > Centre of Mass (CM) of the 2-point system (for example, M and m can > model Earth and an electron). In the corresponding CM inertial system, > let be r the distance between M and m. > From the 27Sep1905 paper we know that The mass of a body is a measure > of its energy-content. In 1905, a body Total Energy E=K+U, where K is > the Kinetic Energy and U the Potential Energy. This is incorrect. What was proposed, in fact, was that the total energy of a body was E = mc^2 + K + U. You have misidentified mc^2 as being part of U, when it is not. And in fact, you should review what the allowed contributions of U are, from classical mechanics. It simply isn't true that just any old thing that is not associated with K are then U. A potential energy requires a connection with a *conservative force*, and one that can generate recoverable work. The term you've suggested -- m(r)c^2 -- does not meet those criteria. The rest of this paragraph then falls. > If the body is at rest, > K=0, being then U measured by the rest mass. In the case we are > addressing, for the body m we have then U(r)=m_0(r) c^2, where m_0(r) > is the body m rest mass and c the constant vacuum light speed. We know > that the gravitational potential energy increases when r increases. > Its limit maximal value when r tends to infinite is then m_0m c^2, > where m_0m is the corresponding limit maximal value of the rest mass > m_0. We have then > U(r)= m_0(r) c^2=m_0m c^2 (GM/r)m_0(r) > Here G is the Newtonian gravitational constant, > and (GM/r) is the gravitational potential owed to M with a supposed > arbitrary value 0 at r infinite. As you can see, U(r) takes the very > definite maximal value m_0m c^2 at r infinite. With some simple > algebraic handling we obtain > m_0(r)=m_0m/(1+GM/rc^2) > We have then derived from 1905R how the rest mass of a small body > changes as a function of its position r in the central gravitational > field of a great mass M body. Note the absent of any constant > arbitrary potential energy and the presence of an ABSOLUTE zero > potential energy point at r=0. If M and m are the Earth and an > electron, m_0m is the ordinary rest mass of a free electron (its > maximal value at r infinite). The frequency emitted by an atomic clock > is proportional to the rest mass of the electron involved in the > change of state. In GR the rest mass is supposed a constant an > intrinsic electron attribute, justifying the change in frequency with > the warp of the space-time provoked by M. As you see, the things in > 1905R are very much simple. > I left to you the verification that the change of frequency predicted > by 1905R coincides with the GR one in all the range of practical r > values in real experiments like the Pound&Rebka one. By the way, the > 1905R formula applies for ALL values of r from 0 to infinite, while > the GR one has a limited range of application owed to the presence in > it of a singularity that is absent in the 1905R one. >
From: PD on 27 May 2010 13:58 On May 26, 3:49 pm, va...(a)icmf.inf.cu wrote: > On 25 mayo, 10:46, PD <thedraperfam...(a)gmail.com> wrote:> On May 25, 10:19 am, va...(a)icmf.inf.cu wrote: > > > The frames you mentioned are not inertial. Circularly moving bodies > > are not in inertial motion. > > Let us revise what kind of reference frames are managing 1905 > Einstein. At the beginning of paragraph 1 (30Jun1905 paper) we can > read: > [Let us take a system of co-ordinates in which the equations of > Newtonian mechanics hold good.] > That type of system is denoted by 1905 Einstein stationary system. > By the way, in all his 1905 text he not use the word inertial, but > taking into account that the equations of Newtonian mechanics hold > good (and then also all its laws, including the first), I consider > safe to interpret that stationary system is what today is named > inertial frame. But in all 1905 text we dont find any mention at > all about a possible movement of the stationary system, and much less > specifying that it must be with uniform velocity following a straight > line. Again, it is dangerous to read a paper in isolation. In circularly moving reference frames, the Newtonian laws of motion do not hold. There are acceleration terms that do not arise as the result of any force that would be consistent with Newton's 3rd law. The Coriolis effect is an excellent example. > At the end of paragraph 4 in the 30Jun1905 paper, we find as the > stationary system what today is denoted GPS ECI, the real Earth > rotating in the inertial frame determined by itself with its centre of > mass at rest (as all the rotating axis, including the poles). We find > there also as the moving system a clock at the equator moving in a > CIRCULAR path. Yes, indeed, and there was a considerable amount of controversy about whether this construction should be considered movement in an inertial frame. It was only later that Einstein treated it better. > Taking into account that in a 1905R (without absolute > space and time) all inertial frames must be derived from the bodies > themselves, the stationary system must be then some CM inertial frame, > being the moving system any of its component bodies that can be moving > in any way compatible with Newtons laws. All this is supported with > the huge GPS ECI experimental evidence. Note that the Solar System > (SS) can be considered at rest as a whole entity (the more exact > inertial frame known by men) for internal affairs, but moving not in > the inertial way as a component of the Galaxy. The Earth, an SS > component, is in a similar situation. Considered at rest for internal > affairs (the GPS very successful function!), but moving not in the > inertial way as a part of SS. All seem to indicate that without > absolute space and time, what characterize the inertial attribute of a > system as a whole is only the rest (v=0) of its CM, not the any > uniform velocity v movement (that seem to belong only to the Newtonian > infinite empty inertial frames already put out in 1905 Relativity). As > the title of this thread states, according to 1905 Relativity, a > single material point (modelling an inertial frame) must be always at > rest. Because already it is not present the absolute space and time > permitting it to move with any uniform v velocity. >
From: valls on 27 May 2010 14:04 On 26 mayo, 19:23, "Sue..." <suzysewns...(a)yahoo.com.au> wrote: > On May 26, 5:24 pm, va...(a)icmf.inf.cu wrote: > [...] > > > > ================ > > > > > > No, because they aren't inertial, are they? > > > > > Of course that they are inertial ones, or maybe consider you that an > > > > inertial frame is only a mathematical entity? The Solar System is the > > > > more exact real inertial system used by men. And we can make a similar > > > > remark referring to the GPS ECI. By the way, they are both centre of > > > > mass inertial systems, supported by huge experimental evidence. > > > > The earth moves on an elliptical path around the moon. > > > A surf fisherman can perform gravito-inertial experiments > > > to detect that motion. (sit still while boots dry) > > > Hello Sue. > > Both Earth and Moon are moving in an elliptical path around their > > common Centre of Mass (CM). By the way, that CM is inside the Earth, > > but not in the Earths own centre of mass. The Moon seems provoking a > > sufficient almost equal acceleration in all GPS clocks (the same > > remark for all the rest of the Universe) to be ignored in the GPS > > function. With a Moon with a larger mass, surely the inertial system > > used in GPS would be de Earth-Moon centre of mass. For a future Solar > > System GPS, its centre of mass inertial frame MUST be used. You get my > > point? > > No... Either you have a description for inertial motion or > you don't. The example you offered seems in conflict with > the principle of relativity. > I only describe a very well-known Earth-Moon motion of today Astronomy. In 1905 Relativity, the rotating Earth (without Moon, the today denoted GPS ECI) appears at the end of paragraph 4 of the 30Jun1905 Einsteins paper as the stationary system (a one in which the equations of Newtonian mechanics hold good, let us denote it Newtonian), being a clock at the equator (part of the ECI) the moving system(by the way, with a CIRCULAR motion). In no part of the 1905 text we find some requirement about the type of frame movement to be Newtonian. More ever, doesnt exist at all any general indication about how we can obtain such type of frame. What conflict with the same Physics laws in all Newtonian frames are you talking about? > > > > > << Einstein's relativity principle states that: > > > > All inertial frames are totally equivalent > > > for the performance of all physical experiments. > > > > In other words, it is impossible to perform a physical > > > experiment which differentiates in any fundamental sense > > > between different inertial frames. By definition, Newton's > > > laws of motion take the same form in all inertial frames. > > > Einstein generalized this result in his special theory of > > > relativity by asserting that all laws of physics take the > > > same form in all inertial frames. >> > > http://farside.ph.utexas.edu/teaching/em/lectures/node108.html > > > > > That text doesnt correspond to 1905 Relativity, the topic in this > > thread (surely it corresponds to 1916 Special Relativity). In the GPS > > ECI inertial frame (used by 1905 Einstein, end of paragraph 4 of the > > 30June1905 text), it is impossible to make any physical experiment > > with the Moon or the Sun (or any other body not taken into account > > when computing the ECI centre of mass). > > Please accept my apologies for interrupting your walk down > memory lane with notions that have the benefit of over > 100 years of review and experiment. > You seem confused in what I am doing. I am not addressing at all post-1905 Relativity (SR and GR), rejecting always any interpretation of what I denote 1905 Relativity with concepts developed in its future (including the 1907 Minkowski view and all the rest of SR and GR). As we are in 2010, your referred 100 years of review and experiment are obviously in the future of 1905 Relativity, and by that simple reason must be maintained out of any valid interpretation of 1905 Relativity. Is that sufficiently clear for you? If 1905 Relativity in its limited historic context (Newtonian mechanics, Cartesian coordinates, Euclidian geometry) has any value at all to contribute the solution of today Physics problems, is another very different story. I am convinced that 1905 Relativity interpreted only in its own historic context can be used successfully for that purposes. > You seem to be saying we can reject the contemporary statement > of the principle of relativity and regain something from views > of the early 1900s but it is certainly not clear to me what > was lost. > > Sue... > I am not talking about any rejection at all of any contemporary statement. But if following 1905 Relativity, results can be obtained of superior scientific quality than similar contemporary ones, by sure the last will be substituted by the first (including your referred Principle of Relativity if that were the case) following the natural logic of science development. Sue, do you remember that (many?) years ago, you helped me putting in a more clear form formulas derived only from 1905 Relativity for the change in the frequency of an atomic clock owed to a change in the gravitational potential? RVHG (Rafael Valls Hidalgo-Gato)
From: Sue... on 27 May 2010 15:34
On May 27, 2:04 pm, va...(a)icmf.inf.cu wrote: > On 26 mayo, 19:23, "Sue..." <suzysewns...(a)yahoo.com.au> wrote: > > > On May 26, 5:24 pm, va...(a)icmf.inf.cu wrote: > > [...] > > > > > ================ > > > > > > > No, because they aren't inertial, are they? > > > > > > Of course that they are inertial ones, or maybe consider you that an > > > > > inertial frame is only a mathematical entity? The Solar System is the > > > > > more exact real inertial system used by men. And we can make a similar > > > > > remark referring to the GPS ECI. By the way, they are both centre of > > > > > mass inertial systems, supported by huge experimental evidence. > > > > > The earth moves on an elliptical path around the moon. > > > > A surf fisherman can perform gravito-inertial experiments > > > > to detect that motion. (sit still while boots dry) > > > > Hello Sue. > > > Both Earth and Moon are moving in an elliptical path around their > > > common Centre of Mass (CM). By the way, that CM is inside the Earth, > > > but not in the Earths own centre of mass. The Moon seems provoking a > > > sufficient almost equal acceleration in all GPS clocks (the same > > > remark for all the rest of the Universe) to be ignored in the GPS > > > function. With a Moon with a larger mass, surely the inertial system > > > used in GPS would be de Earth-Moon centre of mass. For a future Solar > > > System GPS, its centre of mass inertial frame MUST be used. You get my > > > point? > > > No... Either you have a description for inertial motion or > > you don't. The example you offered seems in conflict with > > the principle of relativity. > > I only describe a very well-known Earth-Moon motion of today > Astronomy. In 1905 Relativity, the rotating Earth (without Moon, the > today denoted GPS ECI) appears at the end of paragraph 4 of the > 30Jun1905 Einsteins paper as the stationary system (a one in which > the equations of Newtonian mechanics hold good, let us denote it > Newtonian), being a clock at the equator (part of the ECI) the moving > system(by the way, with a CIRCULAR motion). In no part of the 1905 > text we find some requirement about the type of frame movement to be > Newtonian. More ever, doesnt exist at all any general indication > about how we can obtain such type of frame. What conflict with the > same Physics laws in all Newtonian frames are you talking about? I gave the example that a surf fisherman can detect the elliptical motion of the earth by observing the tides. One violation is sufficient to demonstrate that your new definition of inertial motion fails. > > > > > > > << Einstein's relativity principle states that: > > > > > All inertial frames are totally equivalent > > > > for the performance of all physical experiments. > > > > > In other words, it is impossible to perform a physical > > > > experiment which differentiates in any fundamental sense > > > > between different inertial frames. By definition, Newton's > > > > laws of motion take the same form in all inertial frames. > > > > Einstein generalized this result in his special theory of > > > > relativity by asserting that all laws of physics take the > > > > same form in all inertial frames. >> > http://farside.ph.utexas.edu/teaching/em/lectures/node108.html > > > > That text doesnt correspond to 1905 Relativity, the topic in this > > > thread (surely it corresponds to 1916 Special Relativity). In the GPS > > > ECI inertial frame (used by 1905 Einstein, end of paragraph 4 of the > > > 30June1905 text), it is impossible to make any physical experiment > > > with the Moon or the Sun (or any other body not taken into account > > > when computing the ECI centre of mass). > > > Please accept my apologies for interrupting your walk down > > memory lane with notions that have the benefit of over > > 100 years of review and experiment. > > You seem confused in what I am doing. Ah! We agree on something. :-) > I am not addressing at all > post-1905 Relativity (SR and GR), rejecting always any interpretation > of what I denote 1905 Relativity with concepts developed in its future > (including the 1907 Minkowski view and all the rest of SR and GR). As > we are in 2010, your referred 100 years of review and experiment are > obviously in the future of 1905 Relativity, and by that simple reason > must be maintained out of any valid interpretation of 1905 Relativity. > Is that sufficiently clear for you? > If 1905 Relativity in its limited historic context (Newtonian > mechanics, Cartesian coordinates, Euclidian geometry) has any value at > all to contribute the solution of today Physics problems, is another > very different story. I am convinced that 1905 Relativity interpreted > only in its own historic context can be used successfully for that > purposes. You want to argue that Einstein's 1905 paper is a theory of relativity even tho the author claims otherwise? Your true calling may be in law or politics. Have you studied Michigan vs. Michigan in preparation for your arguments? :-)) http://www.scrp.us/larger3.asp?go=37 > > > You seem to be saying we can reject the contemporary statement > > of the principle of relativity and regain something from views > > of the early 1900s but it is certainly not clear to me what > > was lost. > > > Sue... > > I am not talking about any rejection at all of any contemporary > statement. But if following 1905 Relativity, results can be obtained > of superior scientific quality than similar contemporary ones, by sure > the last will be substituted by the first (including your referred > Principle of Relativity if that were the case) following the natural > logic of science development. Does the 1905 paper even claim to be a theory of relativity? If so that claim is short lived. <<This circularity in the definition of inertia and the inability to justify the privileged position held by inertial worldlines in special relativity were among the problems that led Einstein in the years following 1905 to seek a broader and more coherent context for the laws of physics. In the introduction of his 1916 review paper on general relativity he wrote "The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration." http://www.mathpages.com/rr/s4-07/4-07.htm > Sue, do you remember that (many?) years ago, you helped me putting in > a more clear form formulas derived only from 1905 Relativity for the > change in the frequency of an atomic clock owed to a change in the > gravitational potential? That credit is not mine. I may have offered Lev Okun's paper which is now supported by years of GPS operation. http://arxiv.org/abs/physics/9907017 My memory for every poster's views is not very good and for posters that stopped reading in 1905 it is nearly useless. Sue... > > RVHG (Rafael Valls Hidalgo-Gato) |