The importance of gyroscopes for the construction of real LIFs[i]
“Local inertial frames have a fundamental role in Einstein geometrodynamics. The spatial axes of a local inertial frame along the world line of a freely falling observer are mathematically defined using Fermi-Walker transport (eq. 3.4.25); that is, along … her geodesic they are defined using parallel transport. These axes are physically realized with gyroscopes. … The most advanced gyroscopes … measure the very tiny effect due to the gravimagnetic field of the Earth: the ‘dragging of inertial frames,’ that is, the precession of the gyroscopes by the Earth’s angular momentum, which in orbit, is of the order of a few tens of milliarcseconds/year. There are two main types of gyroscopes … mechanical and optical. The optical gyroscopes … are usually built with optical fibers or with ring lasers.” (6.12)
Fermi-Walker Transport, De Sitter (Geodetic)&Lense-Thirring Effects
For weak gravity fields in the first Einstein 20^{th} Century correction to Newton’s 17^{th} century gravity theory: S^{a} is a spacelike 4-vector outside its local light cone that describes the spin of the test gyroscope about its rotation axis. The test gyroscope travels along a timelike worldline x^{a} (s) with tangent vector u^{a}. S^{a}u_{a} = 0 and the equation for Fermi-Walker transport is
S^{a}_{;}_{b}u^{b} = u^{a} (a^{b}S_{b}) = u^{a}(u^{b}_{;}_{g}u^{g}S_{b}) (3.4.25)
Where a semi-colon “;” always stands for the covariant partial derivative with respect to the Levi-Civita connection that describes fictitious forces on the test gyroscope that are, in reality, real forces on the detector measuring the motion of the gyro. Repeated upper and lower indices are summed through 0,1,2,3. The local observable objectively real proper acceleration first-rank tensor directly measured by accelerometers clamped to the center of mass of the test gyro is
a^{b} = u^{b}_{;}_{g}u^{g}
If the arbitrary timelike world line of the center of mass of the test gyro (remember LIFs have three of them forming a spacelike triad base frame) is a geodesic, then, by definition, the proper acceleration tensor a^{b} = 0. Therefore,
S^{a}_{;}_{b}u^{b} = 0
This is the equation for Fermi-Walker transport.
“A mechanical gyroscope is … made of a wheel-like rotor, torque-free to a substantial level, whose spin determines the axis of a local, nonrotating frame. Due to very tiny general relativistic effects … that is, the ‘dragging of inertial frames’ and the geodetic precession, this spin direction may differ from a direction fixed in ‘inertial space’ that may be defined by a telescope always pointing toward the same distant galaxy assumed to be fixed with respect to some asymptotic quasi-inertial frame (see 4.8).”
Inertial Navigation From ICBMs to Starships
“Mechanical gyroscopes are based on the principle of conservation of angular momentum of an isolated system … with no external forces and torques. … the spinning rotor maintains its direction fixed in ‘space’ (apart from dragging effects as Earth rotates but, however, a vector with general orientation, fixed with respect to the laboratory walls, describes a circle on the celestial sphere in 24 hours, a spinning rotor … describes a circle with respect to the laboratory walls in 24 hours … In a moving laboratory, using three ‘inertial sensors’, that is, three gyroscopes to determine three fixed directions (apart from relativistic effects…) plus three accelerometers to measure linear accelerations and a clock (and possibly three gravity gradiometers to correct for torques due to gravity gradients, one can determine the position of the moving laboratory with respect to its initial position. This can be done by a simple integration of the accelerations measured by the three accelerometers along the three fixed directions determined by the gyroscopes [held by gimbals]. Position can thus be determined solely by measurements internal to the [starship] laboratory … a priori independently of external information is called ‘inertial navigation’ … an onboard computer integrates the accelerations … one is able to find velocity, attitude, and position of the object.”
The word “acceleration” here means off-geodesic proper tensor acceleration not the old Newtonian kinematic acceleration measured by Doppler radar in Einstein’s somewhat misleading popular “happiest thought quote” I discussed earlier whose Siren’s song that has shipwrecked many a wannabe physicist-philosopher Flying Dutchman searching for Ithaca. However, for a starship in free float on a timelike geodesic we can dispense with the gyroscopes to preserve “direction.” “Instead one may use gradiometers …”
“The needs of air navigation have generated a powerful drive for a compact, light weight gyroscopic compass of high accuracy … Today, optical gyros have displaced the mechanical gyro … A wave-guide is bent into a circle. A beam splitter takes light from a laser and sends it round the circle in two opposite directions. Where the beams reunite, interference between them gives rise to wave crests and troughs. If the wave-guide sits on a turning platform, the wave crests reveal the rotation of the platform or the airplane that carries it.
While mechanical gyroscopes are based on the principle of conservation of angular momentum, optical gyroscopes (really optical rotation sensors) are essentially based on the principle of the constancy of the speed of light c in every inertial frame. Therefore, in a rotating circuit and relative to the {LNIF} observers moving with it, the round trip travel time of light depends on the sense of propagation of light with respect to the circuit angular velocity relative to a local inertial frame.” [LIF]
From the general connection of continuous Lie groups[ii] of symmetries of closed dynamical systems to conserved local currents and global “charges” that form the group’s non-commuting Lie algebra[iii], we conclude that the operation of the gyroscope corresponds to the three rotational symmetries of Einstein’s 1905 special relativity’s Poincare group. Therefore, the Sagnac effect[iv] basis of the optical gyros correspond to the three Lorentz boosts of that same Poincare group that formally express the constancy of the speed of light in inertial frames. Newton’s action-reaction third law comes from the three space translation symmetry’s conservation of linear momentum and the conservation of energy comes from the time translation symmetry – if these symmetries are not broken. Does the accelerometer’s operation depend on the Rindler boosts of constant proper accelerating hyperbolic world lines of test particles? These are outside of the Poincare group requiring Roger Penrose’s twistor conformal group.[v] The Poincare group is a subgroup of the conformal group that also includes dilations.
from my bookContrary to popular misconceptions, although the local laws of classical physics have the same “tensor” and/or “spinor” form for all motions of detectors measuring all the observable possessed by the “test particles,” nevertheless, there still are privileged geodesic force-free dynamical motions of the test particles in Einstein’s two theories of relativity special 1905 and general 1916.[i] This was in Einstein’s words “My happiest thought.”
“The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly the thought struck me: If a man falls freely, he would not feel his own weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity. I continued my thought: A falling man is accelerated. Then what he feels and judges is happening in the accelerated frame of reference. I decided to extend the theory of relativity to the reference frame with acceleration. I felt that in doing so I could solve the problem of gravity at the same time. A falling man does not feel his weight because in his reference frame there is a new gravitational field, which cancels the gravitational field due to the Earth. In the accelerated frame of reference, we need a new gravitational field.” [ii]
First note the date 1907. Einstein is using Newton's 1686 theory of gravity not his then future 1916 general relativity way of thinking that he has not yet created. Einstein is struggling with the wrong notion of “acceleration.”
"A falling man is accelerated."
Yes, in Newton, but not in Einstein nine years in the future! The falling man's frame is LIF with zero proper acceleration. In fact it's the surface of static LNIF Earth with proper radial acceleration upward rushing toward the falling man.
Proper acceleration of falling man = Relative 1905 SR kinematic acceleration - Proper acceleration of Earth
Proper acceleration of falling man = D^{2}X/ds^{2}
Relative 1905 SR kinematic acceleration = d^{2}X/ds^{2}
Proper acceleration of Earth = {STATIC LNIF EARTH}(dX/ds)(dX/ds)
X = relative separation test particle to detector on Earth.
{ } = Christoffel symbol used in the Levi-Civita connection
v = dX/ds
In fact when v/c << 1, the 3-vector piece of the above 4-vector equation is:
{STATIC LNIF EARTH}(dX/ds)(dX/ds) ~ -GM_{Earth}r/r^{3}
Proper acceleration of falling man = 0 because an accelerometer pinned to the man shows zero on its pointer. Therefore,
Relative kinematic acceleration = Proper acceleration of Earth
Where a Doppler radar measures the relative kinematic acceleration between the falling man and Earth. In contrast, a second accelerometer clamped to the detector at rest on surface of the Earth measures -GM_{Earth}r/r^{3 }as the weight divided by the mass of the detector.
“A falling man does not feel his weight because in his reference frame there is a new gravitational field, which cancels the gravitational field due to the Earth. In the accelerated frame of reference, we need a new gravitational field.”
That statement by Einstein in 1907 is how Newton would explain it. Einstein put himself in Newton's shoes for a moment. It's not the way his later 1916 matured GR explains it.
0 = Relative 1905 SR kinematic acceleration - Proper acceleration of Earth
This “cancellation”, the “0” on the above word equation is not a cancellation of two real dynamical fields. Einstein's unfortunate informal language in 1907 has no relevance to his, then, future theory.
"In the accelerated frame of reference, we need a new gravitational field."
That's the LIF, which is not accelerated in the sense of 1916 Einstein GR, but is accelerated in the different sense of 1686 Newton. These subtle oft unnoticed paradigm shifts in the meanings of “acceleration,” “inertia,” “inertial frame” cause many people a great deal of confusion even today,
Einstein was still muddled in 1907 as he struggled to make the great breakthrough. Your understanding is trite and superficial based on semantics and exaggeration of an early remark of Einstein's.
[i] This geodesic premise is Newton’s first law of motion most generally expressed.
[ii] On the Relativity Principle and the Conclusions Drawn from It, Albert Einstein,
Jahrbuch der Radioaktivitat und Electronik 4 (1907) – Re-Published in three parts.
Am. J. Phys. 45, Part I - (6), June 1977, pp. 512-517; Part II – (9), September
1977, pp. 811-816, Part III - (Gravitational Part) – (10), October 1977, pp. 899-
902. This paper addresses only Part III – from Peter Brown’s paper.
In this EARLY 1907 quote Einstein (who is still under Newton’s magick without magic spell) means Newton's "accelerated frame", that is, dV(test particle)/ds in Newton's first law (geodesic equation) as written in modern POST-1907 GR language. Suppressing indices:
DV(test particle)/ds = dV(test particle)/ds - {LNIF detector}V^{2}(test particle) = 0
The "cancellation" is precisely
dV(test particle)/ds - {LNIF detector}V^{2}(test particle) = 0
In other words, in the general case that even applies to Newton's 2nd and 3rd laws is:
Einstein's proper tensor acceleration = Newton's apparent acceleration - fictitious LNIF inertial pseudo fictitious forces per unit test particle rest mass = real applied force to the test particle per unit test particle mass
Fictitious forces on test particle = Real forces on LNIF detector of test particle's motion
In the case of Newton's 3rd law, when Alice and Bob form an isolated closed system
DP(Alice + Bob)/ds = DP(Alice)/ds + DP(Bob)/ds = 0
Both must be measured in the same frame by Eve, i.e.,
DP(Alice or Bob)/ds = dP(Alice or Bob)/ds + {Eve}V(Alice or Bob)P
“I continued my thought: A falling man is accelerated. Gravity and inertia are interrelated." Einstein
Here is the source of the confusion.
Einstein is naturally thinking in Newtonian terms.
However, in GR terms that he still had not invented back then in 1907: "acceleration" above means relative kinematical acceleration between test particle and local frame. It does not mean real (proper) acceleration (off-geodesic) as measured by an accelerometer.
The general law is:
Real acceleration on test particle = relative kinematical acceleration between test particle and local frame - real acceleration of local frame.
DP(test particle)/ds = dP(test particle-frame)/ds - DP'(local frame)/ds
P = mV for the test particle under observation by the local frame detector
V = dX/ds
X = relative kinematical displacement between test particle and local frame detector as measured by a Doppler radar clamped to the local frame.
D/ds = d/ds - {LC frame connection}dX/ds
DP(test particle)/ds
= dP(test particle)/ds - {LC frame connection}(dX/ds)P(test particle)
When dm/ds = 0, it follows that
D^{2}X/ds^{2} = d^{2}X/ds^{2 }- {LC frame connection}(dX/ds)^{ 2}
{LC frame connection}(dX/ds)^{ 2 }= M^{-1}DP(frame)/ds
M = mass of frame/detector
{LC frame connection} has dimension 1/Length
ds is the PROPER TIME element along world line of object.
Each term has an independent measurement technique.
Real accelerations are measured by accelerometers attached to the objects.
http://en.wikipedia.org/wiki/Accelerometer
Accelerometers measure off-geodesic "pushes" by real forces.
Doppler radars measure the kinematic acceleration.
http://en.wikipedia.org/wiki/Doppler_radar
Therefore,
DV/ds is measured directly locally by an accelerometer clamped to the test particle - real measurement 1
dV/ds = d^{2}X/ds^{2 }is measured indirectly by the Doppler radar clamped to the local frame detector - real measurement 2
M^{-1}DP(frame)/ds is measured directly by a second accelerometer clamped to the frame-Doppler radar - measurement 3
The BASIC LAW is
Measurement 1 = measurement 2 - measurement 3
Provided that test particle and frame Doppler radar are not far away from each other relative to the smallest local radius of curvature A^{1/2}. The curvature is of order A^{-1}
The geodesic equation is simply Newton's first law when
Measurement 1 = 0
Newton's second law is simply when
Measurement 1 =/= 0
There is never any cancellation of real forces on any one object in this context
The LNIF ---> LIF in measurement 3 simply means removing a real unbalanced force on the frame detector according to Newton's 1st law.
“Then what he feels and judges is happening in the accelerated frame of reference.” Einstein
Einstein's use of "accelerated" here is in Newton's sense - the rest frame of the freely falling man is kinematically accelerated relative to the Earth
i.e. d^{2}X/ds^{2}
The freely falling man's local frame is LIF - though Einstein did not yet discover that in 1907 and his informal language is still Newtonian because the modern GR informal language of 1916 and after is not yet emerged.
“There is a new gravitational field, which cancels the gravitational field due to the Earth.” Einstein
This is Einstein's remark that physics cranks pull out of proper context. Yes, Einstein wrote it back around 1907 before he understood the problem the way he eventually would in 1916 and later.
In fact there is only one gravity field not two.
The point is that there was never a real gravity force field on the test particle to begin with.
Therefore, you don't need a second gravity force field to cancel what was never there!
Indeed, there is no way to measure either of these alleged two real gravity forces to begin with. You can never separate them. Accelerometers on test particles always show zero.
Therefore, like the Maxwellian 19^{th} century mechanical aether that acts without being reacted upon that Einstein eliminated in 1905, these two ghostly independently unobservable-in-principle forces are not independently measurable - they are errors of thinking - excess metaphysical informal language baggage. Even the great Einstein got muddled temporarily on this one, though with good reason. Unfortunately many people today who should know better remain muddled. If gravity is not a real force like the electro-weak-strong forces, then what does it mean to unify them?
The issue before me is how to address them properly in my Stargate book and in my reviews of his book. I will take several weeks pondering this. I will not make Jim's theory a central part of my book as I have plenty of original material myself.
Gentlefolk,
The continuation of last night's comments. Jack and Paul, by the way, have repaired to a shorter list to continue their mathematical discussions. As far as I am concerned, this process has been like tapping a kaleidoscope. I've known about Einstein's predilection for Mach's ideas since reading John David North's history of cosmology back in the '60s.
And with every pass, I learn a bit more -- though a bit less with each pass, at least recently.
As I said yesterday, much of the confusion [leaving aside the silliness about "fictitious" forces] in this business seems to be an outgrowth of the now allegedly mainstream view that gravity is only present when non-vanishing spacetime curvature is present -- a view that seems to have its origins in a neo-Newtonian view that large constant potentials can be gauged away as irrelevant. This comports with the widespread view that the Aharanov-Bohm experiment notwithstanding, potentials in classical situations are not real. Only the fields derived from them are.
This may be true for all other physical fields. But it is not true for gravity. The vector part of the gravitational potential very definitely does depend on the particular value of the scalar potential calculated. There are some formal technical details that complicate this a bit. But the idea that you can ignore cosmic scale matter currents when computing local gravitational effects is still just wrong.
Tonight, what I want to do, however, is talk a bit about a couple of other matters. The first is the "origin" of inertia. You may recall that Jack gave a long list of mechanisms -- the Higgs process, QCD calculations, and suchlike -- that allegedly are the origin of mass, and thus inertia. The fact of the matter is that none of these processes (valid in and of themselves) account for the origin of mass and inertia. Frank Wilczek, after telling you about these processes in his book The Lightness of Being, allows as much (on pages 200 through 202).
Inertia is a universal property of stuff. And the only universal interaction that couples stuff is gravity. It is thus obvious that if gravity produces inertial forces (that is, the relativity of inertia obtains), that gravity should have a lot to do with the origin of inertia. (The origin of inertia was the title of Sciama's first paper on this I note. So I'm not making this up.)
This is more obvious still when you discover that phi = c^2 is the condition that must be satisfied for inertial forces to be due to gravity. You don't even have to fudge with dimensions to get this to work.
The dimension of phi is velocity squared. You may not like this result. Jack it seems doesn't. But it is a simple consequence of GRT. You might think that this means that should the rest of the matter in the universe be made to disappear (or should you screen an object from the gravity of all that matter) the mass of an object would go to zero -- as is assumed in a number of discussions of Mach's principle and the origin of inertia. But that's not what happens. Read chapters 7 and 8.
The last thing I want to comment on is, how the devil did all this get so bolixed up? Recent kaleidoscope tapping suggests that there were two crucial mistakes that are largely responsible for all the confusion. The first mistake was made by Einstein in 1921. By that time, he had been worked over by Willem deSitter and disabused of his naive Machianism (which is why he started talking about spacetime as an "ether" about this time). So the claims he put into his Princeton lectures on Mach's principle were more tentative than they had been previously. One of the things he calculated that he took to be in accord with Mach's ideas was the effect of "spectator" matter (that is, nearby stuff) on the mass of an object. He claimed that piling up spectator matter would cause the mass of the object in question to increase (because of its changed gravitational potential energy). A very small amount. But if the origin of mass is the gravitational influence of cosmic matter, this is just the sort of effect you might expect to see.
It turns out that Einstein was wrong about this. That's what Carl Brans showed in 1962 (as part of his doctoral work at Princeton with Bob Dicke). The EP simply forbids the localization of gravitational potential energy. So, the inference that GRT is explicitly non-Machian regarding inertia and its origin is perfectly reasonable. It's the inference that Brans and Dicke -- and everyone else for that matter -- took away. Brans and Dicke, to remedy this presumed defect of GRT, resuscitated Pasqual Jordan's scalar-tensor version of gravity, hoping the scalar field part could bring in Machian ideas.
The second crucial mistake is the inference everyone made that Brans' EP argument meant that Mach's principle isn't contained in GRT. Indeed, exactly the opposite is the case. Brans' conclusion from the EP is absolutely necessary for Mach's principle to be contained in GRT. It is the conclusion that must be true if inertial reaction forces are always to satisfy Newton's third law, for it guarantees that phi = c^2 ALWAYS when measured locally. But everyone had adopted the false inference that GRT is non-Machian. It's no wonder that issues of Mach's principle in GRT has been so confused. It's no wonder that C+W (really Wheeler I'd guess, for he witnessed the Mach wars of the '50s and '60s) tried to use Lynden-Bell's initial data and constraint equations approach to implement Einstein's parting shot at Mach's principle in the '20s. The origin of inertia is just too important to let go with the sort of "explanations" now floating around.
On a personal note, I've known that phi = c^2 (locally) is the condition to get all of the Mach stuff to work since around 1992. But I was focused on inertial forces and how they might be transiently manipulated. And doing experiments. I won't tell you how long it took for the other aspect of the origin of inertia to sink in -- even though it was staring me in the face. . . .
Keep the faith,
Jim
____________________________________________________________
Z: I think Addinall's remarks and suggestions make sense, with the main exception that Jack is not talking about Einstein's theory, he's talking about Wheeler's theory.
There is nothing wrong with focusing on dynamical acceleration and geodesic structure since these are the features of GR that are physically the most interesting.
However, anyone who wants to understand the Machian interpretation of GR (which doesn't mean that they have to agree with it) needs to recognize the basic
differences between Einstein's version of GR and Wheeler's.
Z,RA: I do understand some of the historical debate, and that at least Einstein's initial interpretation of the equivalence principle is different from the modern one (and that Jim is working from Einstein's original EP).
RA: Jack has expressed an interest in avoiding the historical debate, so my suggestions to him were based on that.That's why in my second set of points (the first set are numbered 1 to 6, the second 1 to 5) I suggested that Jack describe Einstein's theory in point 3, and move on to focusing the reader on dynamical frames in point 5. I could be missing something, but I didn't find Jack's simplified description of Einstein's GR terribly controversial. It's when Jack moves to dynamical frames and electrical contact forces that he's non-Machian.
RA: With regard to the apple falling on Newton's head - well, as far as I can gather it's correct to say that the apple is moving inertially on a timelike geodesic (it is in a LIF) and Newton is accelerating (in a LNIF). Earth's mass determines that the geodesics will be curved.
RA: Frame dragging doesn't seem important over a relatively short distance.
RA: My impression is that the Machian question comes into play in determining why the apple immediately goes into a LIF... is the rest of the matter in the universe the origin of the inertia? Or is it just an intrinsic property - something will move along a geodesic until it is forced off it?
RA: This is different than Mach's and Jim's example of standing in place and then spinning around and having your arms pulled out to your sides - that is a more convincing Machian argument. However, the apple falling seems to work in either a Machian or non-Machian interpretation.
Please explain the distinctions in greater detail if I'm still confused.Rob
On Oct 12, 2013, at 3:17 PM, Paul Zielinski <iksnileiz@gmail.com> wrote:I think Addinall's remarks and suggestions make sense,Yes, unlike most of yours! Addinall is a smart fellow.with the main exception that Jack is not talking about Einstein's theory, he's talking about Wheeler's theory.This is a good example of a trite waste of time quibble. I made it very clear that for my purpose I don't give a damn about the historical ups and downs of Einstein's rocky road to his 1916 final version except for his later clarifications with action-reaction and other issues in the 1920s till his death. The action-reaction idea is key to my work in quantum theory and beyond as well as in geometrodynamics.Wheeler's version is the one most useful for experimental physicists and engineers.
There is nothing wrong with focusing on dynamical acceleration and geodesic structure since these are the features of GR that are physically the most interesting.So why do you waste every one's time with red herrings?
However, anyone who wants to understand the Machian interpretation of GR (which doesn't mean that they have to agree with it) needs to recognize the basic
differences between Einstein's version of GR and Wheeler's.The important points for the proper understanding of Jim's proposal is1) "inertia" means the pattern of zero-g force timelike geodesics for Mach' principle (also light cones). It does not mean computing rest masses of actual elementary particles.2) phi = c^2 is not even wrong in my opinion in the context of modern cosmology.3) Sciama's vector theory is way too simplistic. My bet is that what Jim sees in the lab is a systematic error like the faster than light neutrino at OPERA. I could be wrong, but that is my bet.
On 10/12/2013 2:01 PM, JACK SARFATTI wrote:On Oct 12, 2013, at 12:14 AM, Robert Addinall <beowulfr@interlog.com> wrote:Jack,In terms of audience you seem to have decided to increase your focus on engineers and people from various other fields interested in UFOs or building stargates.rightThis version seems a lot better towards that end – it is a lot simpler/clearer (with the emphasis on avoiding difficult math).rightMy overall comments:1. There were points that emerged from the e-mail discussion of the last week or two which seemed quite clear to me. I suggest that you might include versions of these statements at key points:
2. Z’s description that you want to focus on “dynamical frames, such that all local frame acceleration in GR is defined with reference to the geodesics.”
I keep emphasizing that. The geodesics are physically privileged, i.e. zero g-force weightlessness as measured locally by real accelerometers. They are mathematically not privileged i.e. the classical local differential equations for the natural laws can be written in any real set of possible physical frames on timelike worldlines geodesic or not - makes no difference. That is what the tensor calculus does for any theory including Newton's Galilean limit and the limit of special relativity.Mach's Principle is all about zero g-force geodesics - that's what he means by "inertia" not "rest mass." In Wheeler's language, Mach's principle is "100 % voting power" - however Jim's phi = c^2 is not an adequate consequence of that in my opinion.1916 GR -> 1905 SR -> Newton's mechanics
3. Your response that focusing on dynamical frames is “good physics… physics should be about real things – phenomena and how they are measured… real accelerometers etc.”
Yes, this is the Cornell 1950's 60's disiderata for good physics - keep the math to a minimum.If you look at archive today things going in opposite direction. Mathematics is the opiate of the theoretical physicist.Key theme in my philosophy of physics - from Mach, Einstein, Dirac, Wheeler, Feynman ...
4. You could also use the version of the statement in #3 from another email: “good physics is about real phenomena measured with real instruments… keep it simple stupid… but not simpler than possible.”
5. Your statement that: “Einstein’s proper tensor acceleration = Newton’s apparent acceleration – fictitious LNIF inertial pseudo fictitious forces per unit test particle rest mass = real applied force to the test particle per unit test particle mass.”
This is in words what in Einstein's math is (sans tensor indices)DP/ds = dP/ds - {Levi-Civita connection}VP = F(electro-weak-strong)P = mV (test particle){Levi-Civita connection} describes the detector not the test particleit is zero for a LIF detector - that is EEP.
6. Your observation that: “What is lacking – except in the Wheeler-Thorne books is a clear description of GR measurement theory – how the symbols connect to real lab procedures” is also good – you seem to have started to include this in the latest version of the chapter. I suggest finding more detailed examples to use to round out the chapter as you progress.
Of course.
An observation of my own: I find that, apart from when theory absolutely demands otherwise, it’s easiest to explain things by simply following the historical sequence of events. My phd is interdisciplinary in military history and political science. When I’m writing as a historian this is usually pretty easy. When I’m writing as a political scientist it gets more difficult. Nonetheless, taking the above six points into account, I suggest that you organize things as follows:1. Explain Newton’s theory.
2. Explain how Einstein’s theory is different.
3. Explain, in very basic terms, as you have done below, the question that Einstein’s theory answers as well as how it answers it.
4. Explain the difficulty of the math and how it causes even experienced physicists to have difficulties.
5. Focus on getting the reader to understand: (a) dynamical frames; (b) how to connect GR theory to real lab procedures.
6. Provide detailed examples of lab procedures.
I made a few comments in red and blue as I read over the chapter, based on where I had to stop and re-read something a couple of times. These were either minor stylistic grammar points, or places where as a non-expert in physics I didn’t quite follow the argument. If you find my comments useful let me know and I’ll comment in the future when I have time. If you don’t like my style and don’t find anything useful tell me and I won’t comment again. I’m trying not to interfere with your stream of consciousness “beat” point of view, but simply to provide input on how to organize your comments enough that you can keep the engineers reading.Like some of the engineers, my interest is in ideas that could be experimentally tested within a feasible budget in order to build starships and stargates. That’s why I find Jim’s work interesting. If your book inspires people to test the Bose-Einstein condensate and Einstein-Cartan curvature + torsion with anti-gravitating dark energy term ideas, I’ll be happy.That actually reminds me: in the overall introduction to the book, in addition to the section “what is a stargate?”, you might want to state the BEC and Einstein-Cartan ideas as your two main proposed solutions to the task of figuring out how stargates work. This will get the attention of experimental physicists and engineers who will then keep reading because there is something they might actually get to build and play around with.Right - I have not gotten yet to the actual stargate stuff these are just the preliminaries.Robupdated V2 Oct 12, 2013
Chapter 1 Einstein’s Theory of Relativity in a Nutshell
“I was dissatisfied with the special theory of relativity, since the theory was restricted to frames of reference moving with constant velocity relative to each other and could not be applied to the general motion of a reference frame. I struggled to remove this restriction and wanted to formulate the problem in the general case.” Albert Einstein[i]
“Nowhere has a precise definition of the term ‘gravitational field’ been given --- nor will one be given. Many different mathematical entities are associated with gravitation; the metric, the Riemann curvature tensor, the curvature scalar … Each of these plays an important role in gravitation theory, and none is so much more central than the others that it deserves the name ‘gravitational field.’”[ii]
The physical meaning of Einstein’s relativity, both special (1905) and general (1916) is quite simple in contrast to the mathematics, which quickly gets very difficult. Except for the books by John Archibald Wheeler and his students like Kip Thorne, most books on the general theory get too mathematical leaving the physical meaning obscure.
“The Question is: What is The Question?” John Archibald Wheeler
The question that Einstein’s relativity is the answer to is this: Alice and Bob have measuring instruments and they decide as voyeurs to watch Eve’s dance. How do they compare their data? Relativity is an algorithm, a set of rules, which takes the raw measurement data input and processes it to give a set of “invariant” output real numbers. If Alice and Bob get the same set of invariants, then they can be quite confident, in the sense of Bayesean probability estimates, that they measured the same set of events and that their measurements were good within the accuracy and precision limits of the technology of their measuring instruments. This is basically classical because Heisenberg’s quantum uncertainty principle will provide a barrier when Alice and Bob attempt to measure the same individual quantum events.
Einstein’s 1905 special theory of relativity at first only considered inertial frames of reference. What is a frame of reference? Basically it is a local set of detectors. What kind of detector? It’s necessary that an accelerometer, like the scales we weigh ourselves with, be included along with other devices like telescopes, Doppler radars etc. The test for an inertial frame is simple, the pointer of the accelerometer reads zero. Every object in the inertial frame is weightless in free-float like the astronauts in the International Space Station shown in the movie “Gravity.” In this case of free-float zero g-force, we say that the center of mass of the local inertial frame (LIF) moves on a timelike geodesic world line in Einstein’s four-dimensional spacetime continuum. Therefore, we here on Earth are not in inertial frames. We are in non-inertial frames. Unfortunately, Newton defined the word “inertial frame” differently from Einstein and this continues to lead to much confusion when physicists attempt to communicate with each other because Newton’s theory is in closer accord with our common sense. Einstein’s relativity is counter-intuitive. In Newton’s theory, points on the surface of Earth are approximate inertial frames if we ignore its rotation about the poles. However, in Einstein’s theory, any point on Earth, approximated as an ideal non-rotating spherical surface has a real local objective tensor proper acceleration pointing radially outward from the center of the sphere. Of course, we are not moving relative to the center of the idealized spherical Earth yet we are accelerating and this is counter-intuitive violating common sense. It only makes sense in the curved space non-Euclidean differential geometries of Karl Friedrich Gauss and Bernard Riemann. Proper dynamical acceleration is what accelerometers measure. There is also the apparent kinematical acceleration that Doppler radars measure. Therefore, these two quantities can be measured independently by different kinds of detectors. Ideally in principle must be accelerometers on both the test particle and the detector. In addition, the detector is equipped with Doppler radar to measure both the kinematic velocity and kinematic acceleration of the test particle relative to the detector. The general rule is:
Proper dynamical local acceleration of a test particle = Kinematical nonlocal acceleration of a test particle – Proper local dynamical acceleration of the detector.
With the additional rule:
Proper dynamical acceleration of the detector = Fictitious pseudo-acceleration on the test particle = Levi-Civita connection terms
= Real force on detector per detector mass
Let us consider all four physically interesting possibilities.
1) Accelerometer on test particle shows zero, accelerometer on detector shows zero. This is then a geodesic test particle whose motion is measured by an on-geodesic LIF detector. Of course, these are two different geodesics in general.
2) Accelerometer on test particle shows zero, accelerometer on detector shows not-zero. This is then a geodesic test particle whose motion is measured by an off-geodesic LNIF detector. The LNIF observer looking at his Doppler radar tracks mistakenly thinks that there is some kind of universal force on the test particle proportional to its mass causing it to move in a curve at different speeds along it. Indeed, Newton called this “gravitational force” when he looked at the parabolic orbits of apples falling off trees and cannon balls, especially the latter to see a good parabola. Similarly for the elliptical orbits of the planets about the Sun. The Coriolis and centrifugal motions are essentially the same as Newton’s gravity force field because they too are universal proportional to the mass of the test particle. Newton could not have conceived that his apple was on a timelike geodesic straightest possible world line in Einstein’s future idea of the curved four-dimensional spacetime continuum. Newton could not have conceived that it was him who was really accelerating to the apple, which was not really accelerating at all! Indeed, many engineers and ordinary people – and even some physicists still cannot properly and consistently conceive of it so stuck are they in the persistent illusions of common sense.
Both 1) and 2) correspond to Newton’s first timelike geodesic law of test particle motion:
Proper dynamical local acceleration of a test particle = Kinematical nonlocal acceleration of a test particle – Proper local dynamical acceleration of the detector = 0
We are only interested in the center of mass of the test particle and ignore rotations about some axis through its center of mass.
3) Accelerometer on test particle shows not-zero, accelerometer on detector shows zero. This is then an off-geodesic test particle whose motion is measured by an on-geodesic LIF detector.
4) Accelerometer on test particle shows not-zero, accelerometer on detector shows not-zero. This is then an off-geodesic test particle whose motion is measured by an off-geodesic LNIF detector.
Both 3) and 4) correspond to Newton’s second off-geodesic law of test particle motion whose equation in words is
Proper dynamical local acceleration of a test particle = Kinematical nonlocal acceleration of a test particle – Proper local dynamical acceleration of the detector =
Real local force on test particle per mass of test particle.
The proper tensor acceleration of any object is described by the “covariant derivative of the velocity tensor of the object with respect to proper time along the world line of the object in four-dimensional spacetime.
Einstein’s 1905 special relativity showed that if Alice and Bob were each on different zero g-force timelike geodesics, then they would measure the same invariant speed of light c ~ 3 x 10^{8} meters per second in vacuum. However, Alice looking at Bob’s clock would see it running slow (time dilation) and vice versa. A moving meter stick shrinks along its direction of motion relative to the observer for simultaneous measurements of the edges of the meter stick by the observer. However, a more careful analysis of light rays coming from a fast moving object by Richard Terrell in the 1950’s revealed that the object looks rotated rather than contracted.
We all know about E = mc^{2} and I will not dwell on the details of special relativity here. What is not well known however, even by physicists is that one can use special relativity to deal with properly accelerating frames of reference. However, to do so, one must use the full tensor language of Einstein’s 1916 general relativity. The only difference is that the curvature tensor computed from the “covariant curl” of the Levi-Civita connection with itself vanishes everywhere. Special relativity still works for artificial Newtonian gravity fields without curvature that appear in a rotating space station for example where the normally fictitious centrifugal pseudo force balances a real quantum electrical force in a rigid constraint connecting the test object to its detector.
Alice and Bob working together do the actual measurement of the local spacetime curvature tensor field. It’s important that they are both on timelike geodesics and what they measure is their relative kinetic acceleration from each other (aka “geodesic deviation”) in different spatial orientations to get all ten components of the Weyl tensor in space. The Weyl tensor causes stretch-squeeze elliptical distortions in a set of geodesic test particles initially configured in a circle. There are also ten other components of the Ricci tensor coincident with mass-energy sources, but that is harder for Alice and Bob to directly measure. The Ricci tensor causes the radius of the circle of geodesic test particles to contract for positive mass-energy sources and to expand for the negative mass-energy exotic sources needed for warp-wormhole advanced super-technology. The full Riemann curvature tensor in four-dimensional spacetime is the sum of the Weyl vacuum and the Ricci matter tensors.
Curvature introduces a severe restriction on measurements not found in Minkowski spacetime empty of real gravity fields. When the curvature is not zero Alice and Bob, both watching Eve’s activities, must be “physically coincident” in order to compare their data by calculating invariants. This means that the actual physical separations between Alice and Bob must be less than the smallest radius of curvature in the components of the Riemann curvature tensor. Eve, however, can be arbitrarily far away with Alice and and Bob getting light signals and/or cosmic rays from her. The mathematics of tensor general coordinate transformations only connects physically coincident local frames of reference. In fact there are three groups of these reversible coincident frame transformations.
1) LNIF à LNIF’ general coordinate transformations corresponding to the local translation group T4(x).
2) LIF à LIF’ local Lorentz transformations corresponding to the local Lorentz group SO(1,3)
3) LIF à LNIF tetrad transformations corresponding to Einstein’s equivalence principle (EEP) for cancellation of Newton’s artificial gravity force field. Of course there is no cancellation of Einstein’s real gravity curvature field.
[i] How I created the theory of relativity, Albert Einstein, Translated by
Yoshimasa A. Ono, Physics Today, pp. 45-47 (August 1982) cited by Peter Brown in http://arxiv.org/pdf/physics/0204044v2.pdf
[ii] Gravitation, Misner, Thorne and Wheeler, (W.H. Freeman and Company, 1973)