Classical Mechanics and Gravity From Newton to Einstein
Jack Sarfatti
Excerpted from Stargate
Version 121913
I am taking the contemporary approach, not the historical one. This is a reconstruction of Newton and Einstein’s ideas using modern insights. It’s not exactly how they would have thought of what they did, but what I write does not contradict any essential battletested truths of their ideas.
Newton’s dynamics of particles is based on Euclidean geometry for space with absolute time the same for all observers no matter how they move. Newton had no idea that the speed of light was finite. In Newton’s theory the speed of light is infinite.
Newton’s first two laws are basically a single law.
Law 1. Forcefree motions of test particles are geodesics independent of the mass and internal constitution of the particle.
A test particle is so small that we can neglect the gravity field it generates.
In Newton’s implicit geometry a geodesic is a straight line in space with a test mass moving at constant speed. There is also a state of absolute rest.
Law 2. A real vector[i] force F causes the test particle with velocity vector V and instantaneous position vector r to have a curved motion with varying speed that is not geodesic.
Assume the mass m is constant, that is the calculus[ii] derivative dm/dt = 0.
F = dP/dt = d(mV)/dt = mdV/dt = md^{2}r/dt^{2} = ma
This equation assumes a global inertial frame. A global inertial frame (GIF) is an imaginary cubic lattice of rigid steel rods across the entire universe with a clock at each vertex. All the clocks are synchronized. There are artificial intelligences with each clock that can communicate with each other by light signals. They all have Doppler radars to track the motions of test particles or UFOs. Jim Woodward, in his book Making Starships[iii], uses a simplistic model of the universe by the late Dennis Sciama that implicitly assumes such a global frame. Of course the equations that Professor Woodward proposes as an engine for spaceships will not work  more on this later.
Now in fact, such structures do not exist. We really only have local frames consisting of a finite network of detectors over a small region of space connected by the internet.
Physics is not mathematics. The crackpots I have dealt with do not understand the difference. Theoretical physics is about what detectors measure. We use mathematical models to do that, but the models have an enormous amount of redundant excess baggage that must be factored out in the sense of equivalence relation classes[iv] and homomorphisms[v] preserving essential relevant structure. It’s the same as a compression algorithm[vi] in computer science. Mathematics is like a highresolution image. However, what we need to do real physics is a much lower resolution compressed image in which certain nonessential features are erased because only some small subset is needed for the measurements of interest.
Finally we have Law 3.
Newton’s third law of motion[vii] is very limited in its domain of validity and is a specialized case of the more general actionreaction conjecture.[viii] Newton’s third law assumes only central forces, which act instantly at a distance. Therefore, it’s only good really for contact interactions in his original theory. It can be generalized when fields are added to Newton’s particles. Newton did not really have the concept of extended dynamical fields[ix] that have a reality equal to localized hard massy marblelike particles. Today we have Noether’s theorem[x] that relates conservation laws to the symmetries[xi] of dynamical actions[xii] of systems of particles and fields in classical physics. Everything becomes fields in quantum physics, where the real particles are excited out of a very complicated vacuum that is a seething frothy quantum foam of virtual particles[xiii] in an ever turbulent Dirac sea.[xiv] Newton’s third law corresponds to only two systems forming a closed system. More generally a group of space translation symmetries causes the total linear momentum of closed complex systems of particles and fields to be conserved. Similarly, time translation symmetry causes total angular momentum of closed systems to be conserved and rotational symmetry causes total angular momentum to be conserved. There are also internal symmetries beyond spacetime out of which spring the electromagnetic, weak and strong force fields.[xv] When we go to Einstein’s 1905 special relativity[xvi] where space is fused with time into spacetime, then rotations that mix space and time together correspond to the Lorentz boosts[xvii] causing time dilation, length contraction and the equivalence of mass to energy. We can even go beyond that to Roger Penrose’s twistor[xviii] conformal group[xix] that includes uniformly accelerated local frames (LNIFs) with Rindler horizons[xx] as well as a topological stretching dilation symmetry that is badly broken in our world.[xxi]
[viii] Einstein, the reality of space, and the actionreaction principle
Harvey R. Brown, Dennis Lehmkuhl
(Submitted on 20 Jun 2013)
Einstein regarded as one of the triumphs of his 1915 theory of gravity  the general theory of relativity  that it vindicated the actionreaction principle, while Newtonian mechanics as well as his 1905 special theory of relativity supposedly violated it. In this paper we examine why Einstein came to emphasize this position several years after the development of general relativity. Several key considerations are relevant to the story: the connection Einstein originally saw between Mach's analysis of inertia and both the equivalence principle and the principle of general covariance, the waning of Mach's influence owing to de Sitter's 1917 results, and Einstein's detailed correspondence with Moritz Schlick in 1920.
Comments: 
To appear in "The Nature of Reality", P. Ghose (ed.), Oxford University Press 
Subjects: 
History and Philosophy of Physics (physics.histph); General Relativity and Quantum Cosmology (grqc) 
Cite as: 
arXiv:1306.4902 [physics.histph] 

(or arXiv:1306.4902v1 [physics.histph] for this version) 
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 forcefree 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 LeviCivita connection
v = dX/ds
In fact when v/c << 1, the 3vector piece of the above 4vector 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) – RePublished in three parts.
Am. J. Phys. 45, Part I  (6), June 1977, pp. 512517; Part II – (9), September
1977, pp. 811816, 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 POST1907 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 (offgeodesic) 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 particleframe)/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 offgeodesic "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 frameDoppler 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 unobservableinprinciple 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 electroweakstrong 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 nonvanishing spacetime curvature is present  a view that seems to have its origins in a neoNewtonian view that large constant potentials can be gauged away as irrelevant. This comports with the widespread view that the AharanovBohm 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 nonMachian 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 scalartensor 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 nonMachian. 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 LyndenBell'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
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