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Classical Mechanics and Gravity From Newton to Einstein

 

Jack Sarfatti

 

Excerpted from Stargate

Version 12-19-13

 

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 battle-tested 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. Force-free 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 and instantaneous position vector  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 = md2r/dt2 = 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 high-resolution image. However, what we need to do real physics is a much lower resolution compressed image in which certain non-essential 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 action-reaction 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 marble-like 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 space-time, 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 action-reaction principle

Harvey R. BrownDennis 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 action--reaction 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.hist-ph); General Relativity and Quantum Cosmology (gr-qc)

Cite as:

arXiv:1306.4902 [physics.hist-ph]

 

(or arXiv:1306.4902v1 [physics.hist-ph] for this version)

 

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.


On Oct 20, 2013, at 12:20 AM, "jfwoodward@juno.com" <jfwoodward@juno.com> wrote:

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.  
 
60's - paleolithic times in cosmology and in general relativity. See Feynman's letter to his wife at Warsaw GR meeting - it's online.

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.
 
Jim's writing about fictitious forces in his book is hardly intelligible to me. 
 
Jim also seems to be confused about "potentials"
 
There are superficial formal analogies between Einstein's geometrodynamics and Maxwell's electrodynamics, but one must be very careful in applying them.
 
Jim cites Bohm-Aharonov. OK first look at Maxwell's electrodynamics. I use Cartan's forms
 
We have a potential 1-form A that is a connection for parallel transport of objects in the U1 circle fiber space.
 
The gauge transformations are
 
A --> A' = A + df
 
f = 0-form scalar
 
d^2 = 0
 
It's line integrals of A around closed loops that give the observable quantum phase shifts in the Bohm-Aharonov effect via Stoke's theorem etc.
 
The EM curvature is the 2-form
 
F = dA which is gauge invariant
 
F' = dA' = dA + d^2f = dA = F
 
Maxwell's field equations concern the 3-forms
 
dF = d^2A = 0  these are two of Maxwell's equations - no magnetic monopoles and Faradays EMF law (motors, generators ....)
 
d*F = *J   these are the last two - Ampere's law with displacement current and Gauss's law
 
* = Hodge dual
 
Finally
 
d*J = d^2*F = 0
 
is local conservation of electric current densities
 
this is a 4-form in 4D spacetime dual to a 0-form.
 
This gauge theory extends to the non-Abelian unitary groups SU2 and SU3 that describe the weak and strong forces (Yang-Mills).
 
Jim's vector theory if done correctly has
 
g00 = 1 + phi/c^2
 
g0i = Ai
 
However, the analogy to EM as a gauge theory breaks down completely, because the F to Jim's A is the Levi-Civita connection.
 
In fact the proper analogy is that the Levi-Civita connection is the analog to the EM A and the curvature tensor is the analog to EM's F.
 
Conservation of currents is the Bianchi identity in GR.
 
However, to make the analogy more transparent. General relativity as a local gauge theory is a non-Abelian Yang-Mills theory based on the Poincare symmetry group of Einstein's special relativity.
 
Einstein's 1905 Special Relativity mathematically is the representation theory of the global 10-parameter Poincare group.
 
General relativity is properly named because it is a limiting case (zero torsion) of the local gauge theory of the Poincare group with the real gravity field as the curvature 2-form from the connection 1-form just as in Maxwell's electrodynamics.
 
However, the connection 1-form corresponding to Maxwell's A is not the Levi-Civita connection from the usual 1916 GR tensor formulation, rather it is the six spin connection 1-forms AIJ  = - AJI with two LIF indices, IJ analogous to the internal indices Aa in Yang-Mills theory of the SU2 and SU3 internal groups and the 4 tetrad connection 1-forms eI.
 
There are therefore 10 connection 1-forms one for each "charge" generator of the Poincare group (linear-momentum-energy, rotational momentum, Lorentz boosts)
 
The Levi-Civita connection is derivable from the spin connections and the tetrads.
 
The four eI are the base 1-forms for a geodesic LIF dual to the tangent vector fiber space basis.
 
The spin connection allows coupling of gravity to spinors, the Levi-Civita connection does not.
 
Therefore, Einstein's 1916 geometrodynamics reformulated in modern Cartan-forms has the local gauge structure
 
D = d + SIJ/   Cartan exterior covariant derivative
 
summation convention over repeated indices - I am too lazy to put in the ^ for upper indices.
 
TI = DeI = deI + SIJ/eI = dislocation defect torsion field 2-form
 
RIJ = DSIJ = dSIJ + SIK/SKJ = disclination defect curvature field 2-form.
 
Einstein's 1916 theory requires the ad-hoc constraint
 
TI = 0
 
In that limit:
 
Einstein-Hilbert action density is the 0-form scalar without cosmological constant for simplicity
 
*(eI/eJ/RKL)
 
with Euler-Lagrange equation for vacuum is the 1-form equation
 
*(eI/RJL) = 0
 
in ordinary tensor language this is
 
Ruv - (1/2)guv = 0
 
Including the matter-field sources gives
 
*(eI/RJK) = (8piG/c^4)*(TIJK)
 
More details are in Rovelli's lectures http://www.cpt.univ-mrs.fr/~rovelli/book.pdf
 
 
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.
 
I find above comment by Jim unintelligible - at least at the present time.

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).
 
I am staring at those pages and I see nothing in Wilzcek's text that justifies Jim's extraordinary claim above. Certainly nothing that needs Mach's principle that simply replaces one mystery with another. Again Jim is confounding two different meanings of "inertia" just as he and other confound two different meanings of "gravity field".
 
Mach's Principle only is concerned with how matter affects disclination geodesic deviation (aka curvature). The real gravity field of Einstein's geometrodynamics is the field of "geodesic deviation" corresponding to inhomogeneities in Newton's "gravity field", which is a fictitious force field.
http://en.wikipedia.org/wiki/Geodesic_deviation
http://en.wikipedia.org/wiki/Fictitious_force
Mach's principle is not concerned with the origin of rest masses of elementary particles. Einstein briefly confounded the two, but it led nowhere.  Wilzcek is concerned in those pages 200 - 202 with the cosmic landscape/Anthropic principle issue. Why these particular numbers and not others. http://www.fourmilab.ch/documents/reading_list/indices/book_487.html
 
A word on history. What Einstein may or may not have said in 1907 in his informal language as he groped toward GR is completely irrelevant to the modern understanding of general relativity. This is a normal evolution of all good physical theories. I have no patience with cranks that try to make a big deal over that. Such discussions are a waste of time for me.

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.)
 
Jim's remark above is unintelligible to me. This is what I mean by "inertial force."
 
inertial force  (-nûrshl)
An apparent force that appears to affect bodies within a non-inertial frame, but is absent from the point of view of an inertial frame. Centrifugal forces and Coriolis forces, both observed in rotating systems, are inertial forces. Inertial forces are proportional to the body's mass. See also General Relativity.
 
Newton's gravity force per unit test mass -GMr/r^3 is an inertial force in exactly the same way as centrifugal and Coriolis forces are.
 
They are all part of the Levi-Civita connection which vanishes at the origin of a Local Inertial Frame (LIF).
 
 
The "force of gravity" you feel as weight on Earth is the unbalanced electrical force pushing you off a timelike geodesic of the local curvature real gravity field mostly due to the mass of Earth. You need that unbalanced force on you to keep you still (with respect to Earth) in the curved spacetime we live in. Earth pushes up on you and you push down on Earth etc. - action-reaction Newton's 3rd law.
Therefore, I find Jim's discussion of inertial forces here and in his book unintelligible and not mainstream.

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.  
 
I also find "phi = c^2" unintelligible and not mainstream physics.
 

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.
 
Unintelligible to me still as of this date.

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.
 
OK

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.
 
OK

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.
 
Unintelligible. EEP follows trivially once one understands that Newton's gravity force is simply the fictitious force on the weightless geodesic test particle as seen visually in a static LNIF from real electrical forces pushing the static LNIF off a local timelike geodesic.
 

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. . . .
 
Unintelligible.

Keep the faith,
 
Sorry Jim but the faith required here is not scientific in my opinion.

Jim
____________________________________________________________

 

Common misunderstanding of Einstein's "Happiest Thought" corrected & bogus-bad pseudo-physics critiques of Einstein's Equivalence Principle.
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  • Rata Vuloira likes this.
  • Jack Sarfatti Einstein wrote in ~ 1907: "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.”
    4 minutes ago · Like · 1
  • Jack Sarfatti Those quotes are from early Einstein around 1907 and Jim Woodward repeats what I said repeatedly that Einstein himself was still unclear in his own mind on how to use words like "accelerated frame" back then. He was in middle of breaking away from Newton's GRIP on the mind of how to think about gravity.
    3 minutes ago · Like · 1
  • Jack Sarfatti Here is the source of the confusion.

    Einstein is naturally thinking in Newtonian terms.

    In GR terms still not invented then

    "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 test particle

    V = dX/ds

    X = relative kinematical displacement between test particle and local frame detector.

    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

    D^2X/ds^2 = d^2X/ds^2 - {LC frame connection}(dX/ds)^2

    {LC frame connection}(dX/ds)^2 = M^-1DP(frame)/ds

    M = mass of frame/detector

    {LC frame connection} has dimension 1/Length

    ds is 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.
    en.wikipedia.org
    An accelerometer is a device that measures proper acceleration. The proper accel...See More
  • Jack Sarfatti In contrast, the kinematic acceleration is measured by Doppler radars

    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^2X/ds^2 is measured indirectly by the Doppler radar clamped to the local frame detector - real measurement 2

    M^-1DP(frame)/ds is measured directly by a second acclerometer 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 A^1/2 where A^-1 is smallest local radius of curvature.

    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.
    en.wikipedia.org
    A Doppler radar is a specialized radar that makes use of the Doppler effect to p...See More
  • Jack Sarfatti 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^2X/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.
  • Jack Sarfatti “there is a new gravitational field,
    which cancels the gravitational field
    due to the Earth.”

    This is Einstein's remark that Z and other muddled philosophers and Laputa Scholastics pulls out of proper context. Yes, Einstein wrote it back around 1907 before he understood the problem the way he eventually would in 1916. 

    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 mechanical aether these two forces are not independently measurable - they are errors of thinking - excess metaphysical informal language baggage.

 

Oct 16, 2013 from Jack Sarfatti’s Stargate book under construction
Returning to Wheeler:
1) Equivalence principle
2) Geometry
3) Geodesic equation of motion of point test particles (aka Newton’s 1st Law first-order partial derivatives of the metric tensor field describe fictitious inertial pseudo-forces on the test particle corresponding to real forces on the detector)
4) Intrinsic tensor curvature geodesic deviation (disclinations of vectors parallel transported around closed loops in spacetime) from second order partial derivatives of the metric tensor field describing relative covariant tensor accelerations between two neighboring geodesic test particle each with zero g-force proper acceleration
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  • Jack Sarfatti One must use the LIF to distill the intrinsic geometry of the real Einstein gravity field. The LNIF is fool’s gold, MAYA, illusion, the shadow on the wall of Plato’s Cave that has ship wrecked many a careless mariner including Isaac Newton listening to the wiles of Circe. The LNIF is contingent random noise, all sound and fury a tale told by an idiot, and believed by sorry bastards, a fairy tale, a mask. Only Einstein escaped the Cave that Newton was trapped in. Of course, Newton had a good excuse. Newton’s “gravity force” is simply the real quantum electrodynamic force sustaining the static LNIFs. It is a fictitious pseudo-force as far as the observed test particle is concerned without any intrinsic objective reality, same ontic status as Coriolis and centrifugal pseudo-forces all parts of the LNIF Levi-Civita Christoffel symbols that depend only on first order partial derivatives of the metric tensor field. Einstein’s equivalence principle (EEP) relegates them to Prospero’s phantoms, the illusions of the Wizard of Oz behind the theater curtain of the world stage.
  • Jack Sarfatti There are three levels of the equivalence principle:
    1) Weak – uniqueness/universality of free fall known to Galileo – the motion of any freely falling point test particle (or center of mass of an extended object) in vacuum is independent of its composition and structure. “A test particle is … electrically neutral … negligible gravitational binding energy compared to its rest mass … negligible angular momentum … [negligible] inhomogeneities of the gravitational field within its volume … the ratio of inertial mass to the gravitational passive mass is the same for all bodies.” In every LIF the path of a force-free geodesic test particle is a straight line with constant speed in accord with Einstein’s 1905 special theory of relativity that works increasingly well as the scale shrinks compared to the scale of curvature radii until quantum gravity is reached where the curvature field itself has large random zero point quantum fluctuations. Although this scale is thought to be 10-35 meters, the hologram conjecture combined with cosmology give a quantum gravity scale that is twenty powers of ten larger at 10-15 meters ~ (Planck length x area-entropy of our future dark energy de Sitter event horizon) 1/3.
    2) Medium strong – metric theories of gravity. Einstein went beyond the weak form to the hypothesis that all the non-gravity laws of physics obey special relativity in a LIF in the same shrinking limit as above.
    3) Very strong – replace non-gravity laws of physics with all the laws of physics.
    In this book we assume 3) the very strong form as there is no experimental evidence yet that it is false.
  • Jack Sarfatti Fermi Normal Coordinates for the LIF’s Image of Intrinsic Geometry
    “The metric tensor can indeed be written using the Riemann (curvature) tensor, in a neighborhood of a spacetime event, in a freely falling non-rotating local inertial frame to second order in the separation δxi from the origin” where i,j,k,l are spacelike (outside local light cones with origins at the spacetime event of interest) 1,2,3 indices. The Taylor series expansion to lowest non-vanishing order for the LIF is 
    g00 ~ - 1 – R0i0jδxiδxj for the gravity redshift
    g0k ~ - (2/3)R0ikj δxiδxj for the LIF drag gravimagnetic field
    gkl ~ δkl – (1/3)Rkilj δxiδxj for the curved spacelike 3-geometry
    Next, consider what the physically coincident LNIF metric looks like including the first order terms that are zero in the LIF. Here u,v,w,z = 1’,2’,3’ for LNIF like i,j,k,l = 1,2,3 for the coincident LIF.
    g’0’0’ ~ - 1 – Γu0’0’δxu – R’0’u0’vδxuδxv 
    g’0’v ~ - Γu0’vδxu - (2/3)R’0’uvw δxuδxw 
    g’uv ~ δuv - Γwuvδxw – (1/3)R’uwvz δxwδxz 
    Newton’s gravity force is purely 100% fictitious and corresponds to the first order in separation δxu from the origin of the LNIF Levi-Civita connection Γ terms, which by the equivalence principle, vanish in the physically coincident LIF.
    Kornel Lanzcos in “On the Problem of Rotation in the General Theory of Relativity” proved that in any LNIF for test particle rest mass m:
    1) mg0’0’-1Γu0’0’ independent of the test particle’s velocity corresponds both to Newton’s gravity fictitious force – GMmr/r3 in the particular contingent choice of the static LNIF and to the centrifugal force mwxwxr in the particular contingent choice of a uniformly rotating LNIF with angular momentum pseudo-vector w along the rotation axis. That we are in the slow speed weak curvature limit is understood.
    2) 2mg0’0’-1Γu0’v’dxv/dτ linear in the velocity of the test particle is the Coriolis fictitious force 2mwxv analogous to the magnetic Lorentz force in Maxwell’s electrodynamics and to the vortex force in irrotational hydrodynamics. The Greek symbol τ refers to proper clock time along the world line of the test particle.
    3) Finally, mg0’0’-1Γuvw (dxv/dτ) (dxw/dτ) quadratic in the velocity of the test particle is also a fictitious force that has no name and is usually too small to measure. 
    All of these fictitious forces blow up at horizons where the LNIF g0’0’ vanishes.
    The relative covariant tensor acceleration between two freely-falling geodesic test particles each with zero local proper tensor acceleration, is
    d2δxα/dt2 ~ Rα0μ0δxμ equation of geodesic deviation