Text Size

### Stardrive

Tag » general relativity
Oct 24

## Sarfatti Lecture in Einstein's General Theory of Relativity Oct 23, 2013

Posted by: JackSarfatti |
Tagged in: Sarfatti Lectures in Physics, general relativity, Einstein's happiest thought, Albert Einstein

from my book

Contrary 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.”

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 = D2X/ds2

Relative 1905 SR kinematic acceleration = d2X/ds2

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) ~ -GMEarthr/r3

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 -GMEarthr/ras 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}V2(test particle) = 0

The "cancellation" is precisely

dV(test particle)/ds - {LNIF detector}V2(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

D2X/ds2 = d2X/ds- {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 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.

Therefore,

DV/ds is measured directly locally by an accelerometer clamped to the test particle  - real measurement 1

dV/ds = d2X/dsis 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 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 A1/2. 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. d2X/ds2

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 19th 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?

Oct 20

## Sarfatti Lecture on General Relativity Oct 20, 2013

Posted by: JackSarfatti |
Tagged in: Mach's Principle, James Woodward, Jack Sarfatti, Isaac Newton, Geometrodynamics, general relativity, fictitious force, Eli Cartan, Albert Einstein

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
____________________________________________________________

Oct 18

## Sarfatti Lecture 1 on General Relativity of Oct 18, 2013

Posted by: JackSarfatti |
Tagged in: Laputa, Jack Sarfatti, Isaac Newton, Gravity - the movie, general relativity, Einstein's happiest thought, Doppler radars, Albert Einstein, accelerometers
##### Jack Sarfatti
Common misunderstanding of Einstein's "Happiest Thought" corrected & bogus-bad pseudo-physics critiques of Einstein's Equivalence Principle.
1
Oct 16

## Sarfatti Lecture in General Relativity of Oct 16, 2013 v1

Posted by: JackSarfatti |
Tagged in: Kornel Lanczos, John Archibald Wheeler, Isaac Newton, general relativity, fictitious force, Einstein's curved spacetime gravity

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
• 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
Mar 04

## Notes for my Star Ship book March 4, 2013

Posted by: JackSarfatti |
Tagged in: Unruh effect, general relativity
• Jack Sarfatti The equation in question is (in index-free short hand)

DU/ds = dU/ds + {LC}UU

This is a tensor equation- a geometric object, and the choices of local coordinate patches of differential geometry are irrelevant.

We are interested in this book in the heuristic physical meaning of the equations not in the excess baggage of formalism that only obscures the essential physics leading many physicists astray into purely mathematical dead ends perhaps important to pure mathematics but not to physics. The Cornell philosophy of Hans Bethe, Ed Salpeter, Phil Morrison, Tommy Gold and Richard Feynman himself was to get the most physics with the least possible mathematics. This is in accord with Einstein's remark that any intelligent fool can make the subject more complicated than it need be. Indeed, this is the trend we see in modern theoretical physics today.

DU/ds is what accelerometers measure locally on the test particle being observed.

{LC}UU is what accelerometers measure locally on the detector observer of the test particle.

dU/ds is the apparent or kinematical 4-acceleration of the test particle relative to the detector.

The test particle and the detector are nearly coincident, i.e. their actual space-time separation must be small compared to the local radii of curvature of 4D spacetime for the equation not to break down. For example on Earth surface that curvature radius is about 10^13 cm, so this is not a problem for local experiments.

The Levi Civita connection {LC} in Einstein's GR physically describes the fictitious inertial pseudo forces that appear to act on test particles. These inertial forces are caused by real forces on the local noninertial frames lnifs measuring the motion of the test particle.

Newton's 2nd Law is for rest mass m constant

F = mDV/ds = real 4-force

V = 4-velocity of test particle

ds = proper time differential

DV/ds = dV/ds - {LC}VV = proper 4-acceleration. it is a gct group tensor

In a local inertial frame lif {LC} = 0

This is Einstein's equivalence principle

In a local non inertial frame lnif

{LC} =/= 0

- m{LC}VV = all the inertial fictitious pseudo forces that seem to act on the test particle from the POV of the properly accelerating lnif detector observer, but don't.

Note the g00 in the denominator that is zero at horizons.

Note also that the quantum Unruh effect in which vacuum zero point virtual photons are transformed into real black body radiation photons is proportional to the local tensor proper acceleration DU/ds of the detector accelerometer.