You are here:
Home Jack Sarfatti's Blog Blog (Full Text Display) Tag Geometrodynamics

Tag » Geometrodynamics

Oct
24

Tagged in: Sagnac effect, inertial navigation, ICBMs, gyroscopes, gravity gradiometers, Geometrodynamics, Einstein's curved spacetime gravity, accelerometers, 100 Year Star Ship

**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.

Oct
20

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 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.

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).

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.

Jim's remark above is unintelligible to me. This is what I mean by "inertial force."

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.)

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 -GM**r**/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.

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

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

____________________________________________________________

Jul
15

Tagged in: Matter-AntiMatter Asymmetry, John Archibald Wheeler, Geometrodynamics, David Bohm