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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 20th Century correction to Newton’s 17th century gravity theory: Sa 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 xa (s) with tangent vector ua.  Saua = 0 and the equation for Fermi-Walker transport is

Sa;bub = ua (abSb) = ua(ub;gugSb)  (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

ab = ub;gug

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 ab = 0. Therefore,

Sa;bub = 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.

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.