Quasi-Inertial Coordinates and Gyroscopic Precession Neil Ashby - - PowerPoint PPT Presentation

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Quasi-Inertial Coordinates and Gyroscopic Precession

Neil Ashby Department of Physics University of Colorado, Boulder, CO 80309-0390 USA Affiliate, National Institute of Standards and Technology email: ashby@boulder.nist.gov PURPOSE: To develop “non-rotating” coordinate systems that provide alternative ways to describe gyroscopic precession phenomena. In relativity there are usually several different ways to look at a given phenomenon.

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Outline

• Purpose
• Examples of equivalent points of view
• Normal Fermi coordinates
• Quasi-inertial coordinates
• Applications

– Frame-dragging near ring, disc – precession of satellite orbits – precession of gyroscope in earth orbit

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Equivalence of Lorentz Contraction & Time Dilation

V L0

τ τ τ τ0

0 = Lifetime of unstable

particle when at rest.

2 2

1 V L V V L v c τ τ τ < = > −

To an observer traveling with the particles, the tube appears shortened: Unstable particles travel down a tube and get through before they decay.

2 2 2 2

1 ; 1 v L L c v V L L c τ = − > − =

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A fictitious force-Coriolis force

Plumb line East An object dropped from rest near earth’s equator hits the ground a little to the East

• f the vertical through the drop point.

From the viewpoint of a locally inertial (non-rotating) frame, this is due to conservation of angular momentum. From the viewpoint of an earth-fixed reference frame, it is due to a fictitious Coriolis force induced by rotation:

2 ( )

Coriolis

m = F ω× v

⊗

North

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“+” Gravitational wave in transverse traceless gauge

1 1 ( ) 1 ( ) 1 h ct z g h ct z

µν

−     + −   =   − −    

In the “transverse traceless” gauge, for example, a “+” wave traveling in the z-direction is described by:

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Gravitational Wave in Fermi Coordinates

A coordinate transformation can be constructed such that the metric is

2 2 2 2 2 2 2 2 2 2 2 2

1 ( ) 1 ( ) 1 2 2 1 1 1 ( ) 1 ( ) 1 2 2 x y x y h h c c g x y x y h h c c

µν

  − − − + −       =       − − − +     ɺɺ ɺɺ ɺɺ ɺɺ

In the (x,y) submanifold (z=0), for a test mass at rest the spatial metric is Minkowskian. The 00 component of the metric tensor gives a quadrupolar spring-like force, and provides a way of computing the effect on a clock. The higher the frequency of the wave, the stiffer the spring.

00 2 2

2 (1 ); . f g c f c Φ ∆ ∆Φ = − + =

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Quasi-Inertial Coordinates

Such systems of coordinates (a) don’t rotate (reference axes don’t precess) (b) are as close to inertial as possible. This will be done in the slow-motion, weak-field limit. Calculations are taken to an order that is appropriate for slowly moving objects in weak fields:

3 2

( / ) ( / )( / ) V c V c c Φ ∼

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Notation

Greek indices such as µ, ν, run from 0 to 3. Latin indices run from 1 to 3. (up-down summation convention ignored) Capital letters--Xµ, Gµν , Λµ

(k) -- ICRF or theoretical model

Lower case letters: xµ, gµν −−locally inertial or quasi-inertial coordinates Minkowski metric:

1 1 1 1

µν

η −       =      

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SMWF approximate solution of field equations, including PPN parameter γ

4 00 2 3 5 4 2

2 (1 ) ( ); / 2( 1) ( ) 2 (1 ) ( )

A A A i A A i A A ij ij

GM G O c c GM V c G O c G O c c γ γ δ

− − −

Φ = − + + Φ = − − = − + + − Φ = − +

∑ ∑

X X X X

A -- index on point masses that contribute to the metric.

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Moving, rotating source—e.g., earth

( ) [ ]

3 3 3

; ( ) ; ( ) 2( 1) ( 1)

A e A e A e e A i e e e e i i e e

G GM V G c c γ γ = + × − = − + − − × + + = − − − − V V ω X X X - X X X X X X X J X X X X

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Basis Tetrad for normal Fermi coordinates

(0)

dX ds

µ µ

Λ =

G--timelike geodesic S--spacelike geodesic P0--origin of freely falling reference frame P--general field point

( ),

0,1,2,3 basis tetrad; (i) a label

i i µ

Λ =

Choose:

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Parallel Propagation of Tetrad

For construction of Normal Fermi coordinates, the four members of the tetrad are carried along the timelike geodesic by parallel transport:

( ) ( ) i i

d dX ds ds

µ β µ α αβ

Λ + Γ Λ =

The spatial members of the tetrad can be realized by three mutually

• rthogonal spins.

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Christoffel Symbols of the Second Kind

( )

, , ,

1 2 G G G G

µ µν αβ να β νβ α αβ ν

Γ = + −

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Ordinary Normal Fermi Coordinates--Spins Falling Freely

Let there be three mutually orthogonal vectors falling together along a geodesic with 4-velocity

(0)

dX ds

µ µ

Λ =

The 4-velocity is timelike; the spins are three spacelike vectors labeled by Latin indices: ( );

1,2,3

i

i

µ

Λ =

Take the orthonormality conditions

( ) ( )

G

µ ν µν α β αβ

η Λ Λ =

These four 4-vectors form a “tetrad” which comprise a basis for a locally inertial system

• f coordinates. We solve the equations of parallel transport along the geodesic

to a certain level of approximation.

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Results

( )

( )(

)

( ) ( ) 2 ( ) 2 2 0 , 0 , , , 3 3

2( 1) 1 ; (3) 1 ; (2) 2 1 1 2 1 . 2

l k i i i kl k i k k ki i i ki k i k i i k i k i k k i

V G O c c V V O c c d G G V V ds c dV dV V V c dT dT γ δ γ δ γ + Φ   Λ = + Λ −     Φ   Λ = + + + Ω     + Ω = − − + Φ − Φ   + −    

All quantities are evaluated on the curve G, a time-like geodesic for the case of free fall.

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Accelerated Tetrad

If the origin is accelerated, the tetrad is carried along the path by Fermi-Walker transport:

( ) ( ) ( )

; (shorthand) ; .

i i i

DA dA A U Ds ds dX U ds D DU DU G U G U Ds Ds Ds

µ µ µ α β αβ µ β µ ν µ λ µ λ ν λν λν

≡ + Γ = Λ = Λ − Λ

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Results

2

jk i ijk

c d ds ε Ω Ω = −

( )

( )(

)

0 , 0 , , , 3 3

1 1 2 1 . 2

ki k i k i i k i k i k k i

d G G V V ds c dV dV V V c dT dT γ + Ω = − − + Φ − Φ   + −    

The same form is valid for an accelerated tetrad transported by Fermi-Walker transport. All quantities are evaluated on the time-like curve G. Angular velocity vector corresponding to the rotation matrix:

ijk

ε −

3-index alternating symbol

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Results for tetrad

(0) ( ) ( ) 2 ( ) 2 2

; 2( 1) 1 ; (3) 1 ; (2) 2

l k i i i kl k i k k ki i i

dX ds V G O c c V V O c c

µ µ

γ δ γ δ Λ = + Φ   Λ = + Λ −     Φ   Λ = + + + Ω    

This is for parallel transport; the antisymmetric matrix represents precession of the reference axes.

ik

Ω

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Geodetic Precession--spin in free fall

We can separate the gyroscope’s acceleration into a free-fall part And a non-gravitational part:

,

;

i i i NG

dV A dT = −Φ +

Then

( )

( ) ( )

0 , 0 , , , 3 3

1 1 2 2 1 . 2

ki k i k i i k i k k i i k NG NG

d G G V V ds c V A V A c γ   +   Ω   = − − + Φ − Φ + −

Schiff=geodetic+ free-fall Thomas

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Geodetic Precession--spin in free fall

L-T Geodetic Thomas

= + +

• Total rate of precession:

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Geodetic Precession--spin in free fall

Schiff

L-T Geodetic Thomas L-T Geodetic Thomas(freefall) Thomas(NG)

= + + = + + +

• Total rate of precession:

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Geodetic Precession--spin in free fall

Schiff

L-T Geodetic Thomas L-T Geodetic Thomas(freefall) Thomas(NG)

= + + = + + +

• Total rate of precession:

Coefficients (

1) γ +

Coefficient

1 2

( ) γ +

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Interpretations of Geodetic Precession: The de Sitter precession may be thought of as having contributions from two sources: The first is the effect of mass on the curvature

• f space, which results in locally measured angles differing from

those measured with respect to the fixed stars. The second source, which contributes half as much as the first, is the gravitational analog

• f the spin-orbit coupling of an electron in an atom.
• I. I. Shapiro, Phys. Rev. Letts. 61, 2643 (1988)

(the 3/2 term)…is essentially just the Thomas Precession caused by gravitation.

• S. Weinberg, Gravitation and Cosmology, p. 237

For a particle moving along a geodesic, the Thomas Precession is identically zero.

• S. Vokos, arXiv:hep:ph/9304260 (1993)

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Gravitomagnetic Precessions

0 , 0 ,

1 ( ); 2 . 2

ki k i i k jk i ijk

d G G ds c d ds ε Ω = − − Ω Ω = −

Hold the gyroscope fixed. Velocity and Acceleration are zero. Then

ijk

ε −

3-index alternating symbol

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“Frame-dragging” precession near a rotating ring

2 3

1 2 G c d γ + = −

• J

A B At point A, in the middle of the ring that rotates with angular momentum J, the spin precession rate is At point B, a distance d from the ring’s center,

2 3

1G c a γ + =

• J

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“Geodetic” precession of freely falling gyroscope

2 3

1G c d γ + =

• L

Non-rotating Mass Source gyroscope in

• rbit

L is the orbital angular momentum

• f the mass source relative to the
• gyroscope. Here the gyroscope is

accelerated so there will be an additional Thomas precession.

( ) ( ) ( ) ( ) − − = L = d × MV d × MV

d

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Quasi-Inertial Coordinates

Rather than allow the basis vectors of the tetrad to precess, we construct a non-rotating coordinate system in which the basis vectors are transported along a path, but without rotation. The tetrad is assumed to be:

(0) ( ) ( ) 2 ( ) 2 2

; 2( 1) 1 ; (3) 1 0* ; (2) 2

l k i i i kl k i k k ki i i

dX ds V G O c c V V O c c

µ µ

γ δ γ δ Λ = + Φ   Λ = + Λ −     Φ   Λ = + + + Ω    

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Basis Tetrad for Quasi-Inertial or Locally Inertial (Normal Fermi) coordinates

(0)

; . dX V but no rotation of tetrad ds

µ µ µ

Λ = =

G--timelike geodesic S--spacelike geodesic P0--origin of freely falling reference frame P--general field point

( ),

0,1,2,3 basis tetrad; (i) a label

i i µ

Λ =

Choose:

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Construction of coordinate transformation

Local coordinate time: ; ( );

P O

x ds dX K P ds = ≡

∫

( )

( ) ; ( 1,3).

i i

dX P i d

µ µ

α λ = Λ =

• 1. A spacelike geodesic is dropped from the field point P

along S to the origin P0, where it intersects the basis tetrad with direction cosines αi so that:

• 2. Let λ be the proper distance along S from the field point P

to P0. Then the spatial coordinates are taken to be

.

i i

x λα =

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Coordinate transformation equations

2 3 2 3 2 3 ( ) ( ) ( ) , ( ) ( ) ( )

1 1 ( ) ( ) ... 2 6 1 ( ) 2 1 ... 6

k k l k k l k l m k l m

dX d X d X X P X P d d d X P x x x x x x

µ µ µ µ µ µ µ µ α β αβ µ α β γ αβ γ

λ λ λ λ λ λ = + + + + = + Λ − Γ Λ Λ − Γ Λ Λ Λ +

These are constructed from a Taylor series in powers of the proper distance from P to P0:

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Coordinate transformation--results

( )

2 2 2 2 2 2 2 2 ,0 , 0 , 3 2 , 2 2 2 , 2 2

; 1 ; 2 2 1 2 1 1 ; 2 2 ( ) (1 ) . 2 2

k l kl P j j O j i j j i j j j k k k k k

dX V r x x K ds c c V X K dx G x c c c r x G x x c c x X X P x c c r V c c δ γ γ γ γ γ Φ = = = − + + Φ   = + − + +     Φ − Φ + + Φ Φ = + + + Φ + −

∫

V r V r V r i i i

Linear terms in local coordinates--locally “Minkowskian” metric Quadratic terms—spin precessions Cubic terms—tidal forces Quartic terms—field equations Only quadratic terms are computed here.

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Transformation coefficients

To calculate the metric tensor in quasi-inertial coordinates, we need

. X X g G x x

α β µν αβ µ ν

∂ ∂ = ∂ ∂

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Metric tensor in quasi-inertial coordinates

3 , 00 2 1

( ) 1 ( ),

k k k i j k

A x g O x x c

=

+ Φ = − + +

∑

where Ak is the acceleration of the tetrad:

,

.

k k k k NG

dV A c A dX = = −Φ +

Thus when the origin is in free fall, only quadratic terms (tidal terms) will contribute to this component of the metric tensor.

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Metric tensor in quasi-inertial coordinates

Similarly, the spatial components of the metric tensor are of the form:

( ).

i j ij ij

g O x x δ = +

Only quadratic terms contribute. There are no spatial curvature terms.

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Metric tensor--space-time terms

( ) ( ) ( )

0 , 0 , , , 3 , 3 3

1 1 2 1 . 2

j i m i i j j i i m i m m i i

g G G x V x c x V A V c c γ γ + = − + Φ − Φ Φ + − − V r V r A r i i i

Comments: (a) There are linear terms in spatial coordinates. This is due to using a non-rotating basis tetrad. (b) The acceleration terms (third group) will combine with the second group if the quasi-inertial frame is in free fall: (c) All spin precessions will arise from these off-diagonal components of the metric tensor. (d) The last term doesn’t contribute to spin precessions.

1 ( 1) ( ). 2 γ γ + → +

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Spin precession in the quasi-inertial frame

(1,0,0,0); 1. u u u

µ µ µ

= = −

0. dS S ds

µ µ α α

+ Γ =

0; 0. S u S

µ µ =

=

Place a gyroscope at rest at the origin of the frame. The four- velocity has only one component: Thus, Fermi-Walker transport reduces to parallel transport: For a spin at rest, the Pirani condition is:

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Spin precession in a quasi-inertial frame--summary

0 , 0 ,

0; 1 ( ) 2

k k i i k i k i i k

dS S ds g g + Γ = Γ = −

Note:

( ). ∇Φ − ∇Φ = × ∇Φ × V r r V r V i i

( ) ( )

0 , 0 , , , 3 , 3

1 2

1 2 .

j i m i i j j i i m i m m

g G G x V x c x V c γ γ + = − + Φ − Φ Φ − V r i

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Precession Rate

L-T Schiff

( ) d dt = + × S

• S

Schiff 2

1 2

c γ + = ∇Φ ×

• V

Now for a digression to discuss “effacement.”

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Freely-falling frame centered at the earth

[ ]

( )

3 3 3 , , , 3 3 3

2( 1) ( 1) ( ) ; ; 1 2( 1) 2

i e e e i e e e i e e i m i e m m i m e e i i e m e i

V G GM G c c x V V g G x V V c c c γ γ γ γ γ + Φ + = − × Φ = −     +     Φ + Φ   = − − + Φ − Φ         X - X J X - X X - X

⊙ ⊙ ⊙

The motional contributions arising from earth’s orbital velocity cancel, leaving terms responsible for Lense-Thirring precession, plus Schiff precession. Background metric includes terms that are “effaced.” For example, The singular terms in the metric are effaced; the coordinate transformation is constructed without such terms and applied to the full metric.

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Example-local frame of earth “Effacement”

Leave out the potential contributions that have singularities at earth’s center; derive the transformation to quasi-inertial coordinates; apply the transformation to the complete metric. The results can be expressed as:

00 2 2

2 (1 ) ( ); 2 (1 ) ( );

i j e i j e ij ij

g O x x c g O x x c γ δ Φ = − + + Φ = − +

The quadratic terms are tidal contributions from the Sun. Such quadratic terms occur whether one constructs a locally inertial frame centered at the earth, or a quasi-inertial frame.

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Example: “Geodetic” precession of earth-centered spins

;

• rbital velocity of earth's center of mass

earth's acceleration toward sun

e e

GM R Φ = Φ = − − − = −∇Φ = V A

⊙ ⊙ ⊙

( ) ( ) ( )

, , 3 3 , , 3

1 1 2 1 2

i m i i i e i e m e e e e i m e i e m

g V x A V c c V x c γ γ + = Φ − Φ + − + = Φ − Φ V r V r A r V r i i i i

The model consists only of one contribution to the potential from a non-rotating sun.

2

1 2 ;

e

d dt c γ + = = ∇Φ × S ×S

• V

⊙

The angular velocity is normal to the ecliptic plane, 19.2 mas/yr.

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Two sources--Earth and Sun

( )

3 3 3

; 2( 1) ( 1) ( )

e e i e e i e i e e

GM GM X GM V G G c c γ γ Φ = − − − + + = − + − × − − X X X X J X X X X

⊙

Consider a gyroscope in free fall about the earth, that in turns orbits the sun. No effacement is needed. For this model we take On transforming to the quasi-inertial coordinate frame centered on the gyroscope, the result is as given earlier:

( ) ( ) ( )

0 , 0 , , , 3 , 3 3

1 1 2 1 . 2

j i m i i j j i i m i m m i i

g G G x V x c x V A V c c γ γ + = − + Φ − Φ Φ + − − V r V r A r i i i

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Total velocity and potential

, , ,

; .

e s k k e k

= + Φ = Φ + Φ V V V

⊙

Schiff 2 Averages to 0

1 2 19.2mas/yr 6.6"/yr

e e e s

c γ +     = ∇Φ × + ∇Φ × + ∇Φ × + ∇Φ ×     ↓ ↓

s e

• V

V V V

⊙ ⊙

• The “geodetic” part of the precession, including Thomas Precession

when the gyroscope is in free fall, is In addition the average frame-dragging due to the angular momentum of earth is:

2 3

( 1) ( 3 ( )) 42mas/yr 2

LT

c X γ + = − + =

• J

N N J i

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Aberration

Consider a null vector, tangent to the path of a light ray from a distant star. In the background metric, the null vector is

( )

2

( , ); 1. 1 1 .

i i j ij i i

N N L L L N G L c

µ

δ γ = = + Φ = − +

Transform this vector to quasi-inertial coordinates by:

. x n N X

µ µ ν ν

∂ = ∂

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Transformation Coefficients

The inverse of the matrix of transformation coefficients previously calculated is needed:

( )

2 2 2 2 2 2 2 2 2 2 2

1 ; 2 2 1 1 ; 2 1 ; 2 1 ; 2

i i i i i i i j i ij j j

x V X c c x V V G X c c c x V V X c c c x V V X c c γ γ γ δ ∂ Φ = + + ∂ + Φ   ∂ = − − + −   ∂     ∂ Φ = − − +   ∂   ∂ Φ   = − + − Ω   ∂  

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Null vector in local inertial frame

2 2 2 2 2 2 2 2 2 2

(2 1) 1 1 ; 2 2 (2 1) 1 1 2 2

k k k kl l

V V n c c c c c V V n L L c c c c c γ γ γ γ   + Φ Φ = − − + − +       Φ + Φ   = − − − − + − Ω         V L V L i i

All quantities are evaluated at the origin of the local inertial coordinates where the gyroscope is. The most important correction is aberration, the V/c term in the second equation. Other corrections include Lorentz contraction, resynchronization of clocks, and rescaling of lengths due to external gravitational potentials. Relativistic combination of velocities must be performed if accuracy of O(1/c3) is needed.

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…The heaven is spherical in shape, and moves as a sphere; …the earth is in the middle of the heavens; …the earth does not have any motion from place to place, either. REPHRASED: …the satellite is in the middle of the heavens, the satellite does not have any motion from place to place…. Claudius Ptolemy Almagest, ~ 150 A.D.

A Possible Coordinate System:

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The cloak of authority??

Rome, 1990

Ignazio Ciofolini’s jacket

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END

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Gravity Probe B

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What is Geodetic Precession?

Line of nodes of Moon’s

• rbit lies

parallel to AC

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Defining angular momentum in General Relativity

Consider the antisymmetric combination of two vectors

S X P X P

αβ α β β α

= −

To construct a spin vector from a second-rank tensor, the only tensor objects available are the 4-velocity and the completely antisymmetric tensor that has values

1, 0 ±

, . U µ

αβγδ

ε

1 ; 0. 2 U S S S U c

β γδ α α αβγδ α

ε   = − → =    

This can be inverted:

U S S c

γ αβ αβγδ δ

ε   =    

For example, for a spin at rest, 23 1

pluscylicpermutations;

0; S S S = =

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Geodetic Precession--spin in free fall in a potential

( )(

) ( )

, , 2 2 , 3/2 2 2 2 2 2 2

1 1 . 2 (when in free fall)

ki k i i k i k k i i i i i

d V V dT c dV dV V V c dT dT GM GMX X X Y Z X Y Z dV dT γ + Ω = Φ − Φ   + −       ∂ Φ = − =   ∂ + + + +   = −

2 3

1 ( ) ( ) 2 G M c X γ = + ×

• X

V

Assume that

0;

k

G =