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

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. 1

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 2

Equivalence of Lorentz Contraction & Time Dilation Unstable particles travel down a tube and get through before they decay. τ τ τ τ 0 0 = Lifetime of unstable 0 0 particle when at rest. L 0 To an observer traveling with V the particles, the tube appears shortened: 2 v = − L L 1 ; τ < V L 0 2 c 0 0 τ V 2 τ = > v 0 V L τ > − = V L 1 L 0 2 0 0 v 2 c − 1 2 c 3

A fictitious force-Coriolis force An object dropped from rest near earth’s Plumb equator hits the ground a little to the East line of the vertical through the drop point. East North ⊗ 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: = ω × v F 2 ( m ) Coriolis 4

“+” Gravitational wave in transverse traceless gauge In the “transverse traceless” gauge, for example, a “+” wave traveling in the z-direction is described by:   − 1 0 0 0   + − 0 1 h ct ( z ) 0 0   =  g  µν − − 0 0 1 h ct ( z ) 0     0 0 0 1 5

Gravitational Wave in Fermi Coordinates A coordinate transformation can be constructed such that the metric is   − − 2 2 2 2 1 ( x y ) 1 ( x y ) ɺɺ ɺɺ − + −   1 h 0 0 h 2 2 2 c 2 c     0 1 0 0 =  g  µν 0 0 1 0     − − 2 2 2 2 1 ( x y ) 1 ( x y ) ɺɺ ɺɺ − +   h 0 0 1 h   2 2 2 c 2 c 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. Φ ∆ ∆Φ 2 f = − + = (1 ); . g 00 2 2 c f c 6

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: ∼ Φ 3 2 ( V c / ) ( V c / )( / c ) 7

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-- X µ , G µν , Λ µ (k ) -- ICRF or theoretical model Lower case letters: x µ , g µν −− locally inertial or quasi-inertial coordinates Minkowski metric:   − 1 0 0 0   0 1 0 0   =  η  µν 0 0 1 0     0 0 0 1 8

SMWF approximate solution of field equations, including PPN parameter γ Φ 2 ∑ GM − = − + + Φ = − 4 A (1 ) ( ); G O c 00 − 2 c X X A A i 3 / ∑ GM V c − = − γ + + 5 A A 2( 1) ( ) G O c 0 i − X X A A γ Φ 2 − = δ − + 4 G (1 ) O c ( ) ij ij 2 c A -- index on point masses that contribute to the metric. 9

Moving, rotating source—e.g., earth ( ) = + ω × − V V X X ; A e A e = − + − X - X X X ( X X ) ; A e e A [ ] − × γ + γ + i G ( X X ) J 2( 1) GM V ( 1) = − − e e e e i G 0 i − 3 3 3 − c X X c X X e e 10

Basis Tetrad for normal Fermi coordinates µ Λ = ( ) , 0,1,2,3 basis tetrad; (i) a label i i G--timelike geodesic S--spacelike geodesic P 0 --origin of freely falling reference frame P--general field point Choose: µ dX µ Λ = (0) ds 11

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: µ Λ β d dX µ α ( ) i + Γ Λ = 0 αβ ( ) i ds ds The spatial members of the tetrad can be realized by three mutually orthogonal spins. 12

Christoffel Symbols of the Second Kind 1 ( ) µ µν Γ = + − 2 G G G G αβ να β νβ α αβ ν , , , 13

Ordinary Normal Fermi Coordinates--Spins Falling Freely Let there be three mutually orthogonal vectors falling together along a geodesic with 4-velocity µ dX µ Λ = (0) ds The 4-velocity is timelike; the spins are three spacelike vectors labeled by Latin indices: µ Λ = ( ) ; i 1,2,3 i Take the orthonormality conditions µ ν Λ Λ = η G µν α β αβ ( ) ( ) These four 4-vectors form a “tetrad” which comprise a basis for a locally inertial system of coordinates. We solve the equations of parallel transport along the geodesic to a certain level of approximation. 14

Results   γ + Φ l V 2( 1) Λ = + Λ δ − 0 k 0   G 1 ; O (3) ( ) i 0 i ( ) i kl   2 c c   γ Φ k i V V Λ = δ + + + Ω k k ki 0   1 ; O (2) ( ) i i   2 2 c 2 c ( ) ( γ + Ω ki 1 1 d ) ( ) = − − + Φ − Φ k i G G V V 0 , k i 0 , i k , i , k 3 ds 2 c   i k 1 dV dV + − k i   V V . 3   2 c dT dT All quantities are evaluated on the curve G, a time-like geodesic for the case of free fall. 15

Accelerated Tetrad If the origin is accelerated, the tetrad is carried along the path by Fermi-Walker transport: µ µ DA dA µ α β ≡ + Γ A U ; (shorthand) αβ Ds ds µ dX β = U ; ds µ Λ ν µ D DU DU λ µ λ ν ( ) i = Λ − Λ G U G U . λν λν ( ) i ( ) i Ds Ds Ds 16

Results ( ) ( γ + Ω ki 1 d 1 ) ( ) = − − + Φ − Φ k i G G V V 0 , k i 0 , i k , i , k 3 ds 2 c   i k 1 dV dV + − k i   V V . 3   2 c dT dT 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: Ω jk c d Ω = − ε i ijk 2 ds ε − 3-index alternating symbol ijk 17

Results for tetrad µ dX µ Λ = ; (0) ds   γ + Φ l V 2( 1) Λ = + Λ δ − 0 k  0  1 ; (3) G O ( ) i 0 i ( ) i kl   2 c c   γ Φ k i V V Λ = δ + + + Ω k k ki 0   1 ; O (2) ( ) i i   2 2 c 2 c Ω ik This is for parallel transport; the antisymmetric matrix represents precession of the reference axes. 18

Geodetic Precession--spin in free fall We can separate the gyroscope’s acceleration into a free-fall part And a non-gravitational part: i dV dT = −Φ + i A ; , i NG Then Schiff=geodetic+ free-fall Thomas   1 γ +   Ω ki   d 1 ( ) ( ) 2 = − − + Φ − Φ k i G G V V 0 , k i 0 , i k , i , k 3 ds 2 c 1 ( ) + − k i i k V A V A . NG NG 3 2 c 19

Geodetic Precession--spin in free fall Total rate of precession: � = � + � + � L-T Geodetic Thomas 20

Geodetic Precession--spin in free fall Total rate of precession: � = � + � + � L-T Geodetic Thomas = � + � + � + � ��������� L-T Geodetic Thomas(freefall) Thomas(NG) � Schiff 21

Geodetic Precession--spin in free fall γ + Coefficients ( 1) Total rate of precession: � = � + � + � L-T Geodetic Thomas = � + � + � + � ��������� L-T Geodetic Thomas(freefall) Thomas(NG) � Schiff 1 γ + ( ) Coefficient 2 22

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 of 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 of 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) 23

Gravitomagnetic Precessions Hold the gyroscope fixed. Velocity and Acceleration are zero. Then Ω ki d 1 ( = − − G G ); 0 , k i 0 , i k ds 2 Ω jk c d Ω = − ε . i ijk 2 ds ε − 3-index alternating symbol ijk 24