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VERSION LAST UPDATED NOVEMBER 1997 [GEMAPP,GEMREF Dec 1999]: PRINTED IN USA
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 [0


]
Table of Contents:
(C:/bobfiles/TEXFILES/gem/gem.toc
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 [1

] [2] [3])
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List of Figures:
[4] (C:/bobfiles/TEXFILES/gem/gem.lof)
\tf@lof=\write4
 (C:/bobfiles/TEXFILES/gem/gem0.tex
[5

]
Preface.
[6

] [7])) (C:/bobfiles/TEXFILES/gem/gem1.tex
Chapter 1.
 Introduction
Section 1.1. Motivation: Local Special Relativity plus Rotating Coordinates.
Figure 1.1.
[1



] [2]
Section 1.2. Why bother?.
[3]
Section 1.3. Starting vocabulary.
[4] Table 1.1 [5] awkward??
Section 1.4. Historical background.
[6] [7] [8]
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[9]
Section 1.5. Notation and conventions.
[10]) [11] (C:/bobfiles/TEXFILES/gem/gem2a.tex
Chapter 2.
 The congruence point of view and the measurement process
Section 2.1. Algebra.
[12



]
Subsection 2.1.1. Observer orthogonal decomposition.
[13] [14]
Add electric and magnetic parts of 2-form, curvature 4-form.
Subsection 2.1.2. Observer-adapted frames.
[15] [16]
Edit here: (electric and magnetic fields)
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[17]
Revise this section according to articles:
Subsection 2.1.3. Relative kinematics: algebra.
Figure 2.1.
[18] [19] [20]
TO DO:
Subsection 2.1.4. Splitting along a parametrized spacetime curve.
Subsection 2.1.5. Splitting along a test particle worldline.
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[21]
Subsection 2.1.6. Addition of velocities and the aberration map.
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[22]
Section 2.2. Derivatives.
Subsection 2.2.1. Natural derivatives.
[23]
Subsection 2.2.2. Covariant derivatives.
[24]
Subsection 2.2.3. Kinematical quantities.
[25] check?? [26]
Subsection 2.2.4. Splitting the exterior derivative.
[27]
Subsection 2.2.5. Splitting the differential form divergence operator.
[28] check?? (-1)^p inconsistent
Subsection 2.2.6. Spatial vector analysis.
change this dyad stuff??
[29]
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[30]
Subsection 2.2.7. Ordinary and Co-rotating Fermi-Walker derivatives.
[31]
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[32]
Subsection 2.2.8. Relation between Lie and Fermi-Walker temporal derivatives.
[33] [34] shear and Lie; check check shear relative velocity eq?? [35]
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[36]
Update this section according to articles:
Subsection 2.2.9. Total spatial covariant derivatives.
[37] [38]
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[39]
Subsection 2.2.10. Splitting the total covariant derivative.
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[40]
Section 2.3. Observer-adapted frame derivatives.
Subsection 2.3.1. Natural frame derivatives.
[41]
clean up here??
Subsection 2.3.2. Splitting the connection coefficients.
[42]
Subsection 2.3.3. Observer-adapted connection components.
[43]
Subsection 2.3.4. Splitting covariant derivatives.
[44] [45] check check?? previous too
Subsection 2.3.5. Observer-adapted components of total spatial covariant deriva
tives.
[46] when is Lie transport comp with ON condition??
Change symbol for rotation matrix from kernel letter R
[47]
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[48]
Section 2.4. Relative kinematics: applications.
Subsection 2.4.1. Splitting the acceleration equation.
[49]
Minus??
[50]
comment or not??
Subsection 2.4.2. Analogy with electromagnetism: gravitoelectromagnetism.
[51]
Subsection 2.4.3. Maxwell-like equations.
[52]
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[53]
Subsection 2.4.4. Splitting the spin transport equation.
[54]
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[55]
Subsection 2.4.5. Relative Fermi-Walker transport and gyro precession.
[56]
Update with bcjw.tex.
[57]
MORE HERE ON NESTER??
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[58]
Subsection 2.4.6. The Schiff Precession Formula.
MORE HERE SCHIFF END?? confused??
Emphasize secular effects so that these details are irrelevant to the actual ex
periment.
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[59] [60]
Subsection 2.4.7. The relative angular velocity as a boost derivative.
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[61]
Subsection 2.4.8. Relative kinematics: transformation of spatial gravitational 
fields.

Overfull \hbox (1.86423pt too wide) in paragraph at lines 4150--4165
\OT1/cmr/m/n/10 where the ex-pres-sions in square brack-ets in the grav-i-to-el
ec-tric field trans-for-ma-tion laws are just $\OML/cmm/m/it/10 ^^M[][][][]\OT1
/cmr/m/n/10 (\OML/cmm/m/it/10 U; u\OT1/cmr/m/n/10 )$
 []

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[62]) [63] (C:/bobfiles/TEXFILES/gem/gem2b.tex
>>>>> em2b.tex.
>>>>> 20-nov-1997<-4-june-1997 <-24-feb-1995.
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Update with diagrams illustrating spatial curvature, synchronization defect:
Section 2.5. Spatial curvature and torsion.
Subsection 2.5.1. Definitions.
sign?? sign??
Subsection 2.5.2. Algebraic symmetries.
[64


] check?? finish here
Subsection 2.5.3. Symmetry-obeying spatial curvature.
[65]
Subsection 2.5.4. Spatial Ricci tensors and scalar curvatures.
[66]
Subsection 2.5.5. Pair interchange symmetry.
[67] check??
Subsection 2.5.6. Spatial covariant exterior derivative.
[68]
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[69]
Overfull \hbox (1.85295pt too wide) in paragraph at lines 534--534
[]\OT1/cmr/bx/n/14.4 The sym-metrized curl op-er-a-tor for sym-met-ric spa-tial
 2-tensors 
 []

Section 2.6. The symmetrized curl operator for symmetric spatial 2-tensors.
symmetric parts??
(to finish)
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[70]
Section 2.7. Splitting spacetime curvature.
Subsection 2.7.1. Splitting definitions.
Discuss sign Conventions for electric and magnetic parts of WEYL!
[71]
check this sign problem: weyl electric/magnetic parts.
Subsection 2.7.2. Spacetime duality and curvature.
[72]
Subsection 2.7.3. Evaluation of splitting fields.
check check check
Give FW temporal derivative form also:
[73] check check?? symmetric part of acceleration derivative!!:
check symmetric part of acclereation derivative!!:??
Subsection 2.7.4. Maxwell-like equations.
[74]
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[75]
Section 2.8. Mixed commutation formulas.
Subsection 2.8.1. Splitting the Ricci identities.
[76]
Subsection 2.8.2. Commuting $\hbox {\it \char '44}(u)_{\hbox {$u$}}$ and $\nabl
a (u)$.
warning Ca0b terms?
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[77]
Section 2.9. Splitting the Bianchi identities of the second kind.
Subsection 2.9.1. Spacetime identities.
how does divergence free weyl follow from Bianchi identities? more here:
[78] index level?? [79]
Also give source terms.
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[80]
Subsection 2.9.2. Spatial identities.
warning: Ca0b terms?
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[81]
Section 2.10. ``Time without space defines space without time" and vice versa.
POORLY WRITTEN SECTION??) [82] (C:/bobfiles/TEXFILES/gem/gem3.tex
Chapter 3.
 The slicing and threading points of view
Section 3.1. Introduction.
Section 3.2. Algebra.
Subsection 3.2.1. The nonlinear reference frame.
[83



] Figure 3.1 in text.
Figure 3.1.
Figure 3.2
Figure 3.2.
Subsection 3.2.2. Measurement and the lapse function.
[84] [85]
Subsection 3.2.3. The Shift.
Figure 3.3
Figure 3.3.
[86]
Subsection 3.2.4. Computational frames and the reference decomposition.
[87] Figure 3.4
Figure 3.4.
Subsection 3.2.5. Decomposing the metric.
[88] [89]
Subsection 3.2.6. Relationship between the reference and observer decomposition
s.
Figure 3.5
Figure 3.5.
[90]
Subsection 3.2.7. The slicing, threading and reference representations.
[91]
Subsection 3.2.8. Transformation between slicing and threading points of view.
?? more??
more?? [92] Figure 3.6
Figure 3.6.
Subsection 3.2.9. So far:.
[93]
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[94]
Section 3.3. Derivatives.
Subsection 3.3.1. Evolution.
useful or not??
Subsection 3.3.2. Natural time derivatives.
[95]
Subsection 3.3.3. Natural spatial derivatives.
[96]
Subsection 3.3.4. Gauge transformations of the nonlinear reference frame.
[97] ?? TO DO.
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[98]
Subsection 3.3.5. Observer-adapted frame structure functions and kinematical qu
antities.
[99]
Subsection 3.3.6. Spatial covariant derivative.
[100]
Subsection 3.3.7. Spatial vector analysis.
[101]
Subsection 3.3.8. Partially-observer-adapted frames: connection components.
[102]
Subsection 3.3.9. Total spatial covariant derivatives.
[103] Figure 3.7
Figure 3.7.
argument notation here perhaps not right warning: Ca0b terms? [104]
Subsection 3.3.10. Spatial gravitational forces.

LaTeX Warning: Reference `eq:energy1' on page 105 undefined on input line 1947.


edit here?? !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Subsection 3.3.11. Second-order acceleration equation.
[105] ?? ?? [106] comp 3-spacE?? reference decomp ala forward??
mixed d/dt Lie??
Subsection 3.3.12. The spin transport equation.
edit here?? above and below
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Subsection 3.3.13. Transformation of spatial gravitational fields.
inconsistent with choice of direction of transformation in appendix; check??
[107]
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[108]
Section 3.4. Spatial curvature.
Section 3.5. Initial value problem?.
Subsection 3.5.1. Hypersurface and slicing points of view.
Subsection 3.5.2. Thin sandwich problem.
[109]
Subsection 3.5.3. Congruence and threading points of view.
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[110]
Subsection 3.5.4. Perfect fluids.
develop perfect fluid section??) [111] (C:/bobfiles/TEXFILES/gem/gem4.tex
Chapter 4.
 Maxwell's equations
Section 4.1. Introduction.
REWRITE!
Section 4.2. Splitting the electromagnetic field.
Subsection 4.2.1. Congruence point of view.
[112



]
Subsection 4.2.2. Slicing and threading points of view.
Subsection 4.2.3. Observer Boost.
[113]
Subsection 4.2.4. Reference representation (Landau-Lifshitz-Hanni).
change LL to reference representation? [114]
Section 4.3. Splitting the 4-current.
[115] SSS 4.3.0.0. major work here:?? [116]
Section 4.4. Splitting Maxwell's equations.
Subsection 4.4.1. Congruence point of view.
check??
Subsection 4.4.2. Slicing and threading points of view.
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[117] [118] SSS 4.4.2.0. SSS 4.4.2.0. [119]
Section 4.5. Vector potential.
[120]
Section 4.6. Wave equations.
[121]
Section 4.7. Computational 3-space representations.
Section 4.8. Lines of force.
) [122]
chapter counter fudge??
(C:/bobfiles/TEXFILES/gem/gem5.tex
Chapter 5.
 Stationary spacetimes
Section 5.1. Stationary nonlinear reference frame.
[123



] ??check;develop more more
Section 5.2. Synchronization gap and Sagnac effect.
[124]
Figure 5.1.
Figure 5.2.
[125] [126]
Section 5.3. Rotating spatial Cartesian coordinates in flat spacetime.

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[127] [128] sign?? !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Overfull \hbox (0.1494pt too wide) in paragraph at lines 583--583
[]\OT1/cmr/bx/n/14.4 Stationary axially-symmetric case: ro-tat-ing Minkowski, G
^^?odel
 []

Section 5.4. Stationary axially-symmetric case: rotating Minkowski, G\"odel and
 Kerr spacetimes.
[129] [130] [131]) [132] (C:/bobfiles/TEXFILES/gem/gem6.tex
Chapter 6.
 Perturbation problems
Section 6.1. Linearization about an orthogonal nonlinear reference frame.
Section 6.2. Post-Newtonian approximation.
Section 6.3. The Newtonian limit.
Section 6.4. Friedmann-Robertson-Walker Perturbations.
) [133



] Draft version reset pagenumber for appendices
(C:/bobfiles/TEXFILES/gem/gemapp.tex
Appendix A.
 Formulas from differential geometry
Subsection A.1. Manifold.
Subsection A.2. Frame and dual frame.
[1



]
Subsection A.3. Linear transformations.
Subsection A.4. Change of frame.
[2]
Subsection A.5. Metric.
Subsection A.6. Unit volume $n$-form.
[3]
Subsection A.7. Connection.
[4]
Subsection A.8. Metric connection.
[5]
Subsection A.9. Curvature.
Subsection A.10. Total covariant derivative.
[6]
Subsection A.11. Parallel transport and geodesics.
Subsection A.12. Generalized Kronecker deltas.
[7]
Subsection A.13. Symmetrization/antisymmetrization.
oops??
Subsection A.14. Exterior product.
Subsection A.15. Hodge star duality operation.
[8]
Subsection A.16. Complex Duality Operation.
Subsection A.17. Exterior derivative.
[9]
Overfull \hbox (24.28923pt too wide) detected at line 876
\OML/cmm/m/it/10 dS\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[]; [] ; X[]\OT1/cmr/m/n/
10 ) = [](\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1)[]\OML/cmm/m/it/10 X[]S\OT1/cmr
/m/n/10 (\OML/cmm/m/it/10 X[]; [] ; []; [] ; X[]\OT1/cmr/m/n/10 ) + [](\OMS/cms
y/m/n/10 ^^@\OT1/cmr/m/n/10 1)[]\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 ([\OML/cmm/m/
it/10 X[]; X[]\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 ; X[]; [] ; []; [] ; []; [] ; X
[]\OT1/cmr/m/n/10 )
 []

Subsection A.18. Differential form divergence operator.
[10]
Subsection A.19. De Rham Laplacian.
Subsection A.20. Covariant exterior derivative.
[11]
Subsection A.21. Ricci identities.
[12]
Subsection A.22. Bianchi identities of the first and second kind.
Subsection A.23. Ricci Tensor and Scalar Curvature.
Clean up this section:
Subsection A.24. Contracted Bianchi Identities of the Second Kind and the Weyl 
Tensor.
[13] [14]
Subsection A.25. $n=3$ Structure Functions and Orthonormal Frame Connection Com
ponents.
TO FINISH!!
Subsection A.26. Lie derivative.
[15]) [16] Draft version reset pagenumber for references
(C:/bobfiles/TEXFILES/gem/gemref.tex
Can this first word indentation be corrected in a better way?? [1



]
Overfull \hbox (2.33755pt too wide) in paragraph at lines 269--273
\OT1/cmr/m/n/10 Bruni, M., Matar-rese, S., and Pan-tano, O., 1994,  A Lo-cal Vi
ew of the Ob-serv-able Uni-verse, \OT1/cmr/m/it/10 preprint\OT1/cmr/m/n/10 subm
itted
 []

[2] warning?! [3]
Overfull \hbox (15.97757pt too wide) in paragraph at lines 524--528
\OT1/cmr/m/n/10 Ellis, G.F.R., 1994,  The Co-vari-ant and Gauge In-vari-ant Ap-
proach to Per-tur-ba-tions in Cos-mol-ogy, \OT1/cmr/m/it/10 preprint\OT1/cmr/m/
n/10 (Dept.
 []

[4] warning?! [5]
Overfull \hbox (0.67654pt too wide) in paragraph at lines 788--795
\OT1/cmr/m/n/10 Jantzen, R.T., Carini, P., and Bini, D., 1993,  Grav-i-to-elec-
tro-mag-netism: Ap-pli-ca-tions to Ro-tat-ing Minkowski,
 []

warning?! warning?! [6] warning?! [7] pub city?? warning?! [8] warning?!
[9] warning?!) [10] [11

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(C:/bobfiles/TEXFILES/gem/gem2a.aux) (C:/bobfiles/TEXFILES/gem/gem2b.aux)
(C:/bobfiles/TEXFILES/gem/gem3.aux) (C:/bobfiles/TEXFILES/gem/gem4.aux)
(C:/bobfiles/TEXFILES/gem/gem5.aux) (C:/bobfiles/TEXFILES/gem/gem6.aux)
(C:/bobfiles/TEXFILES/gem/gemapp.aux) (C:/bobfiles/TEXFILES/gem/gemref.aux))

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