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3.2 Fermi normal coordinates

3.2.1 Fermi–Walker transport

Let γ be a timelike curve described by parametric relations zμ(τ) in which τ is proper time. Let μ μ u = dz ∕dτ be the curve’s normalized tangent vector, and let μ μ a = Du ∕dτ be its acceleration vector.

A vector field vμ is said to be Fermi–Walker transported on γ if it is a solution to the differential equation

Dv-μ- ν μ ν μ dτ = (vνa )u − (vνu )a . (112 )
Notice that this reduces to parallel transport if μ a = 0 and γ is a geodesic.

The operation of Fermi–Walker (FW) transport satisfies two important properties. The first is that uμ is automatically FW transported along γ; this follows at once from Equation (112View Equation) and the fact that u μ is orthogonal to aμ. The second is that if the vectors vμ and w μ are both FW transported along γ, then their inner product μ vμw is constant on γ: μ D(vμw )∕dτ = 0; this also follows immediately from Equation (112View Equation).

3.2.2 Tetrad and dual tetrad on γ

Let ¯z be an arbitrary reference point on γ. At this point we erect an orthonormal tetrad (u¯μ,e¯μ) a where, contrary to former usage, the frame index a runs from 1 to 3. We then propagate each frame vector on γ by FW transport; this guarantees that the tetrad remains orthonormal everywhere on γ. At a generic point z(τ) we have

μ De-a- ν μ μ ν μ ν μ ν dτ = (aνea)u , gμνu u = − 1, gμνeau = 0, gμνeaeb = δab. (113 )
From the tetrad on γ we define a dual tetrad (e0,ea) μ μ by the relations
0 a ab ν eμ = − uμ, eμ = δ gμνeb; (114 )
this is also FW transported on γ. The tetrad and its dual give rise to the completeness relations
gμν = − uμuν + δabeμaeνb, gμν = − e0μe0ν + δabeaμebν. (115 )

3.2.3 Fermi normal coordinates

To construct the Fermi normal coordinates (FNC) of a point x in the normal convex neighbourhood of γ, we locate the unique spacelike geodesic β that passes through x and intersects γ orthogonally. We denote the intersection point by ¯x ≡ z(t), with t denoting the value of the proper-time parameter at this point. To tensors at ¯x we assign indices ¯α, β¯, and so on. The FNC of x are defined by

0 a a α¯ ¯α xˆ = t, ˆx = − e¯α(¯x)σ (x, ¯x), σ ¯α(x, ¯x)u (¯x ) = 0; (116 )
the last statement determines ¯x from the requirement that − σ ¯α, the vector tangent to β at ¯x, be orthogonal to uα¯, the vector tangent to γ. From the definition of the FNC and the completeness relations of Equation (115View Equation) it follows that
2 a b s ≡ δabˆx ˆx = 2σ (x, ¯x), (117 )
so that s is the spatial distance between x¯ and x along the geodesic β. This statement gives an immediate meaning to ˆxa, the spatial Fermi normal coordinates; and the time coordinate ˆx0 is simply proper time at the intersection point ¯x. The situation is illustrated in Figure 6View Image.
View Image

Figure 6: Fermi normal coordinates of a point x relative to a world line γ. The time coordinate t selects a particular point on the word line, and the disk represents the set of spacelike geodesics that intersect γ orthogonally at z(t). The unit vector ωa ≡ ˆxa∕s selects a particular geodesic among this set, and the spatial distance s selects a particular point on this geodesic.

Suppose that x is moved to x + δx. This typically induces a change in the spacelike geodesic β, which moves to β + δβ, and a corresponding change in the intersection point ¯x, which moves to x ′′ ≡ ¯x + δ¯x, with δx¯α = uα¯δt. The FNC of the new point are then ˆx0(x + δx) = t + δt and ˆxa(x + δx ) = − ea′′(x ′′)σ α′′(x + δx,x′′) α, with x′′ determined by σ ′′(x + δx,x ′′)u α′′(x′′) = 0 α. Expanding these relations to first order in the displacements, and simplifying using Equations (113View Equation), yields

( ) ¯α β a a ¯α ¯α ¯β ¯γ β dt = μσ ¯αβu dx , d ˆx = − e¯α σ β + μ σ ¯βu σβ¯γu dx , (118 )
where μ is determined by − 1 ¯α β¯ ¯α μ = − (σ ¯α¯βu u + σα¯a ).

3.2.4 Coordinate displacements near γ

The relations of Equation (118View Equation) can be expressed as expansions in powers of s, the spatial distance from ¯x to x. For this we use the expansions of Equations (88View Equation) and (89View Equation), in which we substitute σ¯α = − e¯αaˆxa and g¯αα = u¯α¯e0α + e¯αa¯eaα, where (¯e0α,¯eaα) is a dual tetrad at x obtained by parallel transport of (− u ,ea) ¯α ¯α on the spacelike geodesic β. After some algebra we obtain

1 μ− 1 = 1 + aa ˆxa +-R0c0dˆxcˆxd + 𝒪 (s3), 3

where ¯α aa(t) ≡ a ¯αea are frame components of the acceleration vector, and ¯α ¯γ ¯β ¯δ R0c0d(t) ≡ R ¯α¯γβ¯δ¯u ecu ed are frame components of the Riemann tensor evaluated on γ. This last result is easily inverted to give

a a 2 1 c d 3 μ = 1 − aa ˆx + (aa ˆx ) − -R0c0dˆx ˆx + 𝒪 (s ). 3

Proceeding similarly for the other relations of Equation (118View Equation), we obtain

[ a a 2 1 c d 3] ( 0 β) [ 1 c d 3 ]( b β ) dt = 1 − aaˆx + (aaˆx ) − 2R0c0dˆx ˆx + 𝒪 (s ) ¯eβdx + − 6R0cbdˆx ˆx + 𝒪 (s ) ¯eβdx (119 )
and
[ ] ( ) [ ]( ) dˆxa = 1-Rac0dˆxcˆxd + 𝒪 (s3) ¯e0βdxβ + δab + 1Racbdˆxcˆxd + 𝒪 (s3) ¯ebβdx β , (120 ) 2 6
where R (t) ≡ R ¯¯e¯αe¯γuβ¯e¯δ ac0d ¯α¯γβδ a c d and R (t) ≡ R ¯¯e¯αe¯γe¯βe¯δ acbd ¯α¯γβδ a c b d are additional frame components of the Riemann tensor evaluated on γ. (Note that frame indices are raised with ab δ.)

As a special case of Equations (119View Equation) and (120View Equation) we find that

| | ∂t | ∂ˆxa| a ∂xα-|| = − u ¯α, ∂x-α|| = e¯α, (121 ) γ γ
because in the limit x → ¯x the dual tetrad (¯e0α,¯eaα ) at x coincides with the dual tetrad (− u ,ea) ¯α α¯ at ¯x. It follows that on γ, the transformation matrix between the original coordinates α x and the FNC a (t, ˆx ) is formed by the Fermi–Walker transported tetrad:
| | ∂x α| ∂xα | ----|| = u¯α, --a-|| = e¯αa. (122 ) ∂t γ ∂ˆx γ
This implies that the frame components of the acceleration vector a (t) a are also the components of the acceleration vector in the FNC; orthogonality between ¯α u and ¯α a means that a0 = 0. Similarly, R0c0d(t), R0cbd(t), and Racbd(t) are the components of the Riemann tensor (evaluated on γ) in the Fermi normal coordinates.

3.2.5 Metric near γ

Inversion of Equations (119View Equation) and (120View Equation) gives

[ ] [ ] 0 α a 1 c d 3 1 c d 3 b ¯eαdx = 1 + aaˆx + 2R0c0dxˆxˆ + 𝒪 (s ) dt + 6R0cbdˆx xˆ + 𝒪 (s ) dˆx (123 )
and
[ ] [ ] a α a 1- a c d 3 b 1- a c d 3 ¯eαdx = δb − 6R cbdˆx ˆx + 𝒪 (s ) dˆx + − 2R c0dˆx ˆx + 𝒪 (s ) dt. (124 )
These relations, when specialized to the FNC, give the components of the dual tetrad at x. They can also be used to compute the metric at x, after invoking the completeness relations gαβ = − ¯e0α¯e0β + δab¯eaα¯ebβ. This gives
ds2 = gttdt2 + 2gta dtdˆxa + gabdˆxadxˆb,

with

g = − [1 + 2a ˆxa + (a ˆxa )2 + R ˆxcˆxd + 𝒪 (s3)], (125 ) tt a a 0c0d g = − 2R xˆcxˆd + 𝒪 (s3), (126 ) ta 3 0cad 1 c d 3 gab = δab − -Racbdˆx ˆx + 𝒪 (s ). (127 ) 3
This is the metric near γ in the Fermi normal coordinates. Recall that aa(t) are the components of the acceleration vector of γ – the timelike curve described by ˆxa = 0 – while R0c0d(t), R0cbd(t), and Racbd(t) are the components of the Riemann tensor evaluated on γ.

Notice that on γ, the metric of Equations (125View Equation, 126View Equation, 127View Equation) reduces to g = − 1 tt and g = δ ab ab. On the other hand, the nonvanishing Christoffel symbols (on γ) are t a Γ ta = Γ tt = aa; these are zero (and the FNC enforce local flatness on the entire curve) when γ is a geodesic.

3.2.6 Thorne–Hartle coordinates

The form of the metric can be simplified if the Ricci tensor vanishes on the world line:

R (z) = 0. (128 ) μν
In such circumstances, a transformation from the Fermi normal coordinates (t,xˆa ) to the Thorne–Hartle coordinates (t, ˆya) brings the metric to the form
g = − [1 + 2a ˆya + (a yˆa )2 + R ˆycˆyd + 𝒪 (s3)] , (129 ) tt a a 0c0d g = − 2-R ˆycˆyd + 𝒪 (s3), (130 ) ta 3 0cad g = δ (1 − R ˆycˆyd) + 𝒪 (s3). (131 ) ab ab 0c0d
We see that the transformation leaves gtt and gta unchanged, but that it diagonalizes gab. This metric was first displayed in [58Jump To The Next Citation Point] and the coordinate transformation was later produced by Zhang [64].

The key to the simplification comes from Equation (128View Equation), which dramatically reduces the number of independent components of the Riemann tensor. In particular, Equation (128View Equation) implies that the frame components Racbd of the Riemann tensor are completely determined by ℰab ≡ R0a0b, which in this special case is a symmetric-tracefree tensor. To prove this we invoke the completeness relations of Equation (115View Equation) and take frame components of Equation (128View Equation). This produces the three independent equations

cd cd cd δ Racbd = ℰab, δ R0cad = 0, δ ℰcd = 0,

the last of which states that ℰab has a vanishing trace. Taking the trace of the first equation gives δabδcdRacbd = 0, and this implies that Racbd has five independent components. Since this is also the number of independent components of ℰab, we see that the first equation can be inverted – Racbd can be expressed in terms of ℰab. A complete listing of the relevant relations is R1212 = ℰ11 + ℰ22 = − ℰ33, R1213 = ℰ23, R1223 = − ℰ13, R1313 = ℰ11 + ℰ33 = − ℰ22, R1323 = ℰ12, and R2323 = ℰ22 + ℰ33 = − ℰ11. These are summarized by

Racbd = δabℰcd + δcdℰab − δadℰbc − δbcℰad, (132 )
and ℰab ≡ R0a0b satisfies ab δ ℰab = 0.

We may also note that the relation δcdR0cad = 0, together with the usual symmetries of the Riemann tensor, imply that R 0cad too possesses five independent components. These may thus be related to another symmetric-tracefree tensor ℬab. We take the independent components to be R0112 ≡ − ℬ13, R0113 ≡ ℬ12, R0123 ≡ − ℬ11, R0212 ≡ − ℬ23, and R0213 ≡ ℬ22, and it is easy to see that all other components can be expressed in terms of these. For example, R0223 = − R0113 = − ℬ12, R0312 = − R0123 + R0213 = ℬ11 + ℬ22 = − ℬ33, R0313 = − R0212 = ℬ23, and R0323 = R0112 = − ℬ13. These relations are summarized by

d R0abc = − ɛbcdℬ a, (133 )
where ɛ abc is the three-dimensional permutation symbol. The inverse relation is ℬa = 1ɛacdR b 2 0bcd.

Substitution of Equation (132View Equation) into Equation (127View Equation) gives

( 1 ) 1 1 1 gab = δab 1 − -ℰcdˆxcxˆd − --(ˆxcˆxc)ℰab + -xˆa ℰbcxˆc + --ˆxbℰacˆxc + 𝒪(s3), 3 3 3 3

and we have not yet achieved the simple form of Equation (131View Equation). The missing step is the transformation from the FNC ˆxa to the Thorne–Hartle coordinates ˆya. This is given by

a a a a 1 c b 1 b c 4 ˆy = ˆx + ξ , ξ = − 6 (ˆxcxˆ) ℰabˆx + 3ˆxaℰbcˆx ˆx + 𝒪 (s ). (134 )
It is easy to see that this transformation affects neither gtt nor gta at orders s and s2. The remaining components of the metric, however, transform according to g (THC ) = g (FNC ) − ξ − ξ ab ab a;b b;a, where
1- c d 1- c 1- c 2- c 3 ξa;b = 3δabℰcdˆx xˆ − 6 (ˆxcˆx ) ℰab − 3 ℰacˆx ˆxb + 3ˆxaℰbcˆx + 𝒪 (s ).

It follows that gTHC = δ (1 − ℰ ˆycˆyd) + 𝒪(ˆy3) ab ab cd, which is just the same statement as in Equation (131View Equation).

Alternative expressions for the components of the Thorne–Hartle metric are

[ a a2 a b 3] gtt = − 1 + 2aa ˆy + (aayˆ ) + ℰabˆy ˆy + 𝒪 (s) , (135 ) 2- b c d 3 gta = − 3 ɛabcℬ dˆy ˆy + 𝒪 (s ), (136 ) ( c d) 3 gab = δab 1 − ℰcdˆy ˆy + 𝒪 (s ). (137 )

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