Fermi normal coordinates

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

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 (112 ) 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 (112 ).

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

From the tetrad on $γ$ we define a dual tetrad $(e0,ea) μ μ$ by the relations

this is also FW transported on $γ$ . The tetrad and its dual give rise to the completeness relations

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

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 (115

) it follows that

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 6

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 (113 ), yields

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 (118 ) can be expressed as expansions in powers of $s$ , the spatial distance from $¯x$ to $x$ . For this we use the expansions of Equations (88 ) and (89 ), 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 (118 ), we obtain

and

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 (119 ) and (120 ) we find that

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:

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 (119 ) and (120 ) gives

and

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 (125 , 126 , 127 ) 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:

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 [58

] and the coordinate transformation was later produced by Zhang [64 ].

The key to the simplification comes from Equation (128 ), which dramatically reduces the number of independent components of the Riemann tensor. In particular, Equation (128 ) 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 (115 ) and take frame components of Equation (128 ). 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

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

where $ɛ abc$ is the three-dimensional permutation symbol. The inverse relation is $ℬa = 1ɛacdR b 2 0bcd$ .

Substitution of Equation (132 ) into Equation (127 ) 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 (131 ). The missing step is the transformation from the FNC $ˆxa$ to the Thorne–Hartle coordinates $ˆya$ . This is given by

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 (131 ).

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 )