3.2 Fermi normal coordinates

3.2.1 Fermi-Walker transport

Let $g$ be a timelike curve described by parametric relations $zm(t )$ in which $t$ is proper time. Let $m m u = dz /dt$ be the curve’s normalized tangent vector, and let $m m a = Du /dt$ be its acceleration vector.

A vector field $vm$ is said to be Fermi-Walker transported on $g$ if it is a solution to the differential equation

Notice that this reduces to parallel transport if $m a = 0$ and $g$ is a geodesic.

The operation of Fermi-Walker (FW) transport satisfies two important properties. The first is that $um$ is automatically FW transported along $g$ ; this follows at once from Equation (112 ) and the fact that $um$ is orthogonal to $am$ . The second is that if the vectors $vm$ and $wm$ are both FW transported along $g$ , then their inner product $m vmw$ is constant on $g$ : $m D(vmw )/dt = 0$ ; this also follows immediately from Equation (112 ).

3.2.2 Tetrad and dual tetrad on $g$

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

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

this is also FW transported on $g$ . 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 $g$ , we locate the unique spacelike geodesic $b$ that passes through $x$ and intersects $g$ 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 $a$ , $b$ , and so on. The FNC of $x$ are defined by

the last statement determines $x$ from the requirement that $- sa$ , the vector tangent to $b$ at $x$ , be orthogonal to $ua$ , the vector tangent to $g$ . 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 $b$ . This statement gives an immediate meaning to $^xa$ , the spatial Fermi normal coordinates; and the time coordinate $0 ^x$ 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 $g$ . The time coordinate $t$ selects a particular point on the word line, and the disk represents the set of spacelike geodesics that intersect $g$ orthogonally at $z(t)$ . The unit vector $wa =_ ^xa/s$ 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 $x + dx$ . This typically induces a change in the spacelike geodesic $b$ , which moves to $b + db$ , and a corresponding change in the intersection point $x$ , which moves to $x'' =_ x + dx$ , with $dxa = uadt$ . The FNC of the new point are then $^x0(x + dx) = t + dt$ and $^xa(x + dx) = -ea''(x'')sa''(x + dx,x'') a$ , with $x''$ determined by $s ''(x + dx,x'')ua''(x'') = 0 a$ . Expanding these relations to first order in the displacements, and simplifying using Equations (113

), yields

where $m$ is determined by $-1 a b a m = - (sabu u + saa )$ .

3.2.4 Coordinate displacements near $g$

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 $sa = -eaa ^xa$ and $gaa = uae0a + eaaeaa$ , where $(e0a, eaa)$ is a dual tetrad at $x$ obtained by parallel transport of $(- u ,ea) a a$ on the spacelike geodesic $b$ . After some algebra we obtain

1 m- 1 = 1 + aa ^xa +-R0c0d^xc^xd + O(s3), 3

where $a aa(t) =_ aae a$ are frame components of the acceleration vector, and $a g b d R0c0d(t) =_ Ragbdu ecu e d$ are frame components of the Riemann tensor evaluated on $g$ . This last result is easily inverted to give

a a 2 1 c d 3 m = 1- aa ^x + (aa ^x ) - -R0c0d^x ^x + O(s ). 3

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

and

where $R (t) =_ R eaegubed ac0d agbd a c d$ and $R (t) =_ R eaegebed acbd agbd a c b d$ are additional frame components of the Riemann tensor evaluated on $g$ . (Note that frame indices are raised with $ab d$ .)

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

because in the limit $x --> x$ the dual tetrad $0 a (ea,ea)$ at $x$ coincides with the dual tetrad $a (- ua, ea)$ at $x$ . It follows that on $g$ , the transformation matrix between the original coordinates $xa$ and the FNC $(t, ^xa)$ is formed by the Fermi-Walker transported tetrad:

This implies that the frame components of the acceleration vector $aa(t)$ are also the components of the acceleration vector in the FNC; orthogonality between $ua$ and $aa$ means that $a0 = 0$ . Similarly, $R0c0d(t)$ , $R0cbd(t)$ , and $Racbd(t)$ are the components of the Riemann tensor (evaluated on $g$ ) in the Fermi normal coordinates.

3.2.5 Metric near $g$

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 $0 0 a b gab = - eaeb + dabeaeb$ . This gives

2 2 a a b ds = gttdt + 2gta dtd^x + gabd^x d^x ,

with

[ a a2 c d 3 ] gtt = - 1 + 2aa^x + (aa^x ) + R0c0d^x ^x + O(s ) , (125) 2 c d 3 gta = - 3R0cadx^x^ + O(s ), (126) 1 gab = dab- -Racbd^xc^xd + O(s3). (127) 3

This is the metric near $g$ in the Fermi normal coordinates. Recall that $aa(t)$ are the components of the acceleration vector of $g$ - the timelike curve described by $x^a = 0$ - while $R0c0d(t)$ , $R0cbd(t)$ , and $Racbd(t)$ are the components of the Riemann tensor evaluated on $g$ .

Notice that on $g$ , the metric of Equations (125 , 126 , 127 ) reduces to $gtt = - 1$ and $gab = dab$ . On the other hand, the nonvanishing Christoffel symbols (on $g$ ) are $Gt = Ga = a ta tt a$ ; these are zero (and the FNC enforce local flatness on the entire curve) when $g$ 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 $(t, ^xa)$ to the Thorne-Hartle coordinates $(t, ^ya)$ brings the metric to the form

[ a a2 c d 3] gtt = - 1 + 2aa ^y + (aay^ ) + R0c0d^y ^y + O(s ) , (129) 2- c d 3 gta = - 3 R0cad^y ^y + O(s ), (130) g = d (1 - R ^yc^yd)+ 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

dcdR = E , dcdR = 0, dcdE = 0, acbd ab 0cad cd

the last of which states that $Eab$ has a vanishing trace. Taking the trace of the first equation gives $dabdcdRacbd = 0$ , 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 $R1212 = E11 + E22 = - E33$ , $R = E 1213 23$ , $R = - E 1223 13$ , $R = E + E = - E 1313 11 33 22$ , $R = E 1323 12$ , and $R = E + E = -E 2323 22 33 11$ . These are summarized by

and $Eab =_ R0a0b$ satisfies $dabEab = 0$ .

We may also note that the relation $dcdR0cad = 0$ , 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 $e abc$ is the three-dimensional permutation symbol. The inverse relation is $Ba = 1eacdR b 2 0bcd$ .

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

( 1 ) 1 1 1 gab = dab 1 - -Ecd^xcx^d - --(^xc^xc)Eab + -x^aEbcx^c + --^xbEac^xc + 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 $^xa$ to the Thorne-Hartle coordinates $^ya$ . 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 $gab(THC) = gab(FNC) - qa;b- qb;a$ , where

q = 1d E ^xcx^d - 1(^x ^xc) E - 1-E ^xc^x + 2^x E ^xc + O(s3). a;b 3 ab cd 6 c ab 3 ac b 3 a bc

It follows that $gTaHbC = dab(1 - Ecd^yc^yd) + O(^y3)$ , 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 ^y + (aay^ ) + Eab^y ^y + O(s ) , (135) 2- b c d 3 gta = - 3 eabcB d^y ^y + O(s ), (136) ( c d) 3 gab = dab 1 - Ecd^y ^y + O(s ). (137)