8/3/2019 Winitzki - Fermi-walker Frame and Frenet-serret Construction

http://slidepdf.com/reader/full/winitzki-fermi-walker-frame-and-frenet-serret-construction 1/2

Construction of Fermi-Walker transported frames

Sergei Winitzki

April 2008

Initially motivated by arxiv:0804.2502We work in an (1 + n)-dimensional Lorentzian manifold (sig-

nature + − −...−) and assume that a timelike curve γ (τ ) isgiven, where τ  is the proper time along the curve.

0.1 Frenet-Serret construction

Recall that in classical differential geometry one talks aboutnormals, binormals, etc. So we perform a similar constructionfor the curve γ . First we define the normalized tangent vectoru := γ , g(u, u) = 1, and its covariant derivative along thecurve,

a := ∇uu.

The usual interpretation of a is the proper 4-acceleration of theworldline γ . If nonzero, the vector a is spacelike but not nec-essarily normalized. Hence there exists a normalized spacelikevector e1 such that g(e1, e1) = −1 and

a = −α1e1,

where α1 is a number (the absolute value of the proper accel-eration; the minus sign is a purely cosmetic convenience). If α1 = 0 then the curve γ  is a geodesic and the construction endshere; any orthonormal frame is parallelly transported along γ .So we assume that α1 = 0.

The vectore1 is automatically orthogonal to

usince

g(e1, u) = −1

α1

g(∇uu, u) = −

1

2α1

∇ug(u, u) = 0.

We will eventually construct an orthonormal frame out of {u, e1,...}, so let us define also e0 := u.

We now continue the same construction and compute thecovariant derivative of  e1 along the curve,

b := ∇ue1.

The vector b is orthogonal to e1 but not necessarily to u; let uscompute its projection onto u using the property g(u, e1) = 0:

g(u, ∇ue1) = −g(∇uu, e1) = −g(−α1e1, e1) = −α1.

Hence there exists α2 and a normalized spacelike vector e2 suchthat e2 ⊥ e1, e2 ⊥ u and

b = ∇ue1 = −α1u + α2e2.

The vector e2 is well-defined as long as α2 = 0. Let us brieflycheck what happens if  α2 = 0. In that case, we may choose thevector e2 arbitrarily at any point. as long as it is orthogonal tou and e1. Moreover, we may choose e2 parallelly transported 

along γ ; it is easy to see that the parallel transport will conservethe properties e2 ⊥ e1, e2 ⊥ u. Let us therefore perform this

choice if  α2 = 0.

Now we assume that α2 = 0, continue the construction anconsider the derivative

c := ∇ue2.

Besides c ⊥ e2 we also have c ⊥ u because

g(u, c) = −g(∇uu, e2) = −α1g(e1, e2) = 0.

However, c may have a component parallel to e1:

g(e1, c) = −g(∇ue1, e2) = α2.

Hence c = ∇ue2 =: −α2e1 + α3e3,

where e3 is a normalized spacelike vector orthogonal to u, e1,

and α3 is an appropriate constant. The vector e3 is well-definas long as α3 = 0. If  α3 = 0 then we may define e3 as parlel propagation along γ  of an arbitrary initial vector thatorthogonal to u, e1, e2 at some point on γ .

We can continue this construction until we exhaust the dmensionality of spacetime and define an orthonormal fram{u, e1, ..., en} and the corresponding constants α1,...,αn. Tillustrate the construction, let us consider what happens at tk-th step. At the previous step we have defined orthonormvectors {u, e1,..., ek} and constants α1,...,αk, some of whimay be zero, and we have established the relationships (callthe Frenet-Serret equations)

∇uej = αj+1ej+1 − αjej−1, 1 ≤  j ≤ k − 1.

(The minus sign in the definition of  α1 plays its cosmetic rohere at j = 1.) Now we consider the vector ∇u

ek. This vectis orthogonal to ek and also to u because

g(∇uek, u) = −g(ek, ∇u

u) = 0.

Similarly, for k = j − 1 we have

g(∇uek, ej) = −g(ek, ∇uej) = 0.

However, the vector ∇uek may have a nonzero projection on

ek−1:

g(∇uek, ek−1) = −g(ek, ∇u

ek−1)

= −g(ek, αkek − αk−1ek−2) = αk.

Hence we obtain the relationship

∇uek = αk+1ek+1 − αkek−1,

which defines the vector ek+1 and the constant αk+1.

the constant αk+1 vanishes, the vector ek+1 is chosen as

8/3/2019 Winitzki - Fermi-walker Frame and Frenet-serret Construction

http://slidepdf.com/reader/full/winitzki-fermi-walker-frame-and-frenet-serret-construction 2/2

parallel transport of an arbitrary initial vector orthogonal to{u, e1,..., ek}.

At the last step of the construction, the constant αn+1 mustvanish since no new spacelike vector en+1 can be found inthe spacetime. Therefore, the construction ends by definingan orthonormal frame {e0, ..., en} and an array of constants{α1,...,αn} of which some may vanish. (By construction, if some αj = 0 then αj+1 = ... = αn = 0 as well.) The frame{e0, ..., en} is in some sense “naturally adapted” to the curve γ 

(since it is constructed purely out of  γ  and its derivatives).

0.2 Fermi-Walker transport

Consider an observer who moves along the timelike, non-geodesic worldline γ  in a 3+1-dimensional spacetime. We wouldlike to construct the observer’s reference frame that does nothave any spatial rotation; this will be appropriate if the observerkeeps track of the spatial directions by using strong gyroscopes.

The timelike basis vector is naturally chosen as e0 = γ . Themain problem is to obtain the spacelike frame {e1, e2, e3}.

Suppose some frame {eµ} is given; then we can compute theaccelerations ∇u

eµ and express them as linear combinations of the frame vectors,

∇ueµ =:

3ν=0

f µνeν .

This defines an antisymmetric matrix f µν . The spatial compo-

nents of that matrix (f 12, f 2

3, f 31) describe the purely spatial

rotation of the frame. Therefore, we wish to choose the space-like vectors {e1, e2, e3} such that these components vanish.

If the spacelike vectors {ej} were so chosen, we would have

∇uej = Aju, Aj ≡ f j

0, j = 1, 2, 3.

The coefficients Aj satisfy

Aj = g(u, ∇uej) = −g(∇u

u, ej) = −g(a, ej),

where a := ∇uu is the proper 4-acceleration of γ . Therefore, wemay compute the suitable vectors {ej} by solving the ordinarydifferential equations (along γ ),

∇uej = −g(a, ej),

with arbitrary (but orthonormal) initial vectors ej . The frame{ej} obtained in this way is called the Fermi-Walker propa-

gated frame with respect to the given curve γ .Given the FW propagated frame, one can define the notion

of Fermi-Walker transport (along γ ) of an arbitrary vector vdefined at some point p of  γ . One first decomposes the vectorv into components with respect to the FW propagated frame{u, ej} and obtains the coefficients

v = v0u +

3j=1

vjej ,

v0 = g(u, v),

vj = −g(ej , v), j = 1, 2, 3.

Then one defines the FW transported vector v at all otherpoints of  γ  as the linear combination of  {u, ej} with the same

coefficients v0, ..., v3.

Let us derive the equation for the FW transport directly terms of the vector v and its derivative:

∇uv = ∇u

v0u +

3j=1

vjej

= v0a +

3j=1

vj∇uej

= ag(u, v) + u

3j=1

g(ej , v)g(a, ej)

= ag(u, v) − ug(a, v).

This is the basic equation of the FW transport.

0.3 Computing the Fermi-Walker frame

How can we compute the FW propagated frame {ej} with rspect to a timelike worldlineγ (τ )? The method in Maluf aFaria is to take an arbitrary spacelike frame {ej}, j = 1, 2,

along γ  and compute a local proper rotation Λ(τ ) of the fram{ej},

ej → ej :=

3k=1

Λjk(τ )ek , j = 1, 2, 3,

such that the rotated frame {ej} is FW propagated. (Of coursthe timelike vector e0 is unchanged by the rotation.) We neto determine the τ -dependent matrix Λj

k(τ ). To determine thmatrix, let us write the equations for the covariant derivativof  {ej} and {ej} along γ :

∇uej = Aju +

3k=1

f jkek,

∇uej = Bju.

Since the frame {ej} is given, we may compute the coefficienf j

k(τ ); if these coefficients are nonzero, the frame {ej} is n

FW propagated. Let us now substitute ej = Λej :

Bju = ∇uej = ∇u

3

k=1

Λjkek

=

3l=1

d

dτ Λj

l

el +

3k=1

Λjk

Aku +

3l=1

f klel

=

3l=1

d

dτ Λj

l +

3k=1

Λjkf k

l

el + u

3k=1

ΛjkAk.

The frame {ej} is FW propagated if the terms at el abo

vanish: 3l=1

d

dτ Λj

l +3

k=1

Λjkf k

l

= 0.

This is a differential equation for Λ that can be rewritten in tmatrix form as

Λ−1d

dτ Λ + f  = 0.

Practical solution of this equation is perhaps easier when ouses the orthogonality property Λ−1 = ΛT  and the quaternionrepresentation of rotations.