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Tangent Euler Top in General Relativity

We define the tangent Euler top in General Relativity through a constrained Lagrangian on the orthonormal frame bundle. The corresponding motions are studied to various degrees of approximation, the lowest of which is shown to yield the Mathisson-Papapetrou equations.


Introduction
As is well known [Bor09, Her10, Noe10, Ros47, ST54, NJ59, PW62, Boy65, WE66,WE67], there are two separate sets of problems preventing the consideration of generic rigid bodies in General Relativity. The first is the lack of a global notion of simultaneity, which renders the notion of distance between two particles undefined. Associated to this is the inexistence of arbitrarily fast signals, which would be necessary for a rigid body to keep its exact shape. The second problem is that even if a family of spacelike slices is somehow singled out to define a notion of simultaneity, it is in general impossible to keep the mutual distances of more than four particles constant, due to the varying curvature of the slices.
Nevertheless, it is many times useful to consider approximate notions of rigidity in General Relativity [Dix70,ER77,TG83,Voi88,XTW04]. These are usually derived from multipole or post-Newtonian expansions, which can be quite formidable. A simpler method for obtaining the (approximate) motion of a small (approximately) rigid body from a finite-dimensional Lagrangian, mirroring what is done in Newtonian mechanics, would therefore be desirable.
A possibility (suggested by Turakulov [Tur] in the context of Newtonian mechanics on a curved space) is to approximate the motion of a rigid body by the motion of an orthonormal frame (determined by a naturally defined Lagrangian). This avoids the curvature problem. The simultaneity problem can be overcome by adding the natural constraint that the frame's timelike unit vector should be tangent to the motion of the base point (related ideas in flat spacetime can be found in [HR74,Ell82,Tak82]).
The consequences of this model are explored in the present paper. Section 1 begins by briefly reviewing the Euler top, in order to set up the notation. The Newtonian tangent Euler top on a curved space, which is a toy model for its relativistic counterpart, is investigated in Section 2. This generalizes [Tur], where the discussion was restricted to spherical tops. The relativistic tangent Euler top is defined in Section 3, as well as simpler Lagragians determining the spinning motion only. An approximation scheme for these is devised in Section 4, where the quadratic and cubic approximations are examined. The first (but not the second) is seen to lead to the usual Euler top motion with respect to a Fermi-Walker transported frame.
This work was partially supported by the Fundação para a Ciência e a Tecnologia through the Program POCI 2010/FEDER and by the grant POCI/MAT/58549/2004. Section 5 studies the exact spinning motion when the base point is moving along a geodesic. The corresponding Lagrangian is shown to yield a (new) completely integrable system on SO(3). Finally, Section 6 analyzes the full Lagrangian in the quadratic approximation. It is found that the motion of the frame's base point is determined by the Mathisson-Papapetrou equation of motion subject to the Mathisson-Pirani spin supplementary condition [Mat37,Pap51,Pir56,MTW73,Sem99,Ste04].
The present paper thus succeeds in deriving the Mathisson-Papapetrou equation from a Lagrangian, defined on the (finite-dimensional) orthonormal frame bundle, which is the natural generalization of the Lagrangian for a free rigid body in Newtonian mechanics. This is quite different from other approaches inspired on variational principles [K72,YB93,Por06], where the authors start with the Mathisson-Papapetrou equation and proceed to construct an action (generally with a much less clear physical meaning). Moreover, this Lagrangian formulation allows for the study of the rotational motion (which can be analyzed for arbitrary motions of the center of mass), instead of considering only the evolution of the angular momentum tensor. This leads to interesting new mechanical systems on SO(3).
We use the conventions of [MTW73] (including the Einstein summation convention), except for the meaning of the indices, which is the following: latin indices will always be associated to an orthonormal frame; they will be spacetime indices (ranging from 0 to 3) if they are from the beginning of the alphabet (a, b, . . .), and space indices (ranging from 1 to 3) if the are from the middle of the alphabet (i, j, . . .). Similarly, greek indices will always be associated to a coordinate system; they will be spacetime indices if the are from the beginning of the alphabet (α, β, . . .), and space indices if the are from the middle of the alphabet (µ, ν, . . .) 1 . As is usual, we will not worry about the vertical position of space indices on orthonormal frames.

Euler top
We shall use this brief review of the Euler top to set up our notation. For simplicity we will consider only the case where the Euler top which spins about its center of mass.
Definition 1.1. The reference configuration of an Euler top is a positive finite Borel measure m on R 3 , not supported on any 1-dimensional subspace, and such that The mass of the Euler top is M = m(R 3 ), and its Euler tensor is the symmetric matrix A motion of the Euler top is an Euler-Lagrange curve for the Lagrangian L : 1 This complication is unfortunate but inevitable, as we work in the orthonormal frame bundle and often distinguish between space and time. As is well known [Arn97,MR99], the Euler top is completely integrable: in addition to the total energy H = L, one can obtain commuting first integrals from the conservation of angular momentum. In fact, the Euler-Lagrange equations themselves are equivalent to the conservation of angular momentum.
The formula C, D = − 1 2 tr(CD) defines an inner product on so(3). Since σ, B is conserved for all B ∈ so(3), so is σ.
(1) The matrix σ is related to the angular momentum vector σ through

Tangent Euler top on a curved space
Let OM be the bundle of positive orthonormal frames on an oriented 3 3-dimensional Riemannian manifold (M, ·, · ). A curve E : R → OM is given by where c : R → M is a curve on M and E 1 , E 2 , E 3 : R → T M are vector fields along c which form a positive orthonormal frame at c(t) for each t ∈ R.
Definition 2.1. A motion of a tangent Euler top on (M, ·, · ) is an Euler-Lagrange curve for the Lagrangian L : T OM → R given by where ∇ is the Levi-Civita connection and m is the reference configuration of an Euler top. 3 The hypothesis of orientability is not crucial, as one can always pass to the orientable double cover.

Remark 2.2.
(1) Notice that if (M, ·, · ) is R 3 with the Euclidean metric then this Lagrangian is just the Lagrangian for a free rigid body.
(2) Physically, one can expect a tangent Euler top to represent the motion of an approximately rigid extended body in the limit in which its size is much smaller than the local radius of curvature. (1) whereΩ ij are the curvature forms for the frame {Ê 1 ,Ê 2 ,Ê 3 }, ♯ : T * M → T M is the isomorphism induced by the metric and σ ∈ so(3) is the angular momentum matrix.
Proof. We define a local trivialization For this trivialization we have i are the connection forms associated to our local frame. The Lagrangian can therefore be written as Our task is now to write the Euler-Lagrange equations. For the SO(3) part, we notice that if the frame {Ê 1 ,Ê 2 ,Ê 3 } happens to be parallel transported along c thenω(ċ) = 0, and hence only terms coming from the rotational kinetic energy K survive in the Euler-Lagrange equations. Therefore {E 1 , E 2 , E 3 } rotates with respect to {Ê 1 ,Ê 2 ,Ê 3 } exactly as an Euler top with reference configuration m. For the motion of the base point, there will be terms coming from both the translational kinetic energy T and the Coriolis term C (but not from the frame term F , which is quadratic inω(ċ)). As is well known, where g µν is the matrix of the metric. Since the trace of the product of a symmetric and an anti-symmetric matrices is zero, we can write Consequently, the equations of motion for the base point are is the angular momentum tensor. As noticed by Turakulov [Tur], this equation is similar to the Mathisson-Papapetrou equation for a spinning particle in General Relativity [Mat37,Pap51]. The fact that the angular momentum matrix is constant on the parallel transported frame {Ê 1 ,Ê 2 ,Ê 3 } is translated into the fact that the angular momentum tensor satisfies the parallel transport equation

Tangent Euler top in General Relativity
Let OM be the bundle of positive future-pointing orthonormal frames on an oriented, time-oriented 5 4-dimensional Lorentzian manifold (M, ·, · ). An admissible curve E : R → OM is given by where c : R → M is a future-directed timelike curve on M parameterized by its proper time and E 0 , E 1 , E 2 , E 3 : R → T M are vector fields along c which form a positive orthonormal frame at c(t) satisfying E 0 (t) =ċ(t) for each t ∈ R.
Definition 3.1. A motion of a tangent Euler top on (M, ·, · ) is a (constrained) Euler-Lagrange curve for the Lagrangian L : T OM → R given by where ∇ is the Levi-Civita connection and m is the reference configuration of an Euler top, chosen among all admissible curves, with unspecified final time.
(1) This is the natural generalization of the A simpler problem is obtained by focussing on the spinning motion. To do this, we regard the motion c : R → M of the base point as given (which may or may not be the one arising from Definition 3.1) and look for the positive orthonormal frame {ċ(t), E 1 (t), E 2 (t), E 3 (t)} along c which maximizes the action. Choosing a positive orthonormal Fermi-Walker transported frame {ċ(t),Ê 1 (t),Ê 2 (t),Ê 3 (t)} along c, so that where m is the reference configuration of an Euler top andâ : R → R 3 represents the proper acceleration on a Fermi-Walker transported frame.

Approximate spinning motions
Expanding the integrand in the Lagrangian for a spinning motion in power series of ξ and integrating we can write where L (n) is the integral of the term of order n in ξ.  Proof. One just needs to check that Proof. One just needs to check that which generically is not a total time derivative.
Remark 4.5. L (3) can be thought of as (minus) the potential energy corresponding to the weight of the rotational kinetic energy in the constant "gravitational field" −â induced by the acceleration of the center of mass.

Spinning motions along geodesics
If the spinning motion is along a timelike geodesic, we haveâ = 0, and the Lagrangian becomes SO(3)-invariant. In this case, we can apply Noether's Theorem.
Proof. We can write the Lagrangian as By Noether's theorem, FL(X B ) is conserved along the motion for all B ∈ so(3), where X B ∈ X(SO(3)) is the infinitesimal action of B. Since we see that − 1 2 tr(Bσ) = B, σ is conserved for all B ∈ so(3), and so is σ.
(1) The angular momentum vector σ, related to the matrix σ through for all ξ ∈ R 3 , can be seen to have the familiar expression (2) This system is completely integrable: in addition to the total energy Therefore the Euler tensor is not enough to characterize a relativistic tangent Euler top: the higher multipoles are also required (and become more important as the spinning motion becomes more relativistic).

Quadratically approximate motions
We can extend Definition 4.1 to the general motion of a tangent Euler top.
where K is the rotational kinetic energy,Ω ab are the curvature forms for the frame {Ê 0 ,Ê 1 ,Ê 2 ,Ê 3 }, ♯ : T * M → T M is the isomorphism induced by the metric and σ ∈ so(3) is the angular momentum matrix.
Proof. The spinning motion with respect to a Fermi-Walker transported frame was already shown to be that of an Euler top in Section 4.
To compute the motion of the base point we define a local trivialization OM | U ∼ = U × SO ↑ (3, 1) by choosing a local orthonormal frame {Ê 0 ,Ê 1 ,Ê 2 ,Ê 3 } ⊂ X(U ) on a sufficiently small open set U ⊂ M . For this trivialization we have a are the connection forms associated to our local frame, and hence the Lagrangian can be written as Notice that the conditions S ∈ O(3, 1) and A ∈ so(3, 1) are conveniently written by raising and lowering indices as (S −1 ) a b = S a b and A ab = −A ba . The best way to deal with the constraint while avoiding introducing coordinates on SO ↑ (3, 1) is to to set up our problem as an optimal control problem (see for instance [Law63], [Pin93]). As state variables we choose the local coordinates x α of the base point, the components S b a of the frame vectors and the action θ. As control variables we simply chose 6 independent components of the anti-symmetric matrix (A ab ). The state variables depend on the control variables through the solution of the Cauchy problem Our objective is to maximize θ 1 = θ(t 1 ), i.e. minimize Φ = −θ 1 , where the final time t 1 is not fixed. To do so, we introduce the multipliers λ α , Λ b a , λ (one for each differential equation) and set up the Hamiltonian function According to the Pontryagin Maximum Principle, we must select the values of the control variables which maximize H. The minimizing curve will then be obtained by solving the Hamilton equations together with the Cauchy problem. The multiplier λ is clearly constant; since θ 1 is not specified, we must select (hence λ ≡ 1). Moreover, H is constant along the solution. Since t 1 is not fixed, we must have H ≡ 0. Because we already know what the spinning motion will be, we do not have to write out all the minimum equations. Also, we shall assume that the trivializing frame {Ê 0 ,Ê 1 ,Ê 2 ,Ê 3 } happens to be Fermi-Walker transported along the minimizing curve, withÊ 0 =ċ. As a consequence, we will havê along this curve. In particular, We now write the relevant minimum conditions. Maximizing H with respect to the control variables A 0i yields We have the Hamilton equationṡ We must also consider the Hamilton equatioṅ We now use our freedom in the choice of local coordinates to select Fermi normal coordinates (t, x 1 , x 2 , x 3 ), defined by the parameterization , where exp p : T p M → M is the geodesic exponential map. To first order in x we can then choose 6Ê [Poi04]). In particular, ∇Ê iÊ j = 0 for x = 0, from where we can deduce that ω ij = 0 andω 0i (Ê j ) = 0 along the minimizing curve. Equation (7) then becomes The quantity λ t in this equation can be obtained from Maximizing H with respect to A ij (with i < j, say) yields further relations: These can be used to rewrite the second term in (9), which is tr(SAΛ t ) = 1 2 tr(SAΛ t + ΛA t S t ) = 1 2 tr(Λ t SA − S t ΛA) = 1 2 tr((Λ t S − S t Λ)A).
After the substitution we obtain where K is the rotational kinetic energy of the Euler top. Because ∂ nωlmαÊ α 0 is anti-symmetric in l, m, the quantity multiplying it in (8) can be replaced by Consequently, Equation (8) can be written as (10)λ n = −(M + K)â n + 1 2 ∂ nωlmαÊ α 0 σ lm .
6 Here we use latin indices i, j, . . . for the coordinates, as they are associated to the Fermi-Walker transported frame. By the same token, the coordinate t coincides with the proper time t along the minimizing curve.
In components, Recalling that with our choicesω ij = 0 andω 0i (Ê j ) = 0 along the minimizing curve, we haveΩ ijnt = ∂ nωijt on this curve. Therefore (10) is equivalent tȯ On the other hand, (6) can be written as From these two equations we finally deduce which are the nontrivial components of Equation (2).
(1) As one would expect, the remaining minimum conditions lead to the Euler equation.
(2) We can write Equation (2)  7 Notice however that many authors define the angular momentum tensor to be e s = −s, corresponding to the alternative relation e σ ij = ε ijk σ k between the angular momentum matrix and the angular momentum vector.
8 As remarked in [Sem99], the sign of the last term in this equation is misquoted is some references (e.g. [MTW73,Ste04]), a problem which can be traced to the (+ − − −) signature used in [Pap51,Pir56].