RELATIVE KINEMATICS IN GENERAL RELATIVITY THE THOMAS AND FOKKER PRECESSIONS* E. Massa - C. Zordan** SOM3dARIO: II concerto di sistema di riferimeuto fisico in uoo spazio-tempo Oa viene applicato allo studio del moto relativo di un giroscopio puntiforme in Relativitgz Generale. La precessione di Thomas e/a precessione di Fokker souo ricavate come casi par- ticolari de/risultato generate. SUMMARY: The theo O, of space /ensors is applied to the study of th, motion of a g),roscopically stabilized point compass in a given frame of reference (/', ~*). The Thomas and Fokker precessions are obtained as special subcases of the genera/result. 1. Introduction. The theory of space tensors, thoroughly discussed in Refs. [1 + 3], leads quite naturally to the concept of physical frame of reference in the space-time manifold 04. This is perhaps one of the most suggestive aspects of the entire theory, as it provides the starting point for the formulation of Relative Mechanics and, more generally, of Relative Physics in curved space-time. These arguments were examined in Re['. [2]. The def- inition of physical frame of reference proposed there is a refinement of the one introduced by C. Moiler [4], and later developed by C. Cattaneo [5 + 9], and is sub- stantiaIly equivalent to the one adopted by F. Eastbrook and H. Wahlquist in their "dyadic analysis of space-time congruences" [10]. Basically, it includes a time-like con- gruence _F' and a distinguished spatial tensor analysis (V*, Vr*), known as the standard spatial tensor analyis over t%, _F'). Ahernatively, we may replace (~*, ~r*) by the corresponding standard afflne connection V* [1 ]. The pair (/1, ~*) is then called the physical frame of reference de- termined by the congruence /' in 04. An interesting application of the previous concepts arises in connection with the following problem: let A be a point compass, identifying three mutually orthogonal spatial directions 2~=) (cz = 1, 2, 3). (One may think, e.g., of a small material ellipsoid of negligible size.) The motion of A is then represented in 04 as a world- line a carrying a distinguished orthonormal triad 2¢:), orthogonal to the direction of a at every point. Assume further that .A is gyroscopically stabilized, i.e., that the triad 2¢=) does not rotate with respect to the so- * Research sponsored by the CNR Gruppo Nazionale per la Fisica Matematica ** Istituto matematico dell'Universit~ di Coenova, Italy. called "compass of inertia". This is an absolute property, expressed mathematically by the condition that the vectors )~¢=~ undergo Fermi-Walker transport along a [11]. The problem is then to determine the angular velocity o~ of 3,¢=~ relative to a given frame of reference ([', ~*). Effects like the Thomas precession [4], [12] and the Fokker precession [4], [13] are clearly included in this general context. More precisely, they arise as special sub- cases, occurring respectively when (/', V*) is identified with an inertial frame of reference in flat space-time (Thom- as precession), or when the curve a is a time-like goedesic (Fokker precession). In this paper we propose a detailed analysis of the general problem: the necessary mathematical preliminaries are briefly discussed in Subsection 1.1, while the explicit de- termination of ~ is indicated in Subsection 1.2. The method followed here depends rather strictly on the results shown in Ref. [2], and provides a further il- lustration of the kinematical concepts introduced there. 1.1 Mathematical preliminaries Let a ----a(s) be a world-line in On, described locally in terms of arc length s by the parametric equations x I = = xl(s)(l). We set 2c0) ----- a'(s) = dxiJds (O]Oxi),(sl, and denote by 2¢~) (co = 1, 2, 3) an orthonormal triad of vector fields defined on a, and orthogonal to 2c0~ at every point of a. We assume further that the fields 2¢~) undergo Fermi- Walker transport along a, i.e. D).¢=~ = (g(2¢~), D2c0~ ) Ds ~ ),¢0~ (1.1) D/Ds being the absolute derivative on a determined by the Riemannian connection of 0a. The tetrad 2,~ (i = = 0,... 3) is then called a Fermi tetrad on a. The corresponding dual tetrad ),J*~ is defined by and satisfies the obvious relations (act,, ;to@ = ~ (1.2) (t) Latin indices run from 0 to 3. Greek indices run from 1 to 3. Einstein's summation convention is used throughout. The signature of the metric is -- + + +. The use of natural units (c = 1) is implicitly assumed. The mapping g: ~i_.~ ~1 is defined by g ( X ~ O/Ox I) = X~ dx~ (with X~ = gij -'\'~), and corresponds to the usual process of "lowering the tensor in- dices". MARCH 1975 27