On the relativistic tidal effects in the second approximation

Abstract

Using the concept of the world function, an exact deviation equation has been derived, which describes the relative motion of two accelerated point masses and is valid in the case of their arbitrary 4-velocities, 4-accelerations, parametrizations of their world lines, and arbitrary correspondence between the points of the world lines. The second approximation of the exact deviation equation, with respect to the components of deviation vector, has been elaborated in general coordinates and in the Fermi coordinates of an observer comoving with one of the point masses. The 3-acceleration of a freely falling test particle in the frame of reference of an accelerated observer in the Schwarzschild spacetime has been calculated in the second approximation. It turns out that the quadratic terms involve the observer's 3-velocity relative to the black hole and become significant compared to the linear terms if the observer moves at a relativistic 3-velocity. Moreover, in most cases, if the observer's 3-velocity approaches the velocity of light, the second-order deviation equation is no longer applicable as the relevant Taylor expansions do not converge. It is established that if a freely falling observer starts its motion from spacelike infinity at a nonrelativistic initial velocity, the problem of convergence does not appear. It is proved that free circular motion of an extended body is impossible near the 1.5 Schwarzschild radius as the body would be destroyed by the tidal forces.

Generalized deviation equation and determination of the curvature in general relativity

“…In special cases, cf. also the next section III, our result is in qualitative agreement with previous results in the literature, see in particular [14,18,21,25,41,63] III. SPECIAL CASES Up to this point our considerations were completely general, resulting in the exact form (9) as well as in the second order version (35) -expanded w.r.t.…”

Critical Tidal Currents in General Relativity

“…For a = 0, the Kerr metric reduces to the spherically symmetric Schwarzschild metric, in which case our treatment applies to motion along any radial direction. The corresponding expressions for A and E simplify for a = 0 in Equations (13) and (14), respectively, but our main results remain unchanged. Indeed, sufficiently far from any astronomical source, we expect that its gravitational field is dominated by its mass M and the accelerated outflow can occur above the threshold (34) along any radial direction away from mass M.…”

“…However, for relativistic motion above the critical speed, there is deceleration for k(T) < 0 toward the critical speed and acceleration for k(T) > 0. Various aspects of this circumstance and its implications for astrophysical jets have been explored in previous work [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. For recent work on general relativistic tidal effects in other contexts, see, for example, Refs.…”

Critical Tidal Currents in General Relativity

“…However, for relativistic motion above the critical speed, there is deceleration for k(T ) < 0 toward the critical speed and acceleration for k(T ) > 0. Various aspects of this circumstance and its implications for astrophysical jets have been explored in previous work [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. For recent work on general relativistic tidal effects in other contexts, see, for example, Refs.…”