Critical velocities $c/\sqrt{3}$ and $c/\sqrt{2}$ in the general theory of relativity

Abstract

We consider a few thought experiments of radial motion of massive particles in the gravitational fields outside and inside various celestial bodies: Earth, Sun, black hole. All other interactions except gravity are disregarded. For the outside motion there exists a critical value of coordinate velocity v c = c/ √ 3: particles with v < v c are accelerated by the field, like Newtonian apples, particles with v > v c are decelerated like photons. Particles moving inside a body with constant density have no critical velocity; they are always accelerated. We consider also the motion of a ball inside a tower, when it is thrown from the top (bottom) of the tower and after classically bouncing at the bottom (top) comes back to the original point. The total time of flight is the same in these two cases if the initial proper velocity v 0 is equal to c/ √ 2.

The Hubble flow: an observer's perspective

“…A classical example of this situation is the description of free fall into a black hole. From the point of view of an observer at spatial infinity the motion of falling objects is initially accelerated, then, after reaching its maximum the coordinate velocity (which in the case of free fall from infinity with zero initial velocity is equal to 2c/[3/ √ 3]) is decreasing [9]. It is clear that this coordinate velocity defined as the ratio of distance to the time interval measured at the infinity is important only for description of an observed picture of free fall.…”

Beyond Gravitoelectromagnetism: Critical Speed in Gravitational Motion

“…The critical speeds 1/ √ 3 and 1/ √ 2 have also been discussed in Ref. 21. Nevertheless, it is important to remark here that with respect to the basic class of static (generally noninertial) observers in the exterior Schwarzschild spacetime, the local speed of an infalling particle monotonically increases toward unity, while the local speed of light is always equal to unity.…”

Significance of c /sqrt2 in relativistic physics

“…Finally, let us remark that the proper critical speed c/ √ 2 has also been discussed for one-dimensional relative motion along the radial direction in the context of the exterior Schwarzschild geometry in [17].…”

“…so that at X µ = (τ, 0), equations (18)- (20) reduce to equation (17). The new (Rindler) coordinates are T , X, Y ∈ (−∞, ∞) and Z ∈ (−1/g, ∞), where Z = −1/g is the Rindler horizon and corresponds to a null cone in the inertial frame, since (Z + 1/g) 2 = (z − z 0 + 1/g) 2 − t 2 .…”