The basic principles of General Theory of Relativity historically have been tested in gedanken experiments in rotating frame of references. One of the key issues, which still evokes a lot of controversy, is the centrifugal acceleration. Machabeli & Rogava (1994) argued that centrifugal acceleration reverse direction for particles moving radially with relativistic velocities within a "bead on a wire" approximation. We show that this result is frame-dependent and reflects a special relativistic dilution of time (as correctly argued by de Felice (1995)) and is analogous to freezing of motion on the black hole horizon as seen by a remote observer. It is a reversal of coordinate acceleration; there is no such effect as measured by a defined set of observers, e.g., proper and/or comoving. Frame-independent velocity of a "bead" with respect to stationary rotating observers increases and formally reaches the speed of light on the light cylinder. In general relativity, centrifugal force does reverse its direction at photon circular orbit, r=3M in Schwarzschild metric, as argued by Abramowicz (1990).
Deep Dive into On reversal of centrifugal acceleration in special relativity.
The basic principles of General Theory of Relativity historically have been tested in gedanken experiments in rotating frame of references. One of the key issues, which still evokes a lot of controversy, is the centrifugal acceleration. Machabeli & Rogava (1994) argued that centrifugal acceleration reverse direction for particles moving radially with relativistic velocities within a “bead on a wire” approximation. We show that this result is frame-dependent and reflects a special relativistic dilution of time (as correctly argued by de Felice (1995)) and is analogous to freezing of motion on the black hole horizon as seen by a remote observer. It is a reversal of coordinate acceleration; there is no such effect as measured by a defined set of observers, e.g., proper and/or comoving. Frame-independent velocity of a “bead” with respect to stationary rotating observers increases and formally reaches the speed of light on the light cylinder. In general relativity, centrifugal force does re
Since the conception of the Special and General Theories of Relativity, rotating frames served as conceptual testbed of our understanding of effects of motion and gravitation on measured quantities. For over a century this has lead to a number of paradoxes, most notably the Ehrenfest paradox [1] of the circumference length of a rotating disk. The Ehrenfest paradox involved a discussion between such prominent physicists as Born, Plank, Kaluza, Einstein, Becquerel, and Langevin among others [2]. Kinematics and especially dynamics in rotating frame continues to be a source of confusion. In this article we aim to elucidate one of the "paradoxes", the reversal of centrifugal acceleration.
Following the work [3] on reversal of centrifugal force in general relativity, Machabeli & Rogava [4] suggested that a somewhat similar effect, reversal of centrifugal acceleration, occurs in special relativity. This suggestion we taken up in a number of astrophysically-related works on particle acceleration around rotating black hole and neutron star magnetospheres [5,6,7,8] and others. In this Letter we show that the effect discussed by Machabeli & Rogava [4] is not frame-invariant and disappears if one uses frame-invariant quantities. Thus, in special relativity, there is no reversal of centrifugal acceleration. The effect seen by Machabeli & Rogava is a time dilation, as correctly argued by [9]. It describes an unphysical coordinate acceleration.
The motivation for this work comes from numerous astrophysical cites (e.g., magnetospheres of various black holes and neutron stars), where both strong gravity, magnetic field, and rotation are all important ingredients. The effects of magnetic field on a single particle motion are often approximated as a solid guiding wire, which restricts particle motion across the field. This simple approximation neglecting various cross-field drifts. The key question that we will address is “what is the behavior of the parallel momentum of the particle?”.
To elucidate the key problems, consider a bead on a radial wire inclined at angle π/2 to the rotation axis. Let us first neglect gravitation. Using standard methods of general relativity, we transform to rotating coordinates by changing the azimuthal variable φ → φ ′ -ωt and assume that in rotating coordinates the motion is strictly radial, dφ ′ = dθ = 0. The nontrivial element of the metric tensor is then
The Hamilton -Jacobi equation ∂ t S + H = 0, where H is Hamiltonian and S is generating
(we set G = c = 1, use (-1, 1, 1, 1) sign convention and assume that the mass of a test particle is unity.) Since the two-dimensional motion in r -t plane has a conserved quantity -the product of the particle momentum and the time-like Killing vector (time is a cyclic variable), we can look for separable solutions in a form S = -E 0 t+S(r). After differentiating with respect to E 0 , Eq. ( 2) gives
By differentiating with respect to time, and eliminating constant E 0 , we find expression for coordinate acceleration in terms of coordinate velocity
where v r = ∂r/∂t. (This result can also be heuristically obtained from Newtonian centrifugal
) This is the result of Machabeli & Rogava [4], who argued that at r = 0, for v r > 1/ √ 2 centrifugal acceleration reverses its sign and becomes centrifugal deceleration. Indeed, for v r > 1/ √ 2 we have r < 0. In addition, Eq. ( 4) does have a solution in terms of elliptic sinus function with formal reversal of velocity occurring at the light cylinder.
In the previous section we derived equations of motion of a bead on a wire and obtained fully analytical and mathematically correct solutions. Does it mean that a particle experiences a reversal of centrifugal accelerations and can never leave the light cylinder of a rigidly rotating wire? The answer, which is physically obvious, but given the above derivation is a bit surprising, is no. The key moment missed by Machabeli & Rogava is that observed quantities must be formulated in a frame-invariant, but observer-dependent form. Thus, quantities measured in terms of, e.g., coordinate time are, in some sense, the least physical. On the other hand, quantities measured by a defined set of observers can be cast in a frame-independent form using the four-velocities of those observers (e.g., notion of ZAMOs in [10]). Expression ( 4) is coordinate-dependent, and thus is not physically useful. Physically important are velocities and acceleration measured by a defined set of observers. For example, we can define a set of local stationary observers rotating with the wire. For such
Eqns ( 5) clearly shows that centrifugal acceleration of the bead, as measured by a set of observers stationary with respect to the rotating wire, is always directed away from the axis of rotation.
We can also find equations of motions and acceleration in terms of proper time τ of the bead:
So that proper velocity and proper acceleration are always positive.
As the question under consideration
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