Accelerating frame of reference

Yablonovitch [215] proposed using a medium with a rapidly decreasing in time refractive index ( plasma window ) to simulate the so-called Unruh effect [216] the creation of quanta in an accelerated frame of reference. More rigorous and detailed studies of quantum phenomena in nonstationary (deformed) media have been performed [159,160,217-229]. The case when the dielectric constant changes simultaneously with the distance between mirrors (in one dimension) was also considered [230,231]. Johnston and Sarkar compared the spectra of photons created by the motion of mirrors and by the time variations of the dielectric permeability [232]. An analog of the nonstationary Casimir effect in the superfluid 3He, namely, the friction force on the moving interface between two different phases, was discussed by Volovik [233]. 

With —eE = V(/>ei this expression is twice as large as the spin-orbit term in (99). But there is still another contribution coming from an effect in relativistic kinematics the Thomas precession. As the composition of two boosts with nonparallel velocities contains a rotation, one finds that an accelerated frame of reference performs an additional precession with the frequency... 

Consider a frame of reference in uniform rotation. It is noted that sustained circular motion requires constant acceleration towards the centre. Lorentz transformation dictates the contraction of a mear suring rod in the direction of motion in this rotating frame. In a second, accelerated frame of reference, with the same origin, there is no contraction and the ratio of the circumference to the diameter of a reference circle remains S/2R = tt. For the same circle, observed in the stationary frame, S /2R > tt, since the radial measurement is not affected by the motion. To account for this effect it is necessary to realize that Euclidean geometry does not apply in the stationary frame of reference. 

As a rule, geophysical literature describes the rotation of a particle on the earth surface with the help of the attraction force and the centrifugal force. It turns out that the latter appears because we use a system of coordinates that rotates together with Earth. As we know Newton s second law, wa = F, is valid only in an inertial frame of reference, that is, the product of mass and acceleration is equal to the real force acting on the particle. However, it is not true when we study a motion in a system of coordinates that has some acceleration with respect to the inertial frame. For instance, it may happen that there is a force but the particle does not move. On the contrary, there are cases when the resultant force is zero but a particle moves. Correspondingly, replacement of the acceleration in the inertial frame by that in a non-inertial one gives a new relation between the acceleration, mass, particle, and an applied force ... 

This equation establishes the relationship between the velocities in both frames of reference. Performing one more differentiation we obtain for the acceleration in the inertial system P ... 

Here v, and a, are the velocity and acceleration of the point p in the rotating frame of reference, respectively. Substitution of Equation (2.55) into Newton s second law gives an equation of motion in the non-inertial frame ... 

The third method involves a three pulse sequence, 90 — r — 180° — x — 90°, with a repetition time of tr s. This pulse sequence refocuses the magnetization vector M0 into its equilibrium position within the repetition time, thus representing a pulse driven relaxation acceleration. This technique, known as DEFT NMR [23, 24] (driven equilibrium Fourier transform NMR) can be understood by following the behavior of the magnetization vector Mq under the influence of the pulse sequence in the rotating frame of reference (Fig. 2.17(a-e)). 

For mathematical convenience, boundary conditions and initial conditions must be prescribed. For the simple marine propeller problem, a Lagrangian viewpoint was adopted. The frame of reference was attached to the propeller so that the propeller was fixed but the vessel was rotating. The boundary condition was then a zero velocity on the impeller, while the vessel wall rotated at -Qimpdier- The free surface was considered to be fiat, therefore the normal velocity was zero and a shear-free condition was assumed. It should be noted that in the Lagrangian viewpoint, the frame of reference is in rotation. The fluid is therefore subjected to a constant acceleration and the momentum conservation equation [Eq. (6)] must be modified to account for centrifugal forces and Coriolis forces.An advantage is, however, that the flow can be solved numerically at steady state provided the flow is fully periodic, which limits the computational efforts significantly. 

The fictitious forces are conventionally derived with the help of the framework of classical mechanics of a point particle. Newtonian mechanics recognizes a special class of coordinate systems called inertial frames. The Newton s laws of motion are defined in such a frame. A Newtonian frame (sometimes also referred to as a fixed, absolute or absolute frame) is undergoing no accelerations and conventionally constitute a coordinate system at rest with respect to the fixed stars or any coordinate system moving with constant velocity and without rotation relative to the inertial frame. The latter concept is known as the principle of Galilean relativity. Speaking about a rotating frame of reference we refer to a coordinate system that is rotating relative to an inertial frame. 

A scalar is a quantity associated with a point in space, whose specification requires just one number. For example, the fluid density, mass fraction, temperature, pressure and work are all scalar quantities. Scalars can be compared only if they have the same physical dimensions. Scalars measured in the same system of units are said to be equal if they have the same magnitude and sign. A vector is an entity that possesses both magnitude and direction and obeys certain laws. For example, velocity, acceleration, force are all vectors. Two vectors are equal if they have the same direction and the same magnitude. Moreover, a direction has to be specified in relation to a given frame of reference and this frame of reference is just as arbitrary as the system of units in which the magnitude is expressed. We distinguish therefore between the vector as an entity and its components which allow us to reconstruct it in a... 

In the subsequent treatment the electron coordinate will be measured from the accelerated target nucleus and is the only dynamical variable. Thus the target system is the frame of reference [31,32], In such a noninertial system non-Newtonian forces arise. The corresponding Hamiltonian is... 

The special theory, of 1905, refers to nonaccelerated frames of reference, while the general theory, of 1915, extends to accelerated systems. 

This is the basic principle of Einstein s special theory of relativity. However, Einstein was not content with the apparent absolute status conferred to accelerating frames by the behaviour of bodies within them. Einstein sought a general principle of relativity that would require all frames of reference, whatever their relative state of motion, to be equivalent for the formulation of the general laws of nature. In his popular exposition of 1916, Einstein explains this by describing the experiences of an observer... 

Bottom section (rows g, h, i, j, and k) Summed to yield the absolute acceleration (acceleration with respect to an inertial frame of reference) of point P. 

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