Lorentz velocity

Data Processing. Fraser (21) has shown that, with data measured in array form, each array member may be individually given Lorentz (velocity), polarization and absorption corrections. This avoids the problems incurred in the Photometric Peak Center method and Sum Intensity method in which the corrections are applied as if all intensity is recorded at the peak amplitude position. Fraser (22) has also shown that, by choosing data... 

The source is brought to a. positive poteptial (I/) of several kilovolts and the ions are extracted by a plate at ground potential. They acquire kinetic energy and thus velocity according to their mass and charge. They enter a magnetic field whose direction is perpendicular to their trajectory. Under the effect of the field, Bg, the trajectory is curved by Lorentz forces that produce a centripetal acceleration perpendicular to both the field and the velocity. 

Magnetic Sector Field. In a magnetic field B an ion with the velocity v and the charge q experiences a centripetal force, the Lorentz force P ... 

Before outlining Toffoli s model of a deterministic relativistic diffusion C A model, we motivate the discussion by recalling a simple formal analogy that holds between a circular rotation by an angle 6 in the x,y) plane and a Lorentz transformation with velocity... 

The Hamiltonian is insensitive to the direction of time, 7i(T) = T L(T ), since it is a quadratic function of the molecular velocities. (Since external Lorentz or Coriolis forces arise from currents or velocities, they automatically reverse direction under time reversal.) Hence both T and I1 have equal weight. From this it is easily shown that. (xl/s) = (exlL). 

A filter that combines both a magnetic and electric field is the so-called Wien filter (or velocity filter). In this case, charged ions pass through a region characterized by uniform magnetic and electric fields at right angles to each other and to the direction of incident ions only those particles for which the module of the Lorentz... 

The spectral function thus has a Lorentz shape with a halfwidth at the half distribution height equal to the average reorientation frequency w. If expressed in spectroscopic units (cm 1), the halfwidth Avv2 amounts to cd2nca (c0 designates the velocity of light in vacuo). 

It is now found that (22) is indeed invariant under (24), which is known as the Lorentz5 transformation of Special Relativity. It is important to note that in the limit v/c —> 0 the Lorentz formulae reduce to the Galilean transformation, suggesting that Lorentzian (relativistic) effects only become significant at relative velocities that approach c. The condition t = t which... 

The most important new feature of the Lorentz transformation, absent from the Galilean scheme, is this interdependence of space and time dimensions. At velocities approaching c it is no longer possible to consider the cartesian coordinates of three-dimensional space as being independent of time and the three-dimensional line element da = Jx2 + y2 + z2 is no longer invariant within the new relativity. Suppose a point source located at the origin emits a light wave at time t = 0. The equation of the wave front is that of a sphere, radius r, such that... 

The condition for a time-like difference vector is equivalent to stating that it is possible to bridge the distance between the two events by a light signal, while if the points are separated by a space-like difference vector, they cannot be connected by any wave travelling with the speed c. If the spatial difference vector r i — r2 is along the z axis, such that In — r2 = z — z2, under a Lorentz transformation with velocity v parallel to the z axis, the fourth component of transforms as... 

If X is space-like and the events are designated such that t2 > 11, then c(ti — f2) < z — z2, and it is therefore possible to find a velocity v < c such that ic(t[ — t 2) = X vanishes. Physically the vanishing of X means that if the distance between two events is space-like, then one can always find a Lorentz system in which the two events have the same time coordinate in the selected frame. On the other hand, for time-like separations between events one cannot find a Lorentz transformation that will make them simultaneous, or change the order of the time sequence of the two events. The concepts "future" and "past" are invariant and causality is preserved. That the sequence of events with space-like separations can be reversed does not violate causality. As an example it is noted that no influence eminating from earth can affect an object one light-year away within the next year. 

Since two successive Lorentz transformations for relative velocities [3 and (3 along z3 are rotations in the same plane, the rotation angles simply add. From (30) tan cj) = i/3, and since... 

It is to be expected that the equations relating electromagnetic fields and potentials to the charge current, should bear some resemblance to the Lorentz transformation. Stating that the equations for A and (j> are Lorentz invariant, means that they should have the same form for any observer, irrespective of relative velocity, as long as it s constant. This will be the case if the quantity (Ax, Ay, Az, i/c) = V is a Minkowski four-vector. Easiest would be to show that the dot product of V with another four-vector, e.g. the four-gradient, is Lorentz invariant, i.e. to show that... 

The Lorentz Force Law can be used to describe the effects exerted onto a charged particle entering a constant magnetic field. The Lorentz Force Fl depends on the velocity v, the magnetic field B, and the charge of an ion. In the simplest form the force is given by the scalar equation [3,4,70,71]... 

As we know from the discussion of magnetic sectors, an ion of velocity v entering a uniform magnetic field B perpendicular to its direction will move on a circular path by action of the Lorentz force (Chap. 4.3.2), the radius of which is determined by Eq. 4.13 ... 

The Lienard-Wiechert potentials (12) can also be derived from a rotation-free Lorentz transformation (boost) of the four potential of a static charge (13) to the moving frame at retarded time. For a charge moving at constant velocity the potentials can also be expressed in terms of the current position giving [21]... 

Figure 9. Two trajectories of the periodic hard-disk Lorentz gas. They start from the same position but have velocities that differ by one part in a million, (a) Both trajectories depicted on large spatial scales, (b) Initial segments of both trajectories showing the sensitivity to initial conditions.

The velocity of light passing through a polymer is affected by the polarity of the bonds in the molecule. Polarizability P is related to the molecular weight per unit volume, M, and density p as follows (the Lorenz-Lorentz relationship) ... 

These central concepts of tachyon theory also come out of the present approach. An alternative way to satisfy the condition (8) of Lorentz invariance is thus to replace the form (70) of the velocity vector C by... 

After their creation, positive ions are accelerated through a voltage difference V. They thus acquire a velocity t> that depends on their mass m. Following acceleration, the ions enter a transversal magnetic field of intensity B. The orientation of this field does not modify the ions velocity but forces them on a circular trajectory that is a function of their m/z ratio. The fundamental relationship of dynamics F = ma (a designates acceleration), applied to ions of mass m on which a Lorentz force F = qv A B is exerted leads to the following relationship ... 

The force exerted on an ion carrying a charge q with a velocity v and subjected to a magnetic field of intensity B is given by the Lorentz equation ... 

In equations 5-8, the variables and symbols are defined as follows p0 is reference mass density, v is dimensional velocity field vector, p is dimensional pressure field vector, x is Newtonian viscosity of the melt, g is acceleration due to gravity, T is dimensional temperature, tT is the reference temperature, c is dimensional concentration, c0 is far-field level of concentration, e, is a unit vector in the direction of the z axis, Fb is a dimensional applied body force field, V is the gradient operator, v(x, t) is the velocity vector field, p(x, t) is the pressure field, jl is the fluid viscosity, am is the thermal diffiisivity of the melt, and D is the solute diffiisivity in the melt. The vector Fb is a body force imposed on the melt in addition to gravity. The body force caused by an imposed magnetic field B(x, t) is the Lorentz force, Fb = ac(v X v X B). The effect of this field on convection and segregation is discussed in a later section. 

In physical terms, this can be understood in the following way. Take an electromagnetic field with Poynting vector S = E x B. By a suitable Lorentz transformation [with direction unit vector n and velocity parameter q given by ntanh2ri = 2S/(E2 +B2)], we can change to a frame in which S = 0 at any... 

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