The Lorentz transformation

The interference predicted by the Galilean transformation is impossible, because physical phenomena would experience the two systems in a different way, while they differ only by their relative motions v has to be replaced by —1 ). 

Tb hav i vi iythiiig back in widei, Cnlz aoisuiii ed that, wh ii a body lliv V 

If we insert such a length /, instead of L, in the expression for t, then we obtain 

As we have alreatty shown, in linear transformation (x, t ) (x, t) the diagonal 

To complete determination of the linear transformation we have to calculate the constant C. Albert Einstein assumed, that if Professors Oconnor and O connor began (in their own coordinate systems O and O ) measurements on the velocity of light, then despite the different distances gone (jr and x ) and different flight times (t and t ), both scientists would get the same velocity ofli t (denoted by c). 

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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 Lorentz transformation is an orthogonal transformation in the four dimensions of Minkowski space. The condition of constant c is equivalent to the requirement that the magnitude of the 4-vector s be held invariant under the transformation. In matrix notation... 

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 generation of invariants in the Lorentz transformation of four-vectors has been interpreted to mean that the transformation is equivalent to a rotation. The most general rotation of a four-vector, defined as the quaternion q = w + ix + jy + kz is given by [39]... 

The Lorentz transformation is further applied to the electric and magnetic fields, which become... 

Therefore if Ea> is null in frame K, it is null in frame K. There is a symmetry between the Lorentz transforms of (3) and the hypothetical Ea> ... 

These are mathematically valid results, but physically, the Lorentz transform of B(3> and the null E<3> are governed by the equation... 

These transformations from the stationary to the moving frame are called the Lorentz transformations. The inverse Lorentz transformation is obtained by reversing the sign of v, so that... 

This is also the relation obtained in the hypothetical rest frame. Therefore, the B cyclic theorem is Lorentz-invariant in the sense that it is the same in the rest frame and in the light-like condition. This result can be checked by applying the Lorentz transformation rules for magnetic fields term by term [44], The equivalent of the B cyclic theorem in the particle interpretation is a Lorentz-invariant construct for spin angular momentum ... 

In a strict sense, the classical Newtonian mechanics and the Maxwell s theory of electromagnetism are not compatible. The M-M-type experiments refuted the geometric optics completed by classical mechanics. In classical mechanics the inertial system was a basic concept, and the equation of motion must be invariant to the Galilean transformation Eq. (1). After the M-M experiments, Eq. (1) and so any equations of motion became invalid. Einstein realized that only the Maxwell equations are invariant for the Lorentz transformation. Therefore he believed that they are the authentic equations of motion, and so he created new concepts for the space, time, inertia, and so on. Within... 

In Einstein s special theory of relativity [1,2], the Galilean transformation had to be replaced by the Lorentz transformation, so that the speed of light would be invariant or independent of the relative motion of the observers—in particular, because the assumption f t is no longer correct. In the Lorentz transformation the time is t / t. 

Per-Olov was born in Uppsala on October 28, 1916 as the son of the musician Erik Wilhelm Lowdin and his wife Eva Kristina, nee Ostgren. He showed mathematical proficiency early in his school work, and when he entered Uppsala University in 1935 his plans were to major in mathematical physics. His first scientific paper in 1939 on the Lorentz-transformation and the kinematical principle of relativity was published in Swedish in the journal Elementa. The years of war that followed meant interruptions in communications and research opportunities and Per-Olov, like most young Swedish men, spent time in the military defending his country. 

In this case the probability of the passage of an atom through a layer of matter becomes greater than the one that follows from the usual exponential dependence. This phenomenon, superpenetration of ultrarelativistic A.2e, allows for measurement of the time of conversion of a non-stationary state of e+e, formed in the target, to stationary states and to verify the form of the Lorentz transformations for the time [8]. The theory of superpenetration has been formulated in [9,10,11]. A quantitative calculation shows that even for a film thickness L = 2.5A the deviation from an exponential absorption law reaches 100%. 

When this speed V becomes relativistic, then the Lorentz transformation steps in ... 

Equation (2.13.8) is called the Lorentz-FitzGerald94 contraction of space Eq. (2.13.11) is the Einstein time dilatation A clock advances more slowly in a system moving at a high speed V. When V Lorentz transformation reduces to the Galilean transformation. 

The Lorentz transformation has the following cute property. If two events are measured in coordinate system S as separated by Ax, Ay, and Az, and time At, and they are measured also in coordinate system S as being separated by different amounts of space Ax , Ay, Az, and time At, then the Lorentz invariance requires... 

It can be shown that det C = 1 (Problem 2.13.1). The norm of C, or trace of C, or sum of its diagonal terms, is 2 + 2y. Since det C = 1, we can consider the Lorentz transformation matrix X like the four-dimensional analog of the Eulerian rotation in 3-space. We now seek quantities that are "covariant with the Lorentz transformation"—that is, are "relativistically correct". We next define in this new four-space a few essential quantities ... 

It is important to realise that the Lorentz transformation describes accurately both the relativistic effects which are significant because of the high velocity of the electron and also the retardation effects which occur because of the finite time the field takes to reach the field point from the charge (which is a non-relativistic effect). 

This result is consistent with the Lorentz transformations... 

James Clerk Maxwell died in 1879, the same year that Albert Einstein was born. Sixteen years later Einstein recognized that Maxwell s equations are covariant with respect to the Lorentz transformations between relatively moving inertial frames of reference, that is, reference frames that are in constant relative motion in a straight line. Thus, Einstein recognized in 1895 that the laws of electrodynamics, expressed with Maxwell s held equations, must be in one-to-one correspondence in all possible inertial frames of reference, from the view of any one of them [1]. 

Thus the identification (17) <[)a (/<), ) is not to be understood as form-invariant regarding the dependence (17) of the spinor variables < )a on the tensor variables FpV in any other Lorentz frame. In other words, the Lorentz transformation of... 

For the same reason it is not clear, how to modify the equation for the inclusion of external fields. The principle of minimal coupling p —> p — A, E E + V for the (scalar) square-root Klein-Gordon equation was critizised by J. Sucher [4], who states that there are solutions ip x) and electromagnetic potentials, such that the Lorentz transformed solution is not a solution of the equation with the Lorentz-transformed potentials. Moreover, the nonlocal nature of the equation means that the value of the potential at some point influences the wave function at other points and it is not clear at all how one can interpret this. 

Even if we eschew the Lorentz transformation requirements for P(x) and M(x), which would break the longitudinal/transverse component separation just described, there is still a class of transformations that mixes their transverse components. If we define new polarization fields (P, M by the relations [5]... 

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