Four-vector Lorentz transformation

The (four-vector). au(x) represents the result of some local measurement at the point x performed by a (Lorentz) observer 0. An observer O (related to 0 by a Lorentz transformation x = Ax) describes this measurement by... 

Four-vectors for which the square of the magnitude is greater than or equal to zero are called space-like when the squares of the magnitudes are negative they are known as time-like vectors. Since these characteristics arise from the dot products of the vectors with reference to themselves, which are world scalars, the designations are invariant under Lorentz transformation[17], A space-like 4-vector can always be transformed so that its fourth component vanishes. On the other hand, a time-like four-vector must always have a fourth component, but it can be transformed so that the first three vanish. The difference between two world points can be either space-like or time-like. Let be the difference vector... 

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]... 

Requiring these order parameters to transform in a Lorentz-covariant way, we are led to a particular basis of 4 x 4 matrices , which was recently derived in detail (Capelle and Gross 1999a). The resulting order parameters represent a Lorentz scalar (one component), a four vector (four components), a pseudo scalar (one component), an axial four vector (four components), and an antisymmetric tensor of rank two (six independent components). This set of 4 x 4 matrices is different from the usual Dirac y matrices. The latter only lead to a Lorentz scalar, a four vector, etc., when combined with one creation and one annihilation operator, whereas the order parameter consists of two annihilation operators. 

Einstein s theory of special relativity relying on a modified principle cf relativity is presented and the Lorentz transformations are identified as the natural coordinate transformations of physics. This necessarily leads to a modification cf our perception of space and time and to the concept of a four-dimensional unified space-time. Its basic Mnematic and dynamical implications on classical mechanics are discussed. Maxwell s gauge theory of electrodynamics is presented in its natural covariant 4-vector form. 

After the preliminaries presented above we can now precisely define vectors and tensors in Minkowski space by their transformation properties under Lorentz transformations. Each four-component quantity A, which features the same transformation property as the contravariant space-time vector as given by Eq. (3.12),... 

In contrast to the three-dimensional situation of nonrelativistic mechanics there are now two kinds of vectors within the four-dimensional Minkowski space. Contravariant vectors transform according to Eq. (3.36) whereas covariant vectors transform acoording to Eq. (3.38) by the transition from IS to IS. The reason for this crucial feature of Minkowski space is solely rooted in the structure of the metric g given by Eq. (3.8) which has been shown to be responsible for the central structure of Lorentz transformations as given by Eq. (3.17). As a consequence, the transposed Lorentz transformation A no longer represents the inverse transformation A . As we have seen, the inverse Lorentz transformation is now more involved and given by Eq. (3.25). 

So far we have just defined another four-component quantity Af, but by now it is not clear whether it properly transforms under Lorentz transformations in order to justify the phrase 4-vector. In order to prove the transformation property of the gauge field, we re-express the inhomogeneous Maxwell equations in Lorenz gauge as given by Eq. (2.138) in explicitly covariant form by employment of the charge-current density and the gauge field A, ... 

Here, it is important to understand that is a four-component object consisting of (4x4)-matrices rather than a Lorentz scalar, since is not a Lorentz 4-vector. Moreover, although the equation already seems to be in covariant form, this still needs to be shown because we do not yet know how Y transforms under Lorentz transformations. In the following, we must determine the transformation properties of Y so that Eq. (5.54) is covariant under Lorentz transformations, which is a mandatory constraint for any true law of Nature. 

Similarly to the nonrelativistic situation [cf. Eq. (2.29)], the components of the Lorentz transformation matrix A may be expressed as derivatives of the new coordinates with respect to the old ones or vice versa. However, since we have to distinguish contra- and covariant components of vectors in the relativistic framework, there are now four different possibilities to express these derivatives ... 

From our experience this far with vector lengths and velocities, we do not expect the magnitude of ordinary linear momentum to be invariant under the Lorentz transformations. By analogy with our previous derivation of the four-vector, we can take a cue from the relations for light signals. For photons we know that the relation... 

We start with the potential set up by a moving charge. Having established that A = (a, (p) is a four-vector, we expect it to transform in analogy with the position four-vector, and the Lorentz transformation of (2.17) should apply if we replace r with A and t with 0/c. More specifically— if S is the stationary frame and S is moving along the x axis with velocity v relative to S—we have the transformation equations... 

We know that the four-vector momentum and the four-vector potential are conserved quantities under a Lorentz transformation, so we expect that a is also conserved, as may be shown with the conditions below. 

Comparing with (4.27), we see that we may identify ca with the velocity operator u. With this identification, we see that the velocity four-vector a can be identified with the classical velocity four-vector, which is the time derivative of the position four-vector w. We can also identify p with the inverse of the y factor arising in the Lorentz transformations. 

Here we recognize a situation that is quite similar to that encountered for the 50(3) rotations in section 6.4. We have transformations that conserve the length of the four-vector, and which may be expressed as matrices operating on the set of basis vectors. We may in fact proceed in a completely analogous manner to the SO(3) case. Thus, an infinitesimal Lorentz transformation— that is, v/c 1—may be written as... 

This idea is motivated by the fact that energy and momentum are the components of a four-vector, a relativistic quantity that transforms like a vector when subjected to a Lorentz transformation ... 

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