General Lorentz Boost

We now consider a general Lorentz boost from the inertial system IS to the inertial system IS, which moves with constant velocity v = relative to 

Note that the velocity n is a 3-vector and therefore its components are labeled with subscript indices, e.g., Vi = Vx. This must not be confused with the covariant components of a 4-vector. The velocity vector v may point in any direction, but the coordinate axes of IS and IS must be parallel to each other, i.e., there is no constant rotation between the axes of the two systems. For a particle at rest in IS (for example at the origin r = 0), we have dr = 0 and can therefore immediately write down the Lorentz transformation given by Eq. (3.13) as 

Since this particle moves with velocity —v for an observer in IS, it holds that 

Similarly, consideration of a particle at rest in IS, which consequently moves in IS with velocity v, we find the same relation, i.e.. A = A q. Exploiting the fundamental equation for Lorentz transformations as given by Eq. (3.17) for ji = V = 0, we immediately find a further relation between AP q and N q. 

In order to finally determine the desired transformation matrix A(v) for a general Lorentz boost, we note that we can always first rotate IS to IS by application of A(K) as given by Eq. (3.19) such that the new coordinate axis x is parallel to v, then apply the familiar Lorentz boost A(t ) in -direction as given by Eq. (3.66), and finally rotate the new system IS back to IS by the inverse rotation A(E ), 

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General Lorentz boosts A(z ) are therefore in general not commutative, whereas boosts for parallel velocities commute, cf. Eq. (3.98). 

We now proceed to derive the Dirac states for a freely moving electron of mass Me- Note that the charge of the fermion does not enter the Dirac equation for this fermion being at rest or moving freely with constant velocity v. The solutions in Eqs. (5.72) and (5.73) may now be subjected to a general Lorentz boost as given by Eq. (3.81) into an inertial frame of reference moving relatively to the previous one, in which the fermion is at rest, with velocity (— ). This option, namely that the solutions for a complicated kinematic problem can be obtained from those of a simple kinematic problem in a suitably chosen frame of reference by a Lorentz transformation, cannot be overemphasized from a conceptual point of view. However, instead of this Lorentz transformation a direct solution of the Dirac Eq. (5.23) is easier. For this purpose we choose an ansatz of plane waves,... 

The absence of an E(3) field does not affect Lorentz symmetry, because in free space, the field equations of both 0(3) electrodynamics are Lorentz-invariant, so their solutions are also Lorentz-invariant. This conclusion follows from the Jacobi identity (30), which is an identity for all group symmetries. The right-hand side is zero, and so the left-hand side is zero and invariant under the general Lorentz transformation [6], consisting of boosts, rotations, and space-time translations. It follows that the B<3) field in free space Lorentz-invariant, and also that the definition (38) is invariant. The E(3) field is zero and is also invariant thus, B(3) is the same for all observers and E(3) is zero for all observers. 

The result is obtained that Faraday s law of induction is invariant under a Z boost. Similarly, it can be shown to be invariant under the general Lorentz transformation, and all solutions are invariant. In general, on the U(l) level... 

We finally note that more general Lorentz transformations are now obtained easily because any proper Lorentz transformation can be written in a unique way as the product of a boost and a rotation. 

So far, only general properties of Lorentz transformations have been investigated but no explicit expression for the transformation matrix A has yet been given. We are now going to derive the transformation matrix A for a Lorentz boost in x-direction in a very clear and elementary fashion. For t = t = 0 the two inertial frames IS and IS shall coincide, and the constant motion of IS relative to IS shall be described by the velocity vector v = vCx, cf. Figure 3.2. Since the y- and z-directions are not affected by this transformation, we explicitly write down the transformation given by Eq. (3.12) (for a = 0) for the relevant subspace... 

The derivation above was focused on a Lorentz boost in the direction of the stick, i.e., the relative motion of the two frames IS and IS, described by the velocity vector v, and the stick were parallel. Due to the Lorentz boost given by Eq. (3.67) no length contraction will occur in directions perpendicular to v. This immediately yields the general expression for the length contraction. 

For general, i.e., non-parallel velocities V and vz we cannot simplify the problem by application of a suitable rotation of the coordinate axes. Now all three Lorentz boost matrices occurring in Eq. (3.95) are of the most general form given by Eq. (3.81) with a yet undetermined resulting velocity V. We thus have to evaluate Eq. (3.95) directly. In order to achieve this task, it is convenient to introduce the abbreviations... 

For the general case of an arbitrary Lorentz boost A( ) we note that the inverse transformation A(—i ) is easily obtained from Eq. (3.81) by switching the signs of the velocity components V (cf. Eq. (3.82)). The Minkowski force / in IS is therefore given by... 

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