K9    STR with Four-Vectors


Geometric problems can be treated with many different mathematical techniques. For example, you can calculate the volume of a tetrahedron using basic geometry, using vector geometry or using integral calculus. Certain questions can often be elegantly answered through the appropriate approach, while another approach would be complicated or provide only approximate success.

The best methodology for doing algebraic calculations in the STR is the one with four-vectors! Not only the place and time of an event, but also all other physical quantities are consistently described by vectors with 4 components: There are the four-speed, four-acceleration, the four-force, the four-momentum and the four-current vectors. All of these four-vectors transform themselves according to the Lorentz transformations in the transition to another coordinate system, just as we have seen with location and time coordinates. And for any four-vectors A and B, there is a simple scalar product A • B which yields a constant value, independent of the reference system!

Let us take as examples the four-momentum P and the four-speed V:

P = ( Etot/c, px, py, pz ) = mo/√ •( c , vx, vy, vz ) = mo• V        or as shorthand

P = ( Etot/c, p ) = mo/√ •(c, v) = mo• V

where p is the 3d-momentum vector and v the 3d-velocity vector. The scalar product of two four-vectors (x0, x1, x2, x3) and (y0, y1, y2, y3) is defined by x0 • y0 – x1 • y1 – x2 • y2 – x3 • y3.  Quickly computing P2 and V2 using this defnition of the square of a vector shows:

P2 = (Etot/c)2p2 = (Etot2p2•c2)/c2 = Eo2 /c2 = mo2 following our equations in E5!

This is obviously an invariant quantity. Considering the Epstein diagram in E5, if you divide all sides of the right triangle by c, you see that this calculation is just a variant of the Pythagorean Theorem. I would argue that Epstein diagrams and four-vector arithmetic are closely related!

We determine V2 for the case where v = vx and thus vy = 0 = vz:

V2 = (1/√ )2•(c2 – vx2 – 0 – 0) = (c2 – v2)/(1 – v2/c2) = c2•(1 – v2/c2)/(1 – v2/c2) = c2

Again, this is obviously an invariant, i.e., a value independent of the reference system. This result agrees beautifully with Epstein’s ‘Myth ‘! As a small exercise you might consider what PV means.

The aim of this book was not the algebraic treatment of challenging (and important) examples such as the Compton scattering. Its primary goal was to provide a view on the statements made by the STR and GTR. Or, as Epstein writes: "To understand the Special Theory of Relativity at the gut level, a good myth must be invented" [15-78] . To communicate Epstein’s myth was my main concern. Once this basis has been attained, it is easy, in a second pass, to acquire some of the technical tools. These tools can then be used to address arbitrarily tricky problems. And the tool for the calculations in special relativity is the algebra of four-vectors.

For an initial study of four-vectors [25] is recommended. If you work through chapters 12 and 13, you will already have made a significant start. Also, the presentation in [26] is perfectly accessible for someone with a solid high school background. Its title "Special Relativity for Beginners: A Textbook for Undergraduates" describes the level well. The sections on four-vectors in [14] and [19] are restricted to the energy-momentum vector and do not introduce the full power of this concept.