By J. Foster, J. Nightingale

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**Example text**

This leads us to the following definition: Definition 3 Let K , K be two LCFs which are related by the general Lorentz transformation L(β, E) = L(β)R(E). We say that the spatial axes of K , K are relativistically parallel if, and only if, R(E) = I4 . 42 1 Mathematical Part This concept of parallelism is directly comparable to the Euclidean concept of parallelism in E 3 because the latter can be defined as follows: Definition 4 Let Σ, Σ be two ECF which are related with the EOT E. We say that the axes of Σ, Σ are parallel if, and only if, E = I3 where I3 = diag(1, 1, 1) is the identity matrix.

15 Proposition 3 The sum of timelike or timelike and null four-vectors is a timelike four-vector except if, and only if, all four-vectors are null and parallel in which case the sum is a null four-vector parallel to the null four-vectors. This result is important because it allows us to study reactions of elementary particles including photons. Indeed as we shall see the elementary particles are characterized with their four-momenta, which is null for photons and timelike for the rest of the particles.

107) We call the hyperbolic angle ψ rapidity. 9 The Geometry of the Boost As we have shown in Sect. 7 the Lorentz transformation L(β)24 can be expressed as the product of a boost and two Euclidean rotations. These rotations concern the direction of the relative velocity in the (parallel) axes of the two LCFs related by the Lorentz transformation. This decomposition is helpful in practice because we solve a specific problem for a boost and then we use the Euclidean rotations to get the (usually more complicated) answer.

### A Short Course in General Relativity by J. Foster, J. Nightingale

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