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**Example text**

Hence the name hyperbolic motion can be used synonymously with the term ‘uniformly accelerated’ motion. The three-velocity is x − x0 = |v| = w (t − t0 ) 1 + w2 (t − t0 )2 . 6 World Lines 47 It follows that |v| 1 over a period of time that |t − t0 | w−1 . 279). We now direct our attention to the question: what is the class of all allowable world lines? It is reasonable to assume that smooth timelike curves belong to this class. 254) which becomes inﬁnite at v2 = 1. This precludes acceleration of particles to velocities greater than that of light.

Indeed, n · x is the projection of the radius vectors xi (drawn from the origin to Σ) on this axis. 169), this projection takes a constant value, D, for every xi . We are thus led to an alternative deﬁnition of hyperplanes. 79). Let φ1i , . . , φm i be m linearly independent covectors. We deﬁne a (n − m)dimensional aﬃne manifold An−m by the set of m linear equations φsj xj = C s , s = 1, . . , m . 171). 172) where z0i and V i are some constant vectors. This straight line runs through the point z0i in the direction of the vector V i .

The eigenvalues of these invariants will label irreducible representations. There are two such invariants. 4). To construct another invariant, we deﬁne the so-called Pauli–Luba´ nski vector wκ = 1 2 κλµν Mλµ pν . 244) is the desired second invariant commuting with all generators Pµ and Mµν . 1. 230). 2. 230). 3. 233). 4. 237), show that P 2 commutes with all generators Mµν . 5. 244) commutes with Pµ and Mµν . Hint Derive ﬁrst the relations w·p=0, [Pµ , wν ] = 0, 5 [Mµν , wλ ] = −i (ηµλ wν − ηνλ wµ ) .