By Firk F.W.K.

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We now ask the question: is Einsteinian space-time Pythagorean under L ? Direct calculation leads to (ct)2 + x 2 = γ2(1 + β2)(ct´)2 + 4βγ 2x´ct´ +γ2(1 + β2)x´2 ≠ (ct´)2 + x´ 2 if β > 0. 10) because γ2(1 – β2) = 1. Space-time is said to be pseudo-Euclidean. The negative sign that characterizes Lorentz invariance can be included in the theory in a general way as follows. 11) and xµ = [x0, x1, x2, x3], a covariant vector, where xµ = [x0, –x1, –x2, –x3]. 12) The scalar (or inner) product of the vectors is defined as xµTxµ =(x0, x1, x2, x3)[x0, –x1, –x2, –x3], to conform to matrix multiplication ↑ ↑ row column 50 CLASSICAL AND SPECIAL RELATIVITY =(x0)2 – ((x1)2 + (x2)2 + (x3)2) .

He symmetrized the space-time equations as follows: t´ x´ = 1 –V –V 1 t x . 5) Note, however, the inconsistency in the dimensions of the time-equation that has now been introduced: t´ = t – Vx. The term Vx has dimensions of [L]2/[T], and not [T]. This can be corrected by introducing the invariant speed of light, c — a postulate in Einstein's theory that is consistent with the result of the Michelson-Morley experiment: ct´ = ct – Vx/c so that all terms now have dimensions of length. Einstein went further, and introduced a dimensionless quantity γ instead of the scaling factor of unity that appears in the Galilean equations of space-time.

If G is the dihedral group D3, discussed in the text, where G = {e, a, a2, b, c, d}, where e is the identity, obtain the group multiplication table. Is it an Abelian group?. MATHEMATICAL PRELIMINARIES 30 Notice that the three elements {e, a, a2} form a subgroup of G, whereas the three elements {b, c, d} do not; there is no identity in this subset. The group D3 has the same multiplication table as the group of permutations of three objects. This is the condition that signifies group isomorphism.