By Frank W K Firk
ISBN-10: 1616100338
ISBN-13: 9781616100339
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Additional info for Essential Physics I
Example text
The matrix with zeros everywhere, except along the main diagonal, has the interesting property that it simply scales the vectors on which it acts — it does not rotate them. The scaling values are given by the diagonal elements, called the eigenvalues of the diagonal matrix. The scaled vectors are called eigenvectors. A small industry exists that is devoted to finding optimum ways of diagonalizing large matrices. Illustrate the motion of the system in the x-y frame and in the u-v frame. 1 The Galilean transformation Events belong to the physical world — they are not abstractions.
In space-space, we have the Pythagorean form x2 + y 2 = r 2 (an invariant under rotations). We are therefore led to ask the question: is (kt)2 + x 2 an invariant under G in space-time? Direct calculation gives (kt)2 + x 2 = (k´t´) 2 + x´ 2 + 2Vx´t´ + V2t´2 = (k´t´)2 + x´ 2 only if V = 0 ! We see, therefore, that Galilean space-time does not leave the sum of squares invariant. 3) 48 CLASSICAL AND SPECIAL RELATIVITY a result that agrees with everyday observations. 4) where a and a´are the accelerations in the two frames of reference.
In matrix notation, the invariant is s2 = xT x = x´T x´ . 20) (The transpose must be written explicitly). The primed and unprimed column matrices (contravariant vectors) are related by the Lorentz matrix operator, L x´ = Lx . We therefore have xT x = (Lx)T (Lx) = xTLT Lx . The x’s are arbitrary, therefore LT L = . This is the defining property of the Lorentz transformations. 21) 54 CLASSICAL AND SPECIAL RELATIVITY The set of all Lorentz transformations is the set L of all 4 x 4 matrices that satisfies the defining property L = {L: LT L = ; L all 4 x 4 real matrices; = diag(1, –1, –1, –1}.
Essential Physics I by Frank W K Firk
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