Basic Transformation of Coordinates and Measures, Corresponding Axes Parallel, Co-Linear x-Axes Parallel to β, and t = 0 When Origins Coincide.

Table 1: Transformation of Coordinates of a Space-Time Point

rest coordinates to moving coordinates

moving coordinates to rest coordinates

tβ = t0, Tβ = T0 [subscripts superfluous]

t0 = tβ, T0 = Tβ [subscripts superfluous]

x[beta] = `/`(`*`(`+`(x[0], `-`(`*`(beta, `*`(c, `*`(t)))))), `*`(`+`(1, `-`(`*`(`^`(beta, 2))))))

x[0] = `+`(`*`(x[beta], `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))), `*`(beta, `*`(c, `*`(t))))

y[beta] = `/`(`*`(y[0]), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2)))))))

y[0] = `*`(y[beta], `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2)))))))

z[beta] = `/`(`*`(z[0]), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2)))))))

z[0] = `*`(z[beta], `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2)))))))

`+`(Typesetting:-delayDotProduct(Vector[row](%id = 150632116), Matrix(%id = 150632180)), `-`(Vector[row](%id = 150632244))) = Vector[row](%id = 150632052)
`+`(Typesetting:-delayDotProduct(Vector[row](%id = 150632116), Matrix(%id = 150632180)), `-`(Vector[row](%id = 150632244))) = Vector[row](%id = 150632052)
`+`(Typesetting:-delayDotProduct(Vector[row](%id = 150632116), Matrix(%id = 150632180)), `-`(Vector[row](%id = 150632244))) = Vector[row](%id = 150632052)

`+`(Typesetting:-delayDotProduct(Vector[row](%id = 150632436), Matrix(%id = 150632500)), Vector[row](%id = 150632564)) = Vector[row](%id = 150632372)
`+`(Typesetting:-delayDotProduct(Vector[row](%id = 150632436), Matrix(%id = 150632500)), Vector[row](%id = 150632564)) = Vector[row](%id = 150632372)

In the moving frame we find the angle φ transformed, tan(phi[beta]) = `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2))))), `*`(tan(phi[0]))); if phi[0] = `+`(`*`(`/`(1, 2), `*`(Pi))) then phi[beta] = `+`(`*`(`/`(1, 2), `*`(Pi))). In the rest frame, the absolute span s at any instant of a moving solid rod of length l, (in its proper frame with velocity β,) with the angle φ0 (in the rest frame) between β and the axis of the rod, is given by `and`(s = `/`(`*`(l, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2), `*`(`^`(sin(phi[0]), 2)))))))), `/`(`*`(l, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))), `*`(sqrt(`+`(1, `-`(`*.... In the moving system, a point at absolute rest (at rest in the rest frame) will have velocity v = `+`(`-`(`/`(`*`(c, `*`(beta)), `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))))). Time in all frames of reference is the same as in the rest frame.

Table 2: Transformation of Measure

tan(φ)

tan(phi[0]) = `/`(`*`(tan(phi[beta])), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2))))))), tan(phi[beta]) = `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2))))), `*`(tan(phi[0])))

cos(φ)

`and`(cos(phi[0]) = `/`(1, `*`(sqrt(`+`(1, `/`(`*`(`^`(tan(phi[beta]), 2)), `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))))))), `/`(1, `*`(sqrt(`+`(1, `/`(`*`(`^`(tan(phi[beta]), 2)), `*`(`+`(1, `-`(`*`(`^`(bet..., `and`(cos(phi[beta]) = `/`(1, `*`(sqrt(`+`(1, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))), `*`(`^`(tan(phi[0]), 2))))))), `and`(`/`(1, `*`(sqrt(`+`(1, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))), `*`(`^`(tan(phi[0]), 2...

sin(φ)

`and`(sin(phi[0]) = `/`(1, `*`(sqrt(`+`(1, `/`(`*`(`+`(1, `-`(`*`(`^`(beta, 2))))), `*`(`^`(tan(phi[beta]), 2))))))), `and`(`/`(1, `*`(sqrt(`+`(1, `/`(`*`(`+`(1, `-`(`*`(`^`(beta, 2))))), `*`(`^`(tan(..., `and`(sin(phi[beta]) = `/`(1, `*`(sqrt(`+`(1, `/`(1, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))), `*`(`^`(tan(phi[0]), 2)))))))), `/`(1, `*`(sqrt(`+`(1, `/`(1, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))), `*`(`^`(tan(p...

absolute span of moving length

l[beta] = `/`(`*`(s[0], `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2), `*`(`^`(sin(phi[0]), 2)))))))), `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))),
s[0] = `*`(l[beta], `*`(sqrt(`*`(`+`(1, `-`(`*`(`^`(beta, 2)))), `*`(`+`(1, `-`(`*`(`^`(beta, 2), `*`(`^`(cos(phi[beta]), 2))))))))))

Derivation (with x-axis parallel to β)

velocity

abs(v[beta]) = `/`(`*`(abs(v[0]), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2), `*`(`^`(sin(phi[0]), 2)))))))), `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))),
abs(v[0]) = `*`(abs(v[beta]), `*`(sqrt(`*`(`+`(1, `-`(`*`(`^`(beta, 2)))), `*`(`+`(1, `-`(`*`(`^`(beta, 2), `*`(`^`(cos(phi[beta]), 2))))))))))

ψ dihedral angle between two angles φ1 and φ2

ψ0 = ψβ

θ angle between the vectors β1 and β2 of two inertial frames

`and`(cos(theta[0]) = `/`(`*`(Typesetting:-delayDotProduct(beta[1], beta[2])), `*`(abs(beta[1]), `*`(abs(beta[2])))), `/`(`*`(Typesetting:-delayDotProduct(beta[1], beta[2])), `*`(abs(beta[1]), `*`(abs...,
tan(theta[beta[1]]) = `*`(sqrt(`+`(1, `-`(`*`(`^`(beta[1], 2))))), `*`(tan(theta[0]))),  cos(theta[beta[1]]) = `/`(`*`(cos(theta[0])), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta[1], 2), `*`(`^`(sin(theta[0]), 2)))))))),
tan(theta[beta[2]]) = `*`(sqrt(`+`(1, `-`(`*`(`^`(beta[2], 2))))), `*`(tan(theta[0]))),  cos(theta[beta[2]]) = `/`(`*`(cos(theta[0])), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta[2], 2), `*`(`^`(sin(theta[0]), 2))))))))

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