User:Rjowsey
The Geometry of SpaceTime
http://www.jowsey.org/physics/SpaceTime.pdf
Maxwell's frame: (x, y, z, iλ1, jλ2, kλ3) where ijk = i2 = j2 = k2 ≡ −1
Animation showing a test particle of Planck mass being accelerated to the speed of light, where it has Planck momentum and Planck kinetic energy. The Minkowski diagram at top-left shows the "4D Lorentz rotation" of the moving frame of reference inhabited by the particle. The top-right time/energy projection shows time dilation approaching infinity at light speed, and the bottom-left position/momentum projection shows length contraction in the moving frame of reference.
The Lorentz factor is the inverse-cosine of the phase angle (0 < φ < π/2), i.e. γ = 1/cos(φ), and the ratio of the particle's velocity to light speed is β = v/c = sin(φ). Thus, time dilation and length contraction simplify to τᵩ = t∙cos(φ) and ʀᵩ = r∙cos(φ). The y-axis in the bottom-right projection represents the imaginary component of the particle's kinetic energy, while the x-axis represents the imaginary component of its potential energy mc2, in units of Planck energy (EP).
Thus, the particle's total energy Eᵩ is a function of √((t∙sin(φ))2 + (r∙sin(φ))2). At light speed, the particle's rest-mass energy and its momentum can be seen as inhabiting two imaginary spatial dimensions. A Planck mass at velocity v = c has total energy of √2∙EP, assuming no losses due to gravitational radiation. The particle's matter-wave now has a de Broglie and Compton wavelength λᵩ = Planck length, oscillating at Planck frequency.
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