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=== [[Special Relativity]] in 6D [[Quaternion-Kähler manifold]]: === |
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[[james_clerk_maxwell|Maxwell]]'s frame: ''(x, y, z, iλ<sub>1</sub>, jλ<sub>2</sub>, kλ<sub>3</sub>)'' where ''ijk = i<sup>2</sup> = j<sup>2</sup> = k<sup>2</sup>'' ≡ −1 |
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Animation showing a test particle of [[Planck mass]] being accelerated to the [[speed of light]], where it has [[Planck momentum]] and [[Planck_energy|Planck kinetic energy]]. The [[Minkowski diagram]] at top-left shows the "4D [[Lorentz_factor|Lorentz rotation]]" of the moving frame of reference inhabited by the particle. The top-right projection shows [[time dilation]] approaching infinity at light speed, and the bottom-left projection shows [[length contraction]] in the moving [[frame of reference]].<br \> |
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<p align=center>[[File:6D Special Relativity.gif|Special Relativity in 6 dimensions]]</p> |
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The [[Lorentz factor]] is the inverse-cosine of the [[Phasor|phase angle]] (0 < φ < <sup>π</sup>/<sub>2</sub>), i.e. γ = 1/cos(φ), and the ratio of the particle's velocity to light speed is β = <sup>v</sup>/<sub>c</sub> = 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 mc<sup>2</sup>, in units of Planck energy (E<sub>P</sub>). |
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Thus, the particle's total energy Eᵩ is a function of √((t∙sin(φ))<sup>2</sup> + (r∙sin(φ))<sup>2</sup>). 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∙E<sub>P</sub>, assuming no losses due to gravitational radiation. The particle's [[Matter_wave|matter-wave]] now has a [[Louis_de_Broglie|de Broglie]] and [[Compton wavelength]] λᵩ = [[Planck length]], oscillating at [[Planck frequency]]. |
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Revision as of 02:25, 12 June 2015
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 projection shows time dilation approaching infinity at light speed, and the bottom-left 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.