Energy Equivalence
Transforming Minkowski spacetime into formula used by Einstein, Klein-Gordon and Dirac.
While using 2D Penrose spacetime, I propose to now use a transform that looks at the rates of change of time and space as before with time being the hodge dual of space E.g. \star dt = dx \wedge dy \wedge dz:
  • Temporal: t :\rightarrow \hbar \gamma^{0} \partial_{0}.
  • Spatial: x :\rightarrow \hbar \gamma^{j} \partial_{j}.
Adding a factor of c to change units of momentum to that of energy. You should notice this still allows the whole system to be described by a single \HH quaternion number, mass being defined as the net radius of a 4D circle.
This then gives us the image shown.
We have now highlighted the four quadrants for space-like (virtual) and time-like (real) events, and the black lines represents light-like and vacuum-like events. The blue and red curved surfaces now represent the particles equal energy and momentum levels respectively. One thing that is now shown is that for each energy level, the gap between each level must decrease as they bunch up towards the corners of infinite energy or momentum. We are using the Minkowski metric [+,-,-,-] here to show the energy and time as always real, momentum and space always imaginary. I do this as vectors exactly match those of higher complex forms.
When doing this, the [\hat{t}, i\hat{x}] spacetime co-ordinates are transformed into [\hat{E}, i\hat{p}c] energies. The net event radius s is transposed to be the mass of the particle mc^2. So we are using the conformal transformation tan(u \pm v) = i\hat{x} \pm \hat{t}, with time-like and light-like \infin at +\pi/2, space-like and vacuum-like \infin at -\pi/2.
\hat{E} = +i{\hbar}c \gamma^{0} \partial_{0}
i \hat{p} c = -{\hbar}c \gamma^{j} \partial_{j}
Time +1 Time -1 Space +★ Space -★ +1 Light -1 Light +★ Vacuum -★ Vacuum
i \hat{p} c = +{\hbar}c \gamma^{j} \partial_{j}
\hat{E} = -i{\hbar}c \gamma^{0} \partial_{0}
Energy Equivalence
Pythagorian Hypotenuse:
  • s^2 = t^2 + x^2
When spacetime is flat, energy and momentum are orthogonal with respect to each other. The event length (s) and mass (m) being directly related to the resultant event radius.
  • x :\rightarrow i\hat{p}c = \hbar c \gamma^j \partial_j
  • t :\rightarrow \hat{E} = i \hbar c \gamma^0 \partial_0
  • s :\rightarrow mc^2
Summing Squares (Second Order, Scalar):
  • (mc^2)^2 = (\hat{E})^2 + (i\hat{p}c)^2
  • (mc^2)^2 = (i \hbar c \gamma^0 \partial_0)^2 + (\hbar c \gamma^j \partial_j)^2
  • (\hat{E})^2 = (mc^2)^2 + (\hat{p}c)^2
  • \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial{t^2}} = (\frac{mc}{\hbar})^2
Result: Einstein & Klein-Gordon Equations
Summing Vectors (First Order, Vector):
  • s = t \pm ix
  • i \hbar \gamma^{\mu} \partial_{\mu} - mc = 0
Result: Dirac Equation
Common Shortcuts:
  • t^2 + x^2 = (t + ix)(t - ix)
  • \partial_{0}^{2} = \frac{1}{c^2} \frac{\partial^2}{\partial t^2}
  • \partial_{j}^{2} = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} = \nabla^2
  • \gamma_{\mu} \gamma^{\mu} = I_4