Variables

\( E \)
The relativistic energy of the system or particle
\( c \)
The speed of light
\( m_{0} \)
The invariant rest mass of the system
\( \vec{p} \)
The momentums of the system or particle
\( h \)
Planck's constant
\( f _{l} \)
The frequency of a photon of light
\( \lambda _{l} \)
The wavelength of a photon of light
\( \lambda _{m} \)
The Compton wavelength of a massive particle
\( \lambda _{p} \)
The de Broglie wavelength of a system of matter

Facts and equations

  1. A photon of light has zero rest mass: \[ m_0 = 0 \tag{1.1} \]
  2. The energy of a single photon depends on the frequency. \[ E = h f_{l} \tag{1.2} \]
  3. Wavelength times frequency is the wave velocity. In the case of a photon: \[ c = f_l \lambda_l \tag{1.3} \]
  4. By syllogism of equations, (1.2) / (1.3) gives: \[ \frac{E}{c} = \frac{h}{\lambda_l} \tag{1.4} \]
  5. The Compton wavelength of an electron depends on the rest mass of an electron. \[ m_0 c = \frac{h}{\lambda_m} \tag{1.5} \]
  6. The De Broglie wavelength of the system as a whole depends on the total momentum of the system. \[ \| \sum \vec{p} \| = \frac{h}{\lambda_p} \tag{1.6} \]
  7. In a closed system the total energy \( \sum E \) is constant. \[ ( \frac{\sum E}{c} )^{2} = (m_0 c)^{2} + \| \sum \vec{p} \|^{2} \tag{1.7} \]

    I can choose any inertial frame of reference for my system and (1.7) still holds. Changing the frame of reference changes the energy of the system, because the momentum changes while \( m_0 \) remains fixed.

  8. Another syllogism, I substitute (1.7) with (1.1) and (1.4) to give me (1.6) again, but for photon wavelengths. So, I can generalize de Broglie and photon wavelengths nicely in terms of momentum.
    \[ \| \sum \vec{p} \| = \frac{h}{\lambda} \tag{1.8} \]

Principles illustrated

To break this down into its most elementary form, I consider the path of two photons travelling in different directions. I can pick almost any two photons, as long as they aren't perfect clones from a monochromatic laser beam. I like the thought experiment, because it exposes the mathematics of the situation more plainly, which often gets me closer to the heart of the matter than the equations by themselves.

Some strange syllogisms emerge from this simple case of two photons, merely considering the implications of equations and facts above.

  1. I choose a frame, namely the rest frame, where the summed momentum is zero. \[ \| \sum p \| = 0 \tag{2.1} \]
  2. The chosen frame resets the pair of photons to the same wavelength in exactly opposing direction. \[ \lambda_L = \lambda_{l_1} = \lambda_{l_2} \tag{2.2} \]
  3. Individually, each photon has energy. I can take (2.2) into (1.4), showing that the pair of photons has the total energy of its parts. \[ \frac{\sum E}{c} = \frac{2h}{\lambda_L} \tag{2.3} \]
  4. I can take (2.1) into (1.7), and I see the pair of photons exhibit rest mass \( m_0 \) although neither has any (0) by itself! Collectively, \[ \frac{\sum E}{c} = m_0 c \tag{2.4} \]
  5. Combining (2.3) and (2.4) shamelessly, I see quite clearly the mysterious mass of a system of 2 photons is quite small, except where the photon wavelength is exceptionally small (for example, gamma rays). \[ \frac{\lambda_L m_0}{2} = \frac{h}{c} \tag{2.5} \]
  6. By (1.8), the combined (de Broglie) wavelength of the system is undefined in the rest frame (the frequency is zero).

Deduction of the Lorentz transformation

Einstein famously published a paper deducing that if the speed of light is constant relative to every independent observer, then measurement of time and length must depend on the velocity at which equipment travels when measuring these times and lengths. If interpreting this foundational premise rather coarsely, I can simply say \( \| \vec{v} \| + c = c \), given \( \| \vec{v} \| < c \). This seems absurd mathematically, unless velocities are not simply additive in the first place. Restated without disbelief, the constancy of the speed of light suggests that adding velocities itself does not strictly work.

Einstein and his less famous colleagues had to knock a few bricks out of the very foundations of classical physics, to make room for modern physics. More specifically, to make room for the accumulating empirical evidence that the speed of light simply does not change — light is a wave in a medium which is immovable with respect to any measuring rod or clock.

Only one small problem. Clocks and rods only define time and distance in close proximity to the event so I need a definition of simultaneous and of synchronous.

  1. Measurement requires the measured event to be virtually simultaneous with a standard event, meaning the event to be measured occurs at virtually the same place and time as a standard measuring event.
    \[ \tag{3.01} \]
    • A specific geometric location within a rigid body.
    • A specific position of the hand of a clock in close proximity.
  2. Einstein then defines a synchronize action between separate stationary locations A and B. Simply bounce the light from A to B to A, and then offset the B clock such that \[ t_B = \frac{t'_A + t_A}{2} \tag{3.02} \]
  3. Define light speed as a constant \( c \), so \[ c = \frac{2 \overline{A B}}{t'_A - t_A} \tag{3.03} \]

Equations 3.01 through 3.03 focus on stationary rigid measurements to establish space-time coordinates. Tracing Einstein's special relativity paper further, I introduce two coordinate systems and try to establish a relationship between them.

  1. I love how tensor notation neatly bypasses so much complexity, so I will use it instead of vector notation. One matrix that comes in handy for describing relativity is this diagonal matrix. \[ g_{i j} = \left( \begin{array}{ccc} c & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) \tag{3.04} \]
  2. I have a spherically expanding surface moving outward from an arbitrary initial spatiotemporal origin \( o_i \) at the speed of light \( c \). This light bubble is often called a light cone and may be described by an equation. \[ (s_i - o_i) g_{i j} (s_j - o_j) = 0 \tag{3.05} \] Eight independent variables minus one equation gives us seven degrees of freedom to consider for all light bubbles \(o_i\) from a single coordinate system \(s_i\).
  3. Light speed is constant for all observers, so another observer should be able to find an origin \( p_i \) for each light bubble as measured from \( \psi_i \) coordinates. This new observer should see that all light bubbles have the same equation or spherically expanding shape and speed of progression. \[ (\psi_i - p_i) g_{i j} (\psi_j - p_j) = 0 \tag{3.06} \]
  4. Now I take an inhomogeneous linear transformation of coordinates. \[ \psi_i = A_{i j} s_j + b_i \tag{3.07} \]
    • The \( A_{i j} \) variable is the homogeneous linear portion of the equation which describes a change in vector coordinates (16D).
    • The \( b_i \) variable is the inhomogeneous portion of the equation which describes clock offset (1D) and spatial translation (3D).
  5. Both observers are watching the same light bubbles, so the origins \( o_i \) for the light bubbles must transform to \( p_i \) by equation 3.07 also. \[ p_i = A_{i j} o_j + b_i \tag{3.08} \]
  6. I substitute 3.07 and 3.08 into 3.06 to see what values of \( A_{i j} \) and \( b_i \) preserve the light bubbles. The \( b_i \) subtracts away entirely, so is completely independent of light bubbles in form or function. \[ A_{i j} (s_j - o_j) g_{i k} A_{k l} (s_l - o_l) = 0 \tag{3.09} \]
  7. The origin of the light bubble \( o_j \) in equations 3.05 and 3.09 is an independent variable added to another independent variable, so \( o_j \) can be absorbed (zeroed) without affecting my result. \[ g_{k l} A_{k i} A_{l j} s_i s_j = 0 \tag{3.10} \]
  8. While it may seem complicated, I note again that \(s_j\) is independent, so in order to satisfy both 3.05 and 3.10, I must have term for term equality up to a constant \(D\). \[ D^2 g_{i j} = A_{k i} g_{k l} A_{l j} \tag{3.11} \]
  9. I can set \(D=1\) which is equivalent to normalizing \(A\). I am currently noodling over this equation. \[ g_{i j} = A_{k i} g_{k l} A_{l j} \tag{3.12} \]
  10. If \(i ≠ j \), then \[ g_{i j} = 0 = c A_{0 i} A_{0 j} - A_{1 i} A_{1 j} - A_{2 i} A_{2 j} - A_{3 i} A_{3 j} \]
  11. If \(i = j = 0\), then \[ g_{0 0} = c = c A_{0 0} A_{0 0} - A_{1 1} A_{1 1} - A_{2 2} A_{2 2} - A_{3 3} A_{3 3} \] So \(A_{0 0}^2 ≥ 1\)
  12. If \(i = j ≠ 0\), then \[ g_{i i} = -1 = c A_{0 i} A_{0 i} - A_{1 i} A_{1 i} - A_{2 i} A_{2 i} - A_{3 i} A_{3 i} \] So \(A_{1 i}^2 + A_{2 i}^2 + A_{3 i}^2 ≥ 1\)

I will want to consider some special cases. My question is how can \( A_{i j} \) be composed from simple transformations. I think it describes the following.

  • Spatial rotations (3D) (definitely)
  • Uniform velocity (3D) (probably)
  • Other transformations like uniform acceleration and angular momentum (maybe, tbd)
  1. The special case where this transformation maintains clock synchronization. \[ \forall s_j, \psi_0 = s_0 \] So \[ s_0 = A_{0 j} s_j + b_0 \] But \(s_j\) is independent and \(b_0\) is a constant, so I expect term for term equality which gives me: \[ A_{0 0} = 1, A_{0 1} = A_{0 2} = A_{0 3} = 0, b_0 = 0 \tag{3.16} \]
  2. I want to consider a path measured by \( s_j \) to move with constant velocity \( v_j \) from an origin. This path may be described by a vector equation. \[ s_j = v_j s_0 \tag{3.17} \]