Breakdown of Hannon’s Criticism of Einstein’s Derivation

1. Setup of Einstein’s Derivation

Einstein considers two coordinate systems:

  • Stationary system K: Coordinates (x, y, z, t)
  • Moving system k: Coordinates (\xi, \eta, \zeta, \tau), moving at velocity v along the x-axis.

He assumes that light is emitted from the moving frame’s origin, reflected at some point, and returns to the origin. He also assumes that the transformation equations must be linear.

2. Problem with Infinitesimal Analysis (Misuse of Differentials)

Einstein introduces an infinitesimally small displacement x' in the moving frame and relates it to time \tau(x', y, z, t).

Einstein’s Step:

He writes:

    \[ \frac{d\tau}{dx'} = 0 \]

since he assumes x' is independent of time.

Hannon’s Criticism:

  • Incorrect Use of Differentials: Einstein treats x' as independent but also makes it infinitesimally small, a contradiction.
  • Incorrect Assumption of Partial Derivative: If x' were independent of t, the correct statement should be: \frac{dx'}{dt} = 0 instead of Einstein’s claimed result.

3. Contradiction in Time Transformation

Einstein applies the constancy of the speed of light and sets up the equation:

    \[ \frac{1}{2} (\tau_1 + \tau_2) = \tau_0 \]

which follows from the assumption that light takes equal time to travel forward and backward.

Hannon’s Criticism:

  • Inconsistent Time Dependence: The derivation implicitly makes \tau a function of x', contradicting earlier assumptions.
  • Failure to Justify Function Assumptions: The assumed functional form of \tau(x', y, z, t) is not rigorously derived.

4. Flawed Interpretation of Light Signals and Synchronization

Einstein assumes that clock synchronization follows from light signal exchanges, leading to:

    \[ t' = t - \frac{vx}{c^2} \]

Hannon’s Criticism:

  • Circular Reasoning: Einstein assumes part of the transformation to derive it.
  • Alternative Synchronization Methods: If Einstein’s synchronization method were incorrect, his derivation would be invalid.

5. Final Conclusion from Hannon

Hannon argues that:

  • Einstein’s derivation of the Lorentz transformation is invalid.
  • The Lorentz transformations may still be correct but require a different derivation.
  • The assumptions about differential dependence and clock synchronization need reevaluation.

Counterpoint from Mainstream Physics

Despite Hannon’s criticisms, Einstein’s derivation remains widely accepted because:

  • His assumption of linearity is justified by physical symmetry.
  • Alternative derivations (such as those based on group theory) confirm the Lorentz transformations.
  • Experimental evidence overwhelmingly supports special relativity.