Extended Classical Mechanics (ECM) Corrective Explanation of Einstein’s Time Dilation Metric and Clock Rate:

Soumendra Nath Thakur 

March 02, 2025

1. Correcting Einstein’s Metric Component for Time Dilation

Einstein’s metric component for time dilation is given as g₄₄ = (1 - α/r).

In ECM, this metric must be reinterpreted in terms of effective mass (Meff), apparent mass (-Mapp), and gravitational mass (Mg) rather than relativistic time dilation.

Since black holes operate at the Planck scale, any oscillation—whether a clock oscillation or another form of oscillation—would be unaccountable within their proximity. Such extreme energetic conditions can only be perceived through gravitational interactions involving Mg, Meff, and -Mapp, as described in ECM.

2. Correcting Einstein’s Derived Clock Rate

Einstein’s derived clock rate (1 - α/r)^(-1/2) suggests that clocks oscillate infinitely fast at r = α, which ECM rejects.

ECM replaces this with (1 - α/r)^(1/2) to properly account for the gravitational transition at r = α and align with effective mass principles.

Since no physical clock would survive near a black hole, ECM refrains from referring to clock oscillation and instead considers energetic oscillation, as presented in Planck’s equation (E = hf). The oscillatory behavior near a black hole is therefore an energetic process rather than a measurement tied to a physical clock.

3. Correcting the Presence of Two Singularities in Einstein’s Interpretation

Einstein’s formulation predicts singularities at r = 0 and r = α.

In ECM, r = 0 remains a region of extreme mass-energy density, but r = α is not a true singularity—rather, it marks the transition where the black hole’s gravitational potential flips into anti-gravitational influence.

Since no physical existence is possible within a singularity due to Planck-scale limitations, normal space considerations apply only beyond the extreme gravitational influence of the black hole.

Beyond the immediate proximity of a black hole, a clock may be considered, but its oscillation frequency would be far beyond standard clock frequencies. Any frequency near the event horizon would be so high that it must be described as an oscillation frequency rather than a clock frequency.

4. ECM’s Alternative Interpretation of the Actual Clock Rate

ECM asserts that a black hole’s negative apparent mass (-Mapp) makes it an imperceptible existence, much like dark matter and dark energy.

The corrected clock rate, (1 - α/r)^(1/2), ensures a smooth transition at r = α, eliminating unnecessary singularities and aligning with ECM’s anti-gravitational dynamics.

Since black holes oscillate at the Planck scale, human perception cannot directly account for their time evolution—only effective mass, apparent mass, and kinetic energy calculations can reveal their behavior.

ECM avoids references to clock oscillation in gravitational contexts where normal mass (Mm) is absent. Instead, it focuses on oscillation frequency, relating it to negative apparent mass (-Mapp), negative effective mass (Meff), and gravitational mass (Mg) to maintain coherence with ECM’s gravitational framework.

This ECM corrective explanation provides a more precise, non-relativistic understanding of black holes, their oscillatory behavior, and their anti-gravitational nature.