The Seemingly and Surprisingly Simple Formula from General Relativity for Gravitational Time Dilation

Introduction to Gravitational Time Dilation

The study of gravitational time dilation has been a cornerstone of modern physics, particularly as described by Einstein's theory of General Relativity (GR). A seemingly and surprisingly simple formula has been derived for the ratio of the rates that two clocks, one near a massive object and the other far from any massive object, keep time. This formula, which is central to understanding gravitational time dilation, is given by (t/to 1/sqrt{1 - 2GM/Rc^2}). This article delves into the intricacies of this formula and provides explanations based on Einstein's law of equivalence of inertial and gravitational mass.

The Basis of the Formula

The formula (t/to 1/sqrt{1 - 2GM/Rc^2}) is derived from the principle of equivalence of inertial and gravitational mass. According to Einstein's theory, an applied force on an object, whether caused by an electric or gravitational field, is equivalent to the force caused by an acceleration. This equivalence affects the speed of light at different gravitational potentials, leading to time dilation.

Another critical aspect of this formula is the fact that the speed of light is reduced by the same factor in these applied force fields. This reduction is universal, as the laws of physics are considered universal throughout the universe. Therefore, within each galaxy, the speed of light, whether electromagnetic (EM) or gravitational, is measured relative to the emitter.

The Role of Specific Examples

To further illustrate the application of this formula, consider the case of an object at the center of a spinning disc and at a radius R. Einstein derived the formula by considering the difference in energy (both kinetic and potential) between these two positions. The force in this scenario is centrifugal.

A key point to remember is that the density of potential energy stored in a gravitational or equivalent force field is proportional to the field strength. However, the formula is not as straightforward as it may seem, requiring careful definition and understanding of the reference frame.

Challenges in the Definition of Reference Frame

One of the age-old questions in physics is whether there is a zero-speed reference medium or 'ether' for EM and gravity that is 'dragged' around by massive objects like the Earth, Sun, or Milky Way. However, based on Einstein's 1916 book and the first postulate of relativity, the speed of light must be universal and measured relative to the emitter within each galaxy, regardless of the speeds of other objects in the universe.

Thus, it is hypothesized that EM must change speed by (pm v) somewhere during its propagation from one galaxy to another to arrive at (c) relative to the destination. This explanation aligns with stellar aberration measurements and the observed Doppler red-shifts of distant galaxies as seen by the JWST, Hubble, and other space telescopes.

Classical Hypotheses and Paradoxes

However, this hypothesis is not without its challenges. If EM propagates at (c) from the emitter and changes speed to (c pm v) somewhere in between, it should change speed by the classical Doppler factors. For a red-shift of 14, corresponding to (v approx 0.93c), and for a blue-shift, (v approx 13c), implying an expansion rate faster than (c).

These conclusions contradict Einstein's second postulate, where EM supposedly leaves the emitter at (c) relative to the destination and does not change speed. This limit, derived from the Lorentz factor, becomes imaginary when (c - v 0), suggesting that an observer could be receding faster than the EM is chasing. This, in turn, leads to hypothetical scenarios that are not consistent with the law of causality.

Mathematical Considerations

The 2nd order Lorentz factor is mathematically impossible, given the mutual exclusivity of the classical Doppler factors involving (1 - v/c) and (1 v/c). This deduction is not mainstream; however, it provides a straightforward explanation without the theoretical contractions of space and time.

Gravitational time dilation or red-shift is also a separate issue but does utilize the Lorentz factor (gamma) directly in its derivation. This further demonstrates the complexity and elegance of understanding the interplay between gravity and time in General Relativity.