Presenting a General Relativity Time Dilation Primer

These calculations are completed and their transfer to the web is ongoing and will be forthcoming.

This is not an exhaustive summary of General Relativity by any means. It is merely a few theoretical facts which pertain to gravitational time dilation which results from General Relativity.

The Gravitational Parameter \( GM \)

The product \( GM \) of Newton's gravitational constant \( G = 6.674Eāˆ’11 \) \( m^3 kg^{āˆ’1} s^{āˆ’2} \) and any Solar System body mass \( M \) in kilograms for a given planet in question, is better known (smaller error) than either of the two constants separately. Therefore, for General Relativity calculations it is best to deploy the gravitational parameter \( GM \).

Applying Newton's Second Law ( \( \mathbf{F} = m \mathbf{a} \) ) to Newton's Law of Gravitation \( \mathbf{F} = \frac{ G M m}{ r^2} \), identifying the mass of a planet with \( M \), and \( r \) with the distance between the center of mass of \( M \) and any given object mass \( m \) thus defines the planet's gravitational acceleration \( \mathbf{g} \) by the Second law acceleration \( \mathbf{a} \) in Newton's Law of Gravitation exploiting the following identification:

$$ \mathbf{F} = \frac{ G M m}{ r^2} = m \left( \frac{ G M}{ r^2} \right) \equiv m \mathbf{g} = m \mathbf{a} \hspace{1cm} \rightarrow \hspace{1cm} \mathbf{a} = \mathbf{g} \equiv \frac{ G M }{ r^2} $$

This in turn implies an explicit altitude dependence of a planet's gravitational acceleration \( \mathbf{g} = \frac{GM}{r^2} \) for a point in space a distance \( r \) from the planet center of mass.

Four Different Functions of the Gravitational Parameter \( GM \)

Four equations in which the fundamental gravitational parameter \( GM \) appears are:

  1. Gravitational acceleration for a planet
  2. (see above derivation) $$ \mathbf{g} = \frac{GM}{r^2} $$
  3. Escape velocity from a planet (Initial Energy: \( E_0 = T_0 + V_0 \) equals final Energy at infinity: \( E_{\infty} = T_{\infty} + V_{\infty} = 0 \))
  4. $$ T_0 = \frac{1}{2} m {v_e}^2 = -V_0 = - \Bigl( - \frac{GM m}{r} \Bigr) \rightarrow v_e = \sqrt{ \frac{ 2 GM}{r} } $$
  5. The Schwarzchild Radius of a planet (escape velocity equals speed of light = event horizon)
  6. (used in Schwarzchild metric time dilation) $$ r_s = \frac{2 GM}{c^2} $$
  7. Centripetal acceleration at radius \( r \) from a planet of mass \( M \) rotating with angular velocity \( \omega = v r \) endowing it with angular momentum \( J = I \omega \) where \( I = \epsilon m r^2 \) is the moment of inertia of the planet ( \( \epsilon = \frac{2}{5} \) ) .
  8. $$ a = \frac{v^2}{r} \rightarrow ar = v^2 = \frac{GM}{r} = r^2 \omega^2 $$ Dividing both sides by \( c^2 \) (which normalizes by light speed) relates the square of the light speed normalized velocity \( \beta^2 \) to the Schwarzchild radius: $$ \beta^2 = \frac{GM}{c^2 r} = \frac{1}{2} \frac{r_s}{r} $$

This factor of \( \frac{1}{2} \frac{r_s}{r} \) will be seen to contribute to the gravitational time dilation result derived from the Schwarzchild metric.

Four Exact Solutions to the Einstein Field Equations

There are four exact solutions to the Einstein Field Equations characterized by their own space-time metrics in General Relativity which might be applied to determining gravitational time dilation for us travelling and spending time in the differing gravitational fields in our Solar System:

Non Rotating ( \(J = 0\) ) Rotating ( \( J \ne 0 \) )
Not Charged ( \(Q = 0\) ) Schwarzchild Kerr
Charged ( \(Q \ne 0\) ) Reissner-Nordstrom Kerr-Newman

A planet here might have an angular momentum \( J \) which is a three component vector quantity and/or an electric charge \( Q \) which is a one component quantity called a scalar.

Schwarzchild Metric ( \( J=0,Q=0 \) )

The time dilation result for the Schwarzchild metric depends on the Schwarzchild radius \( r_s \) and the point \( r \) $$ \tau = t \sqrt{1 - \frac{r_s}{r} } $$ where \( t \) is a time in a gravity-free frame at rest (no velocity) and \( \tau \) is time in another rest frame subjected to a gravitational field \( \mathbf{g} \).

If velocity considerations are to be taken into account then there are two types additive before the minus sign under the square root:

  1. velocity \( \beta = \frac{v}{c} \) from point \( r_1 \) off planet 1 to a point \( r_2 \) off planet 2
  2. velocity \( \beta = \sqrt{ \frac{1}{2} \frac{r_s}{r} } \) of the point \( r_2 \) if planet 2 is rotating
  3. (see centripetal acceleration)

The total time dilation relation including General and Special Relativity (the metric and 2 motion contributions) is therefore:

$$ \tau = t \sqrt{1 - \Bigl( \frac{r_s}{r} + \frac{1}{2} \frac{r_s}{r} + \beta^2 \Bigr) } $$ where \( \beta \) is the velocity between the planets and the whole expression reduces to $$ \tau = t \sqrt{1 - \frac{3}{2} \frac{r_s}{r} - \beta^2} $$

This is the origin of the \( \frac{3}{2} \) misleadingly explained in the Circular Orbits expression. Wikipedia states that including 'motion' requires inclusion also of \( \beta^2 \) (which is #3 in the list below their expression) but it is thought here to make a clearer distinction between motion which is 1) velocity between planets as different from 2) velocity of a point rotating with and therefore around a planet. The circular orbits in question are only the latter not including the former, though technically there are the two types of velocity as listed above and both are velocities of motion which contribute to time dilation in different but not opposing ways. They do contribute differently as described in detail here but they cannot oppose each other due to the square of the velocity which is contrary to the use of the word "opposed" in the comment there.

In the limit of infinitesimally small Schwarzchild radius (which is equivalent to planet mass \( M \rightarrow 0 \) and hence elimination of any gravitational field), this expression reduces to the familiar (velocity based) time dilation expression expected from Special Relativity for two gravity-free frames in relative motion \( \beta \).

These calculations are completed for the Schwarzchild metric and their transfer to the web is ongoing and will be forthcoming.

Kerr Metric ( \( J\ne 0,Q= 0 \) )

Reissner-Nordstrom Metric ( \( J= 0,Q\ne 0 \) )

Kerr-Newman Metric ( \( J\ne 0,Q\ne 0 \) )

These calculations are planned for the other three metrics where it is hoped that the 20 nanosecond measurement might be used to deduce either the charge \( Q \) or the angular momentum \( J \) of the Earth. If you would like to see this page completed with the rest of these calculations please send a message and consider subscribing at the Viewer Level. If you would like to contribute the calculations for the time dilation relations from each of the metrics please consider subscribing at the Contributor Level.

More description materials here are forthcoming.
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