Presenting a General Relativity Time Dilation Primer

The MIT-Sunapee time dilation calculations to demonstrate the measured \( 20 \ ns \) in Inside Einstein's Mind are completed but their transfer to the web is yet to be, ongoing and forthcoming.

General Relativity (GR)
Special Relativity (SR)

The following GR Primer is not an exhaustive summary of GR by any means. It is merely listing some theoretical results derived by others from GR which are then sorted, shuffled and reorganized in mere algebraic manipulations here to make distinctions between gravitationally-driven GR time dilation and velocity-driven SR time dilation.

The purpose of the Primer is to present a most general time dilation formula which accounts for both GR and SR, to be used for hypothetical space travel from one planet to another at straight line velocity \( \beta \) followed by remaining in a fixed position off the destination planet at rotational velocity \( \beta_r \) (in a synchronous orbit) around & with it.

The objective is to use this GR & SR machinery to deduce conditions where 1 year of spacecraft time (or less) equates to 25, 50 or 100 (or more) Earth years so hypothetical travellers can embark on one way trips to the future to avail themselves of medical advances to cure the incurable diseases of their own time.

Both the position proximity to the planets (GR) and the various velocities of such journeys (SR) contribute to the total time dilation experienced by hypothetical occupants of such spacecraft.

The Gravitational Parameter \( GM \)

The product \( GM \) of Newton's gravitational constant \( G = 6.674E−11 \) \( m^3 kg^{−1} s^{−2} \) 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 ( \( \mathbf{F} = { G M m}/{ r^2} \) ), 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 \( \mathbf{g} \) as follows:

This in turn implies an explicit altitude dependence of a planet's gravitational acceleration \( \mathbf{g} = {GM}/{r^2} \) for a point in space a distance \( r \equiv R + h \) from the planet center of mass, of planet radius \( R \) and altitude above ground \( h \).

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

Five functions in which the fundamental gravitational parameter \( GM \) appears are:

1. Gravitational acceleration for a planet (see above derivation) $$ \mathbf{g} = \frac{GM}{r^2} $$
2. The Schwarzchild Radius of a planet defines location of its Event Horizon where escape velocity (see #3 below) equals the speed of light \( c \):
(result from GR not derived here but used in calculations below for all 4 GR metrics) $$ r_s = \frac{2 GM}{c^2} $$
3. Escape velocity from a planet (Initial Energy: \( E_0 = T_0 + V_0 = 0 \) equals final Energy at infinity: \( E_{\infty} = T_{\infty} + V_{\infty} = 0 \)) $$ T_0 = \frac{1}{2} m {v_e}^2 = -V_0 = - \Bigl( - \frac{GM m}{r} \Bigr) \rightarrow {v_e}^2 = \frac{2 GM}{r} = \ \ c^2 \ \ \frac{r_s}{r} $$
4. Centripetal acceleration \( a \) from a rotational velocity \( v_r \) is expressed as \( a = {v_r}^2/r \) at radius \( r \) from a planet of mass \( M \) rotating with angular velocity \( \omega = v_r r \) endowing it with angular momentum \( J = I \omega \) where \( I = \epsilon m r^2 \) is the moment of inertia of the planet ( \( \epsilon = \frac{2}{5} \) ) . Using the relation \( a = GM/r^2 \) from #1 above, the constant orbital (rotational) velocity \( v_r \) in terms of the \( GM \) parameter takes the form:
$$ a = \frac{{v_r}^2}{r} \ \ \rightarrow \ \ ar = \left( \frac{GM}{r^2} \right) r = \frac{GM}{r} = {v_r}^2 = \ \ c^2 \ \ \frac{1}{2} \frac{r_s}{r} $$
5. The light-speed-normalized orbital velocity \( \beta_r = v_r/c \) in the vicinity of a gravitational body \( M \) may be related to the \( GM \) parameter and the Schwarzchild radius by dividing the result in #4 by \( c^2 \) and recognizing the relation for \( r_s \) in #2. This describes the orbital velocity or synchronous rotational speed of a fixed position around the planet at a distance \( r \) away from its center of mass: $$ \eqalign{ {\beta_r}^2 &= \frac{GM}{c^2 r} = \frac{1}{2} \frac{r_s}{r} \hspace{0.25in} {\rm orbital \ velocity \ SR \ contribution \ (from \ \#4 \ above)} \\ {\beta_e}^2 &= \frac{2GM}{c^2 r} = \frac{r_s}{r} \hspace{0.32in} {\rm escape \ velocity \ GR \ contribution \ (from \ \#3 \ above)} } $$

   Accordingly \( {\beta_e}^2 = 2 {\beta_r}^2 \) and so this GR contribution to time dilation is twice the magnitude of this SR contribution, or together they introduce a factor \( 3/2 \) to be discussed below, and both will appear in the GR and SR time dilation relations presented below.
   
   As noted, the orbital velocity contribution is an SR contribution to time dilation since it is explicitly velocity driven. Just because it is possible to relate this to the Schwarzchild radius \( r_s \) does not make it a GR effect. The Schwarzchild radius \( r_s \) just enables valuation of its magnitude but since it represents physical motion with velocity it remains an SR effect.
   
   At the same time, the escape velocity contribution is strictly a GR contribution to time dilation since it is gravitationally or planet-mass driven. The escape velocity of a planet is a constant property of the planet determined only by its mass, but does not represent any physical motion of velocity preventing it from being an SR contribution. It therefore remains a GR contribution.

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 travellers spending time in the differing gravitational fields in our Solar System:

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.

Time Dilation Formula for both GR & SR

Purely GR time dilation effects for \( J\ne 0 \), \( Q\ne 0 \)

The Time Dilation Formula for the most general case for both \( J\ne 0 \) and \( Q\ne 0 \) derives from the Kerr-Newman metric as seen in the table above. From this the other three may be deduced by setting either \( J = 0 \) or \( Q = 0 \) for the two intermediate cases, or both \( J = Q = 0 \) for the Schwarzchild case.

The Time Dilation formula for the most general Kerr-Newman metric, depends on a number of \(J-\)dependent & \(Q-\)dependent length parameters for a position \( (r,\theta) \) away from center of mass and polar axis respectively of a gravitational body of mass \( M \), polar angular momentum \( J \) and electrical charge \( Q \): $$ \eqalign{ a &= \frac{J}{Mc} \cr \rho^2 &= r^2 + a^2 \sin^2 \theta \cr \Delta &= r^2 + {r_Q}^2 - r_s r + a^2 \cr \Delta - a^2 \sin^2 \theta &= r^2 + {r_Q}^2 - r_s r + a^2 \cos^2 \theta \\ {r_Q}^2 &= k \ \frac{Q^2 G}{c^2} \ \ {\rm where} \ \ k = \frac{1}{4 \pi \epsilon_0} \ \ {\rm from \ Electrostatics} } $$ The Kerr-Newman Time Dilation Formula under General Relativity alone is expressed as the spacecraft time \( \tau \) as a function of the reference Earth time \( t \): $$ \tau = t \ \sqrt{1 - \frac{r_s r + {r_Q}^2}{\rho^2} } \hspace{1in} {\rm Kerr-Newman} \ J\ne 0 \ Q\ne 0. $$ The square root is always less than unity indicating that \( \tau < t \) which implies that the smaller \( \tau \) becomes the slower the spacecraft time will be relative to Earth time; thus minimal \( \tau \) implies maximal time dilation, which in turn implies that for each unit of spacecraft time, more units of Earth time elapse.

Combining both GR & SR time dilation effects

The above General Relativity (GR) considerations are proximity-only for fixed points \( r \) off gravitational bodies not including any velocities these points (or space craft at these points) may or may not have or have had. Special Relativity (SR) will include explicity these velocities of which there are at least two: $$ \tau = t \ \sqrt{1 - \left( \frac{r_s r + {r_Q}^2}{\rho^2} + {\beta}^2 + {\beta_r}^2 \right) }. $$ For the specific space voyages contemplated here, these two space craft velocities are: i) straight line travel from one planet to another at a constant velocity \( \beta \), and then remaining at a fixed point off the destination planet and so ii) synchronously rotating with it at its rotation speed \( \beta_r \) which depends on its distance \( r \) away from the planet center of mass. Explicit expressions for these two velocities are:

1. velocity \( \beta = {v}/{c} \hspace{0.8in} \) straight line velocity from point \( r_1 \) off planet 1 to a point \( r_2 \) off planet 2
2. velocity \( \beta_r = \sqrt{ {r_s}/(2r) } \hspace{0.29in} \) rotational orbit velocity of a fixed point \( r_2 \) off a rotating planet 2
   \( \hspace{1.9in} \) (see #5 in Five Different Functions section).

The total time dilation relation including General and Special Relativity (the GR metric and 2 SR motion contributions - linear and rotational) is therefore: $$ \tau = t \ \sqrt{1 - \left( \frac{r_s r + {r_Q}^2}{\rho^2} + {\beta}^2 + \frac{r_s}{2r} \right) } = t \ \sqrt{1 - \left( {\beta}^2 + \frac{r_s}{r} \left\{ \frac{r^2}{\rho^2} + \frac{1}{2} \right\} + \frac{{r_Q}^2}{\rho^2} \right) } $$ where the last expression in parentheses is organized into a first planet-to-planet linear velocity term, a second planetary-angular-momentum \( J \) term (itself split into its GR escape velocity and SR orbital velocity contributions respectively), and a third planetary-electric-charge \( Q \) term.

In the limit of infinitesimally small Schwarzchild radius (which is equivalent to no planet at all, or planet mass \( M \rightarrow 0, J=Q=0 \) and hence elimination of any gravitational fields or effects), this expression reduces to the familiar (velocity based) time dilation expression expected from Special Relativity for two gravity-free frames in relative motion \( \beta \).

Time dilation formulae for GR effects alone

And so the GR Time dilation formulae for all four GR metrics together for convenience of contrasting and comparing them are: $$ \eqalign{ \tau &= t \ \sqrt{1 - \frac{r_s}{r} } \hspace{1.5in} \ J=0 \ Q=0 \ \ {\rm Schwarzchild} \cr \tau &= t \ \sqrt{1 - \frac{r_s r}{\rho^2} } \hspace{1.4in} \ J\ne 0 \ Q = 0 \ \ {\rm Kerr} \cr \tau &= t \ \sqrt{1 - \frac{r_s}{r} - \frac{{r_Q}^2}{r^2} } \hspace{1in} \ J=0 \ Q\ne 0 \ \ {\rm Reissner-Nordström} \cr \tau &= t \ \sqrt{1 - \frac{r_s r}{\rho^2} - \frac{{r_Q}^2}{\rho^2} } \hspace{1in} \ J\ne 0 \ Q\ne 0 \ \ {\rm Kerr-Newman}. }. $$ where the \( r_s/r \), and its Kerr generalization \( r_s r / \rho^2 \), represents a purely GR contribution deriving from the escape velocity property not representing any physical motion of, near or around a planet of mass M.

Time dilation formulae combining both GR & SR effects

And so the combined GR & SR Time dilation formulae for all four GR metrics and the two velocity types (linear and rotational), noting that for the case \( J=0 \rightarrow a=0 \) implies \( \rho = r \) giving rise to the \( 3/2 \) factor, listed together for convenience of contrasting and comparing them are: $$ \eqalign{ \tau &= t \ \sqrt{1 - \beta^2 - \frac{3}{2} \frac{r_s}{r} } \hspace{2.25in} \ J=0 \ Q=0 \ \ {\rm Schwarzchild} \cr \tau &= t \ \sqrt{1 - \beta^2 - \frac{r_s}{r} \left\{ \frac{r^2}{\rho^2} + \frac{1}{2} \right\} } \hspace{1.5in} \ J\ne 0 \ Q = 0 \ \ {\rm Kerr} \cr \tau &= t \ \sqrt{1 - \beta^2 - \frac{3}{2} \frac{r_s}{r} - \frac{{r_Q}^2}{r^2} } \hspace{1.7in} \ J=0 \ Q\ne 0 \ \ {\rm Reissner-Nordström} \cr \tau &= t \ \sqrt{1 - \beta^2 - \frac{r_s}{r} \left\{ \frac{r^2}{\rho^2} + \frac{1}{2} \right\} - \frac{{r_Q}^2}{\rho^2} } \hspace{1in} \ J\ne 0 \ Q\ne 0 \ \ {\rm Kerr-Newman}. }. $$ where the \( 3r_s / 2r \), and its Kerr generalization \( (r_s/r) (r^2/\rho^2 + 1/2) \), represents both the purely GR escape velocity contribution \( r_s/r \) and the strictly SR contribution \( r_s/(2r) \) representing orbital synchronous rotational motion of the position point \( r \) around a planet of mass M.

The calculation of the \( 20 \ ns \) was completed a year or three ago for the Schwarzchild metric but its transfer to the web was delayed by the need for web display coding tools not currently in the know. Armed with the full suite of other GR metrics, this calculaton program may now be resumed to investigate whether the orbital speed of Earth has any impact on the \( 20 \ ns \).

This is doubtful since the orbital speed of any point on the Earth is the same no matter at MIT, the highways to NH or at Sunapee, and since only a difference in the two clocks (both of which have the same Earth synchronous rotational velocity at all times) is sought, the Earth orbital speed corrections provided by the Kerr metric will probably not change the Schwarzchild metric results, even though \( J=0 \) true of the the Schwarzchild metric does not apply to the Earth.

Error on the Wikipedia page for Gravitational Time Dilation

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.