Relative ENERGY (Kinetic and Gravitational Potential Energy) Determines Relativistic Effects

LEFT: Jumpy Jill is in a friction-free tunnel and cycles forever. She is "at rest" at the top and bottom of the hole and therefore has zero Kinetic Energy (KE). She does have positive Gravitational Potential Energy (GPE) with respect to the center of the tunnel. Her maximum speed is achieved at the center. 

RIGHT: Far-Out Fred is launched into space at a hair less than Escape Velocity. He travels a very long distance over nearly infinite time before he runs out of KE and is momentarily "at rest". He falls back towards the planet, enters the friction-free hole at a hair less than Escape Velocity, reaches maximum speed at center, decelerates, exits hole at a hair less than Escape Velocity, heads into space, and cycles forever. 

Note: All speeds and relative energy are with respect to an observer "at rest" at the very center of the planet.

Relativistic Effects are Proportional to Kinetic Energy and Gravitational Potential Energy

Does relative TOTAL ENERGY determine the magnitude of Time Dilation and Length Contraction? Here are some equations, arguments and examples that support the idea that these Relativistic Effects are due to Total Energy with respect to a reference observer. I will appreciate comments and corrections.

Time Dilation is a relativistic effect that slows the rate of a clock that is moving and/or accelerating with respect to an “at rest” reference clock. 

"Coordinate" Time Interval between two events, Δt, ("delta tee") measured by a clock "at rest" in a reference frame (zero speed, zero gravity). 

"Proper" Time Interval between the same two events, Δτ, ("delta tau") measured by a clock that is moving and/or accelerating with respect to the reference frame (due to relative gravity and/or speed).

Δτ is equal to or greater than Δt due to Time Dilation. 

Lorentz Factor, γ = Δt/Δτ, equal to or greater than 1.0. (γ  is lower-case "gamma")

Lorentz Factor due to relative speed, γ_K = 1/SQRT(1-v²/c²) where v is the relative velocity between the "at rest" clock and the moving clock, and c is the speed of light in a vacuum. 

Lorentz Factor due to relative gravity, γ_G = 1/SQRT(1-2GM/rc²), where G is the gravitational constant, M is the mass of the spherical planet (or massive object) that is the source of the gravitational field, and r (which must be equal to or greater than the radius of the planet) is the distance of the clock from the center of the planet.

Time Dilation,TD = Δt - Δτ, is the "coordinate" time interval minus the "proper" time interval. 

Time Dilation,TD = (γ - 1)Δτ, an equivalent way to express Time Dilation. Please note that the (γ - 1) factor is common to the equation for Time Dilation as well as the expressions for KE and GPE.

Kinetic Energy of the clock, KE = (γ_K  – 1)m₀c², where m₀ is the "rest mass" of the clock.

Gravitational Potential Energy of the clock with respect to infinity, -GPE = (γ_G  – 1)m₀c² .

Escape Velocity, v_E, from a non-rotating, no atmosphere planet, P, is the speed, perpendicular to the surface, where KE is exactly equal to -GPE. (The minus sign is due to GPE being defined as zero at infinite distance from P, increasing in magnitude to its largest NEGATIVE value at the center, and then increasing asymptotically to zero an infinite distance from P.) A rocket launched at v_E will just break free of the gravity of P and continue into space forever, with no need for further propulsion.  v_E = SQRT(2GM/r), where G is the gravitational constant, M is the mass of P, and r (which must be equal to or greater than the radius of P) is the distance of the object from the center of P.

A clock moving at v_E (the Escape velocity associated with the surface of P), but far from any massive object (zero gravity), will have exactly the same TD as a clock “at rest” (zero speed) on the surface of P. Note that this occurs when the KE of the former is exactly equal to the -GPE with respect to infinity of the latter.

As an Engineer (not a Physicist) these facts scream out that relative ENERGY (KINETIC and/or -GRAVITATIONAL POTENTIAL) causes TD. I find it surprising that these facts are not mentioned at all in the Wikipedia entries for Escape_velocity nor Time_dilation, nor by most internet resources for relativistic effects. In the few instances where mentioned, the exact relationship of ENERGY to RELATIVISTIC EFFECTS is treated as an unimportant coincidence. It seems to me that RELATIVE ENERGY IS THE PRIME CAUSATIVE FACTOR FOR RELATIVISTIC EFFECTS, as confirmed by simple examination of the applicable equations.

Summary

A change in kinetic energy causes a linear change in TD.  A change in speed will cause a change in TD, but not in a way that is linearly related to the Lorentz factor. For example, a 1% increase in speed, from 0.098c to 0.099c will cause a 0.01% increase in TD, while a 1% increase from 0.98c to 0.99c will cause a  41% increase!

A change in GPE causes a linear change in Time dilation. A change in gravity will cause a change in TD, but not in a way that is linearly related to the Lorentz factor. (Indeed, the relationship between gravity and Time dilation is not even monotonic! When a clock is within a massive object, gravity decreases while Time dilation increases! [Thanks to John Rennie at the physics stack exchange for kindly (and with extraordinary patience) corrected my misconceptions about gravitational potential energy, see here.]

The equations for TD, E_K , and -E_G. share a common element: (γ - 1)

TD =  (γ  –  1)Δτ.

 E_K = (γ_K  – 1)m₀c² 
-E_G = (γ_G  – 1)m₀c²

When E_K = -E_G  then (γ_K  – 1)  = (γ_G  – 1) = (γ –  1) 

Therefore, for a clock with unit mass (1 kg), the time dilation per second of "proper" time, is (γ_K  – 1)/m₀c², due to relative speed, and (γ_G  – 1)/m₀c², due to relative gravity.

It should be "obvious" that TD experienced by a clock "at rest" on the surface of P is exactly equal to the TD experienced by a clock in deep space moving at the Escape velocity, v_E, associated with the surface of P.

LINEAR and MONOTONIC Relationship between ENERGY and Relativistic Effects

The Excel graphs below compare ENERGY, Velocity, and Gravity with the Time Dilation.

The analysis is based on the Kinetic Energy resulting from relative velocities from zero to 295,599,350 meters per second (0 to 98.6% of c, the speed of light) in deep space -or- the Gravitational Potential Energy resulting from being "at rest" within the gravitational attraction of either an Earth-like massive sphere or a compact Neutron Star-like massive sphere (with a mass of two Suns and a radius of only 13 km, experienced from an infinite distance down to the center.

Time Dilation is LINEAR with Respect to Kinetic Energy and NON-Linear with Respect to Velocity

The above graphic shows the relationship between Time Dilation, Velocity, and Kinetic Energy. Note that, for all speeds from zero to the speed of light, TD is exactly proportional to KE, and TD is not proportional to Velocity.

Time Dilation is LINEAR with Respect to Gravitational Potential Energy and NON-Linear and NON-Monotonic with Respect to Gravity (Example for Earth-Like Spherical Massive Object)

The above graphic shows the relationship between Time Dilation, Gravity, and Gravitational Potential Energy. Note that, for this example, using a spherical mass with the same mass and radius of the Earth, TD is exactly proportional to GPE. Note also that the relationship between TD and Gravity is not even monotonic!
From Infinity to the Surface, as Gravity INCREASES in (negative) magnitude, TD increases, but non-linearly.
From the Surface to the Center, as Gravity DECREASES in (negative) magnitude, TD increases, also non-linearly.

Time Dilation is LINEAR with Respect to Gravitational Potential Energy and NON-Linear and NON-Monotonic with Respect to Gravity (Example for Neutron Star-Like Spherical Massive Object)

The above graphic shows the relationship between Time Dilation, Gravity, and Gravitational Potential Energy. Note that, for this example, using a spherical mass with the same mass and radius of a Neutron Star, which is much, much more massive than the Earth, and very much smaller, so its density is far greater than the Earth, TD is exactly proportional to GPE. Note also that the relationship between TD and Gravity is not even monotonic!
From Infinity to the Surface, as Gravity INCREASES in (negative) magnitude, TD increases, but non-linearly.
From the Surface to the Center, as Gravity DECREASES in (negative) magnitude, TD increases, also non-linearly.

Conclusion

TOTAL ENERGY with respect to the center of P determines the magnitude of Time Dilation and Length Contraction for the entire journeys of both Fred and Jill. These journeys involve movement at various speeds, including momentary instants of being “at rest”, and various levels of gravitational attraction, including momentary instants of zero gravity.

The equations for TD, E_K , and -E_G. share a common element: (γ - 1)

TD =  (γ  –  1)Δτ. where γ = Δt/Δτ,

 E_K = (γ_K  – 1)m₀c², where   γ_K = 1/SQRT(1-v²/c²)  
-E_G = (γ_G  – 1)m₀c², where   γ_G = 1/SQRT(1-2GM/rc²)
When E_K = -E_G  then (γ_K  – 1)  = (γ_G  – 1) = (γ –  1) 

Therefore, for a clock with unit mass (1 kg), the time dilation per second of "proper" time, is (γ_K  – 1)/m₀c², due to relative speed, and (γ_G  – 1)/m₀c², due to relative gravity.

When E_K = -E_G  then (γ_K  – 1)  = (γ_G  – 1) = (γ  –  1) 

My Excel analysis confirms, for a wide range of relative velocities and gravitational situations, ENERGY remains linearly related to Time Dilation and Length Contraction This is definitely NOT the case for Velocity nor Gravity, each of which are asymptotically limited.

Therefore it is reasonable to conclude that relative TOTAL ENERGY determines the Relativistic Effects of any combination of speed and gravity.

[Edited 24 July 2021. I fixed a couple of typos and changed "ϒ" to "γ".]

Ira Glickstein