**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,

*Δ***, ("delta tee") measured by a clock "at rest" in a reference frame (zero speed, zero gravity).**

*t***"Proper" Time Interval**between the same two events,

**, ("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**

*Δτ***due to Time Dilation.**

*Δt***Lorentz Factor**,

*ϒ =*

*Δ***/**

*t***, equal to or greater than 1.0.**

*Δτ***Lorentz Factor due to relative speed**,

**where**

*ϒ*_{K}= 1/SQRT(1-v^{2}/c^{2})*is the relative velocity between the "at rest" clock and the moving clock, and*

**v***is the speed of light in a vacuum.*

**c****Lorentz Factor due to relative gravity**,

*ϒ*_{G}= 1/SQRT(1-2GM/rc^{2})*,*

**where**

*is the gravitational constant,*

**G***is the mass of the spherical planet (or massive object) that is the source of the gravitational field, and*

**M****(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.**

*r***Time Dilation**,

*TD =*

*Δ**t -*

*Δτ***is the "coordinate" time interval minus the "proper" time interval.**

*,***Time Dilation**,

*TD = (*

*ϒ - 1)***, an equivalent way to express Time Dilation. Please note that the**

*Δτ*

*(***factor is common to the equation for Time Dilation as well as the expressions for**

*ϒ - 1)**and*

**KE***.*

**GPE****Kinetic Energy**of the clock,

**=**

*KE*

*(ϒ*_{K }**–**

*1)***c**

*m*_{0}^{2}, where

**is the "rest mass" of the clock.**

*m*_{0}**Gravitational Potential Energy**of the clock with respect to infinity,

**=**

*-GPE*

*(ϒ*_{G }**–**

*1)***c**

*m*_{0}^{2}.

**Escape Velocity**,

**, from a non-rotating, no atmosphere planet,**

*v*_{E}**, is the speed, perpendicular to the surface, where**

*P***is exactly equal to**

*KE***. (The minus sign is due to**

*-GPE***being defined as zero at at infinite distance from**

*GPE***, increasing in magnitude to its largest**

*P**NEGATIVE*value at the center, and then increasing asymptotically to zero an infinite distance from

**.) A rocket launched at**

*P***will just break free of the gravity of**

*v*_{E}**and continue into space forever, with no need for further propulsion.**

*P***where**

*v*_{E}= SQRT(2GM/r),**is the gravitational constant,**

*G***is the mass of**

*M***, and**

*P***(which must be equal to or greater than the radius of**

*r***) is the distance of the object from the center of**

*P***.**

*P*
A
clock moving at

**(the Escape velocity associated with the surface of***v*_{E}**), but far from any massive object (zero gravity), will have exactly the same***P**as a clock “at rest” (zero speed) on the surface of***TD****. Note that this occurs when the***P***of the former is exactly equal to the***KE***with respect to infinity of the latter.***-GPE*
As an Engineer (not a
Physicist) these facts scream out that relative

**(***ENERGY***and/or***KINETIC***)***-GRAVITATIONAL POTENTIAL**causes**. I find it surprising that these facts are***TD***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**to***ENERGY**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 a change in kinetic energy causes a

**change in**

*linear**. A change in speed will cause a change in*

**TD***, but*

**TD***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

*, while a 1% increase from 0.98c to 0.99c will cause a 41% increase!*

**TD**
A change in

*causes a***GPE****change in Time dilation. A change in gravity will cause a change in***linear**, but***TD***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***, and***E*_{K }**. share a common element:***-E*_{G}*(**ϒ - 1)***(ϒ**

*TD =*_{ }– 1)

**.**

*Δτ*
E

-E

When E

_{K}= (ϒ_{K }– 1)**c***m*_{0}^{2}-E

_{G}= (ϒ_{G }– 1)**c***m*_{0}^{2}When E

_{K}= -E_{G }then (ϒ_{K }– 1) = (ϒ_{G }– 1) = (ϒ_{ }– 1)
Therefore, for a clock with unit mass (1 kg), the time dilation per second of "proper" time, is (ϒ

It should be "obvious" that

_{K }– 1)**c***/m*_{0}^{2}, due to relative speed, and (ϒ_{G }– 1)**c***/m*_{0}^{2}, due to relative gravity.It should be "obvious" that

**experienced by a clock "at rest" on the surface of***TD***is exactly equal to the***P***experienced by a clock in deep space moving at the Escape velocity,***TD***, associated with the surface of***v*_{E}**.***P*

__LINEAR and MONOTONIC Relationship between ENERGY and Relativistic Effects__

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

*, 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.*

**c****Time Dilation is LINEAR with Respect to Kinetic Energy and NON-Linear with Respect 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)**

**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)**

__Conclusion__
TOTAL ENERGY with respect to the center of

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.

**determines the magnitude of Time Dilation and Length Contraction for the entire journeys of both***P***and***Fred***. 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.***Jill*
The equations for

**,***TD***, and***E*_{K }**. share a common element:***-E*_{G}*(**ϒ - 1)***(ϒ**

*TD =*_{ }– 1)

**. where**

*Δτ*

*ϒ =*

*Δ***/**

*t***,**

*Δτ*
E

-E

When E

_{K}= (ϒ_{K }– 1)**c***m*_{0}^{2}, where ϒ_{K}= 1/SQRT(1-v^{2}/c^{2})-E

_{G}= (ϒ_{G }– 1)**c***m*_{0}^{2}, where ϒ_{G}= 1/SQRT(1-2GM/rc^{2})When E

_{K}= -E_{G }then (ϒ_{K }– 1) = (ϒ_{G }– 1) = (ϒ_{ }– 1)
Therefore, for a clock with unit mass (1 kg), the time dilation per second of "proper" time, is (ϒ

_{K }– 1)**c***/m*_{0}^{2}, due to relative speed, and (ϒ_{G }– 1)**c***/m*_{0}^{2}, due to relative gravity.
When E

_{K}= -E_{G }then (ϒ_{K }– 1) = (ϒ_{G }– 1) = (ϒ_{ }– 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.

**Ira Glickstein**
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