VISUALIZING - Using My FREE Excel Spreadsheet for Special and General Relativity

Nowadays it is common to use computer models to help us VISUALIZE and better understand complex situations and systems. Prior to the advent of computer models, we had to use mental models in our "mind's eye", along with physical aids such as paper maps and diagrams, modelling clay, and so on. I've created an interactive Excel Spreadsheet for Special and General Relativity.

User selects a Star, Planet, or Set Angle Option. 

Right side shows Special Relativity Effects due to the Kinetic Energy of moving at half the speed of light in empty Space. 

Left side shows equivalent General Relativity Effects, where Time "curves" due to the Potential Energy of being "at rest" close to a Black Hole.

Albert Einstein was a great physicist, with all the requisite mathematical tools. However, he rejected purely mathematical abstraction and resorted to physical analogy for his most basic insights. For example, as part of the thought process that resulted in his theories of Special Relativity (1905) and General Relativity (1915) he imagined himself riding along a beam of light; or as an observer standing along the tracks as a train zipped by at near-light-speed; or as a scientist sealed in a closed box and not able to tell if the box was stationary on the surface of the Earth, subject to gravity, or in deep space, far from massive objects, but subject to acceleration due to being dragged by a rocket at ever-increasing speeds.

Of course, Einstein and virtually all scientists and technologists use mathematical abstractions to quantify the meaning in our visualization models. We change the initial conditions and run these models to simulate what may or may not happen in different situations.

COMPUTER MODELS FOR VISUALIZATION

As personal computers and the Internet have become endemic, manual typewriters, paper maps, physical books, and so much else has been displaced by automated versions. Similarly, computer visualizations and models have displaced older methods - except for that old reliable "mind's eye" which remains as important as ever.

During my career as a Senior System Engineer at IBM and Lockheed-Martin I made extensive use of computer models and visualizations and have continued to do so since retirement.

VISUALIZING EINSTEIN'S SPECIAL AND GENERAL RELATIVITY

Perhaps the most well-known equation in the world is E = mc2, recognized by virtually every person. But, what does it really mean?

And, many people know about the so-called "twin paradox", where one twin goes off on long mission at high speeds into space, and comes back younger. But why does this happen and exactly what causes it?

If "everything is relative" why isn't the stay-at-home twin also also younger? So, everything is not relative, and perhaps Einstein's original name for his theories "Invariance" is more apt -for the fact all observers, including those moving at different speeds, measure the same speed for light.

If the traveling twin is younger due to experiencing high speed and acceleration, then it is aging that has slowed down, not time, per se.

Furthermore, what, precisely, is TIME? And how is TIME united with SPACE to form SpaceTime?

When you Google any of this stuff you are quickly buried in equations and tensor mathematics that no one (even an engineer like me) can really understand!

Well, all this bothered me for most of my life until, back in 2012, I decided to answer Alan Alda's Flame Challenge "What is Time?" and produce a short video. In the research process for that project, I think I had a critical insight into TIME, SPACE, and RELATIVITY that may help you VISUALIZE this important scientific theory.

Time - the fourth dimension (2013 Flame Challenge) from Ira V Glickstein on Vimeo.

Since that time, I've continued to delve into Relativity and I've come up with what I think is a unique way to visualize and ... perhaps ... even understand it. The following images are screenshots from an Excel spreadsheet I created to provide myself (and you :^) a "hands-on" experience with the relativistic effects of high speed (kinetic energy) and high acceleration (potential energy), including time dilation, length contraction, and the curvature of SPACE and TIME. It is available free.

Free Excel Spreadsheet for VISUALIZATION of Einstein's Special and General Relativity. 

Image is of the Main Panel where user selects a Star, Planet, or Set Angle Option. In the case illustrated, the SpaceTime angle is set to 30º, where velocity is half the speed of light. This causes clocks to slow down by 13.4%, which corresponds to 49 days per year or 482 seconds per hour. Right side shows Special Relativity Effects due to the Kinetic Energy of moving at half the speed of light in empty Space. Left side shows equivalent General Relativity Effects, where Time "curves" due to the Potential Energy of being "at rest" close to a Black Hole.

Free Excel Spreadsheet for VISUALIZATION of Einstein's Special and General Relativity. 

Image is of the SpaceTime view of the right side of the Main Panel (where the vector sum of TRAVEL + AGING = 1) plus the Minkowski-Like SpaceTime view (where the simple sum of TRAVEL + AGING =1).  

Free Excel Spreadsheet for VISUALIZATION of Einstein's Special and General Relativity. 

Image is of the Minkowski-Like view (described above) compared to a Planck view, where both Space and Time are assumed to be discrete, and Each tiny cell is 1 Planck Time (tP ) by 1 Planck Length (ℓP).

THE MAP IS NOT THE TERRITORY!

As my principal PhD advisor, Howard Pattee, taught me, "The MAP is NOT the TERRITORY". That sage statement means that no model is exactly the same as the thing being modeled (else it would be the real thing.)

We make models because the real thing is too complex and difficult for us to visualize, or -like the Global Climate- is not readily available for us to experiment upon.

The MAP is NOT the TERRITORY !

Many a General (or football coach) has moved symbols around on a map of the field of battle, convincing himself and his staff of inevitable victory, only to find his opponent also had a model, perhaps a better one plus superior forces to carry it to victory. 

We generally model only the most important or critical parts of the situation or complex system we are trying to visualize. We consider the model to have been successful if the results match actuality to some level of fidelity, at least for those significant portions. If subsequent testing reveals that the model does not comport with reality, we must improve or discard it.

Ira Glickstein

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