Derivation of Time Dilation from gravity

Time dilation, near mass

While lying in bed, quite not asleep, I wondered how does one measure the amount of time dilation of gravity - exactly?

SunEscape.gif, 36 kB A falling body from infinity can be used to measure an effect from a larger body.

The phenomenon was proven to exist way back in the 20's by the bending of light around the sun during an eclipse; Implying that the speed of light is less near the sun. A process that is similar to bending light in a prism.

Michleson-Morrley exposed an equation of "length-contraction" with v velocity. But where could I find a velocity term from gravity, so as to use this Michleson-Morrley equation?

I then hit upon the idea of "Escape Velocity".

EqLorentz.gif, 2.4kB
The ratio of two orthogonal distances is this...
to maintain c as constant.

But this equation does not just apply to distance. But also, simultaneously to time. For example, v=d/t:
Time dilation in the denominator decreases the speed, and length contraction in the numerator increases it. The two effects cancel out, maintaining c as constant. Space contraction and time dilation are necessarily the same process. Time is NOT a constant when velocity is involved, and opens the door to other forms of Time Dilation - like gravity.

With this "equality" in mind, I could not contain my excitement! This is when I woke up...
ANASTAR5.gif, 1 kB Escape-velocity!
Escape velocity is not only the velocity needed to escape the confines of a planet, but also the velocity of a small body pulled into a planet from infinity. The two velocities may be in apposing directions, but they are equal. The small body keeps a record of the gravity effects of the large planet; manifested in its VELOCITY. Also, escape velocity is specific and unique for every MASS. EscapeM.jpg, 0 kB

EqEnergyPOT.gif, 2 kB
The potential energy is the integral of all gravitational force times distance, taken from infinity to r. (Bound energy (rest energy) is negative.)

EqEnergy.gif, 2 kB
EqEscape.gif, 2.0kB
The large mass is capital M. The small mass is lower case m. We want the time dilation of (of course) the large M, but we will use the velocity of the small m to measure the effects of the large M. The gravity "effects" are totally due to the large M mass. M >> m
The calculation involves another mass, but escape velocity is the property of a single mass. And is assigned as such.

A very fortuitous equation for escape velocity...
The reason is the GM product. The GM product is accurately known.

For example, The normalised earth radius (to a sphere) is 6.37100080E+06 meters
And, GM product for earth is 3.986004418E+14 m+3 s-2

Therefore, The escape velocity is this...
For the earth it is 1.1186134989E+04 m/s
I just had to try out the equation for a test run sence the GM product is so very good.

EqEscapeSub.gif, 2.0kB
We have a representation of a mass: escape velocity.

EqLorentz.gif, 2.4kB
EqRelativ.gif, 1.2kB
EqRelativSub.gif, 2.2kB
But the real heart and sole equation is the Lorentz-Factor equation, and this is where we are headed with our velocity implant.

EqTimeDilation.gif, 3 kB
Eqn1ANAM.gif, 5 kB
Substituting velocity (from above) into the relativistic Lorentz Factor equation...
We will insert it in the Lorentz equation.
And FORCE the Lorentz length expression to be due to the escape velocity (or gravity).
And call it "Time Dilation".

In other words, initially, the way that I have expressed this, Lorentz length is a function of speed.
But ultimately Lorentz length IS Time Dilation itself.

The small m is no longer to be found any where in the equation. The small m, and its kinetic energy, were instrumental in the developement of the equation - in fact, critical! But in the end, sacrificed and silently missing.

As the small m increases its speed, and as viewed from an empty deep space, it's Time dilation increases. Events slow down. This is an isolated m. Now picture a large number of small m s all converging at the spot in empty space where the large M will exist. The conclusion: all of the constituent small m suffered each the same Time Dilation as the isolated m under study. The large M therefor has the same Time Dilation as the large group of individuals.

Now for the second point:
If the small m is bounced back into space from a totally elastic collision, one can say that at the moment of collision Potential Energy is equal to Kinetic Energy. Therefor, Potential Energy of the small m under gravity, at zero velocity, is equal to the small m Kinetic Energy at the Escape Velocity. Therefor, sense the Large M and the small m are equivalent in Time Dilation, the "polarity" is the same too: Large planets have a SLOWER Time Rate, and a greater Time Dilation, as observed from deep space.

BBALLBLU.GIF, 139B I have established the equivalency of Time Dilation under Speed (Absolute magnitude velocity), and under Gravity.

BBALLBLU.GIF, 139B And also, the "direction" is the same: Time Rate slows down.
Arithmetically, subtract 1. If the result(z) is positive: Time Dilation. If negative: Time Compression.

(of Gravity)

Time dilation at this planet's surface (earth) should be this...
The view is from stationary free space, away from the mass, observing activities of the mass from a distance.
Time Dilation values are always greater than "one" from this perspective.

Time between "ticks" appear to have increased; a slow time-rate.
Time rate has "dilated". Every happening on the planets surface is in slow motion.
Lengths appear to be "inflated" on the surface.
Speed-of-light has decreased on, and near, the objects surface.
Originating Radiation has redshirted from the objects surface.

EqTimeCompression.gif, 2.3kB viewer is on the mass
Viewer is at the mass:
Or you could take the reciprocal expression, if you are on the mass, looking out.
Typical values are slightly less than one.

Time between "ticks" appear "out-there" to have decreased; a fast time rate.
Time has speeded up in empty space.
Local lengths appear normal.
Speed-of-light has increased in space.
Local Speed-of-light appears normal.
Originating Radiation from space has blue-shifted in frequency.

In general, these are the time dilations from a mass.
And so simply derived...

Also implied, sense time is used, is the gravitational length variability of any object. For example the meter. The length of the meter becomes a subjective thing, and is now referenced to another subjective thing: the speed of light. If the "length" of things and the "speed of light" are always in the same environment, they will appear correct. Time defines both. The length of the meter is now DEFINED by the speed of light. This works perfectly! As a result, a meter on any planet, massive or not, is still exactly a meter.


I suppose with a more rigorous (formal) development, Inertial-Mass and Gravitational-Mass could be proven equal with this.
The small mass is the "INERTIAL", with its momentum and velocity; and the large mass is the "GRAVITATIONAL".
I have informally reckoned the two together, as equal.
All, from a dream...

RULERMAR.GIF, 1.6kB In addition, there is another forced conclusion:
Below the earths surface acceleration g decreases, and therefore time speeds up again! Or does it?

At the center of the earth there is no acceleration g. You are in free fall. You float around as in free space. And therefor time dilation is the same as freespace. Or is it? The gravity g acceleration is a vector and is subject to cartesian directionality.

This "logical" result is psychologically distasteful to me: to imply with tons and tons of mass all around, and there to be no time dilation? I developed the intire argument according to a force acceleration vector g. I dare not throw away the vector part of g. This is equally distasteful.

On second thought...
The equation says NOTHING about acceleration g!
Yes, acceleration g (a vector) gave rise to the small body accelerating toward the large body. Due to symmetry, the large body does not care where small bodies are coming from; from the east, from the west. And large G is a scalar. Although the equation was born from vector factors, it contains - in it's final form - only a SCALAR value. My thought: Time dilation anywhere inside a body is no less than at its surface!
My thoughts concerning INSIDE a body is my opinion and is only a guess. I do not know. Perhaps, I can have another dream.

Specifically, putting in the numbers:
Time dilation at the earths surface should be this...
An additional time factor 6.96127371E-10 sec .

(You must expand the expression to enable a computer to get a "resolution bite" on it at low velocities. The method that my computer used to get this answer was by substituting v into this iterated expression. My computer does not have the precision the other way.)

See how fast it converges!
BBALLBLU Maclaurin Expansion

By the way, here are the units for time dilation...
EqTimeUnits.gif, 6 kB
Null dimensions
And that is how it should be, this is a dimensionless "factor". Or you can immediately see this by visual inspection. Forget that I even mentioned it...

EqTimeDilation.gif, 3.2kB Eq-Schwarzschild.gif, 2.0kB
There is a discontinuity when the denominator goes to zero, and the expression tends to infinity. When the dilation goes to infinity, TIME STOPS!

What does THAT mean?
All the physical equations that one is so used too - involving time - have no meaning anymore, and are undefined. Talk about a missing toolbox! Anything that is "happening" involves time. There are no Time wrenches to fit anything material. But that world does truly exist. It exists in BlackHoles, the birth place for new material things, and a new Time.


By choosing earth as the place to live, in contrast to outer space, one gets to live 2.1958 Seconds longer after a hundred year life. (Just one more perk of living on this planet.)
But of course, "Slow-Mo" is no gift. We can not partake.
We are continually subject to "normalized", or "proper", time in all of our human endeavors. There is no way to "experience" the difference.

I have discovered a simple Algebraic Solution for gravitational time-dilation. I knew it would work as soon as I woke up.