Gravity and Time Dilation

GRAVITY
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.

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

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

The ratio of two orthogonal distances is this...
to maintain c as constant.

But this equation does not just apply to time. But also, simultaneously to distance. For example, v=d/t:
Space contraction and time dilation are necessarily the same process. Time is NOT a constant when velocity is involved, and as I will show, 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...

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.

1)The small body keeps a record of the gravity effects of the large planet; manifested in its VELOCITY.
2)Also, escape velocity is specific and unique for every big MASS, planet.

As the small body falls, kinetic energy is gained. In an inelastic collision with the planet, with heat thrown away, kinetic turns into potential. The potential energy is the integral of all gravitational force times distance, taken from infinity to r. (Bound energy (rest energy) is negative.)

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.

Specifically:
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.

In any case, we have v, velocity. fAnd we have a representation of a mass: escape velocity.

But the real heart and sole equation is the Lorentz-Factor equation, and this is where we are headed with our velocity implant.

Substituting velocity (from above) into the relativistic Lorentz Factor equation...
We will insert it in the Lorentz equation.
And FORCE the Lorentz expression to be due to the escape velocity (or gravity).
This is "Time Dilation" of the planet as seen from deep space.

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.
We have now a One-Body equation.

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 with velocity, and nothing to do with "Gravity".

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.

I have established the equivalency of both Time Dilation under Speed (Absolute magnitude velocity), and also under Gravity.

And also, the "direction" is the same: Time Rate slows down.
I should point out that the observer is in deep empty space, and incidently at zero speed. Michelson�Morley proved that NO observer can see time dilation of themselves on the planet. The intire point of the experiment was the shocking revelation that NO ONE can see any problem: No time dilation, and no space compression. No change in C.

TIME DILATION
(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 as observed in a lesser gravity field.
Also, and not related to the discussion, Lengths appear to be "deflated" and contracted on the planet.

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.

In general, these are the time dilations from a mass.
And so simply derived...
RULERMAR.GIF, 1.6kB

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.

EQUIVALENCE

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!

Maclaurin Expansion

By the way, here are the units for time dilation...

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...

RULERMAR.GIF, 1.6kB

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.

I have thought about it...
As one approaches the planet, two of "our" dimensions begin to diminish. Those two are the equivalent of one cartesian coordinate, and the other is time. From far away, an object approaching the Earth is getting flatter and flatter. Taken to the extreme with an approach to a Black Hole, at the event Horizon, an object is as flat as a piece of paper. Here there are only two dimensions, and all of the volume-mass is represented on the surface. There is no internal mass or time. The surface "represents" everything inside. The surface is like a Hologram, and only a conscious being can decode it. A Hologram is read by mixing a Reading-Laser on to the Hologram, and then specific information is decoded out and displayed to human eyes. Therefor, a two dimensional Hologram can "look" or appear three dimensional to a human that is used to processing such information. In like manner, a conscious human can "recreate" the mass. The analogy is: consciousness is like the Reading Laser, and consciousness is projected unto the hologram, and we filter out, or "own" "our" familiar three dimensions and time. But I digress...

RULERMAR.GIF, 1.6kB

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. Nor can we even measure or see it.
We are continually subject to "normalized", or "proper", time in all of our human endeavors. There is no way to "experience" the difference. Thank Michelson-Morley.

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

RULERBOW