[MikeFontenot (OP)]

I've just published another paper, entitled

"A New Gravitational Time Dilation Equation".

It is available on the viXra repository:

  https://vixra.org/abs/2201.0015

[Plugging a commercial product prohibited]

Here is the abstract:

In a previous paper, I showed that the gravitational time dilation equation, which has been accepted since Einstein published it in 1907, is incorrect. It is incorrect because it is inconsistent with the required outcome at the reunion of the twins in the famous twin ‘paradox’ of special relativity. In this paper, I describe a new gravitational time dilation (GTD) equation which IS consistent with the required outcome at the reunion of the twins. And my new GTD equation gives the same instantaneous change of the home twin’s (her) age, according to the traveling twin (him), when he instantaneously changes his velocity, as is given by the CMIF simultaneity method, but without requiring the assumption that the CMIF method requires.

[MikeFontenot (OP)]

I discovered a mistake in my latest paper. I had concluded in that paper that my new method says that as the duration "tau" of the constant acceleration "A" goes to infinity, the tic rate ratio "R" of the separated clocks will go to 1.0 ... i.e., eventually the two separated clocks will tic at the same rate.

That result is correct.

But experimental results many years ago found that two (unaccelerated) vertcally-separated clocks stationary at or near the Earth's surface in the Earth's gravitational field run at slightly different rates, in agreement with the existing gravitational time dilation equation (using the version of the equation for a field that varies with height).

I had concluded in my paper that EITHER (1) those experimental measurements were flawed, or (2) the equivalence principle was incorrect, or (3) I had made a mistake somewhere.

I now know that my results DON'T contradict those experimental measurements. My results (which apply to a special relativity situation (without any gravitational fields)) apply to two clocks that each undergo EXACTLY the same constant acceleration "A" for the entire duration "tau". But in the case of the experimental measurements (conducted in the Earth's gravitational field), the two clocks have slightly different gravitational fields applied to them, because of the inverse-square dependence of the field strength on their distance from the center of the Earth. The clock higher in the field experiences a slightly weaker field.

So my special relativity scenario, and the scenario of the experimental measurements, AREN'T equivalent, and therefore the equivalence principle doesn't apply to them. Therefore the results in my paper DON'T contradict the experimental measurements.

Accordingly, I have revised my paper, on both the viXra repository and elsewhere

[MikeFontenot (OP)]

I have just put a 2nd revised edition of my paper on viXra:

https://vixra.org/abs/2201.0015?ref=13239788

and [elsewhere]

I had originally computed the age change of the "Helper Friend" (the "HF"), during the constant acceleration "A" (which lasts "tau" units of time), by doing a numerical integration of the Rate Ratio "R(tau)". But I recently discovered that the integral could be done analytically, and it gives a surprisingly simple result: the age change is just

AC(tau) = tau + { L * tanh(A tau)},

where "*" denotes multiplication. The quantity "tanh(A tau)" is just the velocity "v" of the accelerating observer (the "AO") and the "HF", which makes the AC equation similar to the "delta_CADO" equation that I derived many years ago.

[Bored chemist]

Given that the "old" version of time dilation has been tested and found to give the right answer, why would we want one that gave any different answer?

[MikeFontenot (OP)]

Given that the "old" version of time dilation has been tested and found to give the right answer, why would we want one that gave any different answer?

The "old" version of the gravitational time dilation (GTD) equation,

    exponential {integral from 0 to L [ g(h) dh] },

was tested only for a weak gravitational field.  The approximation of the exponential version, for weak fields:

   (1 + { integral from 0 to L [ g(h) dh ] }

also agrees with the experimental measurements.  But when each of them is converted (via the equivalence principle) to a special relativity version of the equation (with an acceleration rather than a gravitational field), they fail to agree with known results.

The exponential equation says that, if an observer instantaneously changes his speed with respect to a distant person (from zero to some speed directed toward the distant person), then the observer will conclude that the distant person's age suddenly becomes INFINITE.  That contradicts the known outcome at the reunion of the twins in the twin "paradox":  When the traveling twin returns home, his age is finite, and so is the home twin's age.  Her age isn't infinite, as the exponential version of the GDT equation requires.  Therefore the exponential version is incorrect.

The approximation of the exponential version, for weak fields, doesn't produce any infinities, but it also fails to agree with the outcome of the twin's reunion. I was able to modify the approximation of the exponential version, so that it DOES agree with the twin's reunion, but that equation doesn't obey the principle of causality, so it must also be rejected.  In my newly derived GTD method, I found a way to get agreement with the twin's reunion that doesn't violate the causality principle.

This is all spelled out in much greater detail in my previous paper

  https://vixra.org/abs/2109.0076?ref=12745236

and in my latest paper

  https://vixra.org/abs/2201.0015?ref=13239788

[Bored chemist]

if an observer instantaneously changes his speed

You do realise that's impossible, don't you?

[Bored chemist]

The "old" version of the gravitational time dilation (GTD) equation,

    exponential {integral from 0 to L [ g(h) dh] },

was tested only for a weak gravitational field. 

It was tested using the gravitational field of Sirius B which is pretty strong.
How different would the calculations based on your idea be?

[MikeFontenot (OP)]

if an observer instantaneously changes his speed

You do realise that's impossible, don't you?

In my previous paper,

   https://vixra.org/abs/2109.0076?ref=12745236

in Section 3, I show how that LIMITING behavior is determined.  It is determined by doing a sequence of repeated experiments, each of which consists of a FINITE acceleration "A" and a FINITE duration "tau" of the acceleration.  And in each experiment, "A" and "tau" are such that their product (A * tau) equals 1.317 lightseconds per second, abbreviated as ls/s.  That product, called "theta", is the "rapidity" of the two observers at the end of that acceleration. (The rapidity is always zero at the beginning of the acceleration).  The rapidity of 1.317 ls/s corresponds to a velocity "v" of 0.866 ls/s (which I chose because it results in a gamma factor of exactly 2.0).  I also chose a separation L between the two observers of 7.52 ls (which I chose for convenience, because it made the calculations I was doing on a calculator especially easy).

In the first experiment, I apply an acceleration of A1 ls/s/s, for a duration tau1 of 1 second (where A1 is calculated using the above considerations), and I determine by how much the distant person's age changes, according to the observer, during that acceleration.  Call the answer, when I do that, T1 seconds of ageing of the distant person.  T1 equals 20000 seconds, or 2x10sup4.  I.e., T1 is 2 times 10 raised to the 4th power.  (I got that even number of 20000 seconds intentionally, by the way I chose the value of L.  That was purely to simplify the calculations I was going to do on a hand calculator).  So the first experiment says that the "Helper Friend" ("HF") ages 20000 seconds, or 2x10sup4 seconds, during the 1 second duration of the acceleration.

I then repeat that experiment from the beginning, except that I make the acceleration 10 times larger (so A2 = 10 * A1), and the duration of the acceleration 10 times smaller (so tau2 = tau1 / 10 = 0.1 second).  Call the answer, when I do that, T2 seconds of ageing of the distant person (the "HF").  The result is that T2 = 1.02x10sup42.  That's a BIG difference in the age of the HF after the 0.1 second acceleration, compared to the 1.0 second acceleration.  When we made the duration 10 times shorter (and the acceleration 10 times larger), the age of the HF increased by a LOT more than a factor of 10, it increased by a factor of 10 raised to the 42nd power!

Then, I repeat the experiment again, making the acceleration 10 times larger than for the 2nd experiment  (so A3 = 10 * A2 = 100 * A1), and the duration 10 times shorter than for the second experiment (so tau3 = 0.1 * tau2 = 0.01 * tau1).  The result is that T3 = 1.27x10sup428.  That's a GIGANTIC difference in the age of the HF after the 0.01 second acceleration, compared to the 0.1 second acceleration.  This time, when we made the duration 10 times shorter (and the acceleration 10 times larger), the age of the HF increased by a LOT bigger factor that it did previously.  It increased by 10 raised to the 428nd power!

Each time we reduce the duration by a factor of 10, and increase the acceleration by a factor of 10, the exponent of 10 in the age increase of the HF increases by a factor of about 10.  So on each iteration of the experiment, THE EXPONENT OF 10 in the HF's age increase itself increases by a factor of about 10!

It's clear that this sequence is NOT converging to a finite value ... it is clearly diverging to infinity.  THAT'S what is mean when I say that the limit, as tau goes to zero (while keeping the speed change after the acceleration at 0.866 ls/s), of the age change of the HF, is INFINITE.

[Bored chemist]

Did you notice how it's still impossible after you wrote all that?

Never mind; any progress on this bit?

It was tested using the gravitational field of Sirius B which is pretty strong.
How different would the calculations based on your idea be?

[MikeFontenot (OP)]

Did you notice how it's still impossible after you wrote all that?

It isn't actually possible to change your velocity instantaneously.  But it IS a very useful concept, nevertheless.  In practice, there is usually a negligible difference between changing your velocity in absolutely zero time, and changing your velocity in a very small non-zero time.   But on the other hand, in twin-paradox-like problems (and when the CMIF simultaneity method is used), it is usually much EASIER and FASTER to get the answer you want by assuming the speed change is instantaneous.  Dirac knew that ... that's why he invented the Dirac delta function, which has proven very useful in physics and in engineering, and especially in quantum mechanics.
   

[Bored chemist]

Never mind; any progress on this bit?

Quote from: Bored chemist on 27/01/2022 20:23:33
It was tested using the gravitational field of Sirius B which is pretty strong.
How different would the calculations based on your idea be?

[Bored chemist]

In practice, there is usually a negligible difference between changing your velocity in absolutely zero time, and changing your velocity in a very small non-zero time. 

Except that one is possible and...

[MikeFontenot (OP)]

In practice, there is usually a negligible difference between changing your velocity in absolutely zero time, and changing your velocity in a very small non-zero time.

Except that one is possible and...

There is usually a negligible difference IN THE RESULTING AGE CHANGE OF THE HF between ...

[MikeFontenot (OP)]

It was tested using the gravitational field of Sirius B which is pretty strong.
How different would the calculations based on your idea be?

I'm not familiar with that experiment.  Can you give me a link to it?

[Bored chemist]

It was tested using the gravitational field of Sirius B which is pretty strong.
How different would the calculations based on your idea be?

I'm not familiar with that experiment.  Can you give me a link to it?

It hardly matters.
Can you tell us how large a gravitational effect your idea would predict for light leaving the surface of that star?
Can you tell us how big the difference would be between the conventional theory and your idea?

[MikeFontenot (OP)]

It was tested using the gravitational field of Sirius B which is pretty strong.
How different would the calculations based on your idea be?

I'm not familiar with that experiment.  Can you give me a link to it?

It hardly matters.

Sirius B has about the same mass as our sun, but is vastly farther away from us.  It does NOT give a test of the exponential time dilation (GTD) equation for large arguments of the exponential function ... it only tests the (1 + integral{g[h]dh} ) small-argument approximation of the GTD equation, which I have NOT claimed is incorrect.

But the large-argument correctness of the GTD equation HAS been tested theoretically (not experimentally), by studying the case where the gravitational field strength "g" is constant (not variable with L).  In that case, the gravitational source isn't spherical ... it is of extremely large extent horizontally, and "downward" vertically, with no mass above that horizontal plane.  Does that exist anywhere in our universe?  I don't know, but doubtful, I think.  But by converting that scenario to a special relativity scenario (via the equivalence principle), we DO get an analyzable situation, where there is no gravitational field, but rather there are separated observers who are all accelerating with the same acceleration "A".  In that case, I have shown that the exponential equation is incorrect, because it is inconsistent with the outcome of the twin paradox reunion.  And I have given an equation which DOES fully agree with the twin paradox outcome.

[Bored chemist]

Sirius B has about the same mass as our sun,

And it's much smaller so the gradient is much larger.

The thing is that - unlike most circumstances- the experiment can actually be done. The gravitational shift is measurable.
And if you are saying that your ideas predict significantly different shifts then why not do the maths and show how different.

And I have given an equation which DOES fully agree with the twin paradox outcome.

The twin paradox is not a paradox.
Their experiences are different so they have different outcomes.