Actually, I AM starting to doubt its validity. I THINK the original source of that equation was Einstein, in his 2007 paper. In that derivation, Einstein didn't use the exponential exp(Ad), he used 1 + Ad. He did conjecture that the exact expression was exp(Ad), but he didn't use the exponential in his derivation. I may try using 1 + Ad instead of the exponential in my example, and see if the results are more reasonable.
It's possible that the gravitational time dilation equation has only been experimentally confirmed for very small Ad. Maybe it's not valid for large Ad.
Instantaneous Velocity Changes in the Equivalence Principle Version of the Gravitational Time Dilation Equation - Revised Model (the LGTD Model)
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I repeated my previous analysis of the instantaneous increase in the home person's (her) age (according to the accelerating person, AO, him), according to the Equivalence Principle Version of the Gravitational Time Dilation Equation, (the "EPVGTD" equation), and replaced it with the new equation, which I'll call the "Linearized Gravitational Time Dilation Equation", (the "LGTD" equation). I simply replace the exponential exp(A d) with the quantity (1 + A d). (This is the same approximation that Einstein used in his 1907 paper). In what follows below, I'll repeat each affected calculation that I made in my last post, and show the revised calculation.
[...]
[Previous]:
The "EPVGTD" equation says that the acceleration A will cause the HF to age faster than the AO by the factor exp(A d), where d is the constant separation between the AO and the HF.
[Revised]:
The "LGTD" equation says that the acceleration A will cause the HF to age faster than the AO by the factor (1 + A d), where d is the constant separation between the AO and the HF.
(Both of the above are for the case where the AO accelerates TOWARD the unaccelerated person (her).)
[...]
[Previous]:
The change in the age of the HF, caused by an acceleration "A" that lasts "tau" seconds is just
tau [exp(d)] sup A,
because [exp(d)] sup A is the constant rate at which the HF is ageing, during the acceleration, and tau is how long that rate lasts.
[Revised]:
The change in the age of the HF, caused by an acceleration "A" that lasts "tau" seconds is just
tau (1 + A d),
because (1 + A d) is the constant rate at which the HF is ageing, during the acceleration, and tau is how long that rate lasts.
[Previous]:
But we earlier found that A = theta / tau, so we get
tau [exp(d)] sup {theta / tau}
[Revised]:
But we earlier found that A = theta / tau, so we get
tau (1 + [ ( theta d ) / tau ] = tau + (theta d)
[...]
It is still true that d = 7.52 lightseconds and theta = 1.317.
Therefore the revised result is that the change in HF's age during the acceleration is equal to
tau + ( theta d ) = tau + (1.317)(7.52) = tau + 9.904.
So, in the revised model, as tau approaches zero (to give an instantaneous velocity change), the change in the HF's age during the speed change approaches 9.904 seconds from above. So the HF's age increased by a finite amount, unlike the infinite increase that the EPVGTD equation gave.
Before the instantaneous velocity change, the AO, the HF, and the home twin (she) were all the same age. She and the HF were co-located. So after the instantaneous speed change, the AO hasn't aged at all, but the HF is 9.904 seconds older than he was before the speed change, according to the AO. And since she and the HF have been colocated during the instantaneous speed change, they couldn't have ever differed in age during the speed change ... it would be absurd for either of them to see the other have an age different from their own age at any instant. So after the instantaneous speed change, the AO must conclude that she and the HF both instantaneously got 9.904 seconds older than they were immediately before the speed change.
By comparison, the CMIF simultaneity method says that the AO will conclude that her age instantaneously increases by 6.51 seconds, so the LGTD and CMIF don't agree.
