Instantaneous Velocity Changes in the Equivalence Principle Version of the Gravitational Time Dilation Equation
When using the CMIF simultaneity method, the analysis is GREATLY simplified by using instantaneous velocity changes, rather than finite accelerations that last for a finite amount of time. So I decided to try using instantaneous velocity changes in the Equivalence-Principle Version of the Gravitational Time Dilation equation (the "EPVGTD" equation). The result (assuming I haven't made a mistake somewhere) is unexpected and disturbing. My analysis found that the age change of the HF, produced by an instantaneous velocity change by the AO and the HF, from zero to 0.866 lightseconds/second (ls/s), directed toward the home twin (her), is INFINITE!
I'll describe my analysis, and perhaps someone can find an error somewhere.
Before the instantaneous velocity change, the AO (he), HF, and the home twin (she) are all mutually stationary. She and the HF are initially co-located, and the AO (he) is "d" lightseconds away from her and the HF.
I start by considering a constant acceleration "A" ls/s/s that lasts for a very short but finite time of "tau" seconds. That acceleration over tau seconds causes the rapidity, theta, (which starts at zero) to increase to
theta = A tau ls/s,
and so we get the following relationship:
A = theta / tau.
We will need the above relationship shortly.
(Rapidity has a one-to-one relationship to velocity. Velocity of any object that has mass can never be equal to or greater than the velocity of light in magnitude, but rapidity can vary from -infinity to +infinity.)
We want the velocity, beta, to be 0.866 ls/s after the acceleration. Rapidity, theta, is related to velocity, beta, by the equation
theta = arctanh (beta) = (1/2) ln [ (1 + beta) / (1 - beta) ].
("arctanh" just means the inverse of the hyperbolic tangent function.)
So velocity = 0.866 corresponds to a rapidity of about 1.317 ls/s.
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.
Note that the argument in the exponential exp(A d) can be separated like this:
exp(A d) = [exp(d)] sup A,
where "sup A" means "raise the quantity exp(d) to the power "A" ". The rationale for doing that is because the quantity exp(d) won't change as we make the acceleration greater and greater, and the duration of the acceleration shorter and shorter. That will make the production of the table below easier.
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.
But we earlier found that A = theta / tau, so we get
tau [exp(d)] sup {theta / tau}
for the change in the age of the HF due to the short acceleration. So we have an expression for the change in the age of the HF that is a function of only the single variable tau ... all other quantities in the equation (d and theta) are fixed. We can now use that equation to create a table that shows the change in the age of the HF, as a function of the duration of the acceleration (while keeping the area under the acceleration curve constant).
In order to make the table as easy to produce as possible, I chose the arbitrary value of the distance "d" to be such that
exp(d theta) = 20000.
Therefore we need
ln[ exp (d theta) ] = d theta = ln (20000) = 9.903,
and since theta = 1.317, d = 7.52 lightseconds.
If we were creating this table for the CMIF simultaneity method, we would find that as the duration of the acceleration decreases (with a corresponding increase in the magnitude of the acceleration, so that the product remains the same), the amount of ageing by the HF approaches a finite limit. I.e., in CMIF, eventually it makes essentially no difference in the age of the HF when we halve the duration of the acceleration, and make the acceleration twice as great.
But here is what I got for the EPVGTD simultaneity method:
(in the table, "10sup4" means "10 raised to the 4th power".)
tau | (tau) (2000)sup(1/tau)
____________________________
1.0 | 2x10sup4 = 20000
0.5 | 2x10sup8
0.4 | 2.26x10sup10
0.3 | 6.3x10sup13
0.2 | 0.64x10sup21
0.1 | 1.02x10sup42
0.01 | 1.27x10sup428
0.001 | ? (My calculator overflowed at 10sup500)
Clearly, for the EPVGTD simultaneity method, the HF's age goes to infinity as the acceleration interval goes to zero. That seems like an absurd answer to me. And it is radically different from what happens with CMIF simultaneity, where the HF's age quickly approaches a finite limit as tau goes to zero.
