I didn't put the ACD for the finite acceleration case on the webpage. I have little interest in the finite acceleration case, because it's so little different from the instantaneous velocity change case. That's one of the nice features of my method, compared to the CMIF method ... the instantaneous velocity change case is all you need in practice.

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my equation doesn't apply to the finite acceleration case at all. There IS no equation in the finite acceleration case ... it's just a numerical integration.

The two cases use completely different methods. Numeric integration sounds like you're using CMIF for continuous acceleration and F-method for the instant case. That's hardly 'so little different'. The former can have Alice's age going backwards for instance, so the curves are not similar. The latter is only useful for thought experiments, not anything practical where infinite power is not plausible.

Consider Bob accelerating away at 1G indefinitely. Alice is going to age backwards in his frame. We see this now with distant objects that are continuously accelerating away from us due to dark energy. These distant objects have negative ages in our inertial frame, which is why inertial frames cannot be used to describe distant objects. Instead, cosmologists use something very similar to Minguzzi's method (using a reference event rather than a reference frame). They don't call it Minguzzi's method because Minguuzzi isn't consistent about the method (use it all cases, not just from one PoV), plus Minguzzi selects an event that doesn't foliate all spacetime.

If the two velocities are separated by a finite time, you just handle the two changes independently. Suppose v_a is the velocity change when the twins are born, v_b is the initial velocity change at some distance later, and v_c is the velocity change at a still greater distance. Then you get the first slope from the equation using v_1 = v_a and v_2 = v_b, and the second slope using v_1 = v_b and v_2 = v_c.

OK, I didn't see that in the document, maybe only for lack of finding it, so thanks.

We have triplets Alice, Bob, and Charlie. Bob and Charlie take the ship out 10 years to planet Zog to retrieve the famed jewel of Zog. Zog is moving toward's Earth at -0.8c, but rather than waiting for it to pass nearby and let somebody else get the jewel, they're going to go out to it and pick it up first.

The jewel would be damaged by the infinite power of the ship turning around, so the plan is to jettison Charlie near the planet where he matches its speed by the amazing ball pit they've got. He spends 20 minutes retrieving the jewel, and then leaps up as the returning ship scoops him up with a butterfly net, boosting his return speed by a paltry 0.066c. Bob stays on the ship and turns around after a minute or so. They are once again bound for home, but how old is Alice?

They agree she's 5 years aged at first (a few minutes discrepancy maybe), but disagree as to her new rate of aging.

S = (1 / gamma_2) + gamma_2 * (1 - v_2) * (v_1 - v_2)

Bob computes S = .5 + 2 * 1.866 * 1.732 = 6.964

Charlie computes S = .5 + 2 * 1.866 * .066 = 0.746

Hmm... OK, that's two problems. One, they're with each other but in complete disagreement about the rate of her aging. Second problem is that Charlie has her aging too slowly, less than 1 in fact. She's not going to be her actual age in his frame until long after Charlie has passed Earth by.

Did I do that wrong?

I asked Steve Gull, many years ago, if he realized that his method is non-causal. His reply was "Yes, it is blatantly non-causal". I've described his ACD in my webpage. The slope changes from 1/gamma_1 to a slope greater than 1 well before the turnaround is reached. Since we can't know for sure that the traveler actually DOES change his velocity, the slope shouldn't change before velocity actually changes, for a causal method.

OK, you're using non-causal to describe a situation where a calculation is based on actions that have yet to happen, where my take was an effect outside the future light cone of its cause.

Minguzzi's method is also non-causal, because his ACD has a slope of 1.0 on the entire outbound leg. Since we can't know if the traveler will actually change his velocity, the slope should be 1/gamma on the outbound leg, for a causal method. So Minguzzi's method is non-causal.

This assessment doesn't hold water. His method is never based on future intended actions. It is always computed for here and now. I must defend this because cosmologists use an objective variant of this method, the comoving reference frame, which is often asserted as the absolute frame for those absolutists among you.