ralfcis » January 15th, 2018, 4:03 pm wrote:There's still a problem. I can show how much less Alice ages for a single year if she returns to .6c after that year but I can't show how much less Alice ages for a single year if she remains stopped. In fact the STD shows just before Alice turns 4, Bob is equivalently 4 in Alice years but just after 4, Bob is instantaneously 1 yr older than Alice.
I'm not quite clear what you mean by Bob being equivalently 4 in "Alice years". According to Alice, After 4 years of her time, Bob will be 0.8 years younger than she is before the acceleration and 1 year older after.
The lines of present overlap which means the times are instantaneous present, there is no relativity of simultaneity between Alice and Bob because v=0 and the formula for simultaneity is vx/c. So an instantaneous present exists between them when either x=0 or v=0. Age difference cannot be instantaneous, it makes no physical sense. Maybe if I look at it from the perspective that Bob takes off at .6c to match Alice's speed to create 0 relative velocity. Maybe that will get rid of this anomaly.
The "anomaly" is merely the result of making the non-physical assumption of an instantaneous velocity change. When you do this you are saying that Alice undergoes an infinite acceleration over a 0 time duration. In reality, the acceleration can be as high a value as we want for as short an interval as we want, but it will always have to occur over some non-zero time period.
At the onset of this period, Bill will be 0.8 years younger than Alice, and during the acceleration would age 1.8 years over that short, but not zero time length. As a result, you never get a situation where Alice would determine Bob's age as having two different values.
Anytime we assume an "instantaneous" change in velocity, what we are really doing is just assuming an acceleration period that is so short in duration that is doesn't significantly effect the outcome.
For example, if Alice accelerates at ~2.079e16 m/s
2 for 1 nanosecond of her time, she will cause a 0.6c difference in her velocity with respect to Bob. According to Bob, she will have been under acceleration for ~1.1 nano second to make that same velocity change. But when we are dealing with trips that take years, that 0.1 nanosecond difference is insignificant. But it does mean that according to Alice there is a 1 nanosecond period of transistion between Bob being 0.8 years younger than she is and being 1 year older.