( related keywords: simultaneity, time dilation, twin paradox, Lorentz equations )

Suppose you could take a space voyage, in a space ship capable of accelerating at one g for several years at a time. One g of acceleration would allow you to travel comfortably...you would experience the same sensation of weight that you feel at home in the earth's gravity. One g of acceleration, lasting for several years, would also allow you to reach extremely high speeds relative to the earth (almost as fast as the speed of light), and to travel extremely far away from the earth (tens or hundreds or thousands of light-years away).

Imagine that you do take such a trip. During your trip, you will almost certainly wonder how people you left behind are doing, back on earth. At any given instant of your life during the trip, you won't be able to find out what they are currently doing, because radio signals will take many years to reach you from the earth. But you can determine how old they currently are, and thus you will at least be able to imagine what their current existence might be like.

Suppose that you leave earth when you are 10 years old, and your baby sister is one year old. Unless stated otherwise, all quantities given in the following example will be as measured by you, except for two quantities which will be as measured by her (at some specified instant in your life). Those two quantities are the distance between you and your sister, and your speed relative to her.

You accelerate away from her at 1 g for 2 years of your time. At the end of that period of acceleration, you are 12 years old, your sister is 1.94 years old, your speed is 0.968 lightyears/year (ly/y), and your distance from her is 2.9 lightyears. You then coast for 9 years of your time, after which you are 21, she is 4.2, and your distance apart is 37.6 ly. At this instant of your life, your sister is a four-year-old, and so she is probably spending a lot of time in pre-school.

Next, you accelerate at -1 g for 3 years. I.e., you point your spacecraft at your sister, and accelerate at 1 g in her direction. Of course, during the first part of that acceleration, you will still be moving away from her, but your relative speed will be decreasing. Later in the acceleration, you will be moving toward her, and your relative speed will be increasing. At the end of that acceleration, you are 24, she is 76.5, your speed is -0.774 ly/y (you are moving back toward your sister), and your distance apart is about 40 ly. During this interval of acceleration, your sister's age has smoothly but rapidly increased by over 72 years while you aged only 3 years. Your sister is now a senior citizen, and she has perhaps recently retired from a long career as a teacher at her former pre-school. Of course, you can't know if your sister is currently alive (unless you have received a radio message that indicates that she died long ago), but you can determine how old she would currently be if she is still alive.

Next, you accelerate at +1 g for 2 years (in the direction away from your sister). At the end of that acceleration, you are 26, she is 17, your speed is 0.774 ly/y, and you are again 40 ly apart. During this interval of acceleration, your sister's age has smoothly but rapidly decreased by over 59 years while you aged only 2 years. While you got 2 years older, your sister got 59.5 years younger. Your sister is now a teenager, and she is probably a junior in high school.

Next, you accelerate at -1 g for 2 years. After the acceleration, you are 28, and she is 81.2. Your sister isn't a teenager any more...she's now a senior citizen again. You then accelerate at +1 g for 2 years. After the acceleration, you are 30, and she is 21.8. Your sister isn't a senior citizen any more. She is now a young adult, and is perhaps about to graduate from college. Perhaps she is hoping to find a job teaching at a pre-school somewhere.

Now you are ready to go home, so you accelerate at -1 g for 3 years. At the end of the acceleration, you are 33, she is 94.1, your speed is -0.968 ly/y, and your distance apart is 37.6 ly. You then coast for 9 years, after which you are 42, she is 96.3, and your distance apart is 2.9 ly. Finally, you accelerate at +1 g for 2 years, which leaves you at zero speed and zero distance from your sister. You are then 44, and she is 97.2.

The above scenario is completely realistic. The only reason that such a trip can't happen today is that the technology required to accelerate you and your spaceship at 1 g for several years doesn't yet exist.

It should be noted that your sister does not agree with your conclusions about the bizarre correspondence between your ages. She certainly doesn't perceive that the progression of her own life is in any way affected by your accelerations. But the differing conclusions are equally valid: neither of you is "more correct" than the other, and neither of you can adopt the other's conclusions without contradicting your own measurements.

The numerical results quoted in the above scenario were obtained from a computer program named cado, for "current age of a distant object". If you would like to play around with other similar scenarios, I will send you a free copy of the program, provided that you agree not to use the program (or allow the program to be used) for any commercial purposes. Send an email request to mlfasf@comcast.net, and specify linux-PC or microsoft-PC. (I don't currently have a version compiled for the Mac). State in your email that you will not allow the program to be used for commercial purposes.

The above numerical results, and the cado program, are based on results I derived in a paper published in the December, 1999, issue of Physics Essays (Vol 12, no. 4, p. 629). (That issue was not actually published until early February, 2001). Here is the abstract of that paper:

The Lorentz equations of special relativity unambiguously specify the "current age of a distant object" (abbreviated as the "CADO"), according to an inertial observer. This paper demonstrates that:

1. For an accelerating observer, consistency with special relativity requires that the CADO be defined in one and only one way.
2. There exists an equation for the CADO, derivable from the Lorentz equations, which explicitly shows how sudden changes in the relative velocity of the observer affect the CADO. This equation is well-suited for handling arbitrary acceleration profiles, and piecewise-constant accelerations.
3. The CADO thus defined behaves in a very bizarre manner for an accelerating observer. Specifically, the CADO can decrease as the age of the observer increases, even when the observer's acceleration is limited to one g. The CADO is also, to a large extent, controllable by the observer.
4. According to an accelerating observer, for a one g acceleration occurring when the separation is sufficiently great, the object's maximum (in magnitude) rate of ageing is greater than the accelerating observer's rate of ageing by a factor approximately equal to their separation, as measured in the object's frame, in lightyears. If the observer is accelerating toward the object, the object will be getting older at that rate. If the observer is accelerating away from the object, the object will be getting younger at that rate.
5. If an observer undergoes a constant acceleration forever, the object's age will approach a finite limit, according to the observer. If their initial separation has a certain critical value, the object's age will never change at all, according to the observer.
6. An inertial observer can compute the CADO from first principles, using only his own (appropriately synchronized) clocks and measuring sticks.
7. Under a specific definition of "meaningfulness" and "reality", the bizarre behavior of the CADO, for an accelerating observer, must be regarded as being fully meaningful and real.

Comments, criticism, observations, etc., are welcome. Contact:

Michael L. Fontenot or Ann Mallan Fontenot
1755 Gillaspie
Boulder, Colo 80305
USA
303-499-4493
mlfasf@comcast.net

last revised: December 7, 2003