|MadSci Network: Physics|
Dear Dan, I love this question! The idea of the string really gives this old paradox a new twist, so to speak. To answer this, we need to be very precise about our terms. What do we mean by "the age" of something, or "how old" something is. I hope that you will agree that what we mean is how much time passes according to that particular object. For example, the age of the man in the ship is not determined by how much time passes for the man on Earth, but rather how much time passes as measured by the man in the ship. Every object must measure its own time, that is, its own age. This quantity is called the objectfs proper time. To recap, the age of an object is the amount of time that passes as measured by that object, in other words, according to a clock that is being carried by that object. You suggest using a string that "couldn't age". According to our definition, that would mean a string that doesn't move through time. Let's use your second suggestion instead, let's assume the string is indestructible and although it does age, it doesn't show the effects of aging. And may I make one more suggestion, just one letter? Rather than a string, how about a spring? Let's assume that one end of an indestructible, infinitely stretchable spring is attached to Earth, and the other end on the spaceship. That way, as the ship flies away from Earth, the spring stretches out, and as the ship comes back, the spring compresses. I made a diagram for this. Please refer to the site http://www.gpc.edu/~jgui nn/MadSci/SpringShip.pdf . The rocket starts out on the Earth, with the spring compressed, and several points on the spring shown at A, B, and C. The rocket then flies off for a year, as measured in the rocket, at a high velocity; letfs say 99% the speed of light. Point C ends up at point C'. The point B moves to point B' at a speed of 49.5% the speed of light and ends half the distance from Earth as point C' does. Point A stays in place, so A' is the same as A. On the way back, we have C' going to C'' at 99% the speed of light, B' goes to B'' at 49.5% the speed of light, and A' goes to A' with no motion. What do we find now for the "ages"? They can be determined using the standard time dilation equation for special relativity t = to / sqrt(1-v^2/c^2). Point C in the rocket has aged 2 years, point B has aged 2.30 years, and Point A has aged 14.2 years. Since different points on the spring traveled at different speeds, they have aged different amounts. In the end, you have a spring which has aged a different amount for every point on it! What does that mean? Remember that any physical object is made of atoms, and each atom seems to have traveled through a different amount of time than the ones next to it. The bottom line is that we don't need to consider quantum mechanics to get an idea about the effects of special relativity. Thinking about things this way might help you understand special relativistic length contraction, too. Consider a moving meter stick, for example. While you think of the front and back of the stick as being at the same time in your frame, in the stick's frame, the front end that you see is at a slightly earlier time than the back end you see. This means that the front end has not moved ahead quite as much as the back end, and so you perceive the stick as being shorter than the stick is in its own frame. See? Well, Dan, I hope that answers your question. Please let us know if you would like any more information. Thank you for your interest. Sincerely, Jim Guinn Georgia Perimeter College
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