The Trouble with Time
I’ve started working on a series of Web pages about Alleghany County and, in researching its history, I suddenly found myself back at the Big Bang somewhere between 10 and 20 billion years ago in Earth time as we know it. And, once again, my trouble with time reared its ugly head.
My problem can be described in reverse by a old physics joke. A mathematician and an engineer are asked how long it will take to reach a beautiful young girl standing across the room if they approach her by half the remaining distance every 1 second. The mathematician quickly and emphatically declares “Never! You will always be half the distance away from her.” He is, of course, correct, but the pragmatic engineer replies, “Ah, but in 15 seconds I will be close enough to her for all practical purposes!”
The mathematician in our little joke is invoking something called Zeno’s Paradox. The opposite -- let’s call it Lineback’s Paradox for now -- is that such an event can never be completed because it can never start. Let me explain.
A tenant of Newtonian physics is that an object at rest stays at rest until acted upon by a force. Force is defined as the mass of the object times the rate at which that force increases the velocity (distance per unit time) of that object. So, let's say we have a golf ball laying perfectly still on the ground (in accordance with the Rules of Golf) with a velocity of zero yards per second and an acceleration of zero yards per second per second. It ain’t moving, folks. Then, the golfer does his waggle, lifts the club toward the heavens, and brings it back to earth to create the golf ball version of the Big Bang. The next thing you know, that ball is traveling (hopefully straight and through the air) down the fairway with what we have been taught to perceive as great speed.
But, how did it ever manage to get started on its flight? From the force of the club acting on the mass of the ball, you say. Let’s say we are an engineer and buy into that hypothesis. Yet, the velocity of the ball when the club strikes it is zero feet per second, minute, hour, day, week, month, year, decade, century, millennium … you name it … it ain’t moving, folks. But, you say (as Sir Isaac would), the force exerted by the club head on the ball accelerates the ball at some finite value equal to the mass of the ball divided by the force of the club head. So, explain to me how something that has no … zero … velocity can ever change the rate at which it is moving. Zero multiplied by anything and zero divided by anything remains forever zero.
But, you say, it starts out very slowly from zero. How slowly? How far did it travel over what period of time as defined by Newton's Laws? Let’s suppose we use that final remaining increment in Zeno’s Paradox as the distance. And how long does it take reach the end of that last increment of distance? Never, forever, infinity. In Lineback’s Paradox, the first increment of distance is identically the last increment in Zeno’s Paradox. So, starting at zero velocity and zero acceleration, whatever that increment might be in whatever units of measure, it will take us an infinity of time to reach the end of it. And, as we learned in school anything divided by infinity is zero. Accordingly, the velocity of the golf ball must remain zero over all time and can never leave the club head. The ball never begins its flight.
We know from experience that golf balls apparently do violate both Zeno’s and Lineback’s Paradoxes because the ball is observed to change position -- with the golfer’s intervention and perhaps a little luck and skill -- from the tee box to the bottom of the cup. But, if the ball can never change its speed from zero and can never reach the end of its flight in all time, how is it possible for the golfer to start and finish a hole? Or, to put it another way, how can something end that can never start? The only possible explanation is that time simply does not exist.
But, Sam already knows that. And, now you do!
Next: Calculus, Gravity and The Big Bang
My problem can be described in reverse by a old physics joke. A mathematician and an engineer are asked how long it will take to reach a beautiful young girl standing across the room if they approach her by half the remaining distance every 1 second. The mathematician quickly and emphatically declares “Never! You will always be half the distance away from her.” He is, of course, correct, but the pragmatic engineer replies, “Ah, but in 15 seconds I will be close enough to her for all practical purposes!”
The mathematician in our little joke is invoking something called Zeno’s Paradox. The opposite -- let’s call it Lineback’s Paradox for now -- is that such an event can never be completed because it can never start. Let me explain.
A tenant of Newtonian physics is that an object at rest stays at rest until acted upon by a force. Force is defined as the mass of the object times the rate at which that force increases the velocity (distance per unit time) of that object. So, let's say we have a golf ball laying perfectly still on the ground (in accordance with the Rules of Golf) with a velocity of zero yards per second and an acceleration of zero yards per second per second. It ain’t moving, folks. Then, the golfer does his waggle, lifts the club toward the heavens, and brings it back to earth to create the golf ball version of the Big Bang. The next thing you know, that ball is traveling (hopefully straight and through the air) down the fairway with what we have been taught to perceive as great speed.
But, how did it ever manage to get started on its flight? From the force of the club acting on the mass of the ball, you say. Let’s say we are an engineer and buy into that hypothesis. Yet, the velocity of the ball when the club strikes it is zero feet per second, minute, hour, day, week, month, year, decade, century, millennium … you name it … it ain’t moving, folks. But, you say (as Sir Isaac would), the force exerted by the club head on the ball accelerates the ball at some finite value equal to the mass of the ball divided by the force of the club head. So, explain to me how something that has no … zero … velocity can ever change the rate at which it is moving. Zero multiplied by anything and zero divided by anything remains forever zero.
But, you say, it starts out very slowly from zero. How slowly? How far did it travel over what period of time as defined by Newton's Laws? Let’s suppose we use that final remaining increment in Zeno’s Paradox as the distance. And how long does it take reach the end of that last increment of distance? Never, forever, infinity. In Lineback’s Paradox, the first increment of distance is identically the last increment in Zeno’s Paradox. So, starting at zero velocity and zero acceleration, whatever that increment might be in whatever units of measure, it will take us an infinity of time to reach the end of it. And, as we learned in school anything divided by infinity is zero. Accordingly, the velocity of the golf ball must remain zero over all time and can never leave the club head. The ball never begins its flight.
We know from experience that golf balls apparently do violate both Zeno’s and Lineback’s Paradoxes because the ball is observed to change position -- with the golfer’s intervention and perhaps a little luck and skill -- from the tee box to the bottom of the cup. But, if the ball can never change its speed from zero and can never reach the end of its flight in all time, how is it possible for the golfer to start and finish a hole? Or, to put it another way, how can something end that can never start? The only possible explanation is that time simply does not exist.
But, Sam already knows that. And, now you do!
Next: Calculus, Gravity and The Big Bang
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