Quantized Spacetime

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3 Zeno's Paradox

We can see the contrast between the two types of spacetime by using quantized spacetime to solve one of Zeno’s famous paradoxes. The one involving a race between Achilles and a tortoise. The contestants start at the same time, but the tortoise begins ahead of Achilles. By the time Achilles covers that lead, the tortoise has moved ahead a small distance. When Achilles covers that distance, the tortoise has moved ahead a still smaller distance . . .  With every advance by Achilles, the tortoise is always slightly ahead. Achilles cannot possibly win.

The paradox involves a conflict between Platonically assuming a continuous, infinitely divisible spacetime and our way of thinking of length in fixed units like inches and feet. The mathematics of an infinite series of numbers offers one resolution, for those happy with the math. But the quantum as a limit to the division of spacetime provides a simpler and faster solution, without recourse to infinity.

It is easier to understand what happens by imagining spacetime quantized on a much larger scale that ours. This we can define by choosing a suitable maximum speed. Suppose we choose 100 fps (feet per second) as fast enough for a Greek stadium, even with racing chariots and jaguars. We would achieve this by defining the shortest length as one thousandth (0.001) of a foot and the shortest time as one hundredth-thousandth (0.00001) of a second. Homer’s swift-footed Achilles could probably speed along at 30 fps. Magnifico, Zeno’s racing tortoise, might be one hundred times slower, at 0.3 fps. Let’s give it a 30 ft lead and allow both contestants a running start.

In the first second, Achilles runs 30 ft to arrive at 30 ft from the start. Magnifico ambles 0.3 ft, to reach 30.3 ft.

Not pausing for breath, Achilles runs 0.3 ft in the next 0.01 second to reach 30.3 ft.  Magnifico ambles 0.003 ft to reach 30.303 ft. Still ahead. Go Magnifico, go!

According to Zeno, in the next 0.0001 second, Achilles would run 0.003 ft to reach 30.303 ft while Magnifico would cover 0.00003 ft to reach 30.30303 ft. Still ahead! This is possible in continuous spacetime and can continue for an infinite number of shrinking advances. Not so in quantized spacetime.

Magnifico’s move of 0.00003 ft is less than 0.001 ft, the shortest possible length. To make the smallest permissible move, Magnifico must use all of 0.0033 second to cover that 0.001 ft, reaching 30.304 feet.

But Achilles continues at his own speed and covers 0.099 ft in that 0.0033 second. He reaches 30.402 ft and wins! Spectators cheer! Achilles gets the laurel crown, Magnifico enjoys a lettuce leaf. Paradox resolved.

Quantized spacetime blocks non-sensical infinities. So let’s look at it closer.

8/28/2020  7:08   3

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