The math that destroys space and time

(Photo by Alan Cleaver)

(Photo by Alan Cleaver)

Professional astronomers use math routinely. Their journals brim with equations that can resemble the markings on Inca tablets. This holds no attraction for most folks, who often go far beyond mere arithmetical apathy: They hate it. Perhaps it reminds them of school.

Dan Kelson, the Carnegie Institution for Science researcher whose team discovered the farthest-ever galaxy, told me that, “As an astronomer, math is all that I do. All the time. All of it. Not 37 percent, not 83 percent, not even 99.99 percent. It’s 100 percent.”

No matter. My own pathological need to be loved makes me leave out even simple favorite equations like the amazing Lorentz transformation, which shows how time slows as your speed increases. But let’s tackle it just once, right now, and observe what happens. The result is actually astonishing.

Ready? It’s: t√1-v2/c2 – and doesn’t that seem impossibly arcane? Yet this short equation reveals exactly how your time shrinks when you travel at a particular speed.  Einstein loved this formula so much that he adopted it like a kitten.

Let’s give it a try: First we see a “T,” which simply means time. This denotes whatever time period passed for the folks who stayed home. Let’s make it easy and say it’s a single year. So that “T” is 1.

This gets multiplied by the meat-and-potatoes of Hendrik Lorentz’s brilliant equation, which is √1-v²/c². V is the traveler’s velocity, C is the speed of light; and it all sits under the “square root” symbol, the “√” found on even one-dollar calculators. We’re now ready to see why time freezes at light-speed.

You can insert any velocity you like, as “V.” But first let’s see what would happen if you zoomed at light-speed itself. In that case V and C are the same. So v²/c² becomes c²/c², and dividing anything by itself results in “one.” The formula tells us to subtract this (one) from 1, which yields a zero. And that’s all that remains beneath the square root sign. The square root of zero is zero, and now we’re done. Conclusion: At light-speed, zero time elapses. No time at all passes. Time is frozen. Totally weird.

Try some other speed. Say you go twice as fast as a rifle bullet, or one mile a second. So insert “1” for V. Go through the steps and you may again be amazed. When you travel at that speed, the equation reveals that your time passage is one year when your stay-at-home friends experience one year – meaning, nothing changes.

Merely going twice a bullet’s speed is too slow to alter the passage of time. By contrast, time grinds to a total halt when you move at the speed of light. Somewhere between those two extremes are various amounts of time slowdown.

The beauty of this simple formula, which Lorentz created in 1904, is that you can insert any speed you like, as V, and it quickly reveals how time passes. This very same formula also shows how length or distance shrinks as you move faster. Just substitute D (for distance) instead of T for time: same results. It means that the universe has no fixed size. It alters its dimensions depending on your local circumstances. Travel at 99.9999999 percent of light-speed and the equation tells you that the universe is now 22,360 times smaller.

If you could go at light-speed, to see what a photon experiences, there’d be no distance at all between you and the farthest edges of the cosmos. You’d find yourself everywhere at once.

It’s all undeniably juicy. And it all derives from that single short equation – which makes us wonder whether math is “built into” the universe. Or rather, do our minds impose a system of order that lets us make sense of things? There is no clear-cut answer. What do you think?


Want to know more? To read Bob’s previous “Night Sky” columns, visit our Almanac Weekly website at

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