In the far off reaches of space lies the planet Catscreth, home to a technologically advanced race of felines. It’s a peaceful planet where cats have developed machines and robotic kittens to provide for all their needs. Freed from the need to take care of themselves, most spend their days in quiet repose (napping) under sun-filled skies (much like many house cats here on Earth).
And what of those cats who forego the quiet life? They seek to recapture the adventurous spirit of their ancestors through space racing. Nothing excites them more than piloting their one-cat spaceships against one another. Over time, even those cats who prefer that their paws remain firmly on the ground have become fans of these competitions. Races are transmitted back to Catscreth for all to see.
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Now a race in space isn’t like a car race here on Earth. For one thing, space is far bigger (look around the room you’re in and multiply it by a few billion billion) so it’s impossible to see the entire race course. Instead, each spaceship is equipped with a beacon which sends out a signal every 10 minutes (to make the race more suspenseful and to allow for cat naps) to show their position.
Sushline and Daislee, the two most famous racers, have never lost a race and it’s long been the desire of every cat on Catscreth to see them compete against one another. However, Daislee had always refused due to the rumors of Sushline cheating. Of course, no one has ever proven anything, but Daislee didn’t want to risk losing unless she could be certain of a fair race.
It took some time, but Daislee finally figured out a way. She proposed that the two of them race around a black hole, something that had never been done before, and Sushline quickly agreed. Catscreth’s population had never known such excitement and many even skipped their daily naps to hear the latest news and argue over who they thought would win. The anticipation was so great that large numbers of cats who’d never been in space before boarded large space liners to be present at the race’s starting/finish line.
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On the day of the race, the two competitors, the race command ship, and hundreds of massive space liners converged at a point 70 million kilometers (yes it’s metric– they are an advanced race of cats after all) away from a massive black hole that’s a million times the mass of our sun. Even at this distance it’s an awesome sight as matter swirls around emitting bright flashes as it falls into the black hole. The race rules are simple: both racers have to remain at least 70 million kilometers from the black hole; and the first one to complete an orbit around it will win the race.
Both spaceships line up at the starting point and when the signal is given, roar out of sight. (Well, you don’t hear anything since they’re in space, but roar they did.) Everyone on Catscreth and in the space liners turn their attention to the giant view screens showing an outline of the course and await the first beacon signal. Cheers and moans go through the crowds when red (Sushline) and white (Daislee) dots appear with the red dot in front.
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Over the next three hours, the beacon signals work their way around the black hole with the red dot out in front. At the last signal before the finish line, Sushline’s lead look insurmountable. However, as time passes and her ship still hasn’t reached the finish line, everyone in the audience begins looking around anxiously. A worldwide collective sigh goes up as a ship finally comes into sight, but it quickly turns into cries of surprise as Daislee’s ship crosses the finish line.
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Rescue craft are immediately sent out to sweep the race course for Sushline’s ship. However, only twenty minutes later all are relieved when Sushline’s ship comes into sight. Passing the finish line, she transmits a cry of victory, but it sticks in her throat at the sight of Daislee’s ship already docked with the race command ship.
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Poor Sushline’s day didn’t get any better when the rescue ships returned. While searching for her, they’d found a number of beacons set up along the race course set to send out the same signal as Sushline’s ship. It didn’t take long for the full story to come out: the beacons had been set up before the race by Sushline so she could take a route closer to the black hole to ensure her victory.
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The plan seemed perfect. So why didn’t Sushline win? When asked about this, Daislee merely smiled and replied, “well I guess it was just a matter of time.” Seems to me Daislee knows more than she’s letting on. After all, she was the one who suggested racing around a black hole so maybe that has something to do with it.
This narrative has been purloined without the author's approval. Report any appearances on Amazon.
Now what exactly is a black hole? It’s a collapsed star which has so much mass that not even light can escape from it. Larger black holes can have masses over a million times greater that our sun. They were first predicted by Albert Einstein's General Theory of Relativity. Shortly after the General Theory of Relativity was published, Karl Schwarzschild used it to develop a mathematical representation of a black hole (also known as the Schwarzschild metric).
That’s all well and good, but how is any of that relevant to our story? Ah, but there is one more thing a black hole can do: affect the passage of time. Actually, according to the Theory of Relativity any gravity field (even here on Earth) affects how quickly time passes. This effect is known as Gravitational Time Dilationand it predicts that the further away you are from the source of gravity, the faster time passes.
This has even been measured here on Earth. Super accurate atomic clocks at sea level and on the top of mountains (where gravity is slightly less) show the exact time difference (an extremely small value) predicted by general relativity. For a black hole, where the gravitational forces are so much higher, the difference in the speed at which time passes becomes far more pronounced.
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The Schwarzschild metric mentioned above can be used to derive an equation specifying how time changes as a function of how far someone is from a gravitational source. I know, but it's just one little equation. Assuming you have a massive, perfectly spherical object that isn’t rotating (this makes the math a whole lot easier) the equation for Gravitational Time Dilation is:
[https://img1.wsimg.com/isteam/ip/a7fca0a8-d609-42a7-9836-1679fed1b384/eq_1-0003.jpg/:/rs=w:1280]
This equation tells us that for someone near the black hole time will move more slowly. Let’s say we have sleepy two cats; CB who is near the black hole and FA who’s far away. They want to nap for an hour and FA sets his alarm clock (which both will hear when it goes off) for one hour. When the alarm goes off; FA wakes up refreshed from his hour of sleep while CB is tired and upset that she only got a half-hour of sleep. So compared to FA, CB lost a half-hour of sleep by being near the black hole.
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Okay, now let’s look at what happened with Sushline. Instead of making a perfect circle around the black hole as Daislee did, she took what we call an elliptical (think squashed circle) path that took her much closer to it. This took Sushline around the black hole in a little more than an hour while it would take Daislee nearly three.
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However, as Sushline moved closer to the black hole (and she got within 4 million kilometers of it), the slower time passed for her. So even though she took the shorter rote, Gravitational Time Dilation caused her to lose the race.
So how much time did Sushline lose? We can figure that out by using her path around the black hole (distance from the black hole as a function of time) to calculate the time dilation at each point in her journey and adding them all together. Now I could list all the numbers used in the calculation, but for sanity’s sake let’s just skip to the answer. Compared to Daislee, Sushline lost just a little more than a half hour.
Something’s wrong here. Sushline’s route was two hours faster than Daislee’s so how could losing a half hour prevent her from winning the race? This brings up another interesting fact about time dilation: you have no idea it’s happening.
The Theory of Relativity states that all observations are relative (no, not related). What that means is that from Sushline’s perspective, time was passing normally. Her on-board clock showed that it took her an hour to orbit the black hole. Now being clever, she wasn’t about to cross the finish line two hours early, so she stopped far enough away from the finish line to not be seen and waited for an hour and forty-five minutes.
Without time dilation this would have still provided an ample margin of victory, but that lost half-hour did her in. Sushline had no idea what had gone wrong until after the race when her ship was examined and her clock was found to be a half-hour slow.
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On the bright side, this means that Sushline’s now a half hour younger than she was before (although that’s little consolation to her at the moment). Yes, time travel is indeed possible (if you have a spare black hole lying around), but it’s a one way trip.
If you’ve read or seen science fiction stories about black holes, you’ve probably heard the term "event horizon." In simple terms this represents the boundary of the black hole from which nothing can ever escape. However, let’s consider the Gravitation Time Dilation equation to find out what the event horizon really is. Yes, I realize this will involve math, but it's cool math. We're talking black holes here.
We start with the term inside the radical (that funny thing with the check mark connected to a straight line).
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If this term were to equal zero then time for someone near the black hole would stop. The distance from the black hole at which this condition occurs is the definition of the event horizon. Setting the equation above equal to zero, we can solve for the location of the event horizon (also known as the Schwarzschild radius).
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Using the numbers from above we get a Schwarzschild radius of 2.949 million kilometers for the black hole from our story. Wow, Sushline sure got close (she certainly doesn’t lack guts). Glad her ship had energy shields to protect it against the large amount of radiation present that close to a black hole.
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With her victory, Daislee cemented her position as Catscreth’s greatest space racer. Her decision to race around a black hole certainly shows that she had an idea of what Sushline might do. It all goes to show that physics knowledge can be useful.
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For some time after the race, Sushline was shunned by the racing community and no one would even consider racing her. However, the cats of Catscreth are a forgiving bunch and eventually she was welcomed back to the racing world. She’d learned her lesson and never again tried to cheat. In fact, she even had another race against Daislee, but that’s a story for another time.