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This September will be my first since 2003 in which I don’t ritually raise a piece of chalk and write my name in large block letters on an antiquated blackboard. Due to work constraints and the way that the days fall on the calendar, this will also be my first year since 2003 that I don’t perform the music for the Jewish High Holy Days.

I’ve always been fascinated with calendars, clocks, and the passing of time. And of course, that leads to questions.

For instance, when does the year start? As a teacher, my year started in September. Late August was a sad time, during which I tried to make the most of summer’s end. As a member of the “real world”, the year doesn’t so much have a start and an end. The trailing days of August don’t just mean the end of my summer freedom. In a way it’s reassuring to know that I no longer have to mourn Labor Day, but at the same time, vacations are less common and more cherished.

Relativity says that the universe has no concept of “absolute” time. Just because it’s 5:00 here doesn’t mean it’s 5:00 on the surface of the black hole at the center of our galaxy. But considering that I don’t plan to ever travel to said black hole, my years on this Earth are fixed in number. The time I have to live my life is the same regardless of how I live.

“Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. That’s relativity.” — Albert Einstein

But is the time really fixed when there are hot stoves and pretty girls wandering around?

Space.com reported yesterday that NASA astronomers have found the smallest black hole yet. The little guy has a mass of only 3.8 times the mass of the Sun and is only about 15 miles in diameter.

Gravity is the force that causes masses to be attracted to one another. On Earth, our gravity causes objects to fall towards the center of our planet at an acceleration of 9.8 m/s². That is, for every second an object falls, its downward speed increases by 9.8 meters per second. This means that if I throw something up in the air at 9.8 m/s, it will take one second before its velocity is zero again. After two seconds, the object is traveling downwards at 9.8 m/s.

There are two ways to increase the rate at which objects fall on a planet’s surface. They’re both related to the geometry of the planet.

The first approach is to increase the mass of the planet. The more mass, the greater the pull. Makes sense. The other approach is to keep the mass of the planet the same, but compress it into a smaller ball.

The “shell theorem” states that you can treat a uniformly-distributed sphere of mass as if all of the mass were located at a single point at the sphere’s center. Since we’re on the surface of the Earth, we’re exactly one Earth radius away from that single point where all the mass would be compressed:

The force of gravity experienced by both of the little scientists in this picture would be the same. However, one cannot stand on nothingness, and so this isn’t exactly a tenable situation for our little guy.

If we were to keep the mass the same but shorten the distance between us and the center of the Earth, the force would be greater since we’d be closer.

The only way to keep the mass the same but compress it all into a smaller ball is to increase the average density.

A black hole is formed when a large star dies. The star has burned through most of its fuel, and its furnace isn’t producing enough energy to maintain its own structure. Gravity causes the star to collapse in on itself until all the mass is concentrated in a very small volume.

Black holes have such an intense gravitational pull that nothing can ever escape from their clutches. Their “escape velocity”, the speed required to completely escape a planet’s or star’s gravity, is greater than the speed of light. Even though light has no mass, it is still (effectively) subjected to gravitational forces. At least that’s what Professor Einstein has had us believing since he published his theory of general relativity in 1916.

Just for fun, let’s figure out how dense this black hole is. The mass of the Sun is approximately 2.0 × 10³⁰ kilograms. That’s 2,000,000 trillion trillion kilograms for those of you that don’t like scientific notation. The mass of the black hole is 3.8 times this, or 7,600,000 trillion trillion kilograms.

Compress all of this into a sphere 15 miles across (so the radius is 7.5 miles). To find the volume of a sphere, multiply 4/3 times π times the radius cubed. This gives us a volume of 1,770 cubic miles. Or, if you like bigger numbers and smaller units, 7.3 trillion cubic meters. Then divide the mass by the volume to find the density.

This newly-discovered black hole has a density of one million trillion times that of water. For comparison, lead, the densest commonly-occurring material on Earth, has a density of about eleven times that of water.

A spoonful of this black hole would have a mass of fifteen billion tons! If my mass were compressed into a cube of this density, the sphere would be approximately six microns across. This is about one-twentieth the width of a human hair!

Physicists can usually learn a great deal from these extreme conditions. But, since light can’t escape from a black hole, no information can escape either. We have no way of observing the interior of a black hole. We only know of black holes’ existence because of their influence on the stars and nebulæ around them.

What we have learned is a new lower-limit for the size of star that will become a black hole at its death. Astronomers can now look for known dying stars of about this size and perhaps learn something about how stars die and how black holes are formed.

Such a wonderful game concept, I’m upset I didn’t think of it myself: Reference Games‘ Relativistic Asteroids!

Like classic Asteroids, the player controls a spaceship in a deadly asteroid field. The player has to destroy all of the asteroids by breaking them into smaller pieces; when the pieces are small enough, the asteroids disappear. Just for fun, add in the occasional passing alien spaceship for a real challenge. The alien ships shoot back, so the player is forced to dodge their shots.

Asteroids is one of those games that should seem really easy, but takes a while to master. The difficulty isn’t obvious at first. The asteroids move slowly and the player can stand in place while taking them out.

The difficulty comes at the first close call. The novice player panics and causes the ship to accelerate away from the inbound asteroid. The game’s physics is purely Newtonian, and the player’s ship continues along a straight line in accordance with Newton’s first law. The only way to slow the ship is to fire the engine in the opposite direction of the ship’s travel — and that’s easier said than done when dodging giant space-rocks. This is the point in the game when my ship usually turns into a pile of two-dimensional vector-drawn space-rubble.

In Relativistic Asteroids, the player experiences the game as if they were able to accelerate from rest to a large fraction of the speed of light.

The default reference frame is that of an arbitrary outside observer. But in game, be sure to try the other available reference frame by pressing the ‘F’ key. The view will center on the player’s ship. Accelerating the player’s ship causes the asteroids to appear to change velocity, since the view on the screen is following the ship. Flip back and forth a few times and you’ll understand how there really is no preferred frame of reference — all that matters is the relative velocities between the objects.

When the ship starts to move too fast, the player will notice relativistic length contraction: the ship (or asteroids, depending on reference frame) will diminish in length!

More on special relativity in a future post. For now, I need to go work on getting a high score.

(Thanks to Uncertain Principles.)