Tour de Force – (part 1)
The Baseline: 1/r^2
Ah yes, Newtonian gravity. The perfect place to start. This is gravity as we know it in our universe (ignoring pesky things like general relativity for the moment). With the “one-over-r-squared” law, gravity decreases inversely as the square of the distance between the two objects.
That’s a mouthful, but really it’s quite simple: If we get twice as far away from the center of the earth, it’ll pull on us with one quarter of the force it does now. If we triple the distance, gravity will pull just one-ninth as hard.
And with this wonderful, happy force law, we get orbits that look like this:
(As usual, all the gifs you’ll see today are taken right from the Gravity Simulator. Feel free to play around, yourself!)
Experiment 1: 1/r^2.1
In the Gravity Simulator, we can do something you can’t do in the real world: we can change the force of gravity and see what happens! So let’s increase that exponent over ‘r’. So now the force goes as “one-over-r-to-the-2.1.” Doesn’t quite roll off the tongue as easy, eh?
Well, let’s see what happens:
Huh, the orbits no longer match up. That’s called precession — basically the planet is nearly making an ellipse as it goes around the star, but not quite. The path keeps twisting around and around. Turns out 1/r^2 is special — very few force laws will make orbits that don’t precess.
Why does making it 2.1 instead of 2 change things? Well, ‘r’ is raised to a slightly higher power, which means as two planets get further away, the force of gravity drops off faster than it would in our world. And as they get closer, the force of gravity increases faster, too!
So when the planet comes in close, gravity pulls on it stronger (than in our universe at least), deflecting its path even more, which gives it a tighter curve. That tighter curve when it’s close to the star means that the planet swings around faster than we’d expect. So by the time it comes back out to its furthest distance again, it’s gone more than 360 degrees. Matches with what we see above. (Go, science!)
Experiment 2: 1/r^3
Let’s keep going! Let’s raise ‘r’ to an even higher power! We should expect even stronger precession, right? Well, let’s see:
Huh, now we’re getting something different. When an object gets close, gravity starts pulling so strongly that the planet just spirals inward until it collides. And if the object is too far away, it spirals outward… and since gravity gets a lot weaker the further out you go, it keeps spiraling out more and more, never to return.
Experiment 3: 1/r^10
What if we went really crazy with this? (I always go really crazy with this.) Let’s try a force law that decreases extremely fast. 1/r^10!!!! (Excitement, there, not factorials) Now if two objects get twice as far apart, the force between them is about 1000 times smaller! That makes for this weird world:
Objects really don’t notice each other in that weird universe, they mostly travel in straight lines… at least until they get juuuust close enough to the star. And once they do, they’re pulled quickly and without remorse into a collision. Boy, I’m glad we don’t live there! You can imagine how hard it would be to create galaxies in that universe, let along solar systems with stable orbits.
Tune in next week when we take things the other way… what happens for 1/r^1.9? Or 1/r? In the meantime, play around with your own force laws in the Simulator!
-AndyPost a Comment
Black Hole or Bust
How hard could it be to add in black holes, right?
We all know that black holes are extremely massive, extremely dense objects. (Mostly.) Get close enough to them, and gravity pulls so strong, that not even light can escape. Whoa! So we just need to make a big star in the Gravity Simulator — and we get a black hole!
Well, no. You’ll never get a black hole if you’re dealing with boring, old Newtonian gravity, though. In the old simulator, say, you could make a star bigger and bigger and bigger, and all you’d get is a bigger star. Any astroid or planet or star can escape its pull, so long as it’s moving fast enough.
Around black holes, there’s a line of no return called the Event Horizon. If you’re outside of this boundary — you could escape the black hole. But the moment you cross it, you’re sunk. You’ll get swept ever further into the black hole.
This happens because General Relativity contains our old friend Special Relativity. And, if you’ll recall, a key part of Special Relativity is that nothing can travel faster than light. The speed of light is the speed limit for everything. The Event Horizon represents the line near enough to the black hole where, if you wanted to escape, you’d have to travel at light speed. Fall in a bit closer, and gravity gets a bit stronger, and you’d need to go even faster than light to escape. No can do. You’re stuck.
In the pictures above, I draw where the Event Horizon would be on each the star. The smaller the star is, the weaker the gravity it, and the closer you’d have to get to reach the Horizon. In fact, most of the time, you’d actually have to go deep inside the star to find this line. Which means, it isn’t really an Event Horizon. The calculations I used to draw these assumed that all the mass of the star is inside the Horizon. As you can see above, that’s not the case. The stars aren’t dense enough, which means: no Event Horizon and no black hole.
But if we get enough mass in place, the Horizon grows big enough that it swallows up the whole star — and we finally get our black holes! Now let’s have some fun with them!
Black Holes have tons of neat properties, which you’ll all be able to check out in the next update to the Gravity Simulator. Stay tuned for more General Relativistic fun!
-AndyPost a Comment
Thought I’d keep you all apprised of the latest addition I’m working on:
RELATIVITY IN THE GRAVITY SIMULATOR.
At least, *mostly* relativity. You see, we’d talked about adding in Black Holes… which would be awesome. But not just *nom*nom*nom* generic sci-fi ‘sucks-stuff-in’ Black Holes. This is TestTubeGames, after all. So I wanted to at least get stuff approximately right. Maybe so ‘orbits’ become something like this:
A while back, in an earlier chat in the forums, we found a General Relativistic formula for the attraction between two objects. Seems reasonable that we could plug in that force law (after all, we’ve got change-able force laws already). It won’t be precisely right (there won’t be gravity waves…), but it’ll get us close.
Simple, then, slap on a GR Force Law and call it a day! Well, nope. Because Black Holes have this neat feature where once something gets too close — ~~inside the event horizon ~~ — it’ll *never* come back out. Mwahahaha.
Never, that is, unless it travels faster than light. Which, in the real world (as far as we know) nothing does. But in the Gravity Simulator, you can launch stuff at any speed! Black Holes would lose all meaning, objects could escape at will. They’d become just ‘really strong stars’ instead of ‘points of no return.’ Boo, hiss.
That means I need to add more relativity in the sim, to make objects obey the speed limit of light. Now when something accelerates, it can get close to, but never reach the speed of light. And, lo and behold, we get neat orbits like this:
Great. Can we just paint that star black and stop there?
Because once you have objects traveling near the speed of light, well, then E=mc^2 becomes important. Namely, mass is energy, energy is mass. So what? So EVERYTHING. Imagine two stars colliding. They rush inwards to meet one another, then *boom* they combine to form a single, stationary star.
The total energy has to remain the same, which means that Kinetic Energy had to go somewhere. In our sim, there’s only one place that energy can go: into rest mass. Just as two subatomic particles can combine to form something massive (wee protons crashing into each other to make the Higgs boson, anyone?), two stars can combine to make one *huge* star.
What other parts of relativity will come into play down the line? Well, the Schwartzschild radius is important. And relativistically slowed clocks are awesomely fun…
Where does this all end? With a bang? With a whimper? Will the simulation collapse into a singularity under all the weight of the new code? Stay tuned to find out!
-AndyPost a Comment
This week, I want to talk about negative mass. Wait… negative mass? Like antimatter?
Antimatter, in spite of its ‘anti’ name, still has a positive mass. An electron and an antielectron (positron) have opposite charge, but they have the same mass. Charge flips to negative, mass stays the same.
If antimatter is out of the picture — what in the world does have negative mass? Well, nothing really that we know of. It would take a very strange, exotic form of matter to have negative mass. The closest we come is with the complex Casimir Effect. But let’s imagine that a chunk of negative mass does exist. What do we know about it?
First off, negative masses would have… negative energies. When an object is moving, it has a kinetic energy equal to 1/2*m*v^2. If the mass is negative, the kinetic energy is negative. Even when you look at E = mc^2, the rest energy of matter, you’d find that the negative mass would lead to negative energy here, too!
Strange stuff. The faster it goes, the less energy it has. The more you have of it, the less energy you’ve got.
What would a negative mass mean for gravity? Well, we can check this out in the Gravity Simulator. Two planets with positive mass will, of course, attract.
This is because the force between them is:
Where m and M represent their masses. Positive masses mean a positive force, which in this case means the objects pull towards each other.
What about negative masses? Suppose we had two planets with negative mass.
Looking at the equation above, if we change m to -m and M to -M… the equation doesn’t change at all! (Negative*Negative = Positive!) So they should attract as before, right?
There’s an extra bit of the puzzle we’ve left out. The force hasn’t changed, but the acceleration has. To figure out how something will move, we use F=ma. The force is equal to the mass times acceleration. Even if the force hasn’t changed in our example… if the sign of the mass has flipped, the sign of the acceleration has to flip. So instead of the pulling force making the planets accelerate together… they accelerate apart.
With all that in mind, the final puzzle is: what happens when you have a positive mass and a negative mass. Will they attract? Will they repel?
The test is easy to do (Note, you can do all these tests and MORE in the Gravity Simulator):…whoa. Why did that happen? You’ve now got enough information to figure that one out.
As a final note, it would seem that having two planets suddenly zoom off the screen, ever faster, would violate Conservation of Momentum, or Conservation of Energy. But in fact, it doesn’t. As the positive mass gains positive kinetic energy, the negative mass gains negative kinetic energy. They cancel out.
Same with the momentum. Both planets may start moving in the same direction in that gif above, but the planet with negative mass has its momentum moving to the right. (Momentum = m*v, so flipping the sign of the mass flips the direction of the momentum). So even with strange negative masses around, conservation laws stay intact.
Will we ever really encounter a chunk of negative mass? Who can say. But at least we can make some good predictions about how we’d expect it to behave.
-AndyPost a Comment
To Another Dimension!
One of the new visitors to the forums brought up a neat topic the other day: Dimensions. Namely, what would gravity look like, if instead of 3 spatial dimensions, we had 2? Or 4? Whoa.
Let’s start off simple. The world we live in, for all practical purposes, is in three spatial dimensions. You can go up-down, left-right, or forward-backward. And in this world, Newton’s Law of Universal Gravitation tells us that two objects with mass will attract each other, according to:
If you only care about the distance, the force goes as 1/R^2, which is why we call this an inverse-squared law. Double the distance between the earth and the sun, and gravity will pull them together with 1/4 the strength. Inverse square laws are beautiful, because they lead to cool things like closed and stable orbits.
All GREAT things for life!
Unfortunately, Newton’s Law of Gravity can’t help us when we’re in a world with a different number of dimensions. The law only works in 3D.
Instead, our starting point instead is Gauss’s Law. Take a point mass, make an imaginary spherical surface around it. Imagine that instead of gravity, the point mass is just shooting out 100 bullets in random directions.
How many bullets pass through the sphere around the point mass? No matter how big or small you make your sphere, the answer will always be 100. That’s the core of Gauss’s Law. The number of bullets passing through a sphere doesn’t depend on its radius. When we’re talking about gravity, the bullets represent the Flux, which is simply the strength of gravity times the area of your sphere.
If the total flux is always the same, that means the Force of Gravity is proportional to 1/(Total Area of a Sphere). In 3D, a sphere has a surface area of 4PI*R^2. So the Force goes as the inverse of that, or 1/R^2.
Suppose we were like the Flatlanders, and lived on a 2D plane. We could move forward-backward and left-right, but not up-down. How would that change gravity?
Well, Gauss’s Law still holds. But this time our ‘sphere’ that we draw around the point mass is actually just a circle. (Remember, we can’t leave our 2D surface!) The ‘surface area’ of the circle is just its circumference: 2PI*R. Which means that the Force of Gravity goes as 1/R. And we get orbits that look like this:
What if there were an extra dimension? Suppose we lived in a world where we could move up-down, left-right, forward-backward, and… and… uh… 4Up-4Down. We don’t have a word for those last two of course, since we never move through a fourth spatial dimension. But nonetheless, we are armed with all the information we need to figure out how gravity would look.
This time the ‘sphere’ we make around a point charge is in 4D, which is hard to draw, but we can figure out what it must be like with a simple pattern. In 2D, the ‘surface’ of a sphere is a line. In 3D, the ‘surface’ of a sphere is an area. So in 4D the ‘surface’ of a sphere should be a volume. Its units should be length cubed. No surprise, then, that the surface ‘area’ of the 4D sphere is 2*PI^2*R^3. Which means that the Force of Gravity must go as 1/R^3. So we get orbits like this:
You’ve now got everything you need to ponder the force of gravity in any number of dimensions. And with the Gravity Simulator, you can check out what any of these worlds might look like. Try to make a 5D solar system, I challenge you!
-AndyPost a Comment
Gravity Simulator Updates
The Latest Additions:
- -Set the orbit trails to the length you want, they aren’t always infinite, now
- -Snap the planets to the grid, for some easy symmetry
- -You can now fling the ‘fixed’ stars… they still won’t be affected by gravity
- -Edit planets on the move — change their mass, speed, position…
- -Set the density of an object to whatever you like
- -Pick whatever force law you like. The force goes as tan(r)? Sure! Or maybe r^ln(pi*r)? Uh, that’s fine too!
- A bunch more… naturally. Check out the forums to learn more!
-AndyPost a Comment
Cambridge Science Festival
This past weekend, I took part in a Science Carnival here in Cambridge. There were dozens of booths filled with sciencey-stuff. Most of what I saw was in the Games Corner, a room devoted to the intersection between science and games. There was a 3D printer making Minecraft figurines, a game you controlled with a potato (think potato battery), and a giant chessboard controlled by robots.
And of course, there was me, showing off a bunch of stuff.
I brought three of the games that I’m working on right now – which was a fun challenge. Before I’ve always shown off one game at a time… which is difficult enough. It would be *madness* to prepare three games for a single event. Probably true, but boy am I glad I did it.
There were perhaps a hundred people or so that came by the booth and played one or more of the games. And in general, the event skewed a bit younger than I’m used to (more early-elementary schoolers, say). But I got great feedback, and learned quite a few lessons.
1. Thank goodness none of the games needed much explanation! I was worried I’d drive myself crazy splitting my time introducing people to three very different games. But with tutorials at the beginning of each one, people were generally able to simply sit down and play. Phew!
2. The Gravity Simulator has an extremely wide appeal. I’m used to talking with some of the… well… ‘advanced’ users of the simulator in the forums here. People who push the bounds of the sim, setting up and sharing complex creations. But on Saturday, I was able to place the simulation in front of a bunch of kids and adults who’d never heard of it before. (And thankfully, I’d added a bit clearer instructions than in the original iteration.) And, to a person, they found something cool to do. Whether it was an attempt to get two planets in orbit around a star at the same time — or just a kid seeing how many planets she could spam onto the screen — people found ways to entertain themselves.
3. Shocktopus is a work-horse. People really get into that game.
4. This was Bond Breaker’s first time out, and it performed admirably. It kept people engaged, which was nice to see — and I was able to get a bunch of great ideas for ways to improve the game. Mainly: ways to tweak the tutorials and explanations so people would have a better idea what was going on.
All in all a nice way to spend a Saturday afternoon, chatting with people, seeing them enjoy my games, and learning a heck-uva lot myself. A big thanks goes out to the Cambridge Science Festival, and all the people who stopped by the booth, spending a beautifully sunny Saturday playing video games inside.
-AndyPost a Comment
Gravity Simulator A-Go!
If you’re reading this, you’ve probably played around with my original Gravity Simulator, posted way back in 2011. And you’ve probably heard me talk about the upcoming update to the simulator — with tons more bells-and-whistles.
I present: the *New* Gravity Simulator
The simulation has a bunch more power (3000 planets is handled with ease), a bunch more options (change planet colors, track objects, tweak the calculation precision), and a bunch more gravity (citation needed).
There’s a free online version, just like before, that you can play here. And if you really like gravity, you can always download the Full Version, which lets you play in full-screen mode, runs a lot faster, and will have even more features than the online version. (Available on PC/Mac/Linux) And, if that isn’t enough, you’ll be supporting me in developing this simulation.
Both are in alpha still, which means they’re constantly being changed and improved. (In fact, as of this blog post, an update has already come out… Version 0.26.02). So if you find a bug, or see room for improvement, swing by the forums to let us know about it. Your suggestion might just make it into Version 0.26.03!
-AndyPost a Comment
I’ve been working on a brand new game over here, and up to this point I’ve been pretty hush-hush about it. (Mainly because there have been so many other things to talk about!) But get ready for: Bond Breaker.
Neat factoid: this is a game that I’m making for a physics research group out at UC Irvine. They’re called CaSTL (Chemistry at the Space-Time Limit), and their research is on manipulating and breaking individual molecular bonds. Basically, they use lasers and scanning tunneling microscopes to blast apart molecules.
So boring! How could that possibly be a game?
Oh, wait, it’s the perfect material for a game.
You are a proton navigating a world of puzzles and spikes. In order to get to the goal, though, you’ll need to use some atomic and molecular physics. You’ll pick up electrons from the tunneling microscope. You’ll bond with other protons to form H2. You can even zap in some laser light to excite some electrons.
As always, the physics in the game aims to accurately represent the real stuff. (Note: yes, that means that protons are really blue, with white outlines)
The game is still very much a work-in-progress, though I have posted the current (alpha?) version online already. If you’d like to check it out and give some early feedback, swing by the forums to join the discussion.
-AndyPost a Comment
A brief detour from science, here…
Last month, I took part in the Global Game Jam, a challenge where you have a single weekend (48 hrs) to make a game. Game Jams are a great way to practice game-making – because time constraints and sleep deprivation work wonders!
At my particular site, there were about 80 people who came together, splitting up into 4-5 person teams. The theme? “We don’t see things as they are, we see things as we are.” Pretty abstract. I was pulling for ‘particle physics.’ Oh, well.
I had the chance to work with an amazing team, none of whom had met before the weekend. There were 5 of us, and everyone brought some special skills to the team. We had Jennifer Lay doing art; Nick Bergen making the music; John Wolff as designer; and Vinny DaSilva and I doing programming. (See the credits page in the game for a rad picture of us.) We had a lot of laughs, and pulled together a surprisingly good game. (Most game jam games are pretty rough, but ours has a beginning, middle, and an end, and very few bugs. Rad!)
Our game? 14b.
The game is a mystery, where you are the detective. Someone’s been murdered, and it’s up to you to figure out who did it. You have some folders of evidence you can look at, witness reports of the scene, and clues that you can ask each of the witnesses about. Each witness sees the scene in a different way, focusing on different details, and bringing their own perspective. And you need them all to figure out the caper.
We’ve posted the game online, so you can go check it out. Who do *you* think committed the crime? Why?
-AndyPost a Comment