After spending a whole lot of time and effort the past couple days banging away on the computer to redraw the Dyna Soar stuff, I needed a break. So, what could be more relaxing than working out how to rebuild the universe?
If you want an idea for a really big place to live, science fiction can help you out. First, ya got yer Earth-like planets. But then you can have rocky planets that have low-density cores… instead of iron, lets say aluminum. This lets you build a terrestrial world with one-G of surface gravity, but perhaps one and a half to maybe twice the diameter. Then you go to supramundane worlds, where you build a “floor” over gas giant worlds like Saturn. Conceptually tricky, requiring tech beyond what we’ve got, but in principle it’ll let you build a 1-G world wit a surface area of perhaps 100 Earths.
Then you start getting into things like Ringworlds (millions of Earth equivalent area). But those aren’t “worlds” in the regular sense, spheres with actual 1-G of surface gravity. But let’s say you were a Kardashev Type III or Type IV civilization, and you wanted to build a planet to impress your neighbors. Let’s say not just big, but so big that it would kinda-sorta punch a hole in the universe. How big would it have to be?
Start off by assuming a surface gravity of 1 G, 9.81 m/sec^2. The equation g=(G*M)/(r^2) defines this… g=surface gravity (m/sec), G is the universal constant of gravitation (6.672E-11), M=mass (kg) and r=radius (meters). You have to adjust the mass and radius to get the “g” you want. But there’s another factor: escape velocity. There is of course a maximum escape velocity, the speed of light. If you have an item so massive that at some radius from it the escape velocity is equal to the speed of light, you have yourself a black hole. That radius is the Schwartzchild radius, and is given by: r = (2*G*M)/(c^2) where c=speed of light, 299,792,458 m/sec.
If you mash these equations together and solve for mass, you can determine how big of a world you need to be so that the surface gravity is 1 G and the escape velocity is the speed of light. And the answer? A sphere with a mass of 3.08E24 kilograms (about 1.55E11 suns) packed within a radius of 4.58E15 meters (about 0.484 light years) is the answer, giving you a “planet” with the surface area of 5.17E17 Earths.
Interestingly, the density of the massive sphere is only 7.66e-6 kilograms/cubic meter, numerous orders of magnitude less than air. How would you build such a thing? Damnfino. I’m just assuming that by the time you get to Type III, you’ll have that noodled out. My guess would be to have an ordered cluster of supermassive black holes orbiting each other, surrounded by a shell of… I dunno, foamed Unobtainium with a fine matrix of Bolognium fibers providing extra strength.
When you build your nearly lightyear-diameter ball containing something like one-tenth of the mass of the entire Milky Way galaxy, you have a black hole you can walk around on. Unless you have some sort of faster than light propulsion, if you land on it, you’ll never be able to get away from it. But even landing on it would be a neat trick… you’d need a starship capable of attaining nearly the speed of light to get close, and would have to rely on impressive aerobraking for terminal descent.
Another issue would be time dilation. Gravitational time dilation does for observers close to massive objects the same as relativistic velocity does for fast-moving observers: time slows down. The basic equation: dilation = square root of (1-Schwatzchild radius/radius). The end result is that right at the Schwartzchild radius, time *stops.* So for the people and critters wandering around on your giant world, the universe will fade away and the last elementary particle will evaporate into a cold fuzz of nothingness in the time it takes to sneeze. Another effect of this would be that all the light that would fall on this “planet” would, from the viewpoint of an observer on the surface, fall within a a fraction of a second. A billion years of starlight would blast the surface inhabitants like a phaser blast.
There would be some interesting philosophical implications. As designed, the escape velocity at the surface is exactly the speed of light. But what would happen if the contractors working on it made the interior mass ten percent heavier, so that the escape velocity on the surface was *faster* than the speed of light? The time dilation math suggests that time wouldn;t just go to zero, it’d go *imaginary,* being the square root of a negative number. Buh?
And of course, this would be a black hole with no singularity. Indeed, if the “planet” was made of some monolithic extremely low density and insanely strong foam, if you tunneled your way all the way to the center, the gravity field would seem to be flat. Assume something even better: the outer shell is the whole mass of the thing, so that there was the better part of a lightyear of empty space inside the thing. What’s the time dilation effect inside *that?* Shrug. My guess would be that if the shell was effectively vanishingly thin, then the time dilation at the surface of the shell would continue all the way in… so you’d have a sphere a lightyear in diameter that would “freeze” time till the structure decays away.
I’ve done my math in Excel. HERE is the spreadsheet if you want to check my numbers.