Archive for the ‘Uncategorized’ Category

Running a Stirling Engine Using the Night Sky

November 28, 2018

I’m a failed inventor. I get strange ideas, try them out, and usually discover that I either got my physics wrong or built my prototype poorly, or that someone beat me to the punch thirty or three hundred years ago. I created what I thought was a great factoring algorithm for huge numbers and found out that Fermat developed it first. That’s the way it goes.

So here’s the latest invention. It’s not really an invention so much as an interesting application of an existing device.

Is it possible to make a heat engine that runs off the thermal differential between the night sky and the heat radiating from the ground? Stirling engines can run using fairly small temperature differences, such as the ambient air and the heat in the palm of your hand. You can get one of these hand-driven Stirling heat engines from ebay for under $50.  The real question isn’t so much whether you can make a device that runs off the heat sink of the night sky, but how much of a thermal delta you could provide to that engine.

How cold is the night sky? I’ve read that on a clear night, it can provide a radiative heat sink of -70˚C. Yeah, that’s negative 70 degrees centigrade. Pretty cold. Those with a background in heat transfer physics know that the other two forms of thermal transfer are conductive and convective, and with the right glass or plastic covered chamber, you can minimize those thermal paths so that your heat source/sink sees only the -70˚C of the night sky. This is why some telescopes have a problem with their optics freezing up. Really. It also explains why some windshields frost up even though it doesn’t reach freezing temperatures outside, and other related phenomena.

The other side of your Stirling heat engine could be getting its energy from the radiation from the ground, say, about 15˚C. Or if you heated up a tank of water during the day, maybe 40 or 50˚C. You could run your engine easily with a delta of 100 degrees, though as you thermal guys know, the engine would be a lot more efficient at higher temperatures during the day shift.

But, the hypothetical question is if you could run an engine off the night sky. My speculative answer is “yes”. What’s nifty about the ability to do this? Well, instead of dumping HEAT into the atmosphere to generate energy, which every power source on Earth currently does, you are actually dumping COLD (a.k.a, removing heat) into the atmosphere to run your engine. The net heat loss of your engine is NEGATIVE.

So yeah, we could cool down the Earth and generate energy at the same time. Crazy, huh? The biggest problem with the idea is that you could generate a lot more energy a lot more efficiently using a hot solar Stirling engine during the day at the same cost, while dumping more heat into the atmosphere.

And cost seems to drive everything except our self-preservation instinct.

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Newton’s Shell Theorem in an Infinite Solid Universe.

August 15, 2018

Despite the ominous title, you will find no math in this entry. You’re welcome.

Some of you will be familiar with Newton’s shell theorem. Basically, it says this; if you are inside a sphere, or a thin shell, of homogeneous mass, then the gravitational forces of the mass around you will cancel out, and you will float freely no matter where you are inside that sphere; you will not be attracted to the center of the cavity, and you will not be attracted to the inside edge of the sphere.  See Figure 1 below.

I’ve done the math for the basic shell theorem (Fig. 1). It works. This is a fairly common undergrad physics problem. 

newton1

Since you can add additional shells of mass ad infinitum, then you can fill an infinite volume, a universe, with these shells, and the object in the spherical cavity will still be floating freely, since all the forces will balance out. See Figure 2 below.

newton2

This works great if you have a spherical universe.

A while back, while working on an SF story, I speculated on how a universe that was completely solid might work, especially if you had an Earth-sized cavity in it. My conclusion was that you would be attracted to the side of the cavity, which at first seems to contradict Newton’s shell theorem. But let’s take a look a that.

newton3

So the big rectangular gray area in Figure 3 represents the infinite, solid universe of whatever density you like. Marshmallows, quartz, whatever. We’ve carved out two Earth-sized spheres of whatever this stuff is, and discovered that if you stand where the little guy is standing, then the mass is completely symmetric around him, pulling with equal force in all directions. There is no gravitational gradient there.

Now, we fill in one of those spheres.

newton4

In Figure 4, on your LEFT, there’s a volume the size of the Earth, but it’s full of the aforementioned universal solid. On your RIGHT, there is an empty hole the size of the Earth. Since, as we’ve seen in Figure 3, all the rest of the mass in the universe balances out, not pulling you any direction at all, then the only mass pulling on you is that sphere to the left. If the average density of the universal ‘stuff’ is the same as the Earth, then you would feel a force of 1 gravity no matter where you stood on the inside of that hollow sphere. Feel free to imagine Figure 4 without the extra circle on the left. It’s not really necessary except to help with the explanation.

This is an issue when trying to solve problems involving an infinite universe. In this example, the universe would have to either be infinite and eternal, or closed in such a way that the distribution of mass was the same everywhere, such that if you went in a straight line forever, you would end up back where you started (though digging that hole would be very time consuming).

This is an interesting SF universe to play in, however. Pockets of air scattered around would create their own gravity, as described above, little bubbles in an unknowable solid universe. What if gravity in a large pocket became so great that it overcame the structural integrity of the matter that made up the universe? Would it propagate outward, consuming the universe with a growing ball of vacuum?

If you like this speculation, be sure to check out my short stories listed at my website.

 

 

Zero Gradient Gravity Fields, Dark Matter, and the Formation of Stars

May 17, 2018

We’ve mentioned in the past (to ourselves) that the formula for the Schwarzschild radius for a black hole, c2=2GM/rs tells us that no matter how thinly distributed a mass is, (such as 1 atom per cubic centimeter), if you have a large enough sphere of it, it will have a Schwarzschild radius when viewed from outside that volume. You can see this just by shuffling the equation around a little, so that c2/2G, which is a constant, equals M/r, the mass over the radius. For any given density, the mass, M increases with the cube of the radius, so for any given density, you can always find a radius that contains enough mass to equal the value c2/2G. Cute, huh?

I struggled for awhile wondering if an infinite 3D field of particles (which would appear to be flat gravitationally, that is, not have a gradient), would allow for overlapping apparent black hole horizons; everywhere you looked, there would be large, overlapping, spherical volumes that had enough mass to become black holes. Could this be our apparent cosmological horizon? But today (5/12/18) it occurred to me that the key feature of a black hole is that it has a gravitational gradient. You have to work to get out of the gravity well, or the idea of an event horizon is meaningless. But an infinite field of equally distributed mass has no gradient. It appears flat. Ergo, no event horizon, no matter the density.

Cruising along in deep space, there is, in essence, the same amount of mass pulling on you from all sides, that tenuous 1 atom per cm3. It could just as well be 10 atoms, or 100, or a million, with no noticeable effect. Once we attained a velocity, we would maintain that velocity – an object in motion remaining in motion. The interstellar gas would eventually slow you down, but it would take a very ong time.

Working with the 1 atom/cm3 extending to infinity, let’s say we superimpose another huge sphere of 1 atom/cm3 gas on top of that, so huge that it provides you with an event horizon (if I’ve done my math right, it would amount to roughly 1.5×105 light years in radius, or a ball 0.3 million light years in diameter). Now there is a mass and a very small gravity gradient. Is the event horizon based on the 2 atoms/cm3, or the 1 atom/cm3 density? We’ve already seen that the original 1 atom/cm3 field provides no gradient, so it would make sense that the only effect to the observer is to see the event horizon created by the new 1 atom/cm3 superimposed on the existing field; the other previously existing field is completely flat and cancels out.

However, the new field created by the new mass is going to affect both the old mass (1 atom of hydrogen per cm3 everywhere) and the new mass (1 atom per cm3 in the giant sphere). The object will form with twice the mass (in this case) predicted by the theory. When it’s first put in front of us, we will measure a mass represented by the 1 atom/cm3 in that volume. As it collapses and takes the background mass with it, it will finally produce a mass that accounts for the 2 atoms/cm3 that we actually started with. While it’s doing this, it will also be backfilling the area that it vacated with more interstellar gas, as that gas is also being pulled in by the gravity of the developing black hole, so the overall density of the universe will appear mostly unchanged, even around the black hole.

Practically speaking, this would be more likely to happen in a nebula, where the density is much higher.

One of the most interesting things about this process is that if there is an undetectable mass-type in the universe (like dark matter) that only interacts with regular matter through gravitation, and it’s distributed equally everywhere, then objects that form (planets, Suns, black holes) will also pull in this other mystery mass. As described above, the tenuous gas (1 particle per cc) that we currently measure may actually mass 2 or 10 or 100 particles per cc. We wouldn’t know since the field is flat. Since this new mass doesn’t react with normal matter, it will clump in the center of the object (although it may have its own chemistry and volume that prevent excessive density). Small objects existing on a larger mass (like humans), would have the dark matter pulled out of them, and when we performed tests like The Cavendish experiment to measure the gravitational constant, it would give us a good value for G for normal matter, and would give us erroneous results for the masses of the planets and the Sun. We would think the core is made of denser matter than it really is, both in the Sun and Earth. We know the mass of the Earth, but a substantial chunk could be dark matter and we’d never know it. Perhaps the iron core is made of silicon at half the molecular weight (which is interesting, because magma is 50% silicon dioxide, and only 9% ferrous oxide).

However, most objects that have formed in the last few million years are going to have some dark matter as part of their core. They have gravity, and any dark matter out there will be attracted to it just like regular matter, until an object forms which is part dark and light matter. This includes asteroids. Eventually, we’re going to move an asteroid, and when we do, the acceleration is going to leave the dark core behind. We may not notice it unless we’re looking for it, or if it’s a substantial enough part of the mass that we detect a mass-change in the object as it’s propelled. We would end up with two objects; the obvious light-matter asteroid, and the invisible dark-matter asteroid that could only be detected with a gravitational gradiometer. It would change the way we thought about the universe.

Black Hole Evaporation versus CMBR

May 5, 2018

Black holes evaporate. At least, that’s what most physicists tell us.

What I stumbled into recently was a conjecture that they can’t evaporate beyond a certain point if the input is greater or equal to their output. This was mentioned on Quora by some physicist as a response to a related question. I thought it was interesting enough to mention it here.

Really large black holes are colder than the cosmic microwave background radiation (CMBR), which is about 2.73 degrees. The radiation going into a black hole is actually greater than the radiation leaving the black hole. The only way a black hole could radiate is if it’s very small and already radiating hotter than the CMBR (plus whatever particles fall into it, adding to its mass).

The limit where the size/mass of the black hole is equal to CMBR input is about 1% Earth mass, about 4×10^22 kilograms, based on Susskind’s formula and Hawking’s formula. This would create a black hole smaller than a millimeter. But black holes can’t even form without at least three or more Solar masses to begin with.

So, any black hole larger than a millimeter is going to keep growing. Presumably, primordial black holes smaller than 10^11 kg, created during the Big Bang, would have evaporated by now. This leaves a range of possible primordial black holes from 10^11 to 10^22 kg as possible existing evaporators, since they would be hotter than the background radiation.

However….

Primordial black holes would form because of high density and radiation. It would be crazy to think that their mass wouldn’t quickly grow far beyond Earth mass when surrounded by a buffet of dense gas and radiation. Just the nature of the formative process suggests that they will never radiate faster than mass/energy is added to them from their environment, and will always grow larger in size.

I really WANT them to exist, however. My next SF story kind of depends on it.

Black Holes and Those Pesky Event Horizons

October 8, 2017

In Leonard Susskind’s book, The Black Hole War, page 240, he states, “To a freely falling observer, the horizon appears to be absolutely empty space. Those falling observers detect nothing special at the horizon…” In Amanda Gefter’s book, she points out that the distant observer sees the event horizon, while the falling observer detects no event horizon at all. Of course, she took a lot of her ideas from Susskind. In the meantime, Hawking treats the event horizon as a fixed boundary where virtual particles can split apart (Hawking radiation).

black_hole_2013_0

I think none of these is right. The idea between the “escape velocity being faster than the speed of light” is relative to the delta between the gravitation potential of the observer and the potential at the event horizon. From an infinite distance, we observe an event horizon at a certain radius. Should the event horizon suddenly disappear if we are in an inertial frame starting our fall into the black hole? Starting at what distance? A thousand miles? A light year?

The more likely result is that the event horizon moves inward as you approach it. You are in a deeper gravity well as you approach the black hole, thus the difference between your local gravity potential and that of the event horizon, to maintain a high enough value for the escape velocity to equal the speed of light, requires that the event horizon continuously move away from you (toward the singularity) as you move toward the singularity. You never quite catch up with it. There’s a Wikipedia article that says this explicitly, but then, it’s a Wikipedia reference (Event Horizon). Sometimes they’re wrong, but usually they’re dead-on.

An interesting consequence of this is that if you maintain a certain orbit near the event horizon, and your version of the event horizon is closer to the singularity than that of a more distant observer, then a photon just outside your observed event horizon could reach you just fine, even though it cannot reach the more distant observer. Having received that photon, you could transmit the data from it outward, (boosting the frequency) as the distance from your gravity well to the distant observer requires an escape velocity somewhat less than the speed of light. Is this a loophole?

Why, then, do we think that a photon below the event horizon (for the observer at infinity) can’t escape the confines of the black hole? Is it only because it would be red-shifted to a zero frequency? Or is that false?

Escape velocity is merely a calculation of the velocity required to go from one gravity potential to another. If you are already in a gravitational well (like the outer edge of the Milky-Way galaxy) with an escape velocity of 300km/s, this has no effect on the escape velocity from Earth (11km/s), or the velocity needed to orbit Earth (7.5km/s). Likewise, consider a photon trapped just beyond the event horizon as viewed from an observer at infinity. To the guy in orbit around the black hole, the difference in potential is much smaller, and his relative event horizon is closer to the singularity. Won’t he see that photon? Can’t he receive it from the domain outside his apparent event horizon, but inside the event horizon of the observer-at-infinity? And then capture the photon and retransmit it?

So, even though a photon by itself can’t escape the event horizon of the observer-at-infinity, an intermediate process (natural or human) could conceivably pass a photon up through overlapping light cones, even though the light-cones at either end don’t overlap. This might eliminate the question of whether information can escape a black hole or not. The infalling observer can see what’s happening beyond the outer event horizon, and pass the information on, since his own event horizon is even closer to the event horizon.

A Fool’s Physics

September 18, 2017

I’ve read a lot of physics in my life and have a lot more to read, a lot more to learn. It’s hard to read any general physics text without stumbling across some interesting tidbit that makes me sit back and ponder how that tidbit fits in with my mental model of the universe. Some things make sense, some don’t. When I heard of the Unruh effect, I was dumbfounded (to my understanding, this is the emergence of energy out of a vacuum relative to an accelerating object). When I learned that photons are their own antiparticles, I was confused. When I realized that the time component in the spacetime interval produces a hyperbolic curve in the formula, it was an enlightenment years in the coming. When I read that antiparticles are just regular particles going backward in time (Feynman, I think), that, too, messed with my mental models of the universe. To say nothing of dark matter, the accelerating expanding universe, and so on. So, I try to organize all this hodge-podge of apparently related information into a single model that makes sense. As most physicists will tell you, it is an insurmountable task. But, I am not a physicist, really. I have a Masters in Astronautical Engineering, and as Sheldon Cooper would tell you, I’m really just an engineer. I create mental models that make sense to me, but may not have any practical use or truth in the larger sense of things. But we all have to start somewhere. I’ll keep reading, and revise the incorrect bits as I go along.

In this log of ramblings, I’ll offer up a bunch of foolish ideas on physical reality. I’m a big fan of determinism, so be forewarned. I also think of time as an actual, physical dimension. If you happen to join me on this warped (!) journey of speculation, I’d love for you to tear my arguments apart, tell me what’s wrong, and perhaps help shape the speculations into something that makes a coherent sense of reality, or assure their demise.

Causality Paradox? What causality paradox?

September 11, 2017

I can’t call this real physics, this is just pure and wild speculation. I had a funny idea today about whether or not you can go back in time and shoot your grandfather, thus keeping yourself from ever being born. Ethical questions aside, I thought of a possible solution to the whole “paradox” issue.

First, if you aren’t familiar with the Grandfather Paradox, you can read up on the subject at Wikipedia at https://en.wikipedia.org/wiki/Grandfather_paradox.

Start with the idea that we are always travelling at the speed of light. You, the Sun, the Earth, your brother Bob, everything is travelling at the speed of light through the time dimension, all going the same direction so your relative speed is zero. This is a pretty common concept in modern physics, so I’m not going to expand on that here. Just more physics weirdness.

So, let’s say I get the dubious urge to go murder my grandfather at some time in the past, before I was born. Using some fantastic time machine, I go back in time to when Grandpa was just a young fella and shoot him. What happens then?

Well, imagine that space and time are a 4-dimensional matrix, but that changes made in the matrix can only propagate forward at the speed of light. Remember, that’s how fast we move through time. Eventually, the change (where I no longer exist) reaches the point where I would have gone back in time, but the particles that would have made up my body go shooting forward past that point and never go back. Well, they aren’t “shooting forward” so much as redirecting the world line of their old path at the speed of light. Now, instead of making a U-turn and heading toward their fate with my grandfather, the bullet-magnet, they continue forward in time. The old worldline going backwards collapses/disappears at the speed of light and eventually catches up to where Granddad is, and lets him live. I’m born again! And I foolishly decide, again, to go back in time and kill my granddad.

What does this result in? An oscillation. The world line shifts back and forth between the two realities, carrying the data from both possible realities, like a sine wave on a current. Just as a single electrical sine-wave can contain positive and negative values as it propagates through a wire, so can events toggle on a worldline as that worldline propagates through time. Even past the point where I made my fateful decision, the world line is toggling back and forth; both realities are true, taking their turn as the decision I made causes both of them to be real. The duration of this toggling or oscillation would be twice the duration of the time from when I went back in time to when I snuffed Gramps; the duration of the whole loop.

It’s my belief (not necessarily shared by many others) that we live in a four-dimensional space time that exists perpetually as a 4D matrix, and that what we perceive as our consciousness exists at each point along that worldline. There is the version of you that you perceive now, and the version that existed when you kissed your wife for the first time, savoring the moment you’ve forgotten. Kind of a repetitive immortality.

But what I’m suggesting above is that multiple realities can exist on a single worldline; you don’t need multiple universes dividing every time a critical decision is made or a quantum observation collapses a wave function, or a bit of antimatter goes back in time and changes an existing chemical configuration. Both events occur and exist on a changing, fluctuating, dynamic 4D worldline. There’s the version of you that remembers killing your Grandfather, and the version that never existed, propagating through time, one behind the other forever on the oscillating world line.

The normal view of a 4D worldline is of a static deterministic universe, bound by the future and past configuration of an unchanging worldline. Another view is that every decision, human or quantum, splits the universe into a multiverse, a crowded infinity of infinities. This version allows us to stick with one universe, but to modulate our worldline to allow multiple realities to exist along a single timeline.

Possibly, an outside observer could interface with either version of your reality, based on where he encounters your worldline from his own worldline. Could that be the “collapse of the wavefunction” we talk about? Good grief, that would make the Schroedinger’s Cat conundrum actually possible. Dead and alive! I always thought of it as complete nonsense.

One issue with this model is that each worldline, as it moves from one reality to another, may have to move instantaneously from the collapsed worldline to the new worldline. I think. No real way to test it, that I can imagine. Mmm…maybe pick a subatomic decay process that can have multiple durations, then have someone record the decay time data, then take off (with that data) somewhere at high-speed so your worldline is no longer in sync with the experiment’s original timeline, fast enough that the time separation is greater than the decay time variance. Then, come back and see if the recorded time is the same as it was before; you’ll have two sets of readings of the duration of a single decay, and they might not agree. Wouldn’t that be something?

 

Conservation of Linear and Angular Momentum

September 11, 2017

Some Simple Physics; Conservation of Momentum

There are no strange ideas in this entry. In fact, I might call this a boring entry. If you want to read the weird stuff, read one of my other entries.

I was sitting around reading a primer on particle physics (L.B. Okun) today, and got thinking about the conservation of linear and angular momentum.

Conservation of linear momentum means that, when you chuck something out the rear end of your spaceship, then your spaceship moves the opposite direction, so m1v1 = m2v2 . So, if you toss out a small mass of propellant from one end at a really high velocity, then you move the much larger mass the opposite direction at a much slower velocity (along with your remaining propellant). This is an exponential relationship, but that’s not what this blog is about today, so you can forget learning about that useful tidbit of knowledge.

Anyway, to increase your velocity, you have to chuck part of your mass in the opposite direction. Pretty basic. If you just move stuff around inside your ship, the ship won’t move at all (except incrementally for the duration that you move around in the ship, but you won’t acquire a continuous velocity). Anyone who’s been on one of those playground spinners and tried to throw your body one way or the other knows how that works. You throw your body forward a foot, and the disk rotates a foot and stops.

So you can’t change the momentum of an object by moving stuff around inside. Not even if you have the rocket inside an enclosed sphere. The sphere won’t move.

I was recently (foolishly) wondering if that was true of angular momentum, too, if you had a rotating planet or moon, is there some way you can get energy out of the rotation by diddling around with the insides, somehow tapping the angular momentum of the planetoid for energy. Ultimately I realized you cannot in a closed system, but it should have been obvious to me all along. However, as with a rocket, you can change the angular momentum by ejecting part of the object. You can even speed up the spin a lot or slow it down.

Satellites do this sort of thing all the time. Usually they have spin they want to get rid of, and they call the technique “momentum dumping”. Two methods known to me involve extending tethers (like ballerina arms) to slow down the satellite’s spin, then releasing the tethers, or spinning up a high-speed gyro in the opposite direction of your spin (potentially dumping the core of the gyro, though I’m not certain any spacecraft does that – usually they use the gyros to turn the spacecraft both ways, hoping the overall effect will cancel out, and when the spin in one direction gets to be too much, they finally use propellant to dump the angular momentum). These are called Control Moment Gyros, or CMGs, and they usually have a minimum of three on board to cover the 3 axes.

Carrying this concept one step further, since you can eject propellant from a ship to make it go faster in a straight line, you can similarly spin up a chunk of mass from your planetoid in the opposite direction of your planetoid’s spin to make the planetoid spin faster, then eject that spinning mass into space. What amused me about the idea is that it’s essentially the same as a rocket ejecting propellant linearly to increase linear momentum, but here you are ejecting an object with accelerated angular momentum to increase the angular momentum of your planetoid. The difference being, you don’t ever have to eject the mass; it’s rotating in place, like a CMG.

That’s it. Not really that interesting, I guess. The equivalency of the two systems and the idea of “rotational rocketry” just struck me as amusing.

Trying to Accelerate an Infinite Mass

August 29, 2017

Short entry today; an easily digestible bit of physics.

A number of times in physics books and articles, I’ve come across people stating that “as an object approaches the speed of light, its relativistic mass increases, so it’s harder and harder to accelerate the steadily increasing mass, thus, you can’t speed up.”

I’d like to call bullshit on this argument. As your ship’s mass increases, so does the mass of your propellant in equal parts. If your mass doubles, so does the propellant mass being ejected out of your rockets, and you’re still obeying the law that for every action there’s an opposite and equal reaction. It’s just as easy to accelerate as it was before. You still can’t measurably reach the speed of light, but that’s for another entry I’m working on.

See? Easy peasy.

There’s a more sensible way to get the same results. For the outside observer watching the accelerating spacecraft, as it approaches relativistic speeds, time appears to slow down. For the guy ON the spaceship, he/she has the same amount of propellant per second going out his exhaust as before. For the outside observer, since the spaceship’s time is slowed down, the observer sees less propellant-per-second being used by the ship. As the ship gets closer and closer to the speed of light, it’s using less and less propellant. It accelerates slower and slower. It can never reach light speed.

To the person on the ship, his acceleration is the same as always; if he felt 1 gee at the start of the journey, he still feels 1 gee in his reference frame.

Background Gravity – Time Dilation in a Flat Field

August 11, 2017

I got into an argument with a physics buddy not that long ago (a year, maybe), about gravity. We have an intermittent arrangement where we go drink beer and talk about physics at two or three pubs in San Luis Obispo. Usually, the physics becomes a little less coherent as the evening wears on.

One of the discussions centered on whether there is a “background field” of gravity or not, or whether it’s even sensible to discuss such a thing since, in an infinite field of equally distributed mass (or gas, or 1 atom per cubic light year, whatever), all the forces around you seem to cancel out. The mass of the universe to your left is equal in size to the mass of the universe on your right; you feel a net acceleration of zero. I argued that even though the field was “flat”, there was still a field there. He argued that a field implied a gradient; there is always a force.

We did not come to a satisfactory conclusion. It might merely have been the fact that we were defining the same terms in different ways in our heads. I’m not sure. I thought my argument was rock-solid.

So, here is my side of it.

Some of you are probably familiar with Newton’s Shell Theorem. It’s in his Principia Mathematica, and if I remember right, he solved it without using calculus. Basically, what it says is this; if you are inside a spherically symmetric shell of mass, then you feel no gravity pulling you any direction. It’s a bit non-intuitive. Let’s say the Earth is hollow, and the entire planet’s mass has been compressed into a thin spherical shell a few centimeters (or meters, it doesn’t matter) thick. If you are floating around in your Nike Space Suit inside this shell, you will not be pulled toward the center, or the inner surface of the shell, or anywhere else inside the shell. Wherever you are put, you will remain.

Personally, I think this is one of the coolest theorems ever.

It’s also true that if you are outside the surface of a spherically symmetric planet, then it doesn’t matter how dense it is, at a given radius you will feel the pull of a certain amount of gravity. If you are in orbit above the Earth, and the Earth suddenly becomes a black hole of the exact same mass, you will remain in orbit, totally unaffected by that change. That’s pretty cool, too. Given that the gravitational force is based on F=GMm/r2, this should be kind of obvious. Neither your mass (m) or the mass of the Earth (M) has changed, your orbital radius is the same, and G is the gravitational constant. Ergo, the density of the object you are orbiting at a radius “r” from the center of the object is irrelevant.

So, that was me drifting from the actual subject. Shell Theorem—let’s get back to that.

As you might know, the clock of anyone in a gravity field runs slower than that of a clock outside of that gravity field. This is called gravitational time dilation (and is equal to ∆t’ = ∆t √(1-2MG/rc2) for a non-rotating sphere). A person on Earth actually ages slower than a person in deep space, according to relativity. This was verified with clocks flying around the Earth in the Hafele-Keating experiment. Before you ask, yes, they took into account Earth’s rotational speed, the speed of the airplane (in both directions relative to Earth’s rotation) with regard to Special Relativity’s time dilation due to velocity. It was a nice experiment.

Let’s say we’re using a hollow Earth from Newton’s shell theorem. As you get closer to the Earth, you are in a deeper gravity well, and the outside observer sees your clock slow down. There’s a small hole in the planet, and you pass into the planet, where everything is pulling you in opposite directions equally, so you seem to feel no force. And yet, your time dilation effect does not suffer a discontinuity, jumping suddenly to that of the outside observer. You are in a denser gravity field, but a flat gravity field. [to the physics majors out there, for god’s sake, if my terminology sucks, please correct me]. Your time dilation will be just the same as if you were standing on the surface of the planet.

So now you have a flat gravitational field (no “force” pulling you in one direction, that is, all forces pulling you equally in all directions). And yet, even in this apparent lack of gravity, where you can’t actually tell that you’re in a gravity well, your time runs slower than the time of someone far from the planet.

I extended this argument to the rest of the universe. If mass was distributed equally around you, even though you felt no force one way or the other, there would still be a background gravitational field. Gravitational time dilation implies a gradient; for time dilation to be relevant, you need someone in a weaker gravitational field measuring your time. However, both the measured and the measurer can be in locally flat gravitational fields.

Does a flat gravitational field curve spacetime by itself? Or is it only the gradient between two different gravitational fields that curves spacetime? My general opinion is that you don’t need the gradient for the curvature of spacetime. If you have an infinite universe with equally distributed mass, then from some arbitrary center, it will appear to curve spacetime until it closes the dimensions of universe into closed loops, like the inside surface of an event horizon (though other arbitrary centers will have different, yet overlapping event horizons – a subject I will touch upon another day). Likewise, if you have a large, thin flat sheet of soapy water in the air, fluctuations are going to cause it to form bubbles, closing up the edges. In spacetime, there may be a similar tension (gravity?) that closes the edges together into a 4D hypersphere.

How would you test the curvature of space inside a shell? I’m not entirely sure. I think the universe we have is a good test case, however.