Reythia asked me to write something uplifting, so I copied a recent conversation about general relativity and hypothetical sources of energy.

**Written by Dumb Scientist on 2012-07-05**

Excellent comment. Just to take it further, active gravitational mass in general relativity is defined by the stress-energy tensor, which also has pressure components. That implies tension has **negative** gravitational and inertial mass because tension is just negative pressure. Greg Egan uses this concept masterfully in a short story called *Hot Rock*, which is set in the same universe as *Glory*, *Riding the Crocodile*, and *Incandescence*.

**Written by Someone on 2012-07-19**

The pressure of a ball of hot gas (like the sun) is a significant part of gravity because each atom and electron wizzing around and banging into things inside of it, has an energy component which strengthens the gravitational field (this is expecially true in very hot areas, like stars, where particles are moving very fast).

The energy comes from the kinetic energy, or the momentum. The kinetic energy equation is e =.5mv^2 (does that equation remind you of anything?).

Now, while a colder chunk of matter might have less kinetic energy. The MASS energy (e=mc^2), is is still pretty damn huge, and more then enough to have normal gravitational effects. In addition, even if a particle achieves a negative velocity, by moving slower than the expansion rate of the universe, it probably STILL won’t have any negative kinetic energy. Why? Kinetic energy equals half the mass, times THE SQUARE of the velocity. The square of any number — negative or positive — is always a positive number.

**Written by Dumb Scientist on 2012-07-21**

The pressure of a ball of hot gas (like the sun) is a significant part of gravity because each atom and electron wizzing around and banging into things inside of it, has an energy component which strengthens the gravitational field (this is expecially true in very hot areas, like stars, where particles are moving very fast).

You’re describing the energy density, not the pressure. The stress-energy tensor **T** has 16 components, each with a different physical interpretation. The 4 rows and 4 columns represent the time(^{0}), x(^{1}), y(^{2}), and z(^{3}) dimensions, respectively. The energy density is the time-time component (called **T ^{00}**).

The energy comes from the kinetic energy, or the momentum.

The stress-energy tensor does have three components describing momentum density in some volume, and three describing momentum flux through the surface enclosing that volume. But they’re separate from the **T ^{00}** energy density component.

The kinetic energy equation is e =.5mv^2 (does that equation remind you of anything?).

It reminds me of the first non-cancelling term in a Taylor expansion of the relativistic kinetic energy equation. In other words, your equation is only a good approximation at speeds much slower than light speed. Either way, you’re still describing **T ^{00}**. That’s understandable; it’s the only component that has an analog in familiar Newtonian gravity. But note that some of the other 15 components have counter-intuitive (and possibly useful) physical consequences.

Now, while a colder chunk of matter might have less kinetic energy. The MASS energy (e=mc^2), is is still pretty damn huge, and more than enough to have normal gravitational effects.

Of course. But note that in general relativity the gravity of a rope decreases very slightly when it’s put under tension because of the more negative pressure component (**T ^{11}**,

**T**, or

^{22}**T**) in the stress-energy tensor. This effect is independent of temperature and kinetic energy, which are accounted for by the energy density and energy flux components.

^{33}In addition, even if a particle achieves a negative velocity, by moving slower than the expansion rate of the universe,

*Huh?*

… it probably STILL won’t have any negative kinetic energy. Why? Kinetic energy equals half the mass, times THE SQUARE of the velocity. The square of any number — negative or positive — is always a positive number.

That’s the Newtonian approximation, but I agree that kinetic energy is *definitely* non-negative either way. You’re also still describing **T ^{00}**.

Here’s why I’m babbling about the other 15 components. Nuclear fission (or fusion) only releases ~0.1% (or ~1%) of the fuel’s mass-energy. Matter-antimatter annihilation releases * 100%*, but it’s not a

*source*of energy because we haven’t found naturally occurring antimatter. Even if we find an antimatter asteroid, storing some in a magnetic fuel tank would be hazardous.

We might beat fusion’s ~1% yield using a (preferably rotating) black hole, but first we need to acquire that black hole. A more hypothetical (and more portable) idea is the antimatter reactor Geoffrey Landis describes in *Approaching Perimelasma*. In it, a microscopic twist of spacetime parity-reverses The symmetry between matter and antimatter is CPT: charge, parity, and time. Parity reversal alone would turn matter into mirror matter, not antimatter. Also, Landis’s reactor would change the universe’s baryon and lepton numbers, which have been approximately conserved since the GUT epoch ended 10^{-36} second after the Big Bang. However, converting hydrogen to anti-hydrogen wouldn’t change the baryon minus lepton number, which is more rigorously conserved. matter into antimatter as needed. This would eliminate the hazards of storing antimatter *and* make it a real source of energy. Unfortunately, free conversion of matter to antimatter might be impossible. Is there any other way to beat fusion?

*Maybe.* The three pressure components (**T ^{11}**,

**T**,

^{22}**T**) in the stress-energy tensor imply that placing an object under tension decreases its active gravitational mass, which must equal its passive gravitational mass to conserve momentum. The equivalence principle says that passive gravitational mass equals inertial mass. So placing an object under tension decreases its inertial mass, which seems to imply a decrease in its invariant mass. This mass defect should yield energy like fusion’s mass defect does.

^{33}The maximum yield might be even smaller than fusion’s, which would be disappointing. But in *Hot Rock*, Greg Egan explores the consequences of femtotechnology that can turn heavy nuclei into spinning hoops. They spin so rapidly that each hoop’s tension reduces its total energy *much* more than its (rotational) kinetic energy adds, leading to a mass defect of * 90%*.

A civilization using that energy could *outlive the stars*. One day our descendants may find out if that energy *can* be used. Hopefully this discovery happens before they pass the point when they should start trying to turn heavier, brighter stars into longer-lived red dwarfs, just to get a *little more* of fusion’s ~1%.

The circuitous route I took from active gravitational mass to invariant mass was awkward, but it let me think about the principles that link different types of mass. I’m still not sure how inertial mass is linked to the invariant/intrinsic/rest mass involved in a mass defect.

My explanation started with active gravitational mass because that was relevant to earlier comments by ShakaUVM and Chris Burke. Greg Egan avoided these complications by directly calculating a hoop’s total energy.

Egan’s solutions satisfy the weak energy condition, guaranteeing that the hoop’s tension-induced mass defect can’t be greater than 100%. He finds that the speed of sound in the hoop is a stronger constraint: it approaches the speed of light when the hoop is only stretched about 57% as much as the weak energy condition forbids. His wave packet animations suggest that even the group velocities can be superluminal, implying paradoxical superluminal signals. A short conversation in

Hot Rockhints at a way to avoid this serious problem.Could shear stresses Probably not, now that I think about it. The signs of shear stresses don’t seem physical. produce a similar mass defect? What would change if we used the stress-energy-momentum pseudotensor which includes the energy-momentum of gravity?

Nice mass properties graph here.

(Ed. note: These comments were originally posted starting on 2010-04-25. The conversation started when Stephen Hawking warned about the dangers of contacting aliens.)It seems like an alien race with “sufficient technological capability” that evolved on a terrestrial planet would probably prefer to build swarms of O’Neill cylinders rather than nuking and terraforming terrestrial planets. Consider that:

muchmore surface area. Planets are theleastefficient way of using matter to provide habitable surface area, by many orders of magnitude.Rare elements on planets aren’t conveniently placed; they’re at the bottom of a gravity well. For this reason, elements in asteroids are more easily exploited by any civilization with sufficient experience in spaceflight. It’s cheaper for us to mine the dirt under our feet because our space program is in its infancy (so just getting to an asteroid is hard), and because ~99.999% of mined minerals are used to build devices for use at the bottom of this gravity well anyway. But any civilization with a “mature” spaceflight capability would mine asteroids in order to construct spacecraft. In addition to not being trapped in a gravity well, asteroids are already in a zero-g vacuum which is a superior environment for fabricating devices with nanometer-scale features.

Planets are useful heat sinks, though. Industrial processes that generate a lot of heat would benefit from being able to dump that heat into a planet rather than building liquid droplet radiators which seem like the most efficient theoretical radiators at the moment.

Yes, I agree. But there are

manypeople who won’t want to live like that, and even I would want to keep backups of Earth’s biosphere (for nostalgia or the possibility that we still have things to learn from studying it). So biological life probably isn’t going away. I tend to agree with Greg Egan when he suggests that this diversitywithineach species is greater than the differencesbetweenspecies. That is, each species likely has some members (individuals, or sub-hive minds, or whatever) who are willing to download into a computer, and some who wish to live in their ancestral forms.(Ed. note: this comment was copied from here.)I’ve daydreamed about ways to harness the Casimir force. Suppose one plate is made of vanadium [1], a superconductor cooled to delta_T below its critical temperature (T_c = 5.03K), and the other plate is made of niobium (T_c = 9.26K) which is held at (5.03 + delta_T)K. The Casimir force will pull the two plates together, and when they touch the vanadium plate will lose superconductivity as it’s warmed past its critical temperature. This weakens the Casimir effect, allowing the plates to be pulled apart with less force, where the vanadium plate will eventually cool enough to superconduct again. Repeating this process could turn a crankshaft [2] just like a piston in a gas engine. I wish I had time to figure out if this system would actually break the Carnot efficiency limit…

Footnotes[1]Pinto 1999 (patented) considers plates made of semiconductors, which have finite conductivity and a correspondingly large skin depth. Semiconductors are created by doping a silicon crystal lattice with impurities like phosphorous, which also makes their properties non-uniform at the atomic scale. Superconductors made of just one element would address both concerns. Type II superconductors have second order phase transitions, so passing the critical temperature (T_c) doesn’t involve latent heat. There are just two stable elemental type II superconductors: niobium (T_c = 9.26K) and vanadium (T_c = 5.03K). The niobium plate’s heat capacity should be larger than the vanadium’s, so the vanadium plate is warmed past its critical temperature upon contact. The niobium plate remains below its T_c, so it could also be a type I superconductor like lead or beta-lanthanum. ↩ back[2]The plate motion could also be turned into electricity via piezoelectric crystals or molecular motors. The plate temperatures could be controlled with Peltier coolers. Fortunately, vanadium’s critical temperature is higher than 2.73K, so its heat can be radiated to space. In practice, one “plate” should probably be part of a sphere, because aligning parallel flat plates is difficult when they’re very small. ↩ backI emailed this idea to a few scientists, saying: My gut tells me that vacuum energy can’t be made to do work. But I don’t see an obvious, fatal problem with the following scheme. Do you?

In response, Geoffrey Landis pointed out that there

willbe a latent heat at the phase transition, even for type II superconductors. He uses the same reasoning that explains why magnetic fields cause latent heat. In other words, Casimir-induced latent heat will exactly cancel the net work done by the plates, which seems to forbid this free lunch…Here’s a Sage worksheet (PDF) which calculates Newtonian and relativistic travel times under continuous acceleration.

In 2014, “Ascension” depicted a starship that accelerates continuously at 1g, halfway through a 100 year mission to Proxima Centauri, ~4.3 light years from Earth.

The producers should have listened to Carl Sagan. At 1g, a ~4.3 light year voyage would take ~3.5 years as measured on the ship, and ~5.9 years on Earth. The ship’s maximum speed would be ~95% light speed. Not even a nuclear fusion drive would have a high enough specific impulse.

More importantly, there’s a good reason why Sagan said “a few years” rather than a few decades. Suppose Ascension’s crew started decelerating only 14 ship years into that 1g voyage. The voyage would take a total of 28 years on the ship, but they’d have travelled 2,500,000 light years! That’s far enough to leave the Milky Way galaxy altogether and visit Andromeda, hitting a maximum speed of ~99.99999999997% light speed. Even if intergalactic space were completely empty, that’s still fast enough to blueshift the CMBR into lethal radiation.

Crossing the entire observable universe (~93 thousand million light years) would only take a total of ~49 years on the ship, so they should have started decelerating before year 25.

Ascension’s 100 year voyage would actually cross ~10

^{23}light years, which is about a trillion (i.e. million million) times as large as the observable universe. Not a measly ~4.3 light years. In over 50 years, no scientist on Ascension noticed this discrepancy?Ascension’s voyage would take 100 years as measured on the ship, but ~10

^{23}years would pass in the rest of the universe. Barring godlike “stellar rejuvenation” technology, our Sun would be long dead. Even long-lived red dwarfs like Proxima Centauri would have died. Ascension’s crew would be marooned in a dark universe, lit only by Hawking radiation, proton decay (?) and a much fainter CMBR.Ascension was still interesting, aside from that. Sorry for using big “G” rather than the correct little “g” in that tweet.