The Warping of SpaceTime

One moment you are there, hanging out calmly on the fringes of a remote star system, minding your own business in the near-interstellar vacuum, and in the next there is a vast disturbance of space-time all around you. There is no matter here more dense than a handful of atoms of hydrogen per square meter of space, yet space itself folds at a quantum level.

Rather, space and time alike are unfolding around you, flattening back into normality. You aren’t aware of it having been folded, because you were folded along with it, but there is a definite sign that something dramatic has changed in the local volume.

A starship has arrived.

The starship has traveled a long way, twelve-hundred light-years from its home, yet it has done so over the relatively short time of only three years. That sounds suspiciously like faster-than-light travel, yet it can’t be, because that’s impossible, right? Einstein’s theory of special relativity tells us so.

Yet, Einstein’s theory of general relativity tells us something else. We still cannot travel faster than light in flat space, but space (and time) can be curved.

flat and curved planes
image credit: Dale Gray, PhD Physics, University of North Texas

Indeed, space-time is always curved, regularly distorted by the mass of any objects within it, and we all experience that curvature regularly in our daily life. We experience it as gravity.

image credit: user:Tokamac / under CC BY-SA 4.0

Per special relativity, an object with mass cannot move at the speed of light without requiring infinite energy, and at the same time becoming zero mass.

In general relativity, however, in a curved space-time, the equations allow for a warping or folding of space in such a way that a bubble is formed, and as the bubble moves, space in front of it is highly compressed while space behind it is highly expanded.

image credit: user:AllenMcC / under CC-BY-SA 3.0

The bubble appears to an outside observer as if it has moved faster than light, but really it has just warped space around itself. No matter how much warping occurs, anything inside the bubble experiences no acceleration, and what’s more, no funny time dilation effects (more on this in a later post, I think).

This is not to say that there are no inherent problems with warping space to this dramatic extent. For one thing, in the original equations demonstrating this as a possibility, developed by Miguel Alcubierre in 1994, it appeared that a massive negative mass-energy state is required to generate the bubble. Negative energy implies exotic matter, but more troubling is that the amount of mass-energy required to transport a very small ship across the galaxy would be greater than the mass of the observable universe.

That’s a problem. We can’t have starships destroying the entire universe every time they fire up their warp drive.

In 1999, Chris Van den Broeck modified the equations to require the equivalent of just three solar masses. A great improvement! Now firing up the engine just destroys three star systems. With about 250 billion stars in our galaxy, who’s going to notice a few of them disappearing now and then?

Well, presumably the inhabitants of any planets around those stars will notice, and if we assume that the starship consumes the nearest three stars whenever it starts up the drive, then we could extrapolate that our own Sun (and us along with it) would be the first to go. Perhaps ok if we’re fleeing a Sun going nova about five billion years from now, but in the meantime we’d like to keep our home, thank you very much.

There are other problems with the Van den Broeck solution, and also with Serguei Krasnikov’s modification, despite the latter’s reduction of the mass requirement to mere milligrams. Basically, they require keeping the surface area of the warp bubble microscopically small while expanding the interior volume, which seems contradictory at first glance, and even so, they are only able to transport a few atoms of matter in this way. These are interesting modifications of the equation, but not truly useful.

In 2012, all that changed. Harold White showed (on paper) that modifying the shape of the bubble into a torus, or a rounded doughnut, dramatically reduces the mass-energy requirement into the hundreds of kilograms range, and with this the drive could propel a starship of useful (macroscopic) size… more than just a few atoms. No longer do we need to dismantle a gas giant planet — or a sun — for every voyage. White’s equations excited serious physicists enough that NASA is funding lab experiments to prove the concept.

This is where I would love to insert the beautiful image of IXS Enterprise developed by Mark Rademaker and then used in Harold White’s presentation materials as a concept design for how such an Alcubierre-White starship might look. However, all such images are copyrighted with all rights reserved, and at this time I cannot justify the expense of purchasing rights to display them here. Nevertheless, I encourage you to go view the originals.

So, we’ve solved the energy requirement (ignoring for the moment that we’re talking about negative energy and exotic matter), but issues still remain. Stefano Finazzi, Stefano Liberati, and Carlos Barcelo argued in 2009 that a classic Alcubierre warp bubble might be just fine if it is eternally moving at a stable superluminal speed — which makes it rather difficult for travelers to “hop on the bus,” so to speak — but the switching on of quantum effects to spin up such a bubble from flat space (a more realistic and usable scenario) would create enough Hawking radiation inside itself to completely fry any occupants.

Hawking radiation is strange stuff, and so far remains theoretical in nature. Then again, so does pretty much everything we’re discussing here.

Engineers love a good challenge, however, and we shall assume for the moment that the problem of internal radiation is solvable. After all, you’ve just observed an Alcubierre-White starship arriving in your immediate vicinity.

Luckily for you, you were not in the starship’s direct path when it unfolded space-time around itself and allowed the warp bubble to dissolve. Why?

During the ship’s three-year voyage to arrive here, it has folded up and compressed a great deal of space ahead of itself. While the ship’s navigator carefully plotted a course to avoid any significant obstacles along the way, the vacuum of interstellar space is not completely void of matter. Free hydrogen atoms exist between the stars, and while estimates vary, they average anywhere from one-quarter (0.25) to one-thousand per cubic meter of space (though more likely on the lower end of that range). As our warp bubble transited the Sagittarius Arm, it collected all those atoms on its leading edge, folding them into the highly energetic bubble wall itself.

If the torus has a cross-section of about two-hundred meters, then on the low end of the estimate our starship has pushed about fifty atoms for every meter it traveled. Over three years, the starship traveled twelve-hundred light-years; how far is that in meters?

The speed of light is a hair under 300,000 kilometers per second (you may be more familiar with the number 186,000 miles/second). A light-year is the distance it travels in one year, which is about 31,500,000 seconds, and thus roughly 9.46 trillion kilometers (that’s 9,460,000,000,000 in case you are counting the zeroes). Multiply that by 1200 light-years, and our starship traveled 11.35 quadrillion kilometers.

A long distance indeed.

We gathered 50 hydrogen atoms per meter, or 50,000 per kilometer. I think you see where I’m going with this.

Upon arrival at its destination and deceleration to subluminal (please, autocorrect, stop making that subliminal) velocities, the warp bubble has compressed about 568 quintillion hydrogen atoms.

That’s a lot of atoms.

Of course, atoms are very small. It takes 602,000,000,000,000,000,000,000 (602 sextillion) hydrogen atoms to make up one gram of matter. So, all those quintillions of atoms gathered over the course of three years still only amount to about a milligram of matter.

Surely that’s not enough matter to, well, matter, right? Let’s find out.

It’s only a milligram, but it is moving very, very fast. We know the basic equation:


E = energy
m = mass
c = speed of light

One milligram moving right at the speed of light should be pretty simple to solve for. That is 0.001 g x 300,000 kps squared, or 90 million grams… er, hold on, something funny is happening here. Before we go through all that math, we’ll just approximate a few things. If the milligram of hydrogen atoms are moving at 99.9999% of the speed of light — very, very close, but not quite there — then anything they hit will experience the force equivalent to several kilotons worth of a nuclear bomb. It could definitely ruin your day if your spaceship happened to be right there, but it’s not going to pulverize a planet.

But our atoms aren’t moving at 99.9999% of c. They are moving at c. In our equation, when c = 1, we can simplify it to E = m. Put another way, energy and mass are equivalent, and at the speed of light, our mass becomes pure energy.

When the warp bubble suddenly decelerates, the atoms are ejected off the leading edge as extremely short-wavelength, high-frequency, and thus high-energy radiation, rather than as regular but fast-moving matter. In other words, our decelerating starship emits a gamma ray burst in its forward direction of travel, gamma rays which will easily penetrate almost any shielding and wreak havoc upon anything biological.

For the mathematically- and physics-inclined, I recommend perusing The Alcubierre Warp Drive: On the Matter of Matter by Brendan McMonigal, Geraint Lewis, and Philip O’Byrne.

The starship has a crack navigator, however, and she knows her theoretical physics. A little less theoretical in her case, since she is living it. So, she plots her course not only to avoid large obstacles along the way, but to decelerate far out on the edge of the destination star system. She doesn’t want to pulverize anyone or anything there. So, you survive the starship’s arrival.

There is another minor issue which the starship’s designers had to overcome before sending it on its long voyage. Michael John Pfenning demonstrated in 1998 that for a classic Alcubierre warp bubble there is an inverse relationship between bubble wall thickness and maximum superluminal velocity. At ten times c, the bubble wall can be no thicker than 10^{-32} meters. As the Planck length is 1.6x10^{-35} meters, our bubble wall is almost as thin as the thinnest possible measurement.

As a quick aside, the Planck length in quantum mechanics is the smallest size at which gravitational effects behave rationally. Below this scale, Euclidean geometry ceases to have any meaning, and spacetime becomes quantum foam. That’s a colorful term for saying it is no longer continuous, but has holes in it. So, in effect, the Planck length is the smallest size that anything can be and still effectively be part of our universe.

Our bubble wall is approaching the smallest possible size, and we’re moving at 10 c. That seems pretty fast, and indeed it is pretty fast, but for a voyage of 1200 light-years, and without any time dilation effects, that means… yep. 120 years to complete the voyage.

10 c may be sufficient for the nearest stars, but if our starship is to visit Kepler 62f, either our astronauts must be very patient (and long-lived), or be willing to risk cold sleep, or our ship is somehow going to have to go faster.

Pfenning made his observations during his doctoral thesis only two years after Miguel Alcubierre first published his equations. We’ve already explored how others have built upon the original equations in the intervening time, refining them in many ways so as to reduce the huge amounts of energy required, so it remains plausible that by employing the methods of Harold White and reshaping the bubble into a torus we may also find that we can effectively increase the speed of our starship. Can we do it enough so as to travel a light-year per day, as would be required to reach Kepler 62 in just three years?

I don’t know.

Can we effectively shield the interior of the bubble so that the astronauts are not fried by Hawking radiation or extremely blueshifted high-energy particles?

I don’t know.

So this, my friends, is what in hard science fiction terms we call a McGuffin.

I’ve rambled on far too long about the Alcubierre drive now, and doubtless put many of you to sleep, without ever getting on to the rest of the technology utilized by the crew of Aniara in The Silence of Ancient Light. I’ve also certainly made some significant technical or scientific errors in my attempts to explain (or simply understand) the complex physics and mathematics behind the idea of warping spacetime. For those, I apologize, and indeed, I invite discussion in the comments. Educate me!

If you enjoy these pseudo-scientific ramblings of mine, you may enjoy my previous similar posts of this nature:

header image credit: Les Bossinas / NASA under public domain

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