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.

Two-sided_spacetime_curvatures
image credit: user:Tokamac / wikimedia.org 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.

Alcubierre
image credit: user:AllenMcC / wikimedia.org 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=mc^2

Where:
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

In the Year of ’39

In the year of ’39 assembled here the volunteers,
In the days when lands were few;
Here the ship sailed out into the blue and sunny morn,
The sweetest sight ever seen.

Lately I’ve had this song running through my head, pretty much on constant repeat. It’s an old song, first released in 1975 on the album A Night at the Opera by Queen.

And the night followed day,
And the storytellers say
That the score brave souls inside
For many a lonely day sailed across the milky seas,
Ne’er looked back, never feared, never cried.

At first it seems to be telling a relatively ordinary story. Volunteers set sail in a ship for a dangerous journey. Is it 1939? Is this something to do with World War II? It’s not really clear yet.

Don’t you hear my call though you’re many years away,
Don’t you hear me calling you;
Write your letters in the sand
For the day I take your hand
In the land that our grandchildren knew.

Wait, what? The land that our grandchildren knew? Ok, there is something odd going on here. And what’s this about being many years away? The song seems to be playing around with time.

In the year of ’39 came a ship in from the blue,
The volunteers came home that day,
And they bring good news of a world so newly born,
Though their hearts so heavily weigh;

For the Earth is old and grey,
Little darling went away,
But my love this cannot be,
For so many years have gone though I’m older but a year,
Your mother’s eyes, from your eyes, cry to me.

Right, this is definitely not an ordinary ship sailing ordinary seas, and time is certainly being twisted. The volunteers bring news of a new world, while the Earth is old and grey? We’re talking about space travel, aren’t we? In fact, we’re talking about interstellar travel.

It’s definitely not 1939.

Many music lovers might have been confused by this song, but by now it should be obvious to readers of this blog what’s going on here. For astronauts to travel far enough to discover another world (“so newly born”), one capable of replacing the “old and grey” Earth as humanity’s home, and return back with the news “older but a year,” they must have traveled very fast indeed. Perhaps even approaching the speed of light?

At speeds this fast, the theory of special relativity tells us (and experimental research has shown) that odd things happen with time. Time appears to slow down for the traveler, at least relative to the stationary observer, so that by journey’s end the traveler will have aged far less than those who stayed home.

In the year of ’39 assembled here the volunteers

2139, perhaps? 2239? It’s not completely clear.

In the year of ’39 came a ship in from the blue,
The volunteers came home that day

Not the same ’39, but 100 years later. 2239? 2339?

For so many years have gone though I’m older but a year

As it happens, given the parameters of the song, we can calculate how fast the ship was traveling, and thus how far away they went, and perhaps even speculate what star they visited! This is because, despite being such a non-intuitive phenomenon, time dilation due to relativistic effects is well understood, and there is an equation to calculate it.

t'=t\sqrt{1-\frac{v^2}{c^2}}

Where:

t’ = dilated time
t = stationary time
v = velocity
c = speed of light

We want to know the ship’s velocity, so let’s parse this out (like traveling back in time to algebra class!):

t'^2=t^2\times({1-\frac{v^2}{c^2}})

\frac{t'^2}{t^2}=1-\frac{v^2}{c^2}

\frac{t'^2}{t^2}+\frac{v^2}{c^2}=1

\frac{v^2}{c^2}=1-\frac{t'^2}{t^2}

v^2=(1-\frac{t'^2}{t^2})\times c^2

v=\sqrt{(1-\frac{t'^2}{t^2})}\times c

Ok, let’s plug in some numbers! To keep things simple, we’ll express velocity as a percentage of the speed of light, and time in years, even though normally in physics equations velocities would be meters per second and time in seconds. But at this scale, those would be some big numbers, so we’re going to assume that c=1, and that v therefore is a percent of c.

v=\sqrt{(1-\frac{1^2}{100^2})}\times 1

v=\sqrt{1-\frac{1}{10000}}

v=\sqrt{1-0.0001}

v=\sqrt{0.9999}

v=0.99995

In order for 100 years to have passed on Earth while only 1 year passed for the astronauts, the ship had to be traveling approximately 99.995% of the speed of light. That is some extreme time dilation, and so that is some extreme speed. Quite the starship!

Time isn’t the only thing dilating here, as traveling at these speeds does some interesting things to the fabric of space as well. Distances ahead of the travelers will appear to shrink somewhat, though even at this high fraction of the speed of light, it’s a minimal effect. Add a few more 9s to the significant digits, however, and it gets very strange indeed.

Meanwhile, though, our travelers have spent a year journeying at very close to the speed of light. How far have they gone? One light-year?

Oh no. They’ve gone much farther than that. The distance traveled is at a speed relative to time for the stationary observers waiting patiently back on Earth, so our starship has traveled a hundred light-years, though it seems to the astronauts to take only one year to do so.

So, what star might they have visited to find a “world so newly born” to which humanity could relocate? First off, this was a round-trip, so with half the time spent journeying out and half spent returning, that would imply they went no more than fifty light-years away (“no more than,” I say, as if this is no big deal, but fifty light-years is a very big deal). Gliese 163 is 49 light-years away and has one potentially habitable world, but it’s not considered an absolutely prime candidate.

Let’s assume, for a moment, that our starship took a little bit of time to accelerate and then decelerate on its journey, so that instead of 50 light-years, perhaps it really only traveled about 40 light-years away.

They went to Trappist-1.

Artist’s impressions of the TRAPPIST-1 planetary system
image credit: ESO/M. Kornmesser (https://www.eso.org/public/images/eso1805a/) under CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

Trappist-1 is a cool red dwarf star 39.6 light-years away, and it has seven temperate and terrestrial planets, four of which are considered potentially habitable even by conservative estimates. Trappist-1 is obviously a prime candidate for finding life, or at least worlds on which humans could live, and being at about the right distance, is also a prime candidate for our volunteers on their desperate and lonely journey.

Not that this provided much consolation to our narrator, who returns to Earth to find his wife long dead, and only a memory of her in the eyes of his (presumably centenarian) daughter (or granddaughter?).

Your mother’s eyes, from your eyes, cry to me.

’39 was written by Brian May, lead guitarist for Queen, in 1975. May, as some of you may know, is also an accomplished astrophysicist, and while the planets of Trappist-1 had not yet been discovered in 1975, he certainly understood the effects and impacts of time dilation on travel at relativistic speeds. May studied physics and mathematics up through the time when his music career began to skyrocket to success, though due to focusing on music after that point, it took him 37 years to complete his doctoral thesis (A Survey of Radial Velocities in the Zodiacal Dust Cloud), finally earning his PhD in 2008. He was Chancellor of Liverpool John Moores University from then until 2013, and he was a science team collaborator for the NASA New Horizons mission to Pluto.

Not just a fantastic guitarist and songwriter, but a serious scientist!

Don’t you hear my call though you’re many years away,
Don’t you hear me calling you;
All your letters in the sand cannot heal me like your hand,
For my life
Still ahead
Pity me.

’39 (Youtube)

Final note: Although Freddie Mercury is far more well-known as lead vocalist, it was Brian May who sang the lyrics for the studio version of ’39 (though Mercury sang for most of their live performances).


’39 from the album A Night at the Opera by Queen, 1975
Songwriter Brian May, copyright EMI Music Publishing, Sony/ATV Music Publishing LLC

Special thanks to E=mc2 Explained for breaking down the physics of time dilation for us laypeople.

Header image credit: user:Les Chatfield / flickr.com under CC-BY 2.0