Tag Archives: physics

Life, the Universe, and Everything


(Click here for the PDF version of this presentation.)

Math is everywhere, hidden in places where we don’t even expect to see it. For example, take a look at the following image:


What do you see?

Most people say “music.” People who have studied the piano might recognize this as a piano score. And a true enthusiast might recognize it as the third movement of Beethoven’s Moonlight Sonata.

What you’ve probably never thought of before, though, is that a musical score is actually a form of graph. It tells the performer what combination of notes to play at a given moment in time. In other words, it shows sound as a function of time.

In the image below, I’ve added labeled axes to draw attention to this:moonlight_sonata_graph

Now consider a photograph. Below is one of the most spectacular images I found when Googling “photograph.” (Thanks to whoever posted it!) I love how it shows the strings of mucus frozen in time.


Anyway, a photograph itself is also just a type of graph — and not just metaphorically. In fact, even the way images are produced in our brains is just a way of numerically graphing the intensity and frequency of light that falls on different portions of our retinas. In essence, your retina is the x-y plane and the light is the quantity being graphed.

Below is what the photograph looks like when graphed in three dimensions from different angles, with the colors changed to a different color scale:


Now here is the same graph when viewed from directly above, so that the tiger is easier to make out:

Here’s another example of a great photo:

And here it is with the same procedure applied to it. This one works a little better than the tiger because it isn’t filled with little white spots that end up looking like noisy spikes in the graph.frog_photo_graphs_1

Below is the graph when viewed from directly above, just as I did for the tiger. Pretty cool, huh?frog_photo_graphs_2

Now consider something that really seems to have nothing to do with math: a piece of literature. Below is the first paragraph from A Tale of Two Cities, by Charles Dickens.

It, too, can be considered as a type of graph. It’s a graph that tells the reader what words to speak or think as a function of time:Tale_of_2_cities_graphThere are, of course, many other examples of graphs:


What I’m saying is that anything can be thought of as a kind of graph. Really, though, it’s not just graphs that are so powerful, but numbers themselves. This is because numbers encode information. For example, an entire song can be encoded in a single number. So can a photograph, or even a movie.

What’s particularly fascinating is that physicists now believe that physical reality itself is composed of information. In fact, the universe might even be digital. And since numbers encode information, it is possible that the entire universe could be represented by a single number.

Take a minute to meditate on that.


If that’s true, then there’s only one thing we can conclude…


* * * * *

This post is based on a PowerPoint presentation I made for my math students in an attempt to inspire them. Here it is in PDF form:

Math Is Everything (PDF version)

Trespassing on Einstein’s Lawn


Trespassing on Einstein’s Lawn, by Amanda Gefter, is a memoir of a girl on two simultaneous quests. One is to find the answer to the ultimate question about life, the universe, and everything—namely, why is there something rather than nothing? The other is to publish a book about it and become a legitimate, big-time author. She appears to have succeeded to a significant degree on both counts and does a good job telling the story about how she did it.

Gefter’s treatment of the physics is an excellent demonstration of why popular science writing shouldn’t be left to scientists alone. Whereas scientists tend to be wrapped up in their own particular theories (e.g., string theory), a good journalist is better positioned to take a step back and make an unbiased assessment of what all the different theories out there are saying (even if the lack of bias is partly due to a lack of technical understanding). Gefter has attempted to do this, both by studying cosmology extensively on her own and by interviewing the big players in the field.

By taking this approach, she has done a more convincing job than anyone else has yet done—as far as I know—of presenting a satisfying explanation of how everything that we experience (i.e., life, the universe, and everything) can truly come from nothing (and actually be nothing), even though we perceive it to be something. Using qualitative conceptual arguments, she presents a compelling case for how the universe arises from nothing, without requiring any external laws of physics, such as quantum mechanics or general relativity, to exist a priori to govern the behavior of the nothingness.

Her central thesis, as I interpret it, is that something can’t be fundamentally “real” unless it is invariant—that is, unless it exists in all reference frames. In other words, if a reference frame can be found in which a thing doesn’t exist, then that thing is not “real.” She begins with a list of candidate components of reality, such as space-time, particles, fields, and forces; and in the course of her interviews with the most respected physicists of our time, crosses each item off the list.

In the end, with everything crossed off of her list, she concludes that the universe is ultimately made of nothing—which is the only philosophically satisfying conclusion anyway. It is very important to note here that what Gefter means by “nothing” is not space, but actual nothingness—devoid of all properties whatsoever, including any set of governing mathematical rules. With the help of a certain physicist, she even goes so far as to suggest how the mere imposition of a boundary—which is itself nothing—creates information from nothing. This emergence of information evidently initiates a cascade from which everything emerges.

It is reminiscent, Gefter notes, of how, in the field of mathematics, the entire set of real numbers can be constructed from the empty set alone—i.e., from nothing. She also mentions other pleasing analogies with mathematics that can be drawn, not the least of which is the possibility that, in the spirit of Gödel’s incompleteness theorem, no set of physical laws that can be constructed within physical reality could ever give a complete, external description of physical reality.

Gefter doesn’t spend much time talking about Zen, but she does mention it, and one cannot help but think that she has given a rather compelling case for the truth of Zen from an actual physics perspective. Nothing is everything, and everything is nothing. And no system can observe itself, for it would then cease to exist upon making the observation.

My own analogy is that a blank canvas, which contains nothing, also contains everything—in more than one sense. While still blank, the canvas retains the potential to become any of the infinitely many possible paintings that could conceivably be painted on it. Moreover, if you actually do paint every last one of the infinitely many possible paintings on the canvas, it will become blank again (i.e., pure white or black, depending on the medium). Perhaps our universe is just one of the infinitely many possibilities that exist simultaneously on the blank canvas of nothingness.

Though Gefter’s case is conceptually compelling, it is neither academically rigorous nor airtight, and she acknowledges this. Should some component of the universe be shown to be truly invariant, her core thesis would go out the window. And there are still a few points about which I don’t feel satisfied. For example, why can observers (which don’t necessarily have to be conscious) exist within nothing? (For supposedly it is the observer who creates the boundary that gives rise to information.) Can we really talk about “observers” and “boundaries” within nothingness without invoking some sort of governing system of definitions and rules? Are we really talking about nothing then?

The other part of Gefter’s quest—to become a bona fide science writer—is interesting and inspiring in itself. It’s not a rags-to-riches story, but it’s an excellent example of how it can be possible to attain seemingly unattainable goals. That’s what the title is really about. In the beginning, she was neither a scientist nor a writer. And so she was not just literally trespassing on Einstein’s lawn when she visited his old house in Princeton; she was, in carrying out her quest, venturing into territory where she didn’t rightly belong. But in the end, she earned her spot on the lawn.

Gefter’s success comes from four components that I have been meditating on recently: passion, discipline, assertiveness, and luck. Throughout her quest, she maintained a very intense passion, largely instilled by her father, for the ultimate questions about physical reality. She also demonstrated the discipline to stick to that quest over a period of several years—attending conferences, writing articles one at a time, interviewing physicists, keeping a detailed journal, and ultimately sitting down to write out the book itself. Assertiveness played an important role when, in situations where most people would have thought there was no hope trying, she nevertheless called up high-profile scientists and publishers to try to get her foot in the door—and it worked. Finally, of course, she had plenty of luck. More than anything else, she was lucky to have a father who planted and cultivated the seed of her passion and then provided the financial, intellectual, and emotional support necessary for her to set off on her quest.

Throughout the book, Gefter draws many clever parallels between the mysterious physical phenomena that she is investigating and things that are going on in her personal life. For example, physicists’ conclusion that it is inherently impossible to construct a consistent description of the universe that takes into account more than one point of view at a time—i.e., you can only have one observer—was a nice parallel with her reluctant decision to write the book on her own after the publisher rejected her proposal for a book by a father-daughter duo.

I did grow a bit tired of Gefter’s self-deprecating refrain about how she was an impostor, a fake who didn’t deserve to be present at the conferences she attended or at the magazines where she worked. I also grew a bit tired of the “Oh my God, I’m in the presence of one of the greatest physicists who ever lived!” exclamation that seemed to accompany every single interview she did. And I didn’t appreciate the dig she took—which seemed rather mean-spirited to me—at a waiter who said that he had also majored in philosophy of science. She was making an attempt to draw another clever parallel, contrasting the success she hoped to achieve with the waiter’s apparent lack of success; but the net effect was to shatter the image of humility she seemed to be working so hard to create throughout the rest of the text.

Nevertheless, Gefter did a fantastic job documenting her quest, and the overall picture that she paints of the present status of cosmology is far more satisfying than any other I’ve read—precisely because she takes into account many different physicists’ points of view. She begins by saying:

Reality is a huge mystery, and you have a choice to make. You can run from it, you can placate yourself with fairy tales, you can just pretend everything’s normal, or you can stare that mystery in the eye and try to solve it. If you are one of the brave ones to choose the latter, welcome to science. Science is the quest to solve the eternal riddle.

Then, at the end, she is able to say, “Physics isn’t the machinery behind the workings of the world; physics is the machinery behind the illusion that there is a world.” Before reading the book, I would have dismissed this statement as pseudo-philosophical mumbo-jumbo based on feelings that surely have no grounding in an actual understanding of physics. But I am now convinced that it may be a deep and legitimate conclusion indeed—and I am very glad that I took the time to read her story.

(Thanks to my father for giving me the book.)

The Real-Life Search for Aliens

Absorption spectroscopy, from Coel Hellier's blog post

Diagram explaining absorption spectroscopy, from Coel Hellier’s blog post

In a previous series of posts, I presented a simple analysis of our chances of ever contacting aliens. The foundation of my argument was that any planet that does harbor intelligent life will almost certainly be too far away, not just for us to reach by spaceship, but even for us to contact with any type of signal. Thus, although the universe may be teeming with life, we’re never going to see any of it (except what’s right here on Earth).

Today, I ran across this blog post by Coel Hellier, an actual scientist who searches for the signatures of extraterrestrial life (though I think that’s not the only goal of his work). In his blog, he does a great job of explaining in layman’s terms how real-life searches are actually carried out, from identifying stars that have planets orbiting them to analyzing the chemical composition of those planets’ atmospheres. I encourage you to read his original article, but I’ll give a brief summary of it below in case you’re too lazy to click a mouse.

  • We can identify a star that has a planet orbiting it by the periodic dimming of the star as the planet passes between us and the star, blocking part of the star’s light.
  • The planet’s size is determined from how much of the star’s light it blocks.
  • Its mass is determined by measuring the redshift of the star’s light when the planet is behind the star (pulling the star away from us) or the blueshift of the star’s light when the planet is in front of the star (pulling it toward us).
  • From size and mass, we get density, and from density, we get a rough idea of what chemical elements are most abundant on the planet.
  • We can even determine what molecules are present in large quantities in a planet’s atmosphere by looking at the fringe of light that passes through the planet’s atmosphere as the planet passes in front of the star. Different molecules will absorb light from different parts of the spectrum of the star’s light, so all we have to do is compare unimpeded light from the star to light that passed through the planet’s atmosphere to determine what types of molecules are in the planet’s atmosphere. This is called spectroscopy.

Coel writes that, using spectroscopy,

we are beginning to be able to detect the atmospheres of extra-solar planets, despite them being hundreds of light-years away. If we can detect molecules in the atmospheres of exoplanets then, in principle, we might detect “biomarker” molecules that indicate organic activity (such as free oxygen). Thus it is realistic that, within a couple of decades, we will have found other Earth-like planets that we know to bear life.

The beauty of this approach is that with an array of sensitive telescopes that can record light from stars all over the sky, we will be able to use automated analysis software that will process the data from thousands or millions of stars in a relatively short period of time. Thus, it might actually be possible to verify the presence of life on other planets in a relatively short time frame. Exciting!

Having said all of that, it is still the case that even if we do find planets that harbor life, they will almost certainly be hundreds if not thousands of light-years away from us. We can try sending them a signal, but even on the off-chance that the life there is advanced enough to detect and decode our signal, it would be hundreds or thousands of years before their reply ever reached us.

Thus, I maintain for now my pessimistic conclusion that we’ll never contact aliens.

How to Move the Earth

earth-rotationIf you’re like me — and who isn’t? — then you’ve often wondered how traveling around on the surface of our planet affects its rotation. If you haven’t ever wondered this, then I suggest you go in for a psychiatric evaluation, because no sane person should be able to go for a walk without worrying about the cosmic ramifications of every step he or she takes.

The Earth rotates toward the east; and when you’re standing still with respect to the ground, you’re actually moving along with its surface. If, however, you decide to start walking east, the initial steps you take to get moving will push westward against the Earth, slowing its rotation ever so slightly. You’ll be stealing a bit of the Earth’s angular momentum. If you go west, your feet will push eastward against the Earth, and you will actually speed up its rotation.

The question on my mind — and on yours too, I’m assuming — is how much the Earth’s rotation will be affected if you travel eastward all the way around the world and come to a stop right back where you started. During the trip, the Earth will be rotating at a slower-than-normal rate; and when you come to a stop, it will return to its normal rate of rotation. As a result of this period of slowed rotation, each point on the Earth’s surface will now lag behind where it would have been otherwise. Sunrise will happen a little bit later for everyone.

Well . . . how much later?

This question burns, doesn’t it? Well, relax, because we’re going to answer it right here. We just need to find expressions for the total angular momentum of the system (which consists of you and the Earth) for when you’re standing still and when you’re traveling. By conservation of angular momentum, we can set these two expressions equal to each other and solve for the Earth’s reduced angular velocity during the trip. From this, we can then determine how much lag the Earth will accumulate. Piece of cake!

For the curious, I’ve written out the full solution here, in PDF form. Below, I’ll spare you the calculations and just present the results.

For a 60-kg person making the trip, the amount of time by which sunrise will be delayed is 2.17 attoseconds. In case you’re wondering what the heck an attosecond is, it’s 1/1,000,000,000,000,000,000 of a second, which is roughly how long it takes a beam of light to travel the length of three hydrogen atoms lined up against each other. In other words, it’s a very short time.

One interesting thing about this answer is that it is completely independent of how fast you make the trip. Whether you zip all the way around in less than a second or crawl along over a period of several years, the net effect will still be a delay in the Earth’s rotation of 2.17 attoseconds. It’s your mass, not your speed, that determines how big the resulting delay will be.

If you gained a bit of weight, you would have a bigger impact. We might ask, for example, what your mass would have to be in order to delay the Earth’s rotation by a full second. It turns out you would have to weigh a hefty 27.6 quintillion kilograms, which would require a large number of trips to McDonald’s and is not something you should aim for.

So now we know how the Earth’s rotation is affected each time a person circumnavigates the globe. It’s a small effect; but just to be safe, whenever you take an intercontinental trip, you should probably return the way you came rather than going all the way around the world. Keeping track of time is difficult enough with Daylight Saving Time.

Will We Ever Contact Aliens? A Physicist’s Analysis (Part II)

space_laserIn my previous post, I calculated how much power it would take to send out a signal that would be detectable at our nearest neighboring star, Proxima Centauri. It was equivalent to the output of a large power plant. I then pointed out that by the time this signal reaches any stars that are farther away, it will have dissipated to an undetectable level. I concluded by promising to examine other possibilities in the next post. Here they are.

Rather than sending out a signal in all directions, a better strategy would be to use something like a laser, focusing the signal into a beam that doesn’t spread out much over large distances. Then the signal would still be fairly strong when it reached some distant planet.

Whoa! Brilliant idea. Problem solved, right?

Unfortunately, no. The problem with this is that you can only point a laser at one star at a time (or a relatively small group of stars). No one is sure what the chances are that a given star has a life-supporting planet orbiting it, but one thing people agree on is that it’s a pretty tiny probability. And so we run into the problem that if we focus our laser on a single star, chances are almost zero that it’s a star with life orbiting it. In other words, our signal is almost certain to go undetected.

Calculations based on the popular Drake equation suggest that there’s most likely quite a bit of intelligent alien life out there in the universe. That sounds pretty exciting. But it leads naturally to a very obvious question, which Enrico Fermi is famous for asking: “If that’s true, then where is everybody?”

The contradiction between the conclusion that the universe is almost certainly teeming with life and the fact that we haven’t seen any evidence of extraterrestrial life has become known as Fermi’s paradox. Is it just the case that there isn’t life out there, or is there some other resolution to this paradox? People have been asking this question for some time.

Well, our simple calculation suggests that the answer to Fermi’s question is rather simple. “Everybody” else out there is in the same situation we’re in: floating on a rock so isolated by its distance from the rest of the stars that it’s impossible to travel to or communicate with even the nearest neighbor. And so it may turn out that even if the galaxies are positively teeming with life, we might as well be alone for all practical purposes.

But wait. There’s at least one thing we haven’t considered. Using spectroscopy, it may be possible to identify planets that are likely to have life on them. We can do this by analyzing light from distant planets to determine what kinds of chemical compounds are there (that’s what spectroscopy is). Assuming that extraterrestrial life is based on familiar chemical processes, the detection of organic molecules on a planet would indicate a good chance for the presence of life. Then we can aim a laser at it and try to say hello.

There are still two problems here. First, even among planets that have organic molecules on them, only a tiny percentage could be expected to have intelligent life forms that are advanced enough to detect and respond to such signals. And second, most of the candidates are much farther away than our nearest neighbor, which is already over four light-years away. That means that it would take years for any aliens to receive our signal, and then just as many years for us to receive a response.

Even within our own galaxy, the majority of stars are not just a few light-years away, but thousands of light-years away. Thus, if someone does detect and respond to a signal that we send now, the response probably wouldn’t arrive in time for our grandchildren to receive it. In fact, by the time the response arrives, it’s likely that no one on earth would even remember that we sent a signal to begin with.

And so my conclusion about interstellar communication is, sadly, the same as my conclusion about interstellar travel: Barring some truly revolutionary discoveries in physics, it will remain nearly impossible. That is probably why we have never heard from anyone, even if there are countless alien civilizations out there.

Nevertheless, I will still examine in my next post what might happen if aliens ever do happen upon Earth.

Will We Ever Contact Aliens? A Physicist’s Analysis (Part I)

OLYMPUS DIGITAL CAMERAIn a previous post, I calculated how much energy it would take to travel to Proxima Centauri, the nearest star outside of our own solar system, within a reasonable amount of time. The results were rather discouraging; barring any monumental revolutions in physics, energy considerations alone suggest that interstellar travel might be downright impossible.

So let’s set aside thoughts of space travel and consider a far more modest project: merely broadcasting a signal to the stars (in hopes of contacting intelligent alien life, assuming there is any). Surely, simply sending a signal would be much easier than transporting a massive spaceship over such a long distance. So let’s see what it would take.

First, we need to define the problem.

Suppose our goal is just to send out a signal that is detectable on Proxima Centauri, our nearest neighbor. And suppose that we have at our disposal a transmitter that sends a uniformly intense signal out in all directions. The question we wish to answer is how much energy it would take to generate such a signal.

To complete the setup of the problem, there are two things that we need to specify:

  1. What exactly does it mean for a signal to be “detectable”?
  2. What is the nature of this signal? (Visible light? Radio waves? X-rays?)

Let’s tackle the first question first, and let’s be optimistic about aliens’ signal-detection capabilities. We’ll assume that aliens can detect our signal if it consists of at least one photon (a particle of light) per second flowing through one square meter of area when it reaches the aliens’ location. This is actually an über-weak signal, but as I said, we’re going to be optimistic here.

The second question is important because the amount of energy carried by each photon depends on the nature of the signal. If it’s a radio wave, which is low-frequency, then the amount of energy required is relatively low. Microwaves, visible light, and X-rays have higher frequencies and would require more energy. To keep the requirements low, let’s assume our signal consists of radio waves; and to keep the numbers simple, let’s suppose these waves have a wavelength of one meter.

We now have enough information to solve the problem.

Since the signal is being sent out uniformly in all directions, it is essentially a sphere that’s expanding at the speed of light, with the Earth at its center. Assuming the photons are uniformly distributed over this sphere, and keeping in mind that we want there to be one photon per square meter, we simply need to calculate the surface area (in square meters) of this expanding sphere when it arrives at the destination, Proxima Centauri.

Well, Proxima Centauri is 4.24 light-years away from us, so when the signal arrives there, the radius of the sphere is 4.24 light-years. Calling this distance R, the formula for the surface area of a sphere tells us that a total of 4*pi*R^2 square meters must be covered. And if we want one photon to pass through each of those square meters per second, that’s also the number of photons per second that our transmitter must send out. (Note that we have to convert R to meters.)

That’s 2×10^34 photons.

The amount of energy per photon is E = hc/L, where h is Planck’s constant and L is the wavelength of the radio waves (and c, of course, is the speed of light). So the total amount of energy (per second) is just hc/L times the number of photons, which works out to be 4 billion joules. Since that’s the amount of energy per second, the amount of power is 4 billion Watts.

All right. So what?

Well, that’s about the output of a large power plant, which is a lot of power to put into one signal. And that signal will thin out to just one tiny photon per square meter per second by the time it reaches our nearest neighbor in the universe. In reality, it would have virtually no chance of being detected, even if someone were looking for it with highly advanced technology. Farther away, the signal would be even weaker.

So the sad truth is that even if we devoted a huge amount of energy into attempts to contact extraterrestrial life forms, our signals would dissipate to undetectable levels long before they reached any of the distant planets that might harbor life. We can conclude from this analysis that a radio transmitter that sends a signal in all directions just won’t cut it.

There are other possibilities, though.

To be continued …

Will We Reach the Stars? A Physicist’s Analysis (Part II)


In my previous post, I presented some simple calculations showing how much energy it would take to send a space shuttle to the nearest star, Proxima Centauri, in ten years. It turned out that we would need the amount of energy that the world’s largest power plant produces in 820 years (if we could run it for that long). This led me to conclude tentatively that we’re not likely ever to reach other star systems; but I promised that in my next post I would examine potential breakthroughs in science and technology that may one day make interstellar travel possible. So here we go.

Assuming we manage to keep from destroying ourselves in a nuclear war, humanity will certainly accomplish some impressive breakthroughs in science and technology in the future. For example, we can expect significant advances in the miniaturization of electronic and optical devices. And, as has long been predicted, we will almost certainly see the successful integration of biological systems (e.g., the human brain) with artificial systems (e.g., computers). At first glance, these advancements seem unrelated to interstellar travel, but I think it will turn out that, if interstellar travel is at all possible, these things will play an integral role.

In light of my previous post, however, the developments that seem most relevant to space travel will be those related to energy production, including the development of novel energy sources and improved efficiency of existing technology. The question at the heart of our discussion here, then, is this: Will these advancements be enough?

The way I see it, one of three possible developments will be necessary in order to make interstellar travel possible. Let’s consider each one in turn.

The first possibility is simply to find a better source of energy. Thus far, the vast majority of our energy has come from burning hydrocarbons. The burning of fuel, whether it be gasoline or the solid rocket propellant used by the space shuttle, is simply a chemical reaction. Energy is released because the atoms and molecules start out bonded together in one configuration and end up in a different, lower-energy, configuration. The energy difference between the two configurations is the amount of energy that we get out of the reaction and can use to power our devices.

Other means of production include harvesting the energy of mechanical motion, such as the motion of air (wind energy) or water (hydroelectric power), or collecting sunlight. These are all great sources of energy, but the fact is that you have to have a huge number of collection devices spread all over the place in order to get an appreciable amount of energy. That won’t help us with space travel unless we can store all of that energy in a compact battery that can fit on our space shuttle. And once again, the energy stored in a battery is chemical in nature and has a limited density.

What we need is something with a high energy density — a lot of energy packed into a small amount of volume and mass.

Modern physics places a limit on this. The total amount of energy contained in a given amount of mass — and hence the absolute maximum amount of energy that can be extracted from said mass — is given by Einstein’s famous equation, E=mc^2. This equation governs how much energy is produced in nuclear power plants.

Nuclear reactors work by converting a tiny fraction of the fuel’s mass into energy. (In fact, the same is true of chemical fuels as well, but the change in mass is so tiny that nobody ever talks about it.) However, since only a tiny fraction of the mass is converted to energy, nuclear reactors are not very efficient. We need something even better than conventional nuclear power.

According to modern physics, the absolute best that we could ever hope to achieve would be to convert all of a fuel’s mass into energy. The best way to do this is to combine matter and anti-matter so that all of the mass is annihilated, leaving nothing but energy. Producing the amount of energy that we need for our journey to Proxima Centauri would require the annihilation of about 6,500 kilograms of mass, half of which would have to be anti-matter.

So why don’t we do that?

Well, the problem is where to get the antimatter. Producing or even harvesting the antimatter in the first place would take a tremendous amount of energy. So that really puts us in a catch-22: We need energy to get energy.

Thus, barring some absolutely revolutionary breakthrough in our understanding of the nature of matter and energy, it appears as though nature has put an upper limit on how much energy we can extract from a given amount of material. And even if we’re able to reach that absolute limit, we’ll find ourselves hard-pressed to use that energy to send a ship to another star. I therefore conclude that our first option — finding a better source of energy — is not very promising.

Let’s look at the second possibility, then.

The second advancement that might enable interstellar travel would the development of the ability to bend space-time somehow — i.e., create a wormhole or something similar. We’ve all seen this in science fiction movies, and if you’ve read any popular literature about general relativity, then you have some conceptual idea about how wormholes work in principle. The problem here is that even if it is possible to create a wormhole, doing so would probably require more energy than simply sending a ship the required distance.

That doesn’t mean we won’t ever be able to do it. I can imagine, for instance, setting up a huge power plant — perhaps a space station that orbits a star and directly harvests nuclear power from it — dedicated to opening and closing wormholes. It would serve as a sort of interstellar space port that builds up and stores energy and then releases it in huge amounts on occasions for which the creation of a wormhole is desired.

But that is probably something we’ll only be able to do after we already manage to travel to other stars. So let’s keep thinking.

The heart of our problem thus far is finding the means and the energy to transport a certain amount of mass (i.e., our bodies) over a great distance. My third proposal represents not a solution to this problem but a reformulation of the problem: What if, instead of transporting our bodies across space, we first converted ourselves into something much lighter? Then a much smaller amount of energy would be required for the transport.

By our current understanding of reality, we are composed, at the most fundamental level, of information. In principle, you or I could be converted into pure information, which could then be encoded in a beam of light. This would be helpful because light has no mass at all and travels at the maximum possible speed (the speed of light). And according to relativity, if you were converted into light and traveled the 4.24 light-year distance to Proxima Centauri, no time at all would pass for you, while exactly 4.24 years would pass on earth.

There is one problem with this, though. There’s no device on Proxima Centauri that can receive the signal in which you are encoded and convert you from light back into a more preferable form. When you hit Proxima Centauri, your photons will be absorbed by the matter in the star and disappear forever, which is the same outcome that you would get if you just plunged into the star at 39% of the speed of light while riding in a shuttle!


Although I think this is the most exciting possibility, it once again requires that we first find some way to transport mass across distances from one star to another. And so here I come back to my earlier mention of miniaturization and bionics: specifically, the miniaturization of optoelectronics and the development of brain-computer interfaces.

Rather than trying to send people at first, we could begin by sending robots (i.e., computers) to another star as pioneers. This way, we could take advantage of miniaturization of technology to make these robots so tiny and lightweight that a relatively small amount of energy would be needed. (And they wouldn’t need food and water for the journey, either.) Once there, these robots could build the hardware necessary to receive future signals sent from the earth. Then we could begin sending people (and, in principle, anything else) encoded in beams of light. Hence, we could truly realize interstellar travel by means of teleportation.

There is one obvious and very basic philosophical problem here: If you are physically disintegrated at one location and reintegrated at another, is the new you still you? Or did you die, and is the new you just a copy that other people won’t be able to distinguish from you? Or, if you are merely copied without disintegrating the original you, what’s the difference between the new you and the old you?

It’s a disturbing question. My own graduate quantum mechanics professor commented on quantum teleportation by saying that if the technology ever reaches the level at which humans can be teleported, he would never volunteer for it because he couldn’t be sure that what came out of the other end would really be him.


Well, at least it’s a cool idea. And we might be able to watch other people be teleported. (There are, after all, people signing up for the Mars One suicide mission.)

In the end, maybe it’s just that I’m a pessimist, but if I had to make a bet, I’d say we’re much more likely either to blow ourselves up with nuclear weapons or to permanently strand ourselves on this rock by exhausting all of our energy supplies than to make it to another star system. So I have to conclude that in all likelihood, we’re never going to make it to another star system.

I do hope someone will prove me wrong, though.

Will We Reach the Stars? A Physicist’s Analysis (Part I)


If you watch any science fiction at all, you probably take it for granted that humans will be traveling casually from one star system to another in the not-too-distant future. (At least, I’ve always taken it for granted.) And why shouldn’t you? There’s good precedent for science fiction becoming reality; just look at any of the latest military, medical, and communications technology. But in light of the physical requirements of interstellar travel, should we really expect to reach the stars some day?

To answer this question, let’s examine what it would take to travel to the nearest star, Proxima Centauri.

First, consider the distance. Even though it’s our closest neighbor, Proxima Centauri is a whopping 4.24 light-years away from us. Traveling that far would be equivalent to circumnavigating the globe (at the equator) one billion times. If you flew the space shuttle at its maximum speed, it would take you 164,000 years to get there. Over five thousand generations of humans would be born and die on the shuttle during your trip!

No problem, you say. We’ll just have to go a bit faster than that.

Okay. Let’s decide on a speed, then. First, we need to know how long we want the trip to take. I, for one, would like to get there within one generation and still have some time to spare. So let’s say ten years. That’s a long time, but hey, this is a pioneering trip. I mean, how long did it take Columbus to reach America? (Five weeks.)

So ten years it is.

All right. Now that we know both the time and the distance, we can calculate the required speed. That’s just distance divided by time, right?

Well, not quite. The number that we have for the distance (d) to Proxima Centauri was measured in the Earth’s frame of reference, whereas the amount of time (t) that the trip takes will be measured in the ship’s frame of reference (i.e., we want the traveler to age ten years). If we take into account relativistic effects, the required speed is given by:


Plugging in the distance to Proxima Centauri for d and ten years for t (and the speed of light for c), we find that we will have to travel at 39% of the speed of light.*

Okay. So what?

Well, the next question is how we will accelerate the ship up to this speed. Specifically, we need to know how much energy it will take. For this, we need the ship’s mass in addition to its speed.

To get a reasonable estimate of our ship’s mass, let’s suppose that we’re traveling in something like the space shuttle (which is really too small for a ten-year journey, especially if your mother-in-law is on board). Without its usual two million kilograms of fuel (who needs fuel, anyway?), the shuttle weighs a trifling 75,000 kg.

Armed with knowledge of our ship’s speed and mass, we’re now ready to calculate the amount of energy we need. Don’t worry; this won’t hurt. Remember the ol’ kinetic energy formula from high school? No? Well, it doesn’t matter. That formula isn’t valid at high speeds, anyway. We need to use the relativistic formula for kinetic energy, which is this:


After we plug in our numbers and wait for the dust to clear, we find that the amount of energy required is … drum roll please …


In case you’re wondering what the hell a joule is (which is especially likely if you were educated in America), I’ll just put this number in perspective by telling you that this is the amount of energy that the largest power plant in the world (the Three Gorges Dam) produces in 820 years.

It’s time for some reflection, I think.

Even if we could pack the largest power plant in the world onto the space shuttle without adding to the shuttle’s mass, it would still take us 820 years just to reach the desired speed. And we haven’t even considered how we’re going to slow down once we arrive at Proxima Centauri!

Granted, there are little tricks we can use to get an energy boost here and there. The most obvious is what’s known as a gravitational assist, which would involve flying our ship by a planet in such a way as to steal some of the planet’s gravitational energy.

The Voyager 1 probe executed at least one gravitational assist. Launched in 1977, it’s the first human-made object to leave our solar system. But even after over 36 years of travel time, including a boost from a gravitational assist, it has only traveled 1/2100 of the distance to Proxima Centauri. The full distance will take another 75,000 years or so.

Thus, even clever tricks such as gravitational assists probably won’t be enough for our purposes. And so it looks like traveling to another star — even the nearest one within our own galaxy — won’t be that easy. In fact, we’re likely never to be able to do it at all.

You might say that I’m being overly pessimistic. After all, we can surely expect some technological advancements in the future. Won’t they make interstellar travel possible?

Perhaps. We’ll consider some of the possibilities in my next post.

To be continued … 

*For those of you familiar with relativity, this corresponds to gamma = 1.086.

Fat-Water Separation in MRI

A figure from Dixon's original paper on fat-water separation.

A figure from Dixon’s original paper on fat-water separation.

To anyone who is interested in fat-water separation within the field of MRI, I offer the following (click for PDF):

Fat-Water Separation

This is a chapter from my master’s thesis that explains the most common methods of fat-water separation and presents a very handy geometric interpretation of two-point Dixon imaging. I’ve been told that the interpretation is very helpful for newcomers to the field.

If you find it useful at all, please share it.

What is fat-water separation, you might ask?

Good question.

MRI detects hydrogen atoms; and there are two main places in your body where hydrogen atoms hang out: fat molecules and water molecules. So in the context of MRI, we can say that most tissue in your body is either fat or water. (Examples of “water” here include muscle, blood, skin, and pretty much anything except fat.) Fat-water separation is simply the process of distinguishing between the two tissue types in an MRI image.

Doctors often want either to remove the fat from an image so they can see things the fat might be hiding, or to create two separate images, one of fat only and one of water only. Doing so can help them give patients a more accurate diagnosis.

Fat-water separation techniques involve some interesting physics and math. For more info, click on the link provided above.

Introduction to MRI


(Before you complain, see note at end of post.)

Back in 1997, at the end of my tenth grade year, my chemistry teacher explained the principles underlying magnetic resonance imaging. I’m sure it wasn’t a very in-depth explanation, but I still remember thinking, “Wow, that’s complicated. I’ll never understand it.”

Well, a lot has happened since then. I got a bachelor’s degree in physics and math. I worked as a teacher for seven years, five of which were spent in China. Then I went back to graduate school to get a master’s degree in applied physics.

The topic of my research? Magnetic resonance imaging.

As I began studying MRI in graduate school, I still worried that I might not be able to understand it. But I once again rediscovered a truth that I keep finding myself rediscovering: namely, that if you invest enough time and hard work in studying a topic, you’ll eventually get it, no matter how daunting it may at first seem.

In the end, I didn’t just learn the basics of MRI, but I published a thesis on my own contribution to the field: an improved method for distinguishing between fat-based and water-based tissue in magnetic resonance images.

If you’re curious about how MRI works, I offer you this (click the link to download the PDF):

Foundations of MRI

It’s a chapter from my thesis that explains the basic physics of MRI — from how the tissue in your body is magnetized when you are placed in a magnetic field, to how images are obtained with the desired contrast between different tissue types.

I hope you’ll find it to be puddles of fun. Please splash around to your heart’s content.

NOTE: Anyone familiar with medical imaging will immediately notice that the Homer Simpson image at the top of this post is not an MRI. But hey, it’s already a fictional image of a fictional character, and it’s really funny, so please give me a break.