Relevancy of Relativity

This week, rather than posting a piece written specifically for this blog, I am going to go green and recycle. Hopefully the topic is interesting, although it does not perfectly match the format of what I have been posting. I'll be back next week with content written just for you.


This is a paper I wrote to the prompt "Is Einstein's Theory of Relativity relevant in today's culture?"


This is what I wrote:

Einstein's famous theories of special and general relativity live on in our modern culture beyond academia and the fact that “E=mc²” is a wonderfully elegant equation. Technologically, they account for the accuracy of GPS, keep the ISS on time, and relativistic effects between Earth and Mercury was discovered to be the slight perturbation in its orbit, rather than the delightfully mysterious “planet Vulcan” at Earth's L3 point. At least the latter lives on in Star Trek.

While people count on GPS, many don't care what time it is on the ISS, or spend time thinking about what is lurking on the other side of the sun, messing up the Mercurial procession. If you ask (pester?) your friends about relativity, as I have recently done, they will generally have the name Einstein come out of a holster, then mumble something about the speed of light, and maybe even mutter something about time moving slower. The science-y ones may know the name “Lorenz” has something to do with all of it. This is a bit of a shame, because since the early 20^th century there has been a profound change in many part of the scientific community that had led to many breakthroughs, and it all involved leaving your common sense and intuition at the door. Relativity is a large part of this new phenomenon.

Special relativity posits that as you speed up, you shrink down relative the person who stayed still. It also posits that things don't happen at the same time at different speeds, that time moves slower when you speed up and general relativity posits that being in a gravity well changes everything you thought was real as well.

None of this is at all intuitive. Abandoning intuition is an extremely valuable trait among scientists, and can be useful in everyday life. It allowed nearly every great advancement of human scientific thinking. Thinking that there exists an invisible army of creatures living inside you is preposterous enough as to be worried for whomever suggested it, yet it's true. The idea that the earth is hurtling through space at many kilometers per second is immediately dismissed by the evidence that we don't feel it, yet it's true. Opening oneself up to these strange ideas had proven beneficial in the past, and retains its place today.

Relativity is an excellent tool to train yourself to mistrust intuition. Learning about how a 10 meter ladder can fit in an 8 meter barn with both doors closed forces you to rethink the intuitive answer. There are other areas in which this is an invaluable tool. If a loved one faces a difficult medical choice between invasive surgery or an alternative remedy, one's intuition screams to avoid surgery where it may be the only effective option. If vaccinating your children sounds like child abuse to you, abandoning intuition may indeed help your child live longer.

These examples are not direct effects of relativity in modern culture, and the truth of the matter is that few people are interested in what they think of as Einstein's century-old dusty theories and equations. Many people are much more concerned about if they need a second mortgage, or if they need to take away their teenagers keys after a speeding ticket, even if it means shutting them to their high school. Even so, those who have spent some time with their brain in the kitchen mixer of scientific theory and inquiry can influence those around us. Family members asking for advice or friends facing tough decisions can really benefit from a bought of critical thinking and cognitive dissonance. Science is much more than math, equations, and physicists talking about Star Trek's holodeck or Alcubierre's warp drive in a faculty cafeteria, it is a way to move through life, evaluating the best course of action and persuading others to do the same, a way to avoid fooling yourself, or allowing others to fool you.

Einstein’s relativity has a knack for kick-starting this train of thought with just a small initial investment of research required, or a friend willing to talk about it.

Cheers,

  - Scott

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Random Facts

In my research for this blog as well as my own musings and curiosity, I run across some little factoids that aren't substantive enough to make a post all on their own, so I'll talk about a few of those here.

Faster Than Light Poke

I was thinking about the idea that "information can not travel faster than the speed of light," and I thought I was quite clever when I came up with a way to do it, theoretically at least. I imagined a wooden pole, say one light-year long. If you wanted to transmit information across a light year, it would take at least 1 year to send a radio signal. That's where my pole came in. If one person 'poked' in Morse code then the person one light year away would be able to receive that information immediately. I thought I had cracked it, information traveling faster than light.

It turns out I made an incorrect assumption however. I assumed the pole would all move as one, as with a broom. Poke someone with a broom and they feel it immediately. This is not so with exceptionally large brooms (nor is it with small ones, but the delay is so short it is all but immediate).

There is a 'speed of poke' or a 'poke wave' in the broom. The speed of poke turns out to be the speed of sound through that medium. Your poke would travel at about 3500 meters / second if your pole was made of wood, brass, or concrete. This is far less than the speed of light. It turns out it's about how fast GPS satellites orbit the Earth. The 'poke wave' (I love that phrase) comes about because when you move the pole, individual atoms in the pole have to move to alert their neighbors that they’re moving. Each atom must push its neighbor which must in turn push its neighbor and this is how the poke is transmitted through the object. So no faster than light pokes, science won this round.

Helicopter Speed Limit

It turns out that most helicopters have a speed limit imposed not by the FAA or the maker of the helicopter, but by physics. On helicopter, there is a “retreating blade” (#1, left side) and a “retreating blade” (#2, right side) due to the direction of the blades' rotation.

Image: AVStop.com

When the helicopter is flying, the advancing blade is traveling at the speed of the helicopter PLUS the speed of the rotor. The retreating blade is traveling at the speed of the rotor MINUS the speed of the helicopter. This creates something called "dissymmetry of lift." Because one blade has a higher airspeed than the other one, the central hub of the helicopter changes whats called the angle of attack on the retreating blade, giving it more lift.

Image: Johnathan M. Stern

The increased angle of attack compensates for the slower air speed or the retreating blade. This only works up to a point. If the retreating blade requires more lift than can be compensated for by altering the angle of attack, the helicopter pitches back and slows itself down. This gives most helicopters a physics-induced speed limit.

Image: SPC Glenn Anderson

Most helicopters

Melting Ice and Boiling Water

It is really hard to get water to change temperature. This is because of a property called specific heat capacity that I wont get into much here. Suffice it to say that it requires a whole lot of energy. But the true champion of temperature stubbornness is ice. Ice takes more energy to change by a few degrees than water takes to change by dozens of degrees.

The experiment:

Put two glasses in the microwave for about two minutes. Fill one with water and one with ice and start your microwave. Go do it, it only takes two minutes, and your dishes are still clean afterword. As it turns out the glass filled with water will boil before most of the ice has even melted. This means the water changed temperature from around 70°F to 212°F, or 142°F verses the ice which changed from the temperature in my freezer, 15°F to about 32°F, a change of 17°F. Both the ice and the water absorbed the same amount of energy from the microwave.

The way a microwave works is by creating a standing wave that oscillates about 2.5 billion times per second. This means a charged molecule like water...

...will act like a magnet and try to align itself to the current direction of the field, and this effort causes it to vibrate at about 2.5 billion times per second, increasing thermal energy, heating the water up.

Red is slightly negative and blue is slightly positive

The reason ice doesn't play along and do the same is because it's molecules are in a solid crystal arrangement:

Image: Danski14

In order to align with the field, the molecules must be free to rotate, and in a solid crystal structure, the molecules must break their bonds before they can vibrate and increase the thermal energy. This takes a lot of energy and happens very slowly, which is why the ice takes so long to change temperature and melt.

Bonus fact: When writing this I thought to myself “because fat molecules are non-polar they should heat up less in a microwave.” I tried the experiment, and I heated up identical volumes of water and vegetable oil for one minute, and sure enough, the water changed by 61°C and the oil changed by only 32°C. Science wins another round.

Falling Out of Orbit

When something in orbit loses power or is disabled in some dramatic fashion, often it is portrayed to "fall out of orbit." Almost every science fiction franchise is guilty of portraying this. Even if the writers know the physics, portraying something falling out out of orbit is climactic, while reality in this case is rather underwhelming.

Orbits themselves are often misunderstood. Getting into orbit is not about going up quite as much as it is about going sideways. Space is only 100 km away, so getting there is easy. Staying there takes some work. As you can see in this amazing video (click it) a US space shuttle launch doesn't go straight up, but turns at a steep angle a few seconds into it's flight. By the end of the main fuel tanks' supply, the shuttle will be going about 7.8 kilometers per second sideways and it will not be be changing height at all.

If the shuttle were to get shot down by aliens or lose power, it would not dramatically fall back to Earth but simply stay in orbit. The only way to get out of orbit is to change your velocity dramatically so you're going slower relative to the Earth's surface. In order to change your velocity in orbit you have to take some of your mass and accelerate it away from you. This is all rocket engines do; the fuel is the mass, and they accelerate it by igniting it and directing the outflowing material opposite the direction you want to travel. Without this process occurring you would remain in orbit because your velocity didn't change.

Put simply, it takes nearly as much energy to get out of orbit than it does to get into orbit, so losing power does send your craft careening towards the thing you are orbiting. Orbiting isn't an active process, it is a passive one.

Cheers,

   - Scott

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How Could We Have Broken Germany's Enigma Code?

I initially set out to write about the German Enigma Machine and how it worked, but as I was doing some research, two things happened: I found a video that explains it better than I ever could, with an actual WWII-era Enigma Machine, and I wound up doing some math that I thought was somewhat interesting.

So to start, I think you should watch this video about the Enigma Machine:

Did you watch it? If not, I'll give you one more chance:

IF THE VIDEO DOES NOT SHOW UP CLICK

 >  HERE  <

Alright, so in the video we saw that the Enigma Machine was a device used to encrypt internal messages in the German military, and has an extraordinary number of possible arrangements, each of which will produce a different code. The Enigma code was broken through a regimented and careful application of cleverness on the part of Alan Turing and his team at Bletchley Park. I got to thinking however, “what sort of team would have to be assembled to solve the Enigma code through sheer human power and luck?”

This rabbit hole was quite fun to descend into. I started with a few assumptions for this scenario. Every person who was working on the code had their very own enigma machine to work with, they could try a new arrangement each 100 seconds, and they could work for 16 hours a day. I also assumed that the infrastructure to feed, house and give them water was in place. Alright, here is the number we started with:

158,962,555,217,826,360,000

159 quintillion. That is the number of possible arrangements the enigma machine could be in, each would output a different code, and only one of which would be correct. Oh, and it changed every single day.

In order to try every combination in the space of 24 hours, you would need to try 1,840,000,000,000,000, or 1.8 quadrillion arrangements per second.

If one person can try one arrangement in 100 seconds, that means it will take a force of 184 quadrillion people to try every arrangement in the space of 24 hours, but that only if we work them 24 hours a day. Adjusting for the workforce working 16 hour days, we can multiply 184 quadrillion by (4/3) to get 245 quadrillion people.

So we have ~ 2,350,000,000 people in the world (in 1945)

We need 245,000,000,000,000,000.

Keen-eyed readers will note that the second number is longer, so we are now moving into a very hypothetical world. As long as we're not bound by reality, lets go ahead and pack in our workforce across the entire land area of earth (minus Germany), and lets pack them in at the population density of current-day Tokyo (we are ignoring small facets like food water, adequate shelter, etc...).

Land area of Earth: 57.53 million km²

Land area of Germany: 137,903 km²

Population density of Tokyo: 1800 people per km²

1800 people per km² * ( 57.53 million km² – 137,903 km²)

This gives us a population on our Super-Earth of about 100 billion people.

This means we would need 2.45 million Super-Earths to support our workforce.

It looks like we'll need to drain the oceans to beat the Nazis.

Due to the fact that Earth's surface area is ¾ water, draining the oceans gives us 3x the surface area we had using land alone (minus Germany). We can now fit 400 billion people on each Mega-Super-Earth, and now we need a quarter as many, or a little more than 600,000 Mega-Super-Earths. While that many Earth-Sized objects can orbit many stars, we need to restrict our Enigma team to one star in order to transmit the correct code in time to implement a strategy. The nearest star to us is Proxima Centauri, over 4 light years away. That is not close enough to get information back to central in time to help the war effort.

The question now becomes “Can 600,000 Earth-sized object orbit our sun without bad things happening?” I talked to some astronomers and astrophysicists here at CU Boulder, and the consensus seems to be “no.” When you put a lot of objects in similar orbits, most of the bodies are ejected from the system, most of the rest collide with each other, and quite a few will be engulfed by the star in the middle of the system.

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This is the “Nice Model” of the solar system. About 3.8 billion years ago many Kuiper belt objects were ejected from the system and Uranus and Neptune switched places (at 30 seconds in the video). This area was much more sparsely populated than our scenario would be.

This doesn't happen immediately, so to wrap up, if you’re going to use humans alone to solve Enigma, do it fast, and make it worth it.

There are of course a few problems with this plan:

On average, the code will be broken by midday. Sometimes you will get lucky and solve it early on, and sometimes you wont be so lucky, and it will take until late in the evening to break the code. Even with this workforce, there will only be an average of 12 hours to actually use the information obtained.

If you have 245 quadrillion people united to defeat the axis powers, you probably wouldn't even need to bother with the Enigma code. If each person donated one strand of hair (half a milligram), you could bury Germany in 122 billion kilograms of human hair, which would probably deter them. I wanted to figure out how many feet this would be, but there aren't reliable figures for the packing efficiency of average human hair, so I figured a head of hair (a wig) weighs about 100 grams, and I could stuff one in a one liter bottle giving it a density of 0.1 kg/liter. This means 122 billion kilograms of hair could cover Germany to a depth of ~9 meters, give or take a few orders of magnitude.

So now you know that...

If you’re curious about the Enigma Machine itself, as well as the flaw that turned out to be its undoing, there is another video that describes how Turing and his team broke the code:

Cheers,

    - Scott

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