Q: Why is hitting water from a great height like hitting concrete?

Posted on July 25, 2012 by The Physicist

The original question was: I know that if you hit water at a certain speed it is supposed to crush your body like you have hit concrete. Is this the case for all liquids or is something to do with the surface tension of water and if you hit a gas in a confined space fast enough would it feel like hitting concrete?

Physicist: There’s nothing terribly special about water, and even hitting a gas fast enough would “feel like concrete”.  For example, when meteors (which are fast) hit the atmosphere they generally shatter immediately.

A good way to think about high-velocity impacts is not in terms of things (like water) acting more solid, but in terms of things (like people, rocks, Fabergé eggs) acting more fluid.  The more energy that’s involved in a collision, the less important the binding energy (the energy required to pull a thing apart) is.  A general, hand-wavy rule of thumb is: if the random kinetic energy of a piece of material is greater than the binding energy, then the material will behave like a fluid.  A bit more energy, and it will fly apart.

Whether a substance behaves more like a solid or more like a liquid depends on the energies involved.

This shows up on a much smaller scale as well.  For example, the difference between water and ice is that the random kinetic energy of water, better known as “heat”, is greater than the binding energy between the molecules in ice.

So, when you fall from a great height and land in water there’s a bunch of kinetic energy going every which way.  The water continues to behave like water, but since the kinetic energy in different parts of your body are greater than the binding energy keeping them connected, then the body as a whole will act more like a fluid.  That is; it’ll “splash” (in the grossest sense).

Clearly there’s a big difference between something breaking into chunks, and something liquifying, but that difference is mostly just a matter of energy; it takes more energy to make sawdust than wood chips, but the process is more or less the same.  The take away here is, there’s a lot of different kinds of binding energy (molecular, structural, etc.) but they all do similar things.

Professional fluid dynamicists use a value called the Reynolds number to quickly talk to each other about this property in fluids.  It essentially describes whether a fluid is more “inertial” (water-like) for values much larger than 1 or more “viscous” (honey-like) for values much lower than 1.  Being natural comedians, they’ll say things like “usually my half-and-half has a Reynolds number around 20,000, but this morning it smelled weird and had a Reynolds number of 0.3!”.  It’s usually used a ballpark-estimate, short-hand-description kind of thing, but in general the Reynolds number gets larger when things are bigger, faster, and denser, and it gets smaller when things are more viscous.

So, in a very, very hand-wavy way, a fast moving body hitting water (or whatever) has a higher Reynolds number, and is more waterish itself.

The “solid” marshmallow picture is from here, and the Ghostbusters “liquid” marshmallow picture is from Ghostbusters.

Posted in -- By the Physicist, Physics | 33 Comments

Q: How does instantaneous communication violate causality?

Posted on July 20, 2012 by The Physicist

Physicist: You may have heard that if a technique can be found that allows you to send a signal faster than light, then you can send a signal back in time.  The very short answer for why is that relativity causes some problems when you try to define “instant”, and those problems can be used to send signals back in time.

The speed of light is crazy fast, but on large enough distances it’s agonizingly slow.  Talking to people on the Moon involves a net 2 second delay, which is a little annoying.  If you send a signal to a robot on Mars it’ll take between 8 and 40 minutes to get a response, depending on where Mars and Earth are in their orbits, which is crippling (this is a big part of why the rovers were made to be fairly independent).  So why not just create some kind of sub-space, tachyon, ultrawave relay, thingy that allows you to send FTL (faster than light) or even instantaneous signals?

Any physically realizable communication always involves some delay (left), instantaneous communication doesn’t seem to cause any problems (right), but it totally does.  In all of these pictures time moves from the bottom to the top.

Relativity does more than just place a cap on the speed that signals and objects can move.  It also muddles this question in particular, by complicating what’s meant by “instantaneous”.  One of the unfortunate results of relativity is what’s happening “now” depends on how fast (and in which direction) you’re moving.  That is, if one person sees that two events are happening at the same time (“right now”), then someone traveling at a different speed will see the two events not happening at the same time.

In particular, if you send an instantaneous signal to somewhere else, then the sending and arriving both happen “now”, but to someone else (moving relative to you) one happens before the other.

Two objects, as well as a couple moments from each of their perspectives.  (left) The red object is stationary.  (right) The blue object is stationary.  Notice that the “nows” of the two objects are skewed with respect to each other.

The details of why can be found here, but for now just keep in mind that when something passes by, the set of things that are happening “right now” for that thing happen later for you in the direction of motion.  So, if something passes left to right in front of you, then its now will, for you, happen in a moment and to the right as well as a moment ago and to the left (whens and wheres get mixed up a bit in relativity).  It takes a little pondering (done in the link above), but this result actually falls out of the whole “speed of light is the same to everyone” thing pretty quickly.

By the way, that shouldn’t make any intuitive sense, and should be difficult to keep track of, so don’t stress.

So, “instantaneous communication” isn’t as easy to define as one might think.  But, if you accept the basic tenet of relativity, that everything no matter how it’s moving is on equal footing, then one person’s instantaneous signal is merely traveling very fast to someone else, or even slightly backward in time to another someone else.  So, say you have two objects sending instantaneous (from each of their perspectives) messages back and forth.  Because of how they’re moving, while the sender perceives the message being sent and received instantly, the recipient perceives the message arriving a little after or even a little before it was sent.

Since differently moving things have different notions of now, instantaneous signals can zig-zag backwards or forwards in time.

Getting a message after it was sent is no biggie.  I mean; write yourself a letter if you want to see that in action.  But getting a message before it was sent causes issues (see for example; practically every sci-fi franchise).  What those issues are exactly depends on how time travel works (e.g., “Timecop” or “Back to the Future” rules?), and that’s wide open to debate.

There is a “cure” for faster-than-light communication causing causality violations.  There isn’t really a problem with signals going back in time, if they only go back in time somewhere else.  For example,imagine there was a magic post office in the year 1500 that sent letters from Rome (Rome) to Tenochtitlan (Mexico City) and one week back in time.  Since it took 5 weeks to cross the Atlantic, there’s no risk of paradoxes and causality violations (“Dear Ahuitzotl, in a week Giovanni Borgia is going to be killed.  Nothing you can do about it, just thought you’d like to know.”).  The real problems crop up when you can send instantaneous messages in two or more reference frames.  That allows you to bounce signals back and forth, and thus send a message to yourself in the past.  So, the fix is to have only one frame with instant communication (magic post offices only send letters in one direction).

But this cure; picking a special reference frame (a special speed) in which communication can be instantaneous, isn’t really in keeping with the spirit of relativity or observational evidence; that all speeds are equivalent.

Posted in -- By the Physicist, Physics, Relativity | 38 Comments

Q: What is the “False Vacuum” and are we living in it?

Posted on July 15, 2012 by The Physicist

Physicist: The False Vacuum is just another item on the long list of things to worry about, that are not worth worrying about, and that nobody can do anything about.  If you have any other worries, worry about those first.

In any physical system you’ll find that there’s an energy “ground state” that the system tries to approach.  For example, if you pour water into a bathtub there are a lot of ways that the water could arrange itself, but it will rapidly try to assume the ground state: being as low and still as possible.

In any physical system there’s a lowest energy state called the “ground state”.  If you allow the energy to drain out of a system, then it will approach its ground state.

Gravitational potential (the energy that something on a high shelf has more of, and that’s released when things fall) is the easiest example of an energetic system to picture.  Being on the ground is the lowest gravitational potential a thing can have (without digging); thus the name.

But in the tub example, the water in the tub doesn’t “know” about the drain or the area around the tub.  Given the chance the water would flow out of the tub and into a new, lower, ground state.  In fancy-math-speak, you’d say that the water in a tub is in a “local energy minimum”.  Within the tub, water definitely assumes the only ground state it can find.  However, if it jumped out of the tub, or somebody pulled the plug, then it would try to find a new, lower ground state and would find that it gained a bunch of new energy in the process (what with the flowing and splashing and whatnot).

There’s no good way to find out if you’re in a local “false” ground state or a true ground state.

The idea of energy levels, and ground states, and all that, applies to pretty much everything.  That includes electromagnetic fields and even particle fields.  A “particle field” is the quantum mechanical way of describing particles (all smeared out, instead of being all in one point), and each field has a pretty reasonable set of energy levels.  Every new particle elevates the energy level by one step, and the ground level is exactly what you’d expect: zero particles.

The vacuum is the most absolute ground state: no waves, no particles, nothing at all to elevate the energy above zero*.  However, all the dynamics of the universe are governed by differences in energy level.  For example, when you fill a tub it doesn’t matter if the tub is at the top of a mountain, or at the bottom of a mine, the water will behave the same way.  So, the idea behind a false vacuum is that what we consider the ground state of the universe isn’t really the ground state, and it may be possible to drop into an even lower-energy state (drain the tub, so to speak).  What we think is the ground state, the vacuum, may not be the true ground state.  So it’s called a “false vacuum”.

The “danger” of living in a false vacuum is that, under the proper circumstances the false vacuum can drop into the true vacuum.  The cause is usually described as a sufficient burst of energy to get the appropriate fields “over the hump” (picture above).  If the difference in energy between the false vacuum and true vacuum is large enough, then the surrounding space can likewise be tipped into the lower state.  In theory, a “false vacuum collapse” would expand at light speed (or about light speed) from the originating event, and destroy the heck out of everything in the affected, and ever-expanding, region.

It’s worth mentioning that the idea of a false vacuum is wild speculation and that there is no indication, not even a little, that the vacuum of the universe is a false vacuum and not the true ground state.  There’s a long history of spectacular bursts of energy in the universe, and none of them have tripped a collapse.  The ground state of the universe is kinda like a septuagenarian’s testicle; if it hasn’t dropped by now, it probably won’t.

*For subtle reasons, it turns out that you can’t quite reach zero and as a result the vacuum state has slightly more than zero energy.  This tiny difference is known as the “vacuum energy” or “zero point energy”, and it’s responsible for things like the Casimir effect.  By the way, there are some significant issues to work out involving a disagreement between the measured and predicted values of the vacuum energy.

Posted in -- By the Physicist, Particle Physics, Physics | 45 Comments

Q: How would the universe be different if π = 3?

Posted on July 10, 2012 by The Physicist

Physicist: We sometimes get questions about physical constants changing, and those questions make sense because there’s no real reason for the constants to be what they are.  But π is mathematically derivable; it kinda needs to be what it is.  You can’t, through the power of reason alone, figure out what the gravitational constant or the speed of light are, but you can figure out what π is.

π is the distance around any circle, C, divided by the distance across that circle, D.  All of the weird places that π shows up track back to this definition.

So this question is doubly profound!  Unlike other constants, if π were different, then scientists (mathematicians especially) would continually have the sneaking suspicion that there’s something deeply, deeply wrong with the universe.

The definition of π seems pretty innocent (the ratio of the circumference of a circle to its diameter), but it shows up over and over in the middle of calculations from all over the place.  For example, even though \left[\int_{-\infty}^\infty e^{-x^2}dx\right]^2 doesn’t, on the surface of it, have anything to do with circles, it’s still equal to π (there’s a loop floating around halfway through the calculation).  A surprising number of calculations and derivations involve “running in a loop”, so π shows up all the time in electromagnetism, complex numbers, quantum mechanics, Fourier analysis, all over.  In fact, in that last two, π plays a pivotal role in the derivation of the uncertainty principle.  In a very hand-wavy way, if π were bigger, then the universe would be more certain.

Aside from leading almost immediately to a whole mess of mathematical contradictions and paradoxes, if π were different it would change the results of a tremendous number of (one could argue: all) calculations, and the fundamental forces and constants of the universe would increase or decrease by varying amounts.  π shows up in way too many places to make a meaningful statement about the impact on the universe, one way or another.

Mathematician: The idea of giving π a new value of could be interpreted in a few different ways. For example:

1. That circular physical objects, as you make them progressively closer to perfect circles, approach a circumference to diameter ratio of something other than 3.14159… If this were the case, it might indicate something about the geometry of spacetime. If space is not flat, that can change geometric relationships. For instance, imagine drawing a circle on the surface of an orange. If we allow distances to be measured only along the orange’s surface (disallowing paths that penetrate the orange or go into the empty space around it), then the ratio of the circle’s circumference to diameter is no longer going to be π. It will, in fact, depend on the size of the orange itself. If our universe is not flat, but a curved surface, that could distort the geometric relationships that we measure on physical objects resembling circles.

2. That we change what we mean when we say π. Of course, π is just a symbol referencing an idea, so if the underlying idea that it references were to change, that would change the value of the symbol. But this is an extremely boring way to answer this question, reducing it merely to the redefinition of a symbol.

3. That we change which mathematical axioms we use. Most people think of math as a single, coherent set of rules. But when you get down to it, there are different possible sets of axioms that you can use to define mathematical concepts. By switching axioms it becomes possible to prove different things. If we were to choose a set of inconsistent axioms (i.e. axioms that lead to contradictions) then it would be possible to use this system to “prove” any mathematical statement true. In that case, you could show that π = your phone number, if you wanted.

If, on the other hand, you choose a set of axioms that are consistent with each other, but different than our standard math, you have to be more precise about what you mean by π. Usually, it makes no difference whether we define π to be the ratio of a circle’s circumference to its diameter, or whether we define it as

2 \arcsin(1)

or

Log(-1)/i

or

4 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1}.

But when you start messing with axioms, it is going to affect what is provable, and therefore you have to be careful to specify precisely which definition you are using for π, since definitions that normally are equivalent using standard sets of axioms may no longer be. For some sets of axioms, there won’t even exist a mathematical entity that you reasonably could identify as being π.

Answer Gravy: There are a couple of ways to derive the value of π.  Archimedes estimated it by sandwiching the circle between regular polygons that he could find the exact sizes of.

One step in Archimedes’ work-intensive method for estimating π.

However, we can use methods younger than a few millennia to derive cute formulae for π.

\begin{array}{ll}\int \frac{1}{x^2+1}dx\\= \int \frac{1}{tan^2(u)+1}\left(\frac{1}{cos^2(u)}\right)du &\left\{\begin{array}{ll}x=tan(u)\\dx=\frac{1}{cos^2(u)}du\end{array}\right.\\= \int \frac{1}{sin^2(u)+cos^2(u)}du\\= \int 1 du\\= u+ C_1 &\textrm{Where "}\,C_1\,\textrm{" is a constant of integration}\\=arctan(x)+C_1\end{array}

Coming from another angle:

\begin{array}{ll}\int \frac{1}{x^2+1}dx\\= \int \frac{1}{1-(-x^2)}dx\\= \int 1-x^2+x^4-x^6+\cdots dx\\= x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots +C_2\end{array}

So, arctan(x) +C_1 = x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots +C_2

Since arctan(0) = 0, if you plug in zero you find that C1 = C2, so you can get rid of them and arctan(x) = x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots.  Now, tan\left(\frac{\pi}{4}\right) = 1, so arctan\left(1\right) = \frac{\pi}{4}.  Therefore, \pi = 4\left(\frac{\pi}{4}\right) = 4\, arctan(1) = 4\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots\right).  That tan\left(\frac{\pi}{4}\right) = 1 is based on the definition of the radian, which tracks back to the circumference of the unit circle being 2π, but the fact remains:

\pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots and changing the value of π means, among other things, that this summation would need to somehow equal something different.  But it is what it is.

Posted in -- By the Mathematician, -- By the Physicist, Geometry, Math, Philosophical, Physics | 45 Comments

Q: Is it possible for an artificial black hole to be created, or something that has the same effects? If so, how small could it be made?

Posted on July 6, 2012 by The Physicist

Physicist: Not with any current, or remotely feasible technology.  The method in use by the universe today; get several Suns worth of mass into a big pile and wait, is a pretty effective way to create black holes.

In theory, all you need to do to create an artificial black hole (a “black faux”?) is to get a large amount of energy and matter into a very small volume.  The easiest method would probably be to use some kind of massive, super-duper-accelerators.  The problem is that black holes are dense, and the smaller and less massive they are the denser they need to be.

A black hole with the mass of the Earth would be so small you could lose it pretty easy.  Except for all the gravity.

But there are limits to how dense matter can get on its own.  The density of an atomic nucleus, where essentially all of the matter of an atom is concentrated, is about the highest density attainable by matter: about 1018 kg/m3, or about a thousand, million, million times denser than water.  This density is also the approximate density of neutron stars (which are basically giant atomic nuclei).

When a star runs out of fuel and collapses, this is the densest that it can get.  If a star has less than about 3 times as much mass as our Sun, then when it gets to this density it stops, and then hangs out forever.  If a star has more than 3 solar masses, then as it collapses, on it’s way to neutron-star-density, it becomes a black hole (a black hole with more mass needs less density).

The long-winded point is; in order to create a black hole smaller than 3 Suns (which would be what you’re looking for it you want to keep it around), it’s not a question of crushing stuff.  Instead you’d need to use energy, and the easiest way to get a bunch of energy into one place is to use kinetic energy.

There’s some disagreement about the minimum size that a black hole can be.  Without resorting to fairly exotic, “lot’s of extra dimensions” physics, the minimum size should be somewhere around 2\times 10^{-21} grams.  That seems small, but it’s very difficult (probably impossible) to get even that much mass/energy into a small enough region.  A black hole with this mass would be about 10-47 m across, which is way, way, way smaller than a single electron (about 10-15 m).  But unfortunately, a particle can’t be expected to concentrate energy in a region smaller than the particle itself.  So using whatever “ammo” you can get into a particle accelerator, you find that the energy requirements are a little steeper.

To merely say that you’d need to accelerate particles to nearly the speed of light doesn’t convey the stupefying magnitude of the amount of energy you’d need to get a collision capable of creating a black hole.  A pair of protons would need to have a “gamma” (a useful way to talk about ludicrously large speeds) of about 1040, or a pair of lead nuclei would need to have a gamma of about 1037, when they collide in order for a black hole to form.  This corresponds to the total energy of all the mass in a small mountain range.  For comparison, a nuclear weapon only releases the energy of several grams of matter.

CERN, or any other accelerator ever likely to be created, falls short in the sense that a salted slug in the ironman falls short.

There’s nothing else in the universe the behaves like a black hole.  They are deeply weird in a lot of ways.  But, a couple of the properties normally restricted to black holes can be simulated with other things.  There are “artificial black holes” created in laboratories to study Hawking radiation, but you’d never recognize them.  The experimental set up involves tubes of water, or laser beams, and lots of computers.  No gravity, no weird timespace stuff, nothin’.  If you were in the lab, you’d never know that black holes were being studied.

Posted in -- By the Physicist, Astronomy, Particle Physics, Physics | 15 Comments

Q: Do colors exist?

Posted on June 30, 2012 by The Physicist

Physicist: Colors exist in very much the same way that art and love exist.  They can be perceived, and other people will generally understand you if you talk about them, but they don’t really exist in an “out in the world” kind of way.  Although you can make up objective definitions that make things like “green”, “art”, and “love” more real, the definitions are pretty ad-hoc.  Respectively: “green” is light with a wavelength between 520 and 570 nm, “art” is portraits of Elvis on black velvet, and “love” is the smell of napalm in the morning.

But these kinds of definitions merely correspond to the experience of those things, as opposed to actually being those things.  There is certainly a set of wavelengths of light that most people in the world would agree is “red” (rojo, rubrum, rauður, 紅色, أحمر, ruĝa, …).  However, that doesn’t mean that the light itself is red, it just means that a Human brain equipped with Human eyes will label it as red.

You can create an objective definition for green (right), but that’s not really what you mean by “green” (left).

Color is fascinating because, unlike love, its subjectiveness can be easily studied.  We can say, without reservation, that a colorblind person sees colors differently than a colorseeing person.

Different people and animals see color very differently.  The right side is more or less the way most other mammals, as well as red/green colorblind people, see the world.

When a photon (light particle) strikes the back of the eye, whether or not it’s detected depends on what kind of cell it hits and on the wavelength of the light.  We have three kinds of cells, which is pretty good for a mammal, each of which has a different probability of detecting light at various wavelengths.  One of the consequences of this is that we don’t perceive a “true” spectrum.  Instead, our brains have three values to work with, and they create what we think of as color from those.

The three cones cells, and their sensitivity to light of different wavelengths. The dotted line corresponds to the sensitivity of rod cells, which are mostly used for low-light vision.

However, some animals have different kinds of cone cells that allow them to see colors differently, or see wavelengths of light that we don’t see at all.  For example, many insects and birds can see into the near-ultraviolet which is the color we don’t see just beyond purple.  Many birds have ultraviolet plumage, because why not, and many flowering plants use ultraviolet coloration to stand out and direct insects to their pollen.

Left: what people see. Middle: a false-color simulation of what insects may see. Right: a black and white ultraviolet only image

In the deep ocean most animals are blind, or have a very limited range of color sensitivity (it’s as dark as a witch’s something-or-other; what is there to see?).  But some species, like the Black Dragonfish, have taken advantage of that by generating red beams of light that they can see, but that their prey can’t.

The Black Dragonfish cleverly projects red light from those white thingies behind its eyes, which is invisible to its prey.

It may seem strange that some creatures are just “missing” big chucks of the light spectrum, but keep in mind; that’s all of us (people and critters alike).  The visible spectrum (so called, because we can see it), is the brightest part of the Sun’s spectrum.  Since it’s what’s around, life on Earth has evolved to see it (many times!).  But, there is a lot more spectrum out there that no living thing comes close to seeing.

We can see effectively none of the full light spectrum.

Point is, light comes in a lot of different wavelengths, but which wavelengths correspond to which color, or which can even be seen, depends entirely on the eyes of the creature doing the looking, and not really on any property of the light itself.  There isn’t any objective “real” color in the world.  The coloring of the rainbow is nothing more than a shared (reliable, consistent, and kick-ass) illusion.

The lack of objective colorness is a real pain for the science of photography.  Making a substance that becomes (what we call) yellow when it’s exposed to (what we call) yellow light is exactly as difficult as creating a substance that turns magenta when exposed to yellow light.  In a nut-shell, that’s why it took so long for color photography to come along, although there are other theories:

Bill Waterson makes a case for fatherhood.

So, it’s difficult to design film that reacts to light in such a way that we see the colors on the film as “accurate.”  But, in the same sense that yellow may as well be magenta (for all that the film cares), infra-red may as well be red!  You can (were you so motivated) buy infrared sensitive film that photographs light below what we can see, but above what most people call “heat” (the light radiated by warm, but not glowing-hot, objects).

A picture using film that’s sensitive to near-infrared light. This is not a picture of heat (that would use far-infrared), living plants just happen to be infrared-colored.

In fact, most “science pictures” you see: anything with stars, galaxies, individual cells, etc. are “false-color images”.  That is, the cameras detect a form of light that we can’t see (e.g., radiowaves), and then “translate” them into a form we can see.  Which is fine.  If they didn’t, then radio astronomy would be stunningly pointless.

Infrared photo by Richard Mosse.

Posted in -- By the Physicist, Biology, Philosophical | 62 Comments