Q: Is it of any coincidence that mathematics is able to describe physical reality – given that both are inventions of the human mind?

March 5th, 2010

Physicist: There’s a lot of math that doesn’t describe physical reality at all, and even some (few) mathematicians who feel that
“applicability” is just another word for “impurity”.  The ability of math to describe reality is just a consequence of the fact that reality is nice and consistent.

The fact that the math we use (addition, subtraction, geometry, calculus, whathaveyou) works is no coincidence at all.  Mathematics literally evolves in the sense that, if something doesn’t work, then people will ignore it.  So if you have a theory that \pi = 7, great, but no one will use it because it’s patently, provably false.  It doesn’t describe reality (in this case the reality that the ratio of the circumference to the diameter of a circle is \pi), so it goes the way of the Woolly Mammoth.

π=7

I assume that this question is about perceived reality (colors only exist in the brain, whereas in reality there is no “blueness” or “redness”), and not physical reality.  The fact that we can only describe (mathematically and otherwise) the reality we perceive does guide the direction of mathematical research, and as we perceive more we find that the field of math expands accordingly.  For example; number theory wasn’t much more than a hobby before digital communication and RSA encryption, and differential geometry was mostly a nuisance (and anal-retentive over-generalization) until general relativity cropped up.  Now these are both thriving fields of research (in computer science and physics, respectively).

However, just because something works in your head has absolutely no bearing on whether or not it will work in reality (which you would expect if the physical world were created by our minds).  Very good, very reasonable ideas get shot down by experiment every day, and we are constantly surprised.

Philosopher: If we assume the external world exists (independent of our minds), Math’s correspondence to reality is no more coincidental than the correspondence to reality of theories stated in any other language.  This isn’t dependent on the existence of mathematical objects, and it’s not dependent on Mathematical truths existing independently of humans (though I think they do).  If we assume the external world is merely an “invention of the human mind”, then the correspondence of Math to the world is even less coincidental, since the same thing is the author of both.

Posted in -- By the Physicist, Evolution, Math, Paranoia, Philosophical | No Comments »

Q: If you were to break down an average human body into its individual atoms, and then laid the atoms out in a single straight line, how far would it stretch?

March 3rd, 2010

Physicist: Atoms are a little “fuzzy”, so there exact size is a little tricky to define.  So taking their size in terms of bond length, and looking at the most common elements in the human body (by mass: 65% oxygen, 18% carbon, and 10% hydrogen), you’ll find that 1kg of person will stretch about 7 trillion km.  So an average (80kg) human would extend about 550 trillion km, or about 14 billion loops around the equator, or 1.4 billion trips to the moon, or about 58 light years.

So you can fit a rich man through the eye of a needle, but be sure to coil him up after you string him out.  Otherwise the process will take at least 58 years.

Posted in -- By the Physicist, Biology, Brain Teaser | No Comments »

Q: What’s it like when you travel at the speed of light?

March 3rd, 2010

Physicist: From a classical (Newtonian) view point this is a completely solid question.  However, in the context of special relativity the question itself is (unfortunately) non-sense.  For many practical purposes, the speed of light (hereafter I’ll call it “C”) is “infinitely fast”.  If you define infinitely fast as the speed you’ll be going if you accelerate forever, then C is exactly that.

Normally when you want to figure out “the behavior at infinity” you can “take a limit”.  For example; the limit as x goes to infinity of 1/x is 0.  This statement just means that as x gets bigger and bigger 1/x gets closer and closer to zero.  So by looking at the behavior at larger and larger finite values you can talk about what happens at infinity.  C, on the other hand, is fundamentally different from all other speeds.

At a basic level, speed is just distance traveled over time taken (as in “miles per hour”).  Due to the laws of special relativity, movement affects both the relative distances and relative time between two reference frames.

As a quick aside, a “reference frame” is just the set of all things that are moving at the same speed or, equivalently, are stationary with respect to each other.  So if you’re traveling down the highway you’re in the same frame as all the other cars around you (if everyone’s going the same speed), while the repair teams and clean-up crews on the shoulder are in a different reference frame.

It may seem silly to say it, but no matter how fast you move you still see things passing by, and it still takes at least a little time to get where you’re going.  At C however, the distance to your destination is always zero due to length contraction, while the time it takes to get there is also zero due to time dilation.  If you were to calculate your own speed you would say v= \frac{d}{t} = \frac{0}{0} = ?, which makes no damn sense.  I mean, what is that?

The universe: As seen by something traveling slower than C, and something traveling at C.

Also, consider this: at any other speed you can speed up or slow down, but at C you genuinely don’t have time to step on the brakes or the gas.  Literally, “time” and “distance” are phenomena that only make sense if you’re talking about them at speeds slower than C.  Stuff in the universe is divided into two categories: “massive” and “massless”.  Massive objects (anything with mass) always travel slower than C, while massless things must travel at C.

All that being said, you can wave your hands and talk about what life is like for a photon, that can’t exist at sub-light speeds (after all, what speed would you expect light to move at?).  When a photon is generated it immediately takes off at C, and never slows down until it runs into something.  Photons never experience time or distance.  As far as they’re concerned they are emitted and absorbed at the same place and time.  Many of the radio photons hitting you right now (about a third of them), have been traveling for around 15 billion years, but they think that the beginning of the universe just happened (or would, if they could think).

Posted in -- By the Physicist, Physics, Relativity | No Comments »

Q: Is there a real life example where two negatives make a positive?

March 3rd, 2010

Physicist: Although the laws of the universe are very absolute, the equations and terms we use are generally easy to rewrite and rephrase.  For example: it seems natural to describe the motion of a ball in terms of its altitude.  In this case gravity is negative (it decreases altitude).  But if instead you describe the motion of the ball in terms of “distance fallen”, then gravity becomes positive.

The classic example of the “arbitrarity of sign” is Ben Franklin’s horrifying mistake.  At the time that he was working it was impossible to tell where charge came from (in terms of electrons and protons), so he arbitrarily chose negative to be what we now know is the charge on electrons, and positive to be the charge on protons.  It makes no difference to the physical laws, which only care that the charges are different.  But it is annoying to electrical engineers who are haunted by the fact that “current”, which is defined as the flow of positive charge, actually points in the opposite direction in which the electrons move.

The point is this: I can’t think of any example of putting together two negative things and getting a positive thing, that couldn’t equally well be thought of as putting together to negative things and getting another negative thing.  For example: the force between two negative charges is repulsive.  So if you want to define “apart” as positive then two negatives (charges) makes a positive (force).  But if you define “together” as positive then two negatives make a negative.

Feynman summed up the general feeling in physics toward sign error (flipping positive/negative) when he said “If the sign is wrong, change it.”  So if, after lengthy calculation, you find that the moon is  – 238857 miles away, don’t stress about it.

Posted in -- By the Physicist, Philosophical, Physics | No Comments »

Q: Do the “laws” of physics and math exist? If so, where? Are they discovered or invented/created by humans?

March 1st, 2010

The original question was:

Mathematicians sometimes say, “There exists a number such that . . .”  Which provokes me to ask, Where does it exist? For how long has it existed? Did numbers exist before people did? Or did people somehow create (instead of discover) them?

In her “Incompleteness” quasi-biography of Godel (not a bad mathematician), Rebecca Goldstein emphasizes that he was a Platonist about math. What’s the current state of Platonism in math?

And such questions can be extended to the “laws” of physics: Do they exist?  If so, where? And for how long? Are they discovered (implying prior existence) or invented/created by humans?

Some comments relating to such issues would be interesting!

Physicist: Discovered.  Although most of the laws that can be re-arranged and expressed in different ways.  For example you can express “conservation of total momentum” as “the velocity of the center of mass never changes”.

A good physicist (one who pick their words carefully) will avoid saying that one thing or another is “true”.  Physics, and the laws we come up with, don’t exist “out there somewhere”.  Boiled down to its most basic, what we study is “what has worked before, and still seems to work” as opposed to “what is true”.

For example: Einstein showed that Newtonian physics is wrong (so wrong), but it still “works”.  If you learn Newton’s stuff you’ll notice that it’s fairly intuitive (compared to some other sciences at least), and seems to be true.  It was taught as fact for over 200 years, but again: wrong.  Taking this, and dozens of other similar stories as a warning, physicists try to talk only about what works and not what’s true.

That being said, some of the laws that have been found may actually be true, written into the nature of the universe.  I’d like to say that we know at least a few of them for sure, and that if what we know is wrong then the universe is entirely fucked.  However, that has been exactly the case before (I’m looking at you wave-particle duality), so who knows?

I like the hat best.

The pitiable population of "Monopoly". Are the rules they perceive the same as the rules written on the box? They could pass Go forever, and never know.

The laws we have could easily be special cases of the true laws (like Newtonian mechanics in relativistic mechanics), or could be merely the descriptions of the behavior created by those laws.

As far as the physical laws of the universe actually, physically existing in some form somewhere (this is the total extent of my understanding of Platonism): no, I don’t think there are very many scientists who think that.

Posted in -- By the Physicist, Philosophical | No Comments »

Q: Do we have free will?

February 27th, 2010

Physicist: If you want to get into an argument that drags on forever, you can frame a question like this in terms of consciousness, and the nature of choice, or any number of other ill-defined ill-understood ideas.  So consider only the question in terms of determinism;

Q: “does the state of the universe now (and in the past) completely determine the future of the universe and, by inclusion, the future of me?”

Back in the day (classical physics day) the answer could rightly be “yes” or “I don’t know”.  However, with the advent of modern quantum mech we’ve managed to make great strides on questions like this.  Now we can answer: “yes, no, and kinda”!  It’s progress like this that almost makes going back to clipper ships and horse carts worth it.

One of the biggest weirdnesses to come out of quantum mechanics is the idea of “super-position”, which is that a single thing (a particle or whatever) can be in multiple states at the same time (the state of a thing can involve position, speed, orientation, and even how the thing is related to other things).  QM allows us to see how all of those states change in time and interact each other.  However, any direct interaction with an “undetermined state” will reveal it to be in only one (of its many) state(s).  In what follows I’ll use “universe” to mean the universe with just one state (things did happen this way), and multiverse to mean all the states involved simultaneously (with all the interference and what-have-you).

The two ways of looking at this are the “Copenhagen interpretation” (wrong) and the “many worlds interpretation” (right).

“Yes!”: Given complete knowledge of the multiverse’s quantum wave function you can determine the future of that function forever.  Unfortunately, this isn’t particularly useful for those of us who live inside the universe.  The wave function in question encompasses all possibilities simultaneously and involves plenty of self-interference.  For example: when you do the double slit experiment you can calculate exactly what the fringes will look like on the screen, by doing a calculation that assumes that the photons involved go through both slits.  However, if you were to instead look a one of the slits, this doesn’t tell you anything about whether or not you will see the photon go through that slit.

(Just a quick note about the link above.  “The Secret”, and its creepy brainchild “What the bleep”, are both symptoms of a greater douchiness, but despite their culty bent they explain the double slit pretty well.)

In fact what happens is it goes through both slits, but in turn there are different versions of you that see both outcomes.  If you look at the multiverse as a whole (seeing every state) then everything is completely deterministic.  If you look at just one tiny piece at a time (like we seem to), then everything seems random.

Essentially, for every choice you can make, there are a whole mess of versions of you (identical up to the moment of choice) that do make that choice.  In fact, if your wave function is known completely, then how much (many?) of you goes down any road can be derived.  I don’t want to hear anyone saying “but I chose to do that!”, because some (part?) of you had to.  But then, some of you had to do every available choice.

“No!”: Part of the Copenhagen interpretation is fundamental, true randomness.  There’s no multiverse in Copenhagen (so don’t go flying there to look), so any choice you make is unpredictable (or at least, not completely predictable, there are some pretty reliable people out there) in the sense that no matter how good your fore-knowledge of someone’s wave function, you still can’t make perfect predictions.

It’s seems like there’s enough wiggle room in there to fit some free will.

“Kinda!”: Even if you subscribe to the many world hypothesis you could argue that “dude, who cares?”.  You’ll never meet (can’t meet) those other versions of yourself, so what does it matter that, in theory, all of your simultaneous actions are determined in a multiverse-kind-of-way?  Doesn’t.

Posted in -- By the Physicist, Paranoia, Philosophical, Physics, Quantum Theory | 1 Comment »

Q: How did mathematicians calculate trig functions and numbers like pi before calculators?

February 27th, 2010

Physicist: Don’t know.  But if you’re ever stuck on a desert island, here are some tricks you can use.  The name of the game is “Taylor polynomials“.

\sin{(x)} = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} x^{2n+1} = \frac{x}{1} - \frac{x^3}{1 \cdot 2 \cdot 3} + \frac{x^5}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} - \frac{x^7}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} + \cdots

\cos{(x)} = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{1 \cdot 2} + \frac{x^4}{1 \cdot 2 \cdot 3 \cdot 4} - \frac{x^6}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6} + \cdots

All the other trig function are just combinations of sine and cosine, so this is really all you need.  Of course, you can’t add up an infinite number of terms, so if you only go up to the xL term then the error between the sum you have and the actual value of sine or cosine is no more than \frac{x^L}{L!}.  Now x can be pretty big, but you can use the fact that sine and cosine repeat every 2 \pi, as well as the fact that \sin{(x \pm \pi)} = -\sin{(x)} and \cos{(x \pm \pi)} = -\cos{(x)}, to get the “x” down to -\frac{\pi}{2} \le x \le \frac{\pi}{2}.  So if you sum up to the xL term, then your error will be no larger than \frac{1}{L!} \left( \frac{\pi}{2} \right)^L.  The “1/L!” makes this error pretty small.  Summing up to the x10 term will be accurate to within 3 parts in 100,000 at worst.

For example:

\sin{(16)} = \sin{(16 - 2\pi)} = \sin{(16 - 4\pi)} = -\sin{(16 - 5\pi)} \approx -\sin{(0.2920)}

Summing up to the x5 term yields:

\sin{(16)} \approx -\sin{(0.2920)} \approx - \left( 0.2920 - \frac{0.2920^3}{6} + \frac{0.2920^5}{120} \right) = - 0.2879

Which is accurate to at least the first 4 decimal places.

There aren’t a hell of a lot of important mathematical constants out there.  The most important are “e” and “\pi”.

e^x = \lim_{m \to \infty} \left( 1 +\frac{x}{m} \right)^m \approx \sum_{n=0}^L \frac{x^n}{n!} = 1+ \frac{x}{1}+\frac{x^2}{1 \cdot 2}+\frac{x^3}{1 \cdot 2 \cdot 3}+\cdots with an error of no more than \frac{x^{L+1}}{(L+1)!}.  This is another example of a Taylor polynomial.  To calculate only e, just set x=1.

\pi \approx 4 \sum_{n=0}^L \frac{(-1)^n}{2n+1} = 4 \left( 1 - \frac{1}{3} +\frac{1}{5} - \frac{1}{7} + \cdots \right) with an error of no more than \frac{4}{2L+3}.  One way to derive this equation is to take the Taylor series for Arctan, and plug in 1 (\arctan{(1)} = \frac{\pi}{4}).  This is easy to remember but slow to converge (2,000 terms to get 3 decimal places), so here’s a better one:

\pi \approx \sqrt{12}\sum^L_{k=0} \frac{(-1)^k}{(2k+1) 3^k} = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right) with an error of no more than \frac{\sqrt{12}}{(2k+3) 3^{k+1}} \sim \frac{1}{3^k}.

Most people are under the impression that “there is no pattern in pi“, so the fact that we can write down an equation to find pi may seem a little odd.  What is generally meant by “no pattern in pi” is that there doesn’t seem to be any pattern in the decimal representation of pi (3.14159…).

The Taylor series and the approximations of pi and e above may seem cumbersome, but in most sciences you’ll find that it’s rare for anybody to go beyond the second term in a Taylor polynomial (sin(x) = x, cosine(x) = 1-.5x2).  Moreover, due mostly to our crippling sloth and handsomeness, most physicists are happy to say that \pi = e = 3.  So if you’re striving to get things exactly right, you may actually be an engineer.

Posted in -- By the Physicist, Equations, Math | 5 Comments »

Q: How can planes fly upside-down?

February 20th, 2010

Physicist: The narrative that usually leads to this question is something like: “It was the Wright brother’s brilliant wing shape, among other design innovations, that first made manned flight possible”.  So if the wing shape is so important, why does it still work if it’s flipped upside-down?

A wing (or airfoil or whatever) creates lift by taking advantage of a combination of the Bernoulli force and “angle of attack”.  By increasing angle of attack (tilting the nose up) the oncoming wind hits the bottom of the wing more, and pushes the plane up.  However, this force also increases drag substantially.  The Bernoulli force shows up when the air over the top of the wing is faster than the bottom, but it requires a bit of cleverness to get it to work.  Cleverness like the Kutta-Joukowski condition.

The Kutta-Joukowski condition: If air didn't flow faster over the top of the wing, then the air from the bottom would have to whip around the trailing edge with a very high acceleration. Too high, in fact. K-J assures that this singularity at the trailing edge doesn't show up.

When the Wright brothers built “the Flyer” (they were smart, not particularly creative) the engines available at the time were not powerful enough to lift themselves using only an angle of attack approach, so using a slick airfoil shape to take advantage of Bernoulli forces was essential to get off the ground.  Using the engines we have today (jets and whatnot) you could fly a brick, so long as the nose is pointed up.

So to actually answer the question; back in the day planes couldn’t fly upside-down.  But since then engines have become powerful enough to keep them in the air, despite the fact that by flying upside-down they’re being pushed toward the ground.  All they have to do is increase their angle of attack by pointing their nose up (or down, if you ask the pilot).

Posted in -- By the Physicist, Engineering, Physics | 3 Comments »

Q: A flurry of blackhole questions!

February 18th, 2010

Q: How much of the universe’s mass is currently in black holes?

Blackholes fall into two basic categories: stellar mass blackholes which have a mass of 3 to 30 Suns (give or take), and super-massive blackholes which usually have masses of more than 100,000 Suns.  Even in our own galaxy it’s essentially impossible to determine whether or not stellar mass blackholes are present.  I mean… they’re black, and they’re not heavy enough to throw around the nearby stars.  However, the supermassive blackholes do throw nearby stars around.  And that star-chucking property has allowed us to find that they have a mass of roughly 0.1% of the “bulge-mass” of the galaxies they sit in (the bulge is just the part of a galaxy that isn’t a disk).  So if I had to make a flying guestimate, I’d say that somewhere around 0.2% of the mass of any given galaxy is tied up in blackholes.

Q: Is there a graph of the number of black holes created since the big bang?

Probably.  Blackholes form from large stars, and large stars tend to have short lifetimes (a mere several million years).  So there should be a pretty sharp correlation between star formation rates and blackhole formation rates.  However, star formation rates are also notoriously difficult to measure.

Q: When was the first black hole created and when will the last one be?

“Primordial Blackholes“, if they exist, would have formed almost instantly after the big bang.  If the Big Rip happens, then you can expect the last blackholes to form 50 million years before the end of the universe (give or take).  Otherwise, there’s no telling.

Q: How old will the universe be when black holes start to evaporate, if they even do?

Primordial blackholes should be popping right now.  The lightest stellar-mass blackholes (3 suns) won’t start evaporating until after the universe has cooled to below their Hawking temperature, which should be in about 13 billion years, when the universe is twice as old.  However, one age-of-the-universe is chump change compared to the 1069 years (about 10 billion trillion trillion trillion trillion times the age of the universe) it will take for the first stellar-mass blackholes to completely evaporate.

Q: Could all black holes evaporate away in a expanding cooling  universe?

Yup.

Q: What happens to the universe if all the back holes evaporate away?

No more blackholes?

Posted in -- By the Physicist, Astronomy, Physics | 1 Comment »

Q: Why does going fast or being lower make time slow down?

February 13th, 2010

Physicist: Back in the day, Galileo came up with the “Galilean Equivalence Principle” (GEP) which states that all the laws of physics work exactly the same, regardless of how fast you’re moving, or indeed whether or not you’re moving.  (Acceleration is a different story.  Acceleration screws everything up.)  What Einstein did was to tenaciously hold onto the GEP, regardless of what common sense and everyone around told him.  It turns out that the speed of light can be derived from a study of physical laws.  But if physics is the same for everybody, then the speed of light (hereafter “C”) must be the same for everybody.  The new principle, that the laws of physics are independent of velocity and that C is the same for everybody, is called the Einstein Equivalence Principle (EEP).

Moving faster makes time slow down: I’ve found that the best way to understand this is to actually do the calculation, then sit back and think about it.  Now, if a relativistic argument doesn’t hinge on the invariance of C, then it isn’t relativistic.  So ask yourself “What do the speed of light and time have to do with each other?”.  A good way to explore the connection is a “light clock”.  A light clock is a pair of mirrors, a fixed distance d apart, that bounce a photon back and forth and *clicks* at every bounce.  What follows is essentially the exact thought experiment that Einstein proposed to derive how time is affected by movement.

The proper time "τ" is how long it takes for the clock to tick if you're moving with it. The world time "t" is the time it takes for the clock to tick if you're moving with a relative velocity of V.

Let’s say Alice is holding a light clock, and Bob is watching her run by, while holding it, with speed V.  Alice is standing still (according to Alice), and the time, \tau, between ticks is easy to figure out: it’s just \tau = \frac{d}{C}.  From Bob’s perspective the photon in the clock doesn’t just travel up and down, it must also travel sideways, to keep up with Alice.  The additional sideways motion means that the photon has to cover a greater distance, and since it travels at a fixed speed (EEP y’all!) it must take more time.  The exact amount of time can be figured out by thinking about the distances involved.  Mix in a pinch of Pythagoras and Boom!: the time between ticks for Bob.  So Bob sees Alice’s clock ticking slower than Alice does.  You can easily reverse this experiment (just give Bob a clock), and you’ll see that Alice sees Bob’s clock running slow in exactly the same way.

It turns out that the really useful quantity here is the ratio: \frac{t}{\tau} = \frac{C}{d} \frac{d}{\sqrt{C^2 - V^2}} = \frac{C}{\sqrt{C^2 - V^2}} = \sqrt{\frac{C^2}{C^2-V^2}} = \sqrt{\frac{1}{1-\frac{V^2}{C^2}}} = \frac{1}{\sqrt{1-\frac{V^2}{C^2}}}.  This equation is called “gamma”.  It’s so important in relativity I’ll say it again: \gamma = \frac{1}{\sqrt{1-\frac{V^2}{C^2}}}.

It may seem at first glance that the different measurements are an illusion of some kind, like things in the distance looking smaller and slower, but unfortunately that’s not the case.  For Alice the light definitely travels a shorter distance, and the clock ticks faster.  For Bob the light really does travel a greater distance, and the clock ticks slower.  If you’re wondering why there’s no paradox, or want more details, then find yourself a book on relativity.  There are plenty.  Or look up Lorentz boosts.  (The very short answer is that position is also important.)

The lower the slower: Less commonly known, is that the lower you are in a gravity well, the slower time passes.  So someone on a mountain will age (very, very slightly) faster than someone in a valley.  This falls into the realm of general relativity, and the derivation is substantially more difficult.  Einstein crapped out special relativity in a few months, but it took him another 10 years to get general relativity figured out.  Here’s a good way to picture why (but not quite derive how) acceleration causes nearby points to experience time differently:

Redder light at the top, bluer light at the bottom.

Alice and Bob (again) are sitting at opposite ends of an accelerating rocket (that is to say; the rocket is on, so they’re speeding up).  Alice is sitting at the Apex (top) of the rocket and she’s shining a red light toward Bob at the Bottom of the rocket.  It takes some time (not much) for the light to get from the Apex of the rocket to the Bottom.  In that time Bob has had a chance to speed up a little, so by the time the light gets to him it will be a little bit blue-shifted.  Again, Alice sees red light at the Apex and Bob sees blue light at the Bottom.

Counting the blue crests is faster than counting the red crests. However, since it's all the same light beam the number of crests has to be the same to everybody.

The time between wave crests for Bob are short, the time between wave crests for Alice are long.  Say for example that the blueshift increases the frequency by a factor of two, and Alice counts 10 crests per second.  Then Bob will count 20 crests per second (No new crests are being added in between the top and the bottom of the rocket).  Therefore, 2 seconds of Alice’s time happens in 1 second of Bob’s time.  Alice is moving through time faster.

Einstein’s insight (a way bigger jump than the EEP) was that gravitational acceleration and inertial acceleration are one and the same.  So the acceleration that pushes you down in a rocket does all the same things that the acceleration due to gravity does.  There’s no way to tell if the rocket is on and you’re flying through space, or if the rocket is off and you’re still on the launch pad.

It’s worth mentioning that the first time you read this it should be very difficult to understand.  Relativity = mind bending.

Posted in -- By the Physicist, Physics, Relativity | 2 Comments »

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