Rovelli 5: Fundamental Constants of the Universe

This is my fifth post on Carlo Rovelli's new book, Reality Is Not What It Seems. The first three posts were:

• From Democritus to Relativity to Quantum Mechanics
• Spacetime is Quantum
• Quanta of Space
• Time Does Not Exist

1. Chapters 8-11 are fairly short. Some of it covers basic material from cosmology but there are some helpful synthetic thoughts too.

Chapter 8 fills in some gaps with regard to the Big Bang. He gives high praise to Georges Lemaître, a Belgian priest, for supporting the idea of a "primordial atom" even though Einstein strongly disagreed. Einstein's equations seemed to suggest that the universe was expanding, but he didn't want to believe it. In fact he added a "cosmological constant" to his equations to fix it (Λ). Lemaître turned out to be right about expansion.

Then Einstein lamented adding the cosmological constant and wanted to remove it. Again, Lemaître suggested he should leave it. Lemaître proved to be right again over Einstein. So in both cases Rovelli writes, "It doesn't fall to everyone to disprove Einstein" (204).

Then Lemaître stopped the Pope, apparently, from making the Big Bang official church belief. Again Rovelli says, "It is not given to everyone to disprove the pope" (205). This falls under the principle of not inserting God too dogmatically into your scientific theories, because theories change.

2. Rovelli favors something he calls "the Big Bounce." The idea here is that "our universe could be the collapse of a previous contracting universe passing across a quantum phase, where space and time are dissolved into probabilities" (208). This seems to me to be a form of the oscillating Big Bang theory.

I thought that the current sense of things was that there was not enough matter in the universe to pull everything back together and thus that the universe was headed for a "Big Rip" at the end of things. This is also different from the multiverse idea that other books I've read have suggested, namely, that our universe is just one of an infinite number of universe bubbles.

Meanwhile, Rovelli nicely, I think, counters Lee Smolin's sense that the universe is all there is by definition. Rovelli, much more soundly says, "The word 'universe' has assumed another meaning in cosmology: it refers to the spacetime continuum that we see directly around us, filled with galaxies and history of which we observe. There is no reason to be certain that, in this sense, this universe is the only one in existence" (208).

Take that Smolin, who says in the first chapter of Quantum Gravity, "By definition the universe is all there is" (17). Let's just say Rovelli is a much better philosopher, although I don't always agree with him.

The dissolution of spacetime into a cloud of possibilities when you have that much mass at a quantum size is an intriguing idea.

3. Chapter 9 asks if we have any experimental evidence for loop quantum gravity. A number of times he pushes back both against those who say you cannot talk about anything that you cannot now experimentally show and those who wildly speculate detached from current trajectories. To me this positioning makes perfect sense.

On the one hand, a theory should proceed to experimentation. "A theory lacking empirical confirmation is a theory that has not yet passed its exams" (212). On the other hand, he disagrees with wild hypotheses. "Many theoretical physicists are today looking for new theories by picking arbitrary hypotheses... I don't think that this way of doing science has ever produced good results" (215-16).

Rather, all the experimental evidence has been confirming the three cornerstones of modern physics: general relativity, quantum mechanics, and the Standard Model within quantum mechanics. The new findings have brought a complete absence of surprise. Hawking was disappointed.

The three big findings of this decade are 1) the confirmation of the Higgs boson, 2) the cosmic measurements of the Planck satellite, and 3) the detection of gravitational waves.

Also, the fact that CERN has not discovered supersymmetry is a blow to string theorists, which is why Sheldon on Big Bang Theory went looking for something else to study. :-)

4. Mapping of the cosmos has given us a sense of the lay of the background radiation left not too long after the so called Big Bang. The idea here is that it took some time for the universe to cool down enough for the light (photons) of creation to be released.

The remnants of this release are called "cosmic background radiation" (CBR), alleged to have happened some 380,000 years after the Big Bang.

Apparently, if LQG is correct (loop quantum gravity), then there should also be a gravitational background radiation. An experiment called LISA involving three satellites around the sun, might be able to test for these.

5. Chapter 10 looks at quantum black holes. There are black holes at the centers of most galaxies and, in at least one theory, they may account for what it otherwise called dark matter.

The horizon of a black hole is the point where you might stay out. Past that, nothing can get out. Time stops at the horizon. Stephen Hawking's claim to fame was his discovery that black holes slowly evaporate. Eugenio Bianchi showed that loop quantum gravity can also demonstrate Hawking's formula for the heat of a black hole.

What LQG would show is that, since spacetime is not infinitely divisible--since it never can reach a singularity--at some point a black hole should explode in a miniature version of the Big Bang. From our perspective outside a black hole, this would take billions years, even if it is only moments inside the black hole. Since the universe is allegedly 14 billion years old, we might find some of these. Rovelli suggests that some "fast radio bursts" detected by radio telescopes could be such.

6. Chapter 11 is called the end of infinity. The common sense of the suggestion here is so obvious I've thought of it for some time now and I'm not even a scientist. Why didn't Dirac and Feynman? Quantum mechanics and relativity are plagued with infinities. Feynman the pragmatist simply substituted the experimental values for certain infinities to get his equations to work.

But LQG, because it sees space as quantized, eliminates the infinities. This seems so obvious to me that it is surprising it is not a fundamental working assumption of modern physics.

"Putting a limit to infinity is a recurrent theme in modern physics. Special relativity may be summarized as the discovery that there exists a maximum velocity for all physical systems. Quantum mechanics can be summarized as the discovery that there exists a maximum of information for each physical system. The minimum length is the Planck length L_P, the maximum velocity is the speed of light c, and the total information is determined by the Planck constant h" (232).

Now we are getting somewhere. This is what I've been thinking and looking for someone to put succinctly like this. "The existence of these minimum and maximum values for length, velocity, and action fixes a natural system of units. Instead of measuring speed in kilometers per hour... we can measure it in fractions of the speed of light.. In the same way, we can posit L_P = 1 by definition and measure length in multiples of Planck's length. And we can posit h = 1 and measure actions in multiples of Planck's constant. In this way, we have a natural system of fundamental unities from which the others follow" (233).

One more seems to complete Rovelli's set, namely, the cosmological constant (Λ) used in relativity. I have a book called Just Six Numbers that is also on my reading list. I'm hoping it will help me understand the importance of the ratio between the cosmological constant and the Planck length.

My next post should finish the book.

GPS and Relativity

I'm dawdling through a book called, The Science of Interstellar, by the physicist Kip Thorne. Here's an interesting paragraph on GPS (slightly edited):

"Our smart phones rely on radio signals from a set of 27 satellites at a height of 20,000 kilometers.... Each radio signal... tells the smartphone where the satellite is located and the time the signal was transmitted... The scheme would fail if the signal were the true times measured on the satellite. Time at a 20000 kilometer height flows more rapidly than on earth by 40 microseconds each day and the satellites must correct for this. They measure time with their own clocks, then slow that time down to the rate of time flow on Earth before transmitting it to our phones."

Einstein predicted that time moves more slowly relative to locations close to heavy gravitational objects than it does relative to locations that aren't.

Science Friday: Jumping Photons (2)

Last week I (foolishly) started blogging through George Gamow's Thirty Years That Shook Physics. (Foolish because I wonder whether my puny brain will even make it through the first chapter).

When last we met our hero (old quantum theory), Max Planck had just suggested that the energy of light radiation does not exist at every continuous quantity but that the higher the frequency of light radiation, the bigger the "packets" in which that energy existed. Planck didn't like it. The suggestion was an "act of desperation." He did it to make the theory match the experiment. But he hoped it wasn't part of the basic nature of energy but some quirk having to do with atoms.

But Einstein dashed his hopes to pieces, five years later in 1905. Einstein didn't win the Nobel Prize for relativity, special or general. Einstein didn't get the Nobel Prize for E = mc². Einstein won the Nobel Prize for demonstrating that Planck's "quantum" was in fact very much real. Even though light behaved like a wave on a large scale, on the smallest scale, light was little packets or "quanta" of energy, which Einstein called, "photons." The size of these quanta increased with frequency.

What Einstein did was show that Planck's quantum explained one of the puzzles of late 1800s physics: the photoelectric effect. The photoelectric effect is the fact that, below a certain threshold frequency, the electrons in the surface of a metal plate jump free with the same amount of energy no matter how intense of light radiation you shine on it. As you increase the intensity of the light on the surface at a lower frequency, the number of electrons jumping increases but the amount of energy each of them has does not.

However, once you reach the threshold frequency, which is different for different metals, the amount of energy the electrons have as they jump begins to increase. In fact, after not increasing at all, the amount of energy begins to increase evenly with the increase in frequency.

Planck's E_photon = hf explained this phenomenon. The energy of a photon increased evenly with the increase of frequency.

Next week: The Compton Effect

Science Friday: Planck's Quantum

Since I was in high school, I've wanted to understand quantum physics. In my first year at college, I felt called to ministry and so never continued that path. But in mid-life I've tried to get back into some of those old aspirations on the side.

One book I've had for years is George Gamow's, Thirty Years that Shook Physics (it has a Joseph Beth's bookmark in it from Asbury days). This past week I reread the section on Max Planck. Planck's reluctant proposal in 1900 launched modern physics. He didn't want to. He called his proposal "an act of desperation." He was not at all an "outside the box" type.

According to the story, Planck was working on what had come to be known as the "ultraviolet catastrophe" (I don't actually think that's quite right, but this is the best way to tell the story, it seems). If you have a bunch of gas molecules in a chamber, you expect over time that the energy of the molecules will eventually even out as the molecules bump into each other and share energy (the equipartition theorem). It's the same idea as what happens when you open the door between a hot room and a cold room. Eventually the temperature will even out.

(In reality, there is still somewhat of a variation, but most molecules statistically will have a certain average amount of energy at a particular temperature and those with more or less energy will be in decreasing numbers.)

Two scientists from the late 1800s (Rayleigh and Jeans) supposed that the distribution of energy in a "blackbody" (which absorbs all wavelengths of light)--or in a cavity made to mirror the effects of a blackbody--would also spread out similarly into waves of all the frequencies and wavelengths. Following the idea of equipartition, the total energy would begin to distribute itself into to each wavelength of light until every possible wavelength had roughly the same amount of the total energy.

Gamow has a helpful illustration. It would be like playing a note on the piano without a damper between the strings. In such a situation, the energy from playing one note would seep into all the other strings until each string in the piano corresponding to each key had 1/88th of the initial energy.

This isn't what happens, and it's a good thing too. What would happen if the energy of a fireplace soon spread into other wavelengths so that before long, the fireplace was spewing out ultraviolet and x-rays. Gamma rays would follow and you would soon die.

Planck's suggestion was that energy couldn't seep into higher frequencies of radiation as easily as into the lower frequencies. The higher the frequency, the higher the threshold of energy it took to distribute into that part of the spectrum. Light was not behaving like a continuous wave that spread out evenly to all frequencies but more like discrete packages of energy of differing sizes. The size of these packages was different at different frequencies.

I've tried to think of a good illustration, even to help myself understand exactly what Planck is suggesting. Consider this one a work in progress. Let's say there is an auditorium where the seats are arranged somewhat strangely. The front row has lots of cheap seats. The farther back in the auditorium you go, the seats are fewer in each row and they cost more.

So let's say you start off with a certain amount of energy in a blackbody, based on the temperature. It is like having a certain amount of money to spend on seats in the auditorium. The energy is distributed. Most of it goes to the lower frequencies because those seats are cheap. Yes, some of it goes toward the more expensive seats, but not nearly as much.

How much does a seat cost? Planck found he could make the graph in theory look like the real graph by introducing a number that would eventually be called Planck's constant (6.626 x 10^-34 Joule-seconds, given the symbol h). A packet of energy at a particular frequency, E, was h times the frequency:

E = hf

That's how much a seat cost. A seat at a low frequency didn't cost that much. But the higher the frequency, the more expensive the seat.

Einstein would seal the deal on Planck's desperate theory five years later. Light was not a continuous wave. When you break things way down, you get to something like atoms of energy smaller than which you cannot go--photons.

Next Friday: Einstein and the photoelectric effect

Expanding Universes, Collapsing Stars, Cuckoo Einstein, and Steady States

Alas, it looks like I will leave Florida with only 7 chapters of The Perfect Theory finished. So what else is new. Going by my past record, those last 115 pages may never be read.

Here's a much briefer summary of chapters 3-6.  I've already summarized:

Chapter 1: Einstein in 1907
Chapter 2: The General Theory of Relativity Born

Chapter 3 is called "Correct Mathematics, Abominable Physics"
I feel a bit sorry for Einstein. He had a few very significant "outside the box" thoughts in very early 1900s. He generally seemed to have them bouncing ideas off of genius friends and acquaintances. His college buddy Marcel Grossman helped him bounce his way into special relativity in 1905. Then David Hilbert helped him find his way to general relativity in 1915.

But after that, he pretty much became a celebrity "has been" and eventually a "cuckoo," as Oppenheimer once called him. One of the things I find striking about these chapters is how closed minded the greats were. They rose to fame on thinking outside the box but then became part of the establishment that pretty much ignored new ideas that didn't fit with their sense of things. It's all pretty straightforward Thomas Kuhn stuff.

So, in this chapter, we hear how Einstein and Eddington basically ignored a string of relativity enthusiasts who came to them showing how there were possible solutions to the general relativity functions that might point to an expanding universe. Alexander Friedmann was a Russian who showed that, according to Einstein's functions, the universe had either to expand or contract. Einstein mistakenly corrected him in publication when it was Einstein's mistake.

Einstein and Eddington, for whatever reason, just didn't like the idea of an expanding universe. Georges Lemaître, a Roman Catholic priest, was another who showed this to Einstein. Einstein's response was that his calculations were correct mathematically but that his "physics was abominable."

Eventually, Einstein would have to eat dirt, as would Eddington. In 1925, Hubble showed that there were galaxies beyond our Milky Way galaxy. Then by January of 1929, Hubble and Humason had both shown that the redshifts of these far away nebulae were larger than those closer to us. In other words, the universe was expanding.

Lemaître had been one of Eddington's own students and he had ignored him. But in the end, both Eddington and Einstein would repent and thrust him into center stage. He would become the world's leading cosmologist. Although Lemaître came to his conclusions scientifically, he of course believed that his findings, that the universe expanded from a beginning, fit with his faith in God.

Chapter 4 is called "Collapsing Stars"
Einstein and Eddington also found the idea of a burned out star that might collapse in on itself "absurd." In other words, the idea of a black hole didn't fit with their sensibilities. Eddington had written a classic book in 1926 called The Internal Constitution of Stars, but he couldn't bring himself to see a star becoming so dense that not even light could get out, so much so that the inside of the star became permanently shut off from the outside world.

The work of several "relativists" pointed in this direction. The Russian Karl Schwarzschild died prematurely of illness in 1916, but he had simplified some aspects of Einstein's theory, explained some loose ends in the prediction of Mercury and other planets' orbits. But his work had also curiously predicted the phenomenon of black holes.

Subrahmanyan Chandrasekhar (Chandra for short) integrated some of the developments in quantum mechanics into the relativity of stars and supported Schwarzchild's conclusions from a different angle. But when he presented it, Eddington's clout ruled it out. Mathematically possible but not something that would take place in the elegant universe as he saw it. Chandra would then abandon his research on the subject of white dwarfs, even though he was pretty much correct.

The last part of this chapter is about Robert Oppenheimer, father of the atomic bomb and leader of the Manhattan Project. He set up quite the physics team at Berkeley. He and one of his students published a paper in 1939 arguing for black holes. Of course it came out the day that the Nazi's invaded Poland, and it would disappear for a good while.

Chapter 5 is called, "Completely Cuckoo"
This was Oppenheimer's description of Einstein in his later years at Princeton. Einstein could never reconcile himself with the quantum physics of Heisenburg's uncertainty principle. He spent his last years more or less as a recluse trying to find a grand unified theory that would never come.

These were apparently years when general relativity was viewed somewhat like string theory is today in many circles. Without any clear way experimentally to test it, those who work with it seem to be playing idiosyncratic games with math without any real pay off in the real world.

In the 50s, Princeton played home to a number of famous thinkers, the "Institute for Advanced Study." But before Ferreira, the author of the book, gets there, he reviews how the Nazi's opposed Einstein's theory as "Jewish physics." In the USSR, there was similar opposition by materialists to the seemingly idealized world of Einstein. Of course things like the atomic bomb and the nuclear arms race were too important for the Nazis or the Soviets in the end to let these "fundamentalists" win.

One friend Einstein did have at Princeton in these years was Kurt Gödel. He played with Einsteins general relativity equations and asked what would the universe be like if it were rotating on a central axis. The result? Spacetime would loop back on itself and you could actually travel back in time. Einstein's reaction to his friend's work was predictable--mathematically interesting but completely unrelated to the real world.

Oppenheimer would eventually move from Berkeley to Princeton to head the Institute. He and Einstein had a cordial relationship. Oppenheimer respected Einstein even if he considered him cuckoo and more of a landmark than a beacon in the story of physics. He would later say that Einstein "did no good" in his later life. However, Einstein supported Oppenheimer when he hit on hard times for being opposed to the Hydrogen bomb. Oppie would come to regret the Manhatten project and, in time, he lost his national security clearance. But Einstein supported him and was untouchable in the public eye. Einstein was a pacifist.

Chapter 6: "Radio Days"
This chapter has to do with the discovery of the quasar, which gives off massive radio waves. A lot of the chapter deals with the charismatic Fred Hoyle, who pioneered the "steady state" theory of the universe. Hoyle found the idea that the universe had a beginning and started with a "big bang" a detestable idea. He suggested that the universe was constantly generating enough matter to keep going. It thus wouldn't need a beginning.

Because Hoyle was able to get the idea out to the public, it was taken quite seriously by the British masses, even though few scientists thought the data supported him at all. Suffice it to say, they repaid him by refusing to publish his papers for two or three years.

Hoyle's theory would eventually be shown incorrect.

I don't know if I'll get around to blogging any more from this book. I do hope to finish it in the next couple months. If I find anything I think is really interesting, I may be back...

2. General Theory of Relativity Born

I continue my Spring Break posts on The Perfect Theory, a book on the general theory of relativity and its footprint on the last century.  So far:

1. Einstein in 1907

1. I might have titled these thoughts on the second chapter, "Einstein in 1915." That is the year that Einstein delivered a short four page paper to the Prussian Academy of Sciences explaining how gravity fit with relativity and thus truly gave birth to the general theory of relativity (as opposed to the special theory he set out in 1905). The paper consisted of 10 field equations.

There is some debate whether Einstein actually put these equations in their final form first or whether David Hilbert did. Einstein was not a math-lover, particularly. To be sure, he was way ahead of most of us, but he considered math "superfluous erudition."

I have a hunch I know where he was coming from. He did not mean the calculus or algebra that we find so useful in engineering. I suspect he meant the drive to prove things that seem obvious. From what I can tell, Einstein was more of an intuitive soul, and the drive to mathematical minutia probably did not suit him well.

David Hilbert (University of Göttingen) was quite the opposite, with an agenda to solve 23 problems in the twentieth century. A quick perusal of the list and you might agree with Einstein. Hilbert wanted to "reduce every single mathematical fact in the universe from no more than half a dozen axioms" (76). Kurt Gödel would later demolish Hilbert's goal in 1931 with his incompleteness theorem.

Hilbert was extraordinarily more gifted at math than Einstein and Einstein ended up turning to him to help with non-Euclidean geometry, which was less than 100 years old. Euclid was Greek and had proposed, using common sense, that two parallel lines never meet.  In the 1820s, Carl Friedrich Gauss had explored the rules of geometry on, say, curved sheets of paper, where Euclid's assumptions don't hold. On a sphere, parallel lines intersect and the angles of a triangle add up to 270 degrees. Bernhard Riemann in the 1850s had explored the rules of geometry into all sorts of non-Euclidean obscurity.

So in 1915, Einstein and Hilbert mailed back and forth, as Einstein tried to use Riemann's math to express gravity as a result of the curvature of space rather than as a force per se. It's still debated whether Hilbert beat Einstein, although Hilbert yielded to Einstein and Einstein usually gets the ultimate credit. It might be fair to say that neither would have come up with the answer without the other.

Einstein came up with 10 equations of 10 functions of the geometry of space and time in which "gravity is nothing more than objects moving in the geometry of spacetime. Massive objects affect the geometry, curving space and time" (21).

2. There is a second part to the chapter that relates to Arthur Eddington. There is often a politics and a history to science, as to any discipline, and it is seen in this book. In 1915, Europe was at war. In World War I, the Germans were fighting the English. Both many English scientists and German scientists went irrational, as war tends to make all of us. 93 German scientists (not Einstein) signed "An Appeal to the Cultured World" in support of the German government and rather bad on its facts. Meanwhile, Eddington's colleagues in England wanted to dismiss all German scientific thought as obviously inferior.

Eddington managed to convince the right people to let him head to the island of Principe in 1918 instead of to the war. His task? To measure where a certain set of stars appeared to be as their light passed by the sun during an eclipse. As Einstein's theory had predicted, their apparent position was off approximately what Einstein's theory said it would be due to the effect of the sun curving space because of its great mass.

In 1919 Eddington presented the results to the Royal Astronomical Society, "the most important result obtained in connection with the theory of gravitation since Newton's day," J. J. Thompson said (discoverer of the electron). Einstein was now a celebrity and Eddington the foremost authority on Einstein's theory in the English-speaking world.

The general theory of relativity has been substantiated time and time again ever since. Without taking such things into account, things like GPS wouldn't work.

Einstein in 1907

There are two items on my life's bucket list that I fear I will never attain--a general understanding of quantum mechanics and relativity. There's nothing more humbling to me than my repeated attempts to start up these mountains.

I came across a book a few weeks ago, The Perfect Theory, that I'm giving a little time to this week on vacation. It's about Einstein's theory of general relativity. I doubt I'll get too far into it but I thought I might blog a little to jog the memory when I retire at 70 and return to that bucket list with all that free time. :-)

The first chapter starts interestingly enough in 1907. That's further than I've ever gotten before. Einstein came up with his theory of special relativity in 1905. That was the year he published his paper arguing that clocks move more slowly and objects appear to shrink as they approach the speed of light.

He was working as a patent clerk from 8-6 every day at the time and talking through the problems of modern physics with his old friend Marcel Grossman on the side. Einstein didn't play the game of academics very well. He did what he wanted and didn't accept the principle that professors assign grades which have something to do with getting jobs. So patent office it was, a mercy job even at that.

Einstein set out to resolve the consequent contradictions from two claims of the physics of his day: 1) the laws of physics work the same in any inertial frame and 2) the speed of light always has the same value. These two principles contradicted each other because if you shine a light from the front of a moving train, you would think that light would move faster than some light you shined from a flashlight on the ground in the same direction.

If I have it right, Einstein's famous solution was to suggest that, from your perspective standing on the ground, the time for the light shining from the front of the train moves more slowly than it does for you with the light shining from you the ground. But on the train, time moves the same as always.

Thus the grandfather paradox. If I am on a train moving close to the speed of light, time proceeds normally for me. But if my son does not get on that train. He may grow up and have children who, by the time I return to the speed he is moving, are older than I am.

In 1907, Einstein was asked to summarize his theory and give implications, which he did in the Yearbook of Electronics and Radioactivity. His series was called, "On the Relativity Principle and the Conclusions Drawn from It." There was one very significant addition.

The Special Theory of Relativity only works for frames of reference that are moving at a constant speed in relation to each other. It does not apply to frames of reference that are speeding up in relation to other frames of reference. In other words, it only works if the train is moving at a constant speed in relation to the ground.

Meanwhile, gravity involves acceleration. Einstein didn't have it all figured out (and he would need help), but he had an insight in 1907 that would later bear the appropriate fruit. "If a person falls freely he will not feel his own weight." What I take this insight to be is that, in the frame of reference of the person falling, there is a constancy that is different from that perceived by a person looking on, much as that experienced by a person on a train who does not feel the speed an observer on the ground observes.

With a little help, this seed would lead to a general theory in 1915 that could accommodate gravity and accelerating frames of reference.

Science Friday: Einstein's Relativity 1

Albert Einstein, Relativity: The Special and the General Theory, trans. by R. W. Lawson (New York: Bonanza, 1961).

Part 1: The Special Theory of Relativity
I. Physical Meaning of Geometrical Propositions
Geometry reduces to assumptions, "axioms."  Propositions are built out of these axioms.  We prove things based on whether they proceed logically from the axioms.

At the same time, the basic ideas of geometry undoubtedly derived from nature originally. "Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas" (2).

In fact, we can turn geometry into physics if we add the following proposition: "Two points on a practically rigid body always correspond to the same distance..., independently of any changes in position" (3).  So Einstein has now connected abstract geometry to physical objects.  We have connected what was an idea (geometry) to the real world (physics).

As the book continues, Einstein will go on to show that the "truth" of this proposition is actually limited.
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Note: The idea that the ideas of math originate in nature seems sound.  Pythagoras believed that numbers were the greatest reality, and that the world played them out in some way. Plato believed that the physical world was a copy of an ideal original. They had it backwards.  Aristotle probably came closer: numbers and mathematical "realities" are simply abstractions of the "real world."
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II. The System of Coordinates
We can identify the location of something with reference to the body it is on or we can take a rigid measuring rod from a body to it.  So we can locate a place on the earth by measuring off how many of the units on this rod it takes to get to it from some reference point or we can use a measuring pole to get to a point in a cloud above that point on the earth.

So we imagine locating any point by a number of units of some rigid measuring body to get to it from some point of reference. We don't always have to use a physical pole, since we can use other means to measure.  The Cartesian system of reference imagines three planes (x, y, z) from which we can construct perpendiculars to any point in space.

So in Euclidean geometry, "Every description of events in space involves the use of a rigid body to which such events have to be referred" (8).

III. Space and Time in Classical Mechanics
"The purpose of mechanics is to describe how bodies change their position in space with 'time'" (9). But the concepts of "position" and "space" are somewhat ambiguous. If I drop a rock straight down from a train, it looks like it falls in a straight line to me, but it looks likes it falls in the shape of a parabola to someone sitting on the ground.

First, let's do away with the notion of space ("of which, we must honestly acknowledge, we cannot form the slightest conception," 9) and replace it with "motion relative to a practically rigid body of reference." And by "rigid body of reference," we are thinking of a "system of coordinates" such as was defined in the previous chapter.

There is thus "no such thing as an independently existing trajectory... but only a trajectory relative to a particular body of reference" (10).

A complete description of the motion of a body includes how its position relative that frame of reference changes in relation to time.  The person dropping the rock off the train has a clock and the observer on the ground both have identical clocks measuring "ticks" on the clock as the rock drops.

IV. The Galilean System of Coordinates
The fundamental law of mechanics in physics is the law of inertia set down by Galileo and Newton.  A body at rest tends to stay at rest, and body in motion tends to stay in motion.  This law, however, only relates to a particular frame of reference, an "inertial frame of reference."  [A body at rest on the earth stays at rest on the earth, but it is constantly accelerating in relation to the sun because the earth is spinning.]

"A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called 'a Galilean system of co-ordinates'" (11).

V. The Principle of Relativity (in the restricted sense)
A "uniform translation" is when something is moving at a constant velocity and direction in relation to some frame of reference.  It is not rotating, for example.

"If K is a Galilean co-ordinate system, then every other co-ordinate system K' is a Galilean one, when, in relation to K, it is in a condition of uniform motion of translation" (13).  Accordingly, the mechanical laws of Galileo and Newton will hold good in K' just like they do in K.  In other words, the same physical laws work in K and K'.  This is the principle of relativity (in its restricted sense).

Developments in the study of electrodynamics in the late 1800s had called into question the principle of relativity. [Einstein's work would demonstrate that it could still hold.] But there were strong reasons to think it might hold. For example, "it supplies us with the actual motions of the heavenly bodies with a delicacy of detail little short of wonderful" (13). Why would it work in mechanics but not in electrodynamics?

Another complication if the principle of relativity didn't hold would be that we would have to have some basic frame of reference where the laws of mechanics hold most simply, but the rules would change somewhat in other frames of reference moving in relation to it.  So the laws of motion would apply straightforwardly in coordinate system K but they would be complicated by the motion of K' in relation to K.

Take the earth, for example, rotating around the sun.  Because it is moving in a circle, we would expect its movement to alter its velocity in relation to some absolute frame of reference throughout the course of the year.  Accordingly, we would expect the laws of motion to change in some way throughout the year as we moved in relation to the "absolute state of rest."

"The most careful observations have never revealed such anisotropic [different properties when something is moving in a different direction] properties in terrestrial physical space" (15). For Einstein, this was a very powerful argument for the principle of relativity, that the laws of motion apply the same way in every inertial frame of reference.
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Note: It would be very interesting to trace the history of rhetoric against relativism.  I've never heard anyone in Christian circles speak against relativity, but I can imagine some preachers in the early 20th century doing so.  Although the notion of relativism in ethics has been around forever, I have wondered if rhetoric against relativism in any way was affected or triggered by Einstein's theory of relativity at the turn of the twentieth century.