Friday Science: The Euclidean metric (1.14.1a)

About every other Friday I want to move through another section of Peter Collier's A Most Incomprehensible Thing: Notes Toward a Very Gentle Introduction to the Mathematics of Relativity.

Here is my first post in this series.

1. The brilliant idea of Collier's book, one to which I have long subscribed, is that theory is most effectively learned on a "need to know" basis. There are many gifted "abstracticians" among us who do not need an answer to the question, "What do I need to know this for? When am I ever going to use this?" Indeed, I was one of them in high school and college. I still affirm those who are wired this way. Go for it!

But if the goal is actual learning, most of us best learn theory as we are engaged in practice. This was the guiding principle of Phase 1 of Wesley Seminary at Indiana Wesleyan University, of which I was a co-founder. Indeed, philosophically, I have adopted the epistemological stance of a pragmatist/nominalist. That is to say, the abstraction of theory in fact reduces to a useful game humans play in order to operate more effectively in the concrete world of realia. In other words, ideas are useful abstractions of reality.

I thus mock those who say, "You need to know the theory in order to do the practice effectively." Poppycock! The theory is abstracted from effective practice. Since I operate in the world of academia, you can imagine how often I whisper, "Numbnuts" under my breath. The number of virtual Platonists around me is a constant source of frustration.

2. So my previous post was in 4.1 of Collier's book, introducing the curvature of space. Section 2 moves on to Riemannian manifolds and "the metric." But to understand the "metric" of general relativity, Collier reaches back into the metric of special relativity (3.5), which reaches back into the Euclidean metric (1.14.1). So if I am to follow my principle of "theory on a need to know basis," I must now post on the Euclidean metric. Then in two weeks time, the Minkowski metric of general relativity (3.5).

The Euclidean metric
3. In high school geometry, we basically learned "Euclidean" geometry, named after the ancient Greek mathematician. So when we refer to Euclidean space, we mean the way space would behave if it were exactly like what we learned in high school.

In three dimensional Euclidean space, the Pythagorean theorem applies: l² = a² + b² + c².

Don't let the expanded form mess you up. For three dimensions, we've just added an extra square. We've used L for the hypotenuse for "line," meaning the "line" that results from these three components.

4. When we begin to talk about space in advanced physics, we have to make some modifications. For one, we begin to talk about incredibly small increments of space. In calculus (which I won't review here), we talk about "infinitesimals," meaning almost infinitely small increments.

So when we talk about dx or dy or dz we are talking about almost infinitely small increments on the x axis or y axis or z axis. These are called coordinate differentials. So now we might say that

dl² = dx² + dy² + dz²

We can call this version of L the line element, a really small increment of the line that results from these components (the "resultant).

5. Now I'm no fan of matrices. But they are all over relativity and quantum mechanics. I don't know why they work and this frustrates me because 2018 Ken is not 1984 Ken. So here I will simply suspend my questions and present the "game" of matrices as it relates here.

A metric or metric tensor is a matrix that presents the coefficients of the differential equation above.

So the first 1 in the top left reflects that there is a 1 in front of dx². The second one in the middle reflects that there is a 1 in front of the dy². The third 1 in the bottom right indicates that there is a 1 in front of the dz².

"Ours is not to question why. Ours is just to memorize or die."

g_ij (which should be in brackets, sorry) is a way of referring to a matrix. The matrix is g, and the elements of the matrix are in i rows and j columns. So the dy component is in the position 2, 2 (row 2, column 2).

6. To explain the lay out a little more, think of it this way:

                       dx    dy    dz

Here you can see that the first 1 is in the cross of dx, etc.




7. Collier goes on to formulate this matrix in terms of polar coordinates, but I don't need them to understand the matrices of general relativity yet, so I'm going to pass for the moment.

Friday Science: Susskind 4a: Unitarity

Seventh installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)

Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors
Chapter 3c: Hermitians and Fundamental Theorem of QM
Chapter 3d: Principles of Quantum Mechanics
Chapter 3e: 3-Vector Operators
Chapter 3f: Spin Polarization Principle

Now that I'm done reviewing Hawking, I thought I would return to wending my way through Susskind's book. Here beginneth chapter 4.

4.1 Classical Reminder
A state is the way something is at a particular point in time. The main rule for how states change in classical mechanics is deterministic. If you know the formula, you know what the next state is going to be. The second rule is reversibility. If you know the state now and the formula for change, then you know what the previous state was too.

If two identical systems have the same state at some point in time, then their past and future is the same as well. We call this "unitarity."

4.2 Unitarity
So consider a closed system. Let's use the Greek letter psi to indicate the quantum state of something: |ψ⟩ . To say that "the state was |ψ⟩ at time t, we will use the notation |ψ(t)⟩ . In a sense, this notation |ψ(t)⟩ represents the entire history of the system.

Assuming a system has unitarity, we can use the operator U to say this too:

|ψ(t)⟩ = U(t)|ψ(0)⟩ 

The entire history of the states of a system is this "time-development operator" producing a series of states that start at time 0.

4.3 Determinism in Quantum Mechanics
The development of a state vector in quantum mechanics is deterministic just like in classical mechanics but with one very significant difference. In classical mechanics, determinism tells us the result of the next experiment with certainty. In quantum systems, it tells us the probabilities of the outcomes of later experiments.

4.4 Closer look at U(t)
1. This time-development operator in quantum mechanics must be linear. That means that for every time you put in, you get one quantum state out and the relationship between the two develops at a constant ratio.

2. The unitarity operator also implies that if two basis vectors are orthogonal (are distinguishable), then they will always be orthogonal. This is called "the conservation of distinctions." This means that, for example:

⟨ψ(t)|Φ(t)⟩ = 0

if these two functions are orthogonal.

3. Susskind then shows that for unitary operators

U^†U = I

where U^† is the Hermitian conjugation of U [1] and I is the "unit matrix." The unit matrix is one where, when multiplied by something, results in the same matrix. It is a matrix with all ones down its diagonal and zeros everywhere else.

It has the equivalent result to the Kronecker delta δ_ij, which yields the value 1 when two things with the same basis vector are multiplied but 0 when orthonormal basis vectors are multiplied.

4. This adds a fifth principle to quantum mechanics. The evolution of state-vectors with time is unitary.

[1] As a reminder, an operator is Hermitian if the matrix version and its transposed version (where you interchange the rows for the columns) yield an equivalent result.

Friday Science: Hawking 10 (Unification)

It always feels good to finish a book. Here is the last review of Stephen Hawking's A Brief History of Time.

Chapter 1: Heliocentric
Chapter 2: Spacetime
Chapter 3: Expansion of the Universe
Chapter 4: Uncertainty Principle
Chapter 5: Elementary Particles and the Forces of Nature
Chapter 6: Black Holes
Chapter 7: Black Holes Ain't So Black
Chapter 8: The Origin and Fate of the Universe
Chapter 9: The Arrow of Time

Chapter 10: The Unification of Physics
As the title suggests, Hawking in this chapter is looking for a grand unification theory (GUT). He spends a little time on string theory, which is probably where he put his bet at one point. My sense is, however, that enthusiasm for string theory has waned these last few years as some of the particles it predicts have not been discovered during tests that should have produced some.

• The difficulty is to combine general relativity with quantum mechanics.
• He mentions old problem of renormalization. The math says "infinity," but we know what it should be from experiment. So you just substitute the experimental value and keep going.
• String theory: open string, closed string, two strings join. Two strings separate. Graviton's cross. Strings, strings, strings.
• String theory suggests there may be either ten or twenty-six dimensions. We don't see the others because they're two small.
• He goes into the anthropic principle again. Life can only exist when three dimensions predominate, so we're just lucky.
• It would take too much energy to find out what it would have been like near the big bang.
• Even a GUT would not predict everything about the future--uncertainty principle, some equations are just too hard to solve.

Here endeth Hawking.

Friday Science: General Relativity 1

A little over four years ago, I found a great book on relativity. Peter Collier's A Most Incomprehensible Thing: Notes Toward a Very Gentle Introduction to the Mathematics of Relativity. I've gone through a little less than half of it.

His concept is to introduce all the math needed for special and general relativity in the first 100 pages or so. He tries not to assume that you've had any of the math beyond algebra. He does pretty well although I think you probably need to have done some of it before to get it.

BTW, I first saw an approach something like this in the summer of 1983 at Rose-Hulman. I was given a physics textbook by Marion and Hornyak that interspersed calculus lessons with the physics. I thought it was brilliant--introduce the background math as you need it. It's something like problem-based learning.

I've basically finished Hawking. So on Fridays I hope more or less to alternate between Susskind's book on quantum mechanics and Collier. I don't entirely have down everything from his 58 pages on special relativity. Maybe I'll go back at some point. But I want to move forward through his 180 or so pages on general relativity.

4.1 Introducing the Manifold
a. Special relativity functions on the basis of what is called "Minkowski" space, which is flat.

• 3.2.2 Time for a flashback. In chapter 3, he introduces Minkowski space or spacetime. In Newtonian mechanics, we talk about three-dimensional space, Euclidean space. 
• For special relativity, Einstein drew on the idea of four-dimensional space, with time as the fourth dimension (spacetime). 
• This is named for the German mathematician, Hermann Minkowski (1864-1909).
• In Minkowski space, parallel lines never meet, so it is still flat space.

In general relativity, space curves, so we need some new math. Einstein, with the help of David Hilbert, found this math in the work of the German Bernhard Reimann (1826-66).

b. In general relativity, matter and energy curve spacetime. Gravity is not considered a force but a property of the curvature of spacetime. The idea of a "Riemannian manifold" is used to model this. A manifold is a smoothly curved space that is locally flat.

It would be like an ant walking on an apple. The ant thinks it is walking straight, but it is curving around the apple. Such a path on a sphere or curved surface is called a geodesic.

A circle is a one-dimensional manifold. If you walk on the perimeter and the circle is large enough, it just seems like you are walking straight. A sphere is a two-dimensional manifold. We can speak of a manifold as n-dimensional when locally it can be described by n dimensions.

Friday Science: Hawking 9 (The Arrow of Time)

Friday reviews of Stephen Hawking's A Brief History of Time so far.
Chapter 1: Heliocentric
Chapter 2: Spacetime
Chapter 3: Expansion of the Universe
Chapter 4: Uncertainty Principle
Chapter 5: Elementary Particles and the Forces of Nature
Chapter 6: Black Holes
Chapter 7: Black Holes Ain't So Black
Chapter 8: The Origin and Fate of the Universe

Chapter 9: The Arrow of Time
Getting close to the end. The problem we are dealing with in this chapter is the fact that, on the quantum level, nothing prevents a forward or backward movement in time. In the macro-universe, time only can move in one direction. In the micro-universe, this simply is not the case.

The first reason for this is what Hawking calls the "thermodynamic" arrow of time. We easily identify with a cup shattering on the floor. We do not identify with a cup unfalling and unshattering.

Another arrow is the "cosmological" arrow. The universe is expanding. My sense is that Hawking, writing this book in the late 80s, hoped that eventually this expansion would stop and recontract, making possible an oscillating big bang of sorts. That view has largely been eliminated in the last twenty years

A third arrow he mentions is the "psychological" arrow. This one I am less convinced of. It seems to be related to the anthropic principle. Basically, he argues that our brains are just wired to see time moving in only one direction.

Short chapter. I'm sure I don't fully know the depth of some of what he is saying. But I think I know enough to know that subsequent developments have trashed some of what he said.

The universe has a prevailing arrow of time, based on the second law of thermodynamics and its expansion. On the micro-level, this may not always be the case.

Friday Science: Hawking 8 (Universe Origins)

Friday reviews of Stephen Hawking's A Brief History of Time so far.
Chapter 1: Heliocentric
Chapter 2: Spacetime
Chapter 3: Expansion of the Universe
Chapter 4: Uncertainty Principle
Chapter 5: Elementary Particles and the Forces of Nature
Chapter 6: Black Holes
Chapter 7: Black Holes Ain't So Black

Chapter 8: The Origin and Fate of the Universe
Here are some points of interest in this chapter:

• Hawking presented a paper at the Vatican in 1981 apparently arguing that the universe was finite but had no boundary, meaning no beginning.
• He recounts the path I've been trodding a lot lately. The universe started at a point, virtually infinitely hot. Then it cooled a little to where there were mostly electrons, photons, and neutrinos. About a hundred seconds protons and neutrons would start binding into deuterium and helium...
• George Gamow suggested in 1948 that we should be able to detect background radiation from this beginning. This was discovered in 1965.
• Then he builds to Alan Guth's idea of inflation. Why is the universe so uniform, but with significant fluctuations?
• He mentions two versions of the anthropic principle. He does not like the strong one, although I find it hard to distinguish the two versions. What he calls the strong one basically argues that the universe is the way we see it because otherwise we would not be here. The weak one seems more to say that in a universe there is bound to be life developing somewhere.
• He gets to Guth and inflation. In the hottest time of the universe, all the forces would have coalesced into a grand unification. Then gravity would separate out, then the strong force, then the weak force leaving the electromagnetic force working.
• He shares a little about some papers in Moscow. He's reminiscing. Aww.
• He ends the chapter with some suggestions toward a grand unified theory. This was in the late eighties so I'm not sure how helpful they are. Mainly, they have to do with imaginary time. I don't know enough to follow completely.
• "If Euclidean space-time stretches back to infinite imaginary time... One could say, 'The boundary condition of the universe is that it has no boundary' It would neither be created nor destroyed" (136).
• Hawking suggests that the imaginary time may actually be the real time. He suggests that while this universe looks like it had a beginning and will have an end, maybe this is an illusion. 
• Of course he ends the chapter asking then why we would need God.
• He seems to look to a big crunch. He was wrong.

Friday Science: Just Six Numbers (book review)

I have pretty much finished Hawking, but will post the rest of his book next Friday, dv.

1. I took my son to Clinton, Iowa Tuesday for him to meet in person some online friends of his that have played video games together for about six years. (Interesting development in this new world, where you go to meet some of your best friends for the first time after years of playing together. I have a nephew that first met a friend in person for the first time this spring... as best man in his wedding.) So while I was sitting in a hotel room, Starbucks, parking lots, etc, I finally read/skimmed Martin Rees' Just Six Numbers.

In the last few decades, a strong argument for the existence of God has emerged called the "fine tuning" argument. It falls under the category of an argument for design. There are a number of ratios and constants in the universe that are necessary for us to be here. An atheist at this point invokes the anthropic principle--we wouldn't be here to discuss them if there weren't. So if there are universes in the "multiverse" that do not have these precise ratios, there is no one there to talk about them. In other words, we are just the lucky ones.

By contrast, the theist says, "We are fearfully and wonderfully made." "Oh the depth of the riches of the knowledge and wisdom of God!"

Rees' book is about these constants. I am now working on my third Gabriel novel. This one is called Gabriel's Diary: The Creation. It is going to be truly spectacular and probably a bit controversial. I am hypothesizing what creation might look like from a Christian point of view that engages with contemporary science. I'm not saying it's right. But it will be a book for people who are convinced about the science but not so convinced about God. The first three chapters will embody the fine tuning argument and engage my feeble apprehension of modern cosmology.

2. Here is a summary of my take-aways from Rees' book. I have put them in the order that is most helpful for my writing. Fine tuning arguments are in bold.

Chapter 1: The Cosmos and the Microworld

• Rees goes with the anthropic principle and the multiverse model: "An infinity of other universes may well exist where the numbers are different. Most would be stillborn or sterile. We could only have emerged (and therefore we naturally now find ourselves) in a universe with the 'right' combination" (4).
• He uses an "ouraborus" to picture the scale of the universe. The breadth of size is 10⁶⁰. The smallest size imaginable is about 10^-33 cm. The universe is about 10²⁸ cm across. (These are my numbers, not his.)

Chapter 3: The Large Number N: Gravity in the Cosmos

• He calls the constant discussed in this chapter, N. I've never heard of it but he is referring to the ratio between the electromagnetic force and the force of gravity. It turns out that the electromagnetic force is about 10³⁶ times more powerful. 
• Gravity is always attractive, while electromagnetic forces can be either attractive or repulsive. So with large objects, gravity accumulates a large attractive force, while the electromagnetic forces more or less even out.
• There is an inverse square law that applies to these two forces. The force weakens as the square of the distance increases. More on the significance of this fine tuning in chapter 10.
• Gravity makes objects as big as the Moon and larger spherical.
• If the ratio were less, everything would be smaller--smaller stars, smaller planets, potentially smaller life. Galaxies would form quickly and would be miniaturized. They would be more densely packed.
• Stellar lifetimes would be much shorter, which according to Rees would not have given enough time for complex life to evolve. 
• A weaker gravity might have allowed more elaborate and longer-lived structures to develop, but a stronger gravity would not have allowed enough time for humanity to emerge.
• A little on Einstein - the speed of light is the speed limit of the universe. Near large masses time slows down relative to elsewhere.
• Millions of black holes in our galaxy, the remnants of large stars that have already burnt out. Slightly smaller stars become neutron stars. Our Sun will become a white dwarf when it burns out.
• Some explanations of black holes, event horizons, etc., atomic sized black holes.

Chapter 10: Three Dimensions (And More)

• The number he discusses in this chapter is 3, three-dimensions of space.
• In a three dimensional world, forces like gravity and the electromagnetic force obey an inverse-square law mentioned above.
• William Paley used the inverse square law as part of his argument from design. If it were an inverse cube law, there could be no orbiting of planets or electrons around a nucleus.
• There is an asymmetry of the arrow of time. "No such asymmetry is built into the basic laws governing the microworld" (153). The asymmetry is linked to the expansion of the universe. [Hawking calls this the cosmological arrow. Entropy is another basis for the arrow, which Hawking calls the thermodynamic arrow.]
• The expansion of the universe was fast enough to end nuclear reactions before they could convert more than 23% of the hydrogen into helium. This left fuel for suns.
• There was just the right asymmetry in the earliest phase to leave a slight excess of matter over antimatter. Otherwise, nothing but energy would be here.
• He also talks a little about Planck units. The smallest length is 10¹⁹ times smaller than a proton, 10^-35 the length of a meter. The smallest unit of Planck time is 10^-43 seconds. Space and time are arguably granular, not continuous. Take that, Zeno.
• He mentions superstrings. I believe this approach is increasingly discredited.

Chapter 9: Our Cosmic Habitat III: What Lies Beyond Our Horizon?

• Helium was formed at about the three minute threshold.
• grand-unified (all forces united) to quarks (strong from electroweak) to leptons (electro from weak)
• magnetic monopoles?

Chapter 8: Primordial Ripples: The Number Q

• Q is the ratio between the rest mass energy of matter and the force of gravity. It is 1 to 100,000. 
• It has to do with the "roughness" of space, the "ripple amplitude" of the gravitational waves of cosmic inflation.
• It has to do with the energy that would be needed to break apart galaxies.
• The slight asymmetry of the universe seems to relate in some way, enabling things to form structures.
• If Q were smaller, galaxies wouldn't form. If Q were larger, galaxies would crunch much sooner and the universe would be a rougher place.

Chapter 6: The Fine-Tuned Expansion: Dark Matter and Ω

• What Rees calls Ω is the ratio between the force of universe expansion and the force of gravity. This ratio determines whether the universe will expand forever, expand steadily, or eventually contract again. These correspond to whether the ratio is less than 1, exactly 1, or greater than one.
• Because of gravity, if there were five atoms for every cubic meter in the universe, it would contract one day. As it is, there only seems to be 0.2 atoms per cubic meter, at least as far as ordinary matter is concerned.
• All the indications are thus that this number is less than 1. But it is likely that there is "dark matter" out there, stuff we can't see. It is thought that about 26.8% of the universe is dark matter.
• Candidates for dark matter include brown dwarfs (suns less than 8% of our sun's mass), neutrinos, black holes, "axions," but more likely something we don't yet know about.
• At about one second after creation, Ω could not have differed from 1 by 1 in 10¹⁵. If the expansion force were greater, there would have been no time for stars and galaxies to develop. If the mass were greater, the universe would have collapsed too soon for life as we know it to develop.
• Another fine-tuned factor is the slight asymmetry between matter and antimatter. If there were perfect symmetry in the early universe, they both would have emerged in equal amounts and completely annihilated. But there must have been a slight asymmetry.
• There are different suggestions for the asymmetry. Rees suggests the K° decay, associated with the weak nuclear force, may be the reason. What if, for every billion quarks and antiquarks generated in the earliest universe, one extra quark were produced?

Chapter 7: The Number Λ: Is Cosmic Expansion Slowing or Speeding

• As far as I can tell, Λ doesn't contribute much more than the discussion of omega in the previous chapter. 
• Einstein added Λ to his general relativity equations with the hope of a universe that wasn't expanding. He regretted that when Hubble showed it was. But there does seem to be an unknown force that is affecting comic expansion. This was apparently confirmed in 1998.
• It is relatively small, about 0.7. It is a force driving expansion.
• [It seems to relate to what scientists now are calling "dark energy."]

Chapter 5: Our Cosmic Habitat II: Beyond Our Galaxy

• Galaxies are the building blocks of the universe. Stars and their solar systems collect together to form galaxies. Galaxies often have huge black holes at their centers. Galaxies cluster (our cluster is the "Local Group").
• Galaxies crash into each other. The kind of galaxy known as elliptical galaxies may be the result of galaxies that have crashed into each other. [Hawking had a different thought here in the 80s.]
• There are bigger aggregates like the "Great Wall."
• At every point we look in space, everything is speeding away from us, often faster than the speed of light, which suggests that space itself is expanding, since nothing can move faster than the speed of light in its own reference frame.
• The expansion of space has been well established in the last fifty years. [When Hawking wrote, he hoped it might crunch again but we seem rather headed for a cosmic rip from accelerating expansion.]
• Cosmic Microwave Background radiation (CMB) discovered in 1965 points to a Big Bang. Together, the fact that the universe had a beginning coupled with the fact that it won't re-compress fits well with the notion of creation.
• CMB comes not from the creation itself but from some 380,000 years after the beginning (13.8 billion years ago) when the universe cooled down enough for electrons and protons to form neutral atoms, releasing a massive amount of energy in photons.

Chapter 4: Stars, the Periodic Table and ɛ

• A third number is ɛ, which I've never heard called that, but which is the percentage of mass released as energy when hydrogen is fused into helium. 0.007 or 0.7%
• This has to do with the strength of the strong nuclear force that binds protons and neutrons together in a nucleus. This force is the strongest of all the forces but it only works within the space of a nucleus. It is thus just strong enough to hold a nucleus together without interfering with the electromagnetic forces that are essential for the overall working of an atom or the weak nuclear force that comes into play with large atoms with atomic numbers over about 50.
• Helium is fused in two stages. First, a proton and a neutron fuse together to form deuterium (heavy hydrogen). Then two deuterium atoms fuse together into helium. 
• If the percentage converted to energy were any more, no hydrogen would have survived the big bang. It would have all become helium or heavier, leaving no fuel for stars. If it had been less, no helium would have formed and the universe would just consist of hydrogen.
• Carbon only forms from a helium and beryllium nucleus because the carbon nucleus has a resonance with a very specific energy that can fuse just before primitive beryllium decays. Without carbon, life as we know it would not exist.
• The Earth is thought to be about 4.5 billion years old. The universe about 13.8 billion.
• When a star's hydrogen has all been converted to helium, the core pulls inwards. Prior to that time, the energy from the fusion pushed back against the gravity of the mass.
• When it contracts, it heats up more and heavier nuclei are formed. Iron is the most tightly bound nucleus. When it gets to a critical size, it implodes to a neutron star and supernovas the overlying material. In this material are the trace elements of heavier elements.

Chapter 2: Our Cosmic Habitat I: Planets, Stars and Life

• Stars start as warm blobs ("protostars"). They contract over millions of years under their own gravity. 
• Any slight spin is amplified under a collapse, like a skater pulling in arms. The resulting disks are the precursors of planetary systems. (14)
• Small wobbles in the orbits of stars may indicate planets. Christiaan Huygens in 1698 suggested every star might have planets around them. These are now considered certain.
• A "barycenter" is the center of mass of an orbiting pair like the sun and Jupiter.
• The early history of a solar system is filled with crashes. (A huge crash 65 million years ago (crater underwater in Gulf of Mexico near Chicxulub is thought to have killed the dinosaurs.) This event paved the way for mammalian life to emerge as winners.
• The Moon is thought to have been formed from the earth by a collision with another protoplanet. Uranus' weird axis spin also explained by such collisions.
• For life to exist on a planet like Earth, gravity must pull strongly enough to prevent the atmosphere from evaporating into space but it can't really be any stronger than Jupiter (cf. 32).
• For water to exist on the surface, planets must be neither too hot or too cold.
• The orbit must be stable, not crossed by a Jupiter-like planet in an eccentric orbit.
• The oxygen of our environment is thought to come from primitive bacteria early in earth's history.
• Cf. p32. Gravity makes objects Moon sized and larger spherical.

Chapter 11: Coincidence, Providence - Or Multiverse

• Rees goes with the multiverse theory. It is a logical option for someone who doesn't believe in God as creator. It doesn't preclude God, although it might push creation back further. It seems more philosophical in some ways rather than scientific per se.
• He suggests that the values of these constants might be difference in other universes.

Friday Science: Hawking 7 (Black Holes Evaporating)

Friday reviews of Stephen Hawking's A Brief History of Time so far.
Chapter 1: Heliocentric
Chapter 2: Spacetime
Chapter 3: Expansion of the Universe
Chapter 4: Uncertainty Principle
Chapter 5: Elementary Particles and the Forces of Nature
Chapter 6: Black Holes

Chapter 7: Black Holes Ain't So Black
Here we get quite a bit of Stephen Hawking's distinctive work. Some points of interest:

• Black holes are defined as the set of events from which it is not possible to escape, which basically begins the black hole at the event horizon.
• The paths of light at the event horizon must be parallel to each other but never meet. That also means a black hole can never decrease in area.
• This non-decreasing property is similar to entropy. Jacob Bekenstein in fact suggested that the area of the event horizon was a measure of the entropy of the black hole.
• Entropy has to do with the second law of thermodynamics. The entropy of an isolated system always increases. That is, disorder increases.
• If a black hole has entropy, it should have a temperature and it ought to emit radiation. But a black hole can't omit anything.
• So space isn't really empty. Particles and antiparticles emerge and annihilate. Near the edge of the event horizon, some get separated before they annihilated and go into the black hole. This gives the appearance of a black hole emitting a particle. 
• Meanwhile, a flow of negative energy into the black hole would reduce its mass. The universe is too young, but this process could eventually disintegrate a black hole into nothing.
• There may be some primordial black holes (very small). Some of them might be disintegrating about now. Some scientists are looking for final bursts of their disappearance.

Friday Science: Hawking 6 (Black Holes)

Friday reviews of Stephen Hawking's A Brief History of Time so far.
Chapter 1: Heliocentric
Chapter 2: Spacetime
Chapter 3: Expansion of the Universe
Chapter 4: Uncertainty Principle
Chapter 5: Elementary Particles and the Forces of Nature

Chapter 6: Black Holes
I hate it when I don't finish things. I'm about half way through Hawking's book but got sidetracked by the end of the semester.

• "Black hole" is a term coined by John Wheeler in 1969.
• The concept goes back even further. In 1783 John Mitchell suggested a sufficiently massive and compact star would not allow light to escape its gravitational pull.
• The Marquis de Laplace suggested the same thing just a few years later.
• In 1928 Subrahmanyan Chandrasekhar, on a boat to Cambridge to study with Sir Arthur Eddington, calculated how big a star could get after burning out.
• The principle here is the balance between the Pauli exclusion principle and the fact that nothing can move faster than the speed of light. When a star gets sufficiently dense, the repulsion of the exclusion principle becomes less than the gravitational attraction and the cold star would collapse in on itself.
• A cold star about 1.5 times the size of our sun would do so, a mass now known as the Chandrasekhar limit. (Lev Davidovich Landau came to similar conclusion about about the same time.)
• Stars less than the limit become white dwarfs, with a radius of a few thousand miles.
• Slightly more massive neutron stars can be supported by the exclusion repulsion between neutrons and protons. Radius of about 10 miles. 
• A pulsar is a special kind of neutron star that emits regular pulses of radio waves. Pulsars were discovered in 1967 by Jocelyn Bell at Cambridge.
• Above the Chandrasekhar limit, might such stars reduce to a singularity, a point. Chandresekhar faced strong opposition to the idea from people like Eddington and Einstein. 
• Oppenheimer before WW2 did some work on this in relation to light. At a boundary known as the event horizon, light cannot escape the black hole.
• "God abhors a naked singularity." In cosmic censorship, we can't see what goes on inside a black hole (Penrose).
• Roger Penrose and Stephen Hawking did a lot of work on black holes between 1965-1970, which are a little like space before the Big Bang. 
• Some solutions to the general relativity equations suggest wormholes through black holes to the other side of the universe or perhaps even passage back in time. But a human probably wouldn't survive.
• In 1967, Werner Israel, some non-rotating black holes end up spherical (a solution Karl Schwarzchild suggested in 1917). Penrose and Hawking showed that all non-rotating stars collapsing into a black hole end up spherical.
• In 1963 Roy Kerr postulated Kerr black holes, rotating black holes that end up in various shapes.
• "A black hole has no hair." Their final shape depends only on mass and rate of rotation, not on the shape the star had before collapse. As they collapse, they give off gravitational waves until they settle down.
• Black holes are detected by the gravitational pull they have on other visible stars under certain conditions (e.g., the emission of massive amounts of energy from the nearby star). Cygnus X-1 is a system that seems to have such a black hole. There's probably one at the center of our galaxy as well.
• Quasars are systems that emit enormous amounts of energy and may have massive black holes at their center.
• There could be some primordial black holes out there, formed in the early universe, smaller than our sun.
• Hawking discovered that black holes glow like a hot body. See next chapter.

Friday Science: Hawking 5 (Particles)

Friday reviews of Stephen Hawking's A Brief History of Time so far.
Chapter 1: Heliocentric
Chapter 2: Spacetime
Chapter 3: Expansion of the Universe
Chapter 4: Uncertainty Principle

Chapter 5: Elementary Particles and the Forces of Nature

Particle Discoveries

• Aristotle--matter can be divided endlessly (300s BC)
• Democritus--matter is made up of atoms (400s BC)
• John Dalton--revived atomic theory (1800s)
• Albert Einstein--Brownian motion confirms atoms (1905)
• J. J. Thomson--demonstrated electrons (late 1800s)
• Ernest Rutherford--atoms have a nucleus (1911)
• James Chadwick--discovered the neutron (1932)
• Murray Gell-Mann--Discovered protons and neutrons made up of quarks (1964)
• Six flavors of quark: up, down, strange, charmed, bottom, and top.
• P. A. M. Dirac--antiparticles and combined with special relativity (1928)
• Four categories of force carrying particles:
• The graviton is proposed as the basis for gravity, and gravitational waves have been discovered since Hawking wrote this book. Personally not sure that gravity is based on particles. Einstein didn't look at it that way.
• The photon is the basis for the second force, the electromagnetic force. It is much stronger than the gravitational force but can be either attractive or repulsive and works over smaller distances.
• The particles known as the W and Z mediate what is called the weak nuclear force, which is the basis of radiation. Abdus Salam and Steven Weinburg worked out a way to unify the electromagnetic and weak force into an electroweak force, moving toward grand unification.
• The gluon is the basis for the final force, the strong nuclear force. I believe that the unification of this force with the electroweak force has also been achieved since Hawking wrote his book. Gravity is what is left to unify.

Other Nuclear Characteristics

• Particles have spin. The options are 1/2, 0, 1, and 2. Particles of matter are particles of spin 1/2. Particles of 0, 1, and 2 give rise to the forces between matter particles.
• Pauli's exclusion principle says that two particles cannot be in the same state at the same time.

Friday Science: Hawking 3 (Expansion of Universe)

I started Friday reviews of Stephen Hawking's A Brief History of Time.
Chapter 1
Chapter 2

Chapter 3: The Expanding Universe
1. I'm trying to figure out why I found this book so hard to read back in the nineties. My brain is a strange thing. Sometimes nothing will go in and then at other times a mess of complicated stuff goes right through. My breakthrough book was The Perfect Theory.

So chapter 1 talks about the earth at the center of the universe. We hear about Copernicus and Newton. He mentions relativity and quantum mechanics. Chapter 2 then gives Einstein's special and general theories of relativity. Chapter 3 is about the Big Bang.

2. This chapter makes it clear that Hawking sees how the Big Bang theory could play into belief in God but of course he rejects that. I'm always a little puzzled by Christians who have a really negative view of a big bang, because it seems to serve in strong support of the cosmological argument for the existence of God.

It must be the fact that most scientists think it took place a little less than 14 billion years ago. But that's not the same as evolution. John Piper, for example, is very open to an old earth even though he does not believe in evolution.

3. He talks a little about the process of discovering that we are inside a galaxy of stars and that there are millions of other galaxies of stars. Edwin Hubble in 1924 was the first to discover other galaxies. The two factors in the brightness of a star are its luminosity and its distance from us. Hubble's logic went like this:

• If we know the luminosity of what we're observing, we can figure the distance from us.
• We know the distance of some stars near us using geometry. Since we know the distance, we can determine the luminosity of various type of stars.
• [We can type them by analysis of the light they emit, their spectra.]
• Now, knowing the luminosity of these types, we can figure out the distances of distant ones.

4. Hubble discovered that the spectra of all the stars everywhere was shifted to the red end of the spectrum. That is to say, using the Doppler effect, they were moving away from us. In short, every point, everywhere in the universe, is moving away from every other part of the universe. The universe is expanding.

Indeed, the farther away a galaxy is, the faster it is moving away! The speed of expansion suggests whether 1) it will eventually slow down and pull back in on itself (cosmic crunch), 2) it will continue at a steady speed, or 3) it will expand faster and faster until there is a cosmic rip. The third seems to be the case. Alexander Friedmann in 1922, four years before Hubble, had predicted the expansion.

Einstein hated the idea of cosmic expansion, actually introduced a "cosmological constant" into his formulas to make it stop. :-) Ironically, he was right about the constant, but wrong about expansion.

5. In 1965, two men working for Bell Labs picked up a background radiation that was theoretically explained by a big bang. If the universe was infinitely old, it would have already dissipated. The universe must have been fantastically hot to begin with, but it has been cooling down ever since. If we extrapolate back, the universe would have begun as something like a "singularity," a single point.

"Dark matter" has also been proposed to explain why galaxies swirl as they do. The outer rims of the galaxies swirl at the same basic speed as other parts.

A lot of people didn't like the idea of an expanding universe. It played too easily into an argument for God. Fred Hoyle in England was a very charismatic propagator of a "steady state" theory, where matter is constantly being created. He's a study in how people believe as much because of the person selling stuff as on the basis of truth. He was wrong. Go away, charmer.

6. The chapter ends with Roger Penrose's discovery of the probability of black holes, and of course Hawking's work built off of him. Together they proposed that the universe had begun with such a singularity that then expanded in a big bang. Again, this idea faced a lot of resistance but, in the end, the math was the math.

The problem--still unsolved--is that general relativity and quantum mechanics don't fit together. This is no problem today because the cosmos is really big and the atom is really small. But before the big bang, the universe was immensely dense and immensely small. This is the biggest problem in modern physics.

Friday Science: Hawking 1

1. I can't say that I am a big Stephen Hawking fan. Obviously I never met him. I enjoyed the movie based on his life. I like a lot of the same stuff he did. He was funny on the Big Bang Theory.

I always got the impression that he was a bit of a donkey. Perhaps that's unfair. I imagine it must be hard when you're that much smarter than everyone else not to think that everyone else is an idiot. Of course so much smartness in one area can also entail immense density in other areas. I'll let God handle all that.

2. I bought A Brief History of Time a long time ago, probably from Joseph Beth Bookstore in Lexington, Kentucky in the early nineties. It came out in 1988. For some reason, I just couldn't get into it. I never made it out of the first chapter. It neither grabbed my attention nor did it get past the atrium of my thick head.

One of the benefits of a slowing metabolism is that I can read more and more. And with the death of Hawking this week, I sense it's time for me to buckle down and read this thing. There was a second edition in 1998 with an extra chapter, which I downloaded on Kindle to be able to get any revised thoughts he might have had.

The last ten years have not smiled on Hawking's intuitions. He had bet against the Higgs boson. He lost. In fact, he made one big discovery that, in retrospect, doesn't seem so startling at all. He was just the one to put it together. He concluded that black holes "evaporate" as it were. Yeah Hawking.

3. The first chapter is called, "Our Picture of the Universe." Some in this chapter is well known to those who are interested in these things. But there are a few surprises.

Here are the main points:

• Aristotle (300s BC) and Ptolemy (200s AD) both thought the world was a sphere, but they thought that the sun, stars, and planets revolved around the earth.
• Copernicus, Galileo, Kepler, and finally Newton came up with equations that worked a whole lot better, supposing that all these things revolved around the sun, with the moon only going around the earth.
• No one seems to have asked if the universe was expanding until the twentieth century. There was an assumption of a static universe. Why then did the universe not collapse under gravity? Newton thought by supposing an infinite universe, the pull would be equal in every direction. But this apparently is not how infinity works in this instance.
• Heinrich Olbers in 1823 then asked the question about line of sight. In a static universe, there should be stars everywhere we look, a completely lit sky.
• In 1929, Edwin Hubble discovered that the universe everywhere was moving away from us (red shift). This brought the question of the universe's beginning into science.
• The beginning had always been there in religion. Augustine suggested that God created time when he created the world. So it makes no sense to talk about time before the creation. As a side-note, this is the current convergence between science and faith--both believe that the universe had a beginning.
• The last part of the first chapter has two main points of interest. The first is the incompatibility of general relativity with quantum mechanics, the physics of the very large and the very little. Hawking longed for a "grand unified theory" or, as his biographical movie was titled, "a theory of everything."
• He also endorses Karl Popper's philosophy of science. Science should be oriented around falsifiability. A good theory is one that has not yet been falsified. You can never finally prove a scientific theory. A good scientific theory thus has two characteristics: 1) it must accurately describe a large class of observations and 2) it must make definite predictions about future observations that are not falsified.

Friday Science 3f: Spin Polarization Principle

Seventh installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)
Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors
Chapter 3c: Hermitians and Fundamental Theorem of QM
Chapter 3d: Principles of Quantum Mechanics
Chapter 3e: 3-Vector Operators

Finishing up notes on chapter 3.

3.8 The Spin-Polarization Principle
Any state of a single spin is an eigenvector of some component of the spin.

I wish I more fully understood this principle, but I will do my best. It seems to me it is saying that you're going to get a +1 somewhere.

〈σ_x〉² + 〈σ_y〉² + 〈σ_z〉² = 1

This type of bracket indicates what is called an "expectation value" or the average value of a measurement. The square of the expectation value is the probability of finding a 1 there. So there has to be a 1 somewhere or the probability has to total one.

So, given any state ∣A〉 = α_u∣u〉 + α_d∣d〉

There is some direction 𝜎 ⃗∙𝑛 ̂ ∣A〉 = ∣A〉

3.7 An example
If I had fully followed the matrix analysis of the previous sections, I'm sure this section would be delightful. I get the general sense that he is playing out probabilities in a spherical framework. I generally understand spherical coordinates and the chart on p.89. But I think I'll skip summarizing this section and call chapter 3 concluded.

Friday Science 3e: Three-Vector Operators

Seventh installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)
Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors
Chapter 3c: Hermitians and Fundamental Theorem of QM
Chapter 3d: Principles of Quantum Mechanics

Much is still not clicking but let me finish what I can of chapter three.

3.5 A Common Misconception
1. Measurements in quantum mechanics correlate with operators. However, the two are not exactly the same. Measurements come up with definite answers. For example, if you are measuring a particular spin vector, it will either be 1 or -1.

By contrast, the operator has to do more with the probability of a certain outcome. Operators are mathematical rather than actual. They are the tools used to calculate eigenvalues and eigenvectors. We use them on state vectors like "up" "down" "right" "left" "in" and "out." And the result of his operation is another state vector combination that may involve square roots and imaginary numbers--things you will never get in an actual measurement. The actual measurement is always either 1 or -1.

3.6 3-Vector Operators Revisited
2. So Susskind distinguishes three types of vector in this section. The first is a 3-vector space like we use in ordinary directions in life (two miles south, then a mile east, on the sixth floor).

Then he's been talking about state vectors like up, down, right, left, in, out. These are metaphors, I think.

Now he speaks of spin components x, y, and z. He calls these operators, written as matrices. They are the three measurable components of spin. He calls them a new kind of 3-vector, a 3-vector operator. I don't seem to understand, but I'm going with it.

3. Now what if we want to measure spin in any direction sigma n, where n is the direction? Then we can break down the spin in this direction to

σ_n = σ_xn_x + σ_yn_y + σ_zn_z

So if we use the Pauli matrices from the previous post for the components of sigma, we can express the spin in that direction as:

Susskind does some matrix voodoo to combine all these into one big matrix.

Apparently, if we know the eigenvectors and eigenvalues of this particular σ_n, we can use this matrix to calculate all the probabilities for all the outcomes of our measurements of the spin.

Friday Science 3d: Principles of Quantum Mechanics

Sixth installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)
Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors
Chapter 3c: Hermitians and Fundamental Theorem of QM

1. Again there is the sense that if I can just make it a little further, he'll connect this stream of math to something concrete so that all the rest will click. I feel like I'm reading 1 John.

Principle 1: Observable quantities in quantum mechanics are represented by linear operators. These have to be Hermitian as well.

Principle 2: The possible results of a measurement are the eigenvalues of the operator that relates to that observable. If a system is in the eigenstate ∣λ〉 , the result of a measurement has to be λ .

Principle 3: Distinguishable states are orthogonal vectors.

Principle 4: The probability of observing a value λ is 〈A∣λ〉² That is the probability of observing a particular eigenvalue is the square of the overlap between the eigenvalue and that state in general.

2. So Susskind uses the spin operator as an example. A spin operator provides information about the spin component in a specific direction. There is a spin operator for each direction in which the measuring apparatus can be oriented.

So he asks what an appropriate "spin operator" might be for the "up-down" aspect of spin. For up, the value will be one for up and zero for down. For down, the value will be zero for up and -1 for down. This corresponds to the following matrix:

z matrix (up down)

This satisfies the conditions: 1) it represents one component of the spin, 2) the possible results are +1 and -1. These are the eigenvalues of this matrix. 3) up and down are orthogonal.

3. He derives the matrices for the "right left" and "in out" components as well. These three matrices constitute the "Pauli matrices."

x matrix (left-right)

y matrix (in out)

Friday Science 3c. Hermitians and Fundamental Theorem of Quantum Mechanics

Fifth installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)
Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors

More on chapter 3. I increasingly get the sense that this book should have been written somewhat in the reverse order that he did. Typical linear, building block thinking. Most human minds--especially those this book is allegedly written for, like me--work on a "need to know" basis. That's how this book should be written.

1. Made some progress this week in the book. Think I'm further than I've ever been in it, understanding more than I ever had. Probably could handle a re-read. Since I'm making good progress, I'll try just to jot down some notes.

A "Hermitian" conjugate is like the complex conjugate of a matrix. You do two things to a matrix to find its Hermitian conjugate:

• Interchange the rows and columns (so m₂₃ becomes m₃₂)
• Complex conjugate each matrix element.

A Hermitian conjugate is denoted by a dagger. So the Hermitian conjugage of M is M^† . The matrix might have a T in its upper right hand (for "transposed").

• So you might say that M^† = [M^T]*    (transposed and conjugates).
• So if M∣A〉 = B then 〈A∣M^† = 〈B∣

2. A Hermitian operator is one that is equal to its Hermitian conjugate: M = M^†

The eigenvalues of a Hermitian operator are all real.

3. We now get to what Susskind calls the fundamental theorem of quantum mechanics. It amounts to this: "Observable quantities in quantum mechanics are represented by Hermitian operators" (64). Another way to put it is that "The eigenvectors of a Hermitian operator form an orthonormal basis."

Here is my interpretation of how he unpacks it:

• The possible vectors for a Hermitian operator are all of its eigenvectors and their sums.
• The unequal eigenvalues of a Hermitian are orthogonal.
• Even equal eigenvalues can be analyzed as orthogonal. In other words, two eigenvectors can have the same eigenvalue. This is called "degeneracy."
• If a space is N-dimensional, there will be N orthonormal eigenvectors.

4. The Gram-Schmidt procedure is a procedure for teasing out orthonormal sets that relate to degenerated eigenvectors with the same eigenvalues. Here is the procedure:

• Divide vector one by its own length to get the first orthonormal basis of unit length.
• "Project" the second vector onto that unit vector by taking the inner product with it. 〈V₂∣v₁〉. 
• Subtract this from the second vector.
• Then divide the result by the length of the second vector to get an orthonormal basis for the second vector of unit length.

I don't entirely follow, but I'm making progress.

Friday Science: Quantum States (chapter 2)

Second installment reviewing Susskind's, Quantum Mechanics: The Theoretical Minimum. Here was the first.

1.  I've read this chapter of Susskind several times. I have some sense of it but continue to find the writing frustrating. I have the sense that it would not be hard to make this material more comprehensible to an invested lay person like me, but it seems written by and for Sheldon. At some point it will more fully click and I will be able to add the necessary paragraphs.

2. Section 2.1 So I understand what he's saying in this first section and can already add the necessary background. There is an almost century old debate in quantum physics about whether quantum uncertainty is due to there being hidden factors or "hidden variables" that would make the quantum world predictable. The majority don't think there are. They just think the quantum world has a fundamental uncertainty built into it.

3. Section 2.2 This section sheds a little light on the first chapter and the bras and kets of linear algebra. Still, it feels like Men in Black where Tommy Lee Jones is mid-conversation with Will Smith after flashing him with the memory thingee.

We still don't know what "spin" is but it is apparently the most fundamental quantum characteristic. For you chemistry buffs, I believe it relates to the final options in the 1s2, 2s2, 3p6 stuff. The two electrons in the s orbital, for example have two different spins. One is said to have a +1/2 spin and the other a - 1/2 spin. Why couldn't he have told us this? Give us something to hold on to, man.

"All possible spin states can be represented in a two-dimensional vector space" (38). That's how he puts it. My interpretation: the final quantum description has only two possible states.

4. So here is how they describe this sort of state, apparently:

∣A〉 = α_u∣u〉 + α_d∣d〉

First impression is of course that this is unnecessarily complicated but I'm sure it's helpful. And I like Dirac so I'll stomach it. But it sure would be nice if someone gave a straightforward explanation. As best I can tell, here's the explanation he never really gives.

a. ∣A〉 is a ket. It is a box in which we put one characteristic of the quantum situation. 

b. There are two possible states for that characteristic. Say it is spin. We might say that spin can be up or down. ∣u〉 is the place we check the "up" box. ∣d〉 is where we check the "down" box." 

These are "basis vectors." They are like the x, y, and z axes in normal geometry, but we can't picture the nature of basis vectors in the quantum world.

I believe up and down are "orthogonal." That is to say, it can't be both. If the state is up, it cannot be down. If it is down, it cannot be up. 

c. α_u and α_d is the value, the component that relates to the up and down. These apparently are complex numbers (that is, they have an imaginary component). I have a hunch they relate to the values of Schrodinger's equation, but making such connections would be far too helpful for Susskind to mention.

I am making the connection because he calls these values, "probability amplitudes," and mentions that their squares are probabilities. I know from elsewhere that this notion relates to Schrodinger's equation, which is about the possible states an electron can be in.

d. The total probability that the spin is either up or down has to equal 1. It is something. α_u*α_u + α_d*α_d has to equal 1, where the * version is the complex complement.

e. "The state of a system" ∣A〉 "is represented by a unit vector" α_u "in a vector space of states" ∣u〉. "The squared magnitudes of the components of the state vector" (α_u*α_u), "along particular basis vectors, represent probabilities for various experimental outcomes" (40).

5. Susskind uses the analogy (I think) of x, y, and z axes. They aren't really spatial coordinates like this. It's an analogy I think to help us understand. What he is trying to picture are quantum categories that are orthogonal to each other just like the x, y, and z axes are orthogonal to each other.

So say the first "axis" we measure is the z axis and then we want to measure the x axis. When we measured the z axis, it had to be either up or down. If we multiply the probability of up times itself and the probability of down times itself and add these two together, it has to equal 1.

If we then move from the z to the x axis, there is half a chance that we will move from it being up to it being right and there is half a chance that it will move from being down to being right. So what value, when multiplied by itself, will yield a half probability of it being right after it being up or down?

∣r〉 = 1/√2∣u〉 + 1/√2∣d〉

Then multiplying this by the equivalent probability for left has to equal 0 because left and right are orthogonal to each other, suggesting the probably for left then has to be:

∣l〉 = 1/√2∣u〉 - 1/√2∣d〉

6. So now he moves on to the y axis. There is a half probability of moving from any of these components to any of the others. For example, there is a half probability that we would go from an "up" state to a "left" state or from a left state to an "in" state. So the probability for "in" times the probability for "left" has to equal 1/2 just like the probability from "up" to "right" needs to be 1/2. As before, the probability of it being both in and out is zero.
I don't quite see how the math works out, but he suggests this means that the probabilities of in and out then turn out to be.

∣i〉 = 1/√2∣u〉 + i/√2∣d〉

∣o〉 = 1/√2∣u〉 - i/√2∣d〉

The reason this seems peculiar to me is because it seems to me that 〈i∣o〉 turns out to be 1 rather than 0, and all the other probability multiplications still have an i in them. Obviously I don't understand something here yet.

7. Another thing I don't understand is what he is calling a phase factor (e^iθ). He says it has unit value and that the vectors can be multiplied by it without changing their values. I'm tucking it away until at some point we see why the heck he's telling us about it.

Once again, he describes a lot of trees but has given us no sense of why any of these things are important or that they relate to anything. Waiting to see the relevance...

Friday Science: Let There Be Space 1

"In the beginning, GOD created space and matter. And the matter was formless and unconnected and there was darkness over the face of the soup. And the Spirit of God moved throughout the face of the chaos. Then God said, 'Let there be photons,' and the cosmic background radiation was released. And God saw the light, that it was good." Genesis 1:1-4

1. In the beginning, GOD created space and matter. YHWH said, "Let there be space." And there was space. And the LORD saw the space, that it was good.

God created the smallest cloud of possibilities within a framework of certainties. God created that smallest of nothings with everything in it, and delighted to watch it enfold.

"Before" that moment, there was nothing. There was God of course. "Where" God was--we have no point of reference to say. "When" God was--we have no categories to express. We have no point of reference even to understand these questions.

And when we say there was nothing "here," we mean there was not even emptiness. There was not even zero. There was no space. There was only empty set, which is different than zero. Empty set does not even have zero in it.

We often assume, unknowingly, that God has to follow the constraints of this universe. We think God has to follow our understanding of logic. We act like YHWH had to learn math in universe school just like everyone else.

But God invented our math. YHWH invented our logic. It is difficult for us to imagine how 1 + 1 could not equal 2. It is difficult for us to imagine a world where the syllogism does not work.

What we need to understand is that when God made the universe out of nothing, he made it out of nothing. He invented the rules of math for this universe. They did not exist before. He invented the rules of logic for this universe. They did not exist before.

It was not like someone who stumbles into a kitchen and starts mixing things together. Maybe our creation will taste good even if we mix things with complete randomness. After all, we did not invent the chemical rules that determine how things mix. We did not invent the way our taste buds taste.

Creation was not like that. In creation God not only created the ingredients. YHWH set in motion the laws for how those ingredients combine. The LORD designed and invented all of that. God created all the options.

2. And GOD said, "Let there be everything." YHWH spoke this command at the same instance that he said, "Let there be space." And the LORD saw everything created, that it was good.

It was not yet in heaven and earth form. In fact it wasn't even in space and matter form. In that first moment--whatever we want to call it--what God created was smaller than anything even angel eyes could see. It was smaller than anything science could see. It was a point full of the universe.

That point held the possibility of space. It was not yet exactly space yet. It was all the possibilities of the universe somehow piled on top of each other in a point. It is not a situation that can continue for more than the smallest of instants, a moment about 10^-43 seconds long. In such a situation, the very fabric of everything is unstable and unsituated.

In that moment, all at the same "time," God said several things. When YHWH said, "Let there be space," God was also saying, "Let there be length." "Let there be area." "Let there be volume."

GOD also said, "Let there be one." YHWH did something no mathematician can do. The LORD divided nothing by zero and got one. He pressed the omega button and created one from empty set.

So God created length. That length of one was infinitesimal by our understanding. Call it the "Planck length," L_P. It amounts to 0.0000000000000000000000000000000016 meters. We abbreviate it as 1.6 x 10^-33, which is 1/100,000,000,000,000,000,000th the size of a proton.

3. In that very same moment, GOD said, "Let there be interaction," which allowed not only for two, but for three, four, and all the numbers. He created the possibility for addition and subtraction. He created multiplication and division, which are forms of addition and subtraction. He created exponents and roots, which are forms of multiplication and division. All of math came into existence in that moment, including exotic numbers like e and i and π.

God dictated that these interactions would take place in units. You cannot divide space infinitely. You cannot divide anything infinitely. Space reduces finally to small units of Planck length, and the interactions between things reduce to Planck "packets" of interaction. Light interacts in packets. Space interacts in packets. Gravity interacts in packets.

The key to these packets is another fundamental number. Call it the "Planck constant," h. It is the fundamental unit of information exchange between the smallest units of reality. It is the unit for a quantum of action. It is, by one reckoning, 6.6 x 10^-34 Joule-seconds, where a joule is a unit of work.

4. In that very same moment, GOD said, "Let there be the speed of light," c. That is to say, YHWH set a limit to the rate of interaction between things. "Let space contract as necessary for the speed of light to be the same in any framework. Let the past and the future be constrained by the time it takes for light to get from one place to another."

And the LORD saw the Planck length, and the Planck constant, and the speed of light, the three fundamental constants of the structure of the universe. God saw that it was good.

In that first moment, time itself did not quite exist. What we call time comes from the rate at which things interact and change, governed mostly by the speed limit of light. However, Time as we know it did not yet exist. Time as we know it only comes to exist when things become irreversible. We can only speak definitively of the future when we cannot go back to the past.

Adam and the Genome 7: Four Principles for Bible Reading

Chapter 5 begins the second set of chapters, by Scot McKnight in Dennis Venema and Scot McKnight's book, Adam and the Genome: Reading Scripture after Genetic Science.

Previous posts
Personal Preface
Forward and Introduction
1. Evolution as a Scientific Theory
2. Genomes as Language, Genomes as Books
3. Adam's Last Stand?
4. Intelligent Design?

Chapter 5
Chapter 5 is titled, "Adam, Eve, and the Genome: Four Principles for Reading the Bible after the Human Genome Project."

1. Scot begins the chapter with his own story, which is so similar to so many of us. Indeed, he mentions a student who came up to him after a lecture at North Park University and said, "Thank you. This lecture saved my faith" (104). I don't know if this claim is true, but McKnight claims that "The number one reason young Christians leave the faith is the conflict between science and faith, and that conflict can be narrowed to the conflict between evolutionary theory and human origins as traditionally read in Genesis 1-2" (104-15).

This brings me to a request for the majority of Christians who do not struggle with this issue or who have come to the conclusion that the science for creationism is clear cut. I dare say that we all have thoughts about God and the Bible that are wrong. How could we not? It is sobering to see the issues where Christians have vehemently said, "It's this way, no other," only for others to agree later that they were completely wrong.

I don't believe that our precise understandings save us. I don't believe the Bible teaches that our understanding saves us. We are saved on the basis of our faith in Jesus Christ. You will not find a verse that says we can only be saved if we have a particular understanding of the Bible or science. Rather "the one who comes to God must believe that he exists and rewards those who diligently seek him" (Heb. 11:6).

So Christians need to leave a space for other Christians who struggle on the questions of science or who have reached different conclusions on matters of science and faith. Souls are at stake. Woe to anyone who puts a stumblingblock before one of these little ones! Now that's actually something the Bible says, Jesus in fact.

2. Here are some quotes from the early part of the chapter:

• "There is a better way, one that permits each of the disciplines to speak its own language but also requires each of the voices to speak to one another" (94).
• A defining moment for Scot was asking himself the question "whether traditional interpretations of Genesis 1-2 were perhaps well intended but misguided and in need of rethinking" (95).
• "Every statement about Adam and Eve in the Old Testament, in Jewish literature, and in the New Testament is made from a context and into a context" (97).

3. Scot sets out four principles in this discussion:

A. Respect - "To understand what someone is telling us, we must respect that person as a person" (98). In terms of Genesis 1-11, "we must respect that text as designed for the ancient Near Eastern culture" (99). "It is disrespectful to Genesis 1-11 to think it somehow should understand modern science, DNA, and the Human Genome Project, or for that matter the science of any generation after the writing down of Genesis 1-11!

"Genuine respect begins when we let Genesis 1-11 be Genesis 1-11, which means letting Genesis 1-11 be ancient Near Eastern and not modern Western science" (100).

B. Honesty - "Face the facts and do not fear the facts" (100). "Are you willing to face the facts--the facts of the Bible and the facts of science?"

McKnight makes the controversial claim that some of the most ardent defenders of certain interpretations of the Bible on these issues might actually be afraid. They are afraid to open the door to such questions because they are afraid it will lead them to lose their faith.

McKnight does not only ascribe this kind of uncharitable spirit to creationists but to some Christian evolutionists as well. He quotes Ron Osborn: "I have often been equally dismayed by the attitudes evinced by some individuals on the other side of the debate over creation and evolution... how quickly some are prepared to write off people of sincere faith who are at different places in their intellectual and spiritual journeys" (101).

C. Sensitivity to the Student of Science
I already mentioned the student whose faith McKnight says was saved by him opening up the window to the possibility of having faith and yet not concluding God created the universe 10,000 years ago. "The student is in my rearview mirror in all I have to say in my section of this book" (105).

D. Primacy of Scripture
Scot is an evangelical, so the Bible always comes first. The investigation of truth can go to other bodies of knowledge in addition to the Bible, but it always starts with the Bible: "Scripture first.'

4. He ends the chapter with a sense of some of the complexity these discussions can take on. For example, what do we mean by the "historical" Adam and Eve? Another view of Adam and Eve is the "archetypal view," Adam and Eve as representatives of all humanity. His sense of a "literary" view does not so much draw conclusions on the historical, but looks at them within the text of the Bible.

More to come...