My Summary of General Relativity

As you may know, Einstein’s theory of relativity is actually two related theories.  I wrote about the Theory of Special Relativity here, and that’s where most of the counter-intuitive weirdness is (time dilation, length contraction, and the unification of space and time into spacetime).  In contrast, General Relativity has only one big weirdness, the assertion that the “force” of gravity is an illusion, and that what we observe is just a manifestation of the fact that our spacetime is curved.  Hopefully you’ll be able to get your head around this by the end of the essay.  The math of general relativity is pretty advanced and hairy, but luckily it’s not necessary to understand this to get the gist of the theory, so I’m going to gloss over it.  It’s also not necessary to grok the concept of “spacetime” for our purposes here, so I’m only going to refer to “space”.

General Relativity gets its name from the fact that it describes all kinds of motion, while Special Relativity only describes the special case of inertial motion, which is motion that doesn’t include acceleration.  Einstein’s great insight about acceleration and gravity is that they’re the same thing. Therefore, General Relativity (GR) is a theory about accelerated motion and gravity.  

Before we get into all that, let’s review the theory that GR replaces, Newton’s Law of Universal Gravitation, which is usually given by this equation:

However, this equation is useless on its own.  To understand how gravity acts on massive bodies, we need to add Newton’s Second Law, which can be written this way:

You may be more familiar with the simple F=ma form, but since acceleration is just the change in velocity over time, the middle term should be understandable to anyone with a passing familiarity with calculus.  (For bonus points, understand that acceleration is the second derivative of position).  The reason to emphasize this is to show that the law of motion involving forces is a differential equation, which is an equation that relates quantities and their rates of change.  Einstein’s equations will also take the form of differential equations.  

The two equations above work together to tell us what to expect from gravity.  The first one says how to combine the masses of two objects with the distance between them and the mysterious constant G to calculate the force exerted on the body of interest.  We then plug the force into the second equation to calculate how the body of interest will move under the influence of this force.  

Simple and effective, Newton’s laws reigned for hundreds of years, and were utilized to calculate everything from the paths of celestial bodies and cannonballs to the interactions of billiard balls.  However, there were anomalies that Newton’s equations couldn’t explain.  In particular, the orbit of Mercury was slightly different from what was predicted by Newton’s laws.  The orbit precessed, which means the axis of rotation moved in a regular way, and nobody could explain this.

It took Einstein ten agonizing years to work his way from the Theory of Special Relativity to the Theory of General Relativity.  Reportedly, when he used his completed equations to calculate Mercury’s orbit and the precession was perfectly predicted, he was so excited he couldn’t sleep for days.

Einstein’s equations have the same two-part structure as Newton’s, but instead of a “force” being the currency traded between the two parts, Einstein’s equations use the curvature of space.  While curved space is a mental hurdle we’ll attack shortly, this is actually more intuitive, since it’s very hard to say exactly what a gravitational “force” is, beyond vacuous definitions like “the thing that makes masses move”.  Without further ado, here are the equations:

The first equation plays the role of Newton’s first equation, but instead of an “other mass” and a distance, the equation describes the curvature of space in terms of the matter density (expressed by the T term).  As I said, I’m not going to attempt to explain the math, but the quantities with the Greek indices are special matrices called tensors that contain factors for every dimension of space.  The second equation describes how objects move in a curved space.  This is clearly a differential equation, equivalent to F = ma, which is of course a special case solution when the space is flat and the second term vanishes.  In Newton’s universe, masses create forces and these forces make other masses move.  In Einstein’s universe, masses make space curve, and curved space makes masses move on paths called geodesics, which follow this curvature unless a force compels them otherwise.

Visualizing curved spaces is actually pretty easy if you limit yourself to two dimensions.  The surface of a sphere (such as the earth) is a great example you can intuitively explore.  If you compare a flat plane to the surface of a sphere, some pretty stark differences arise immediately.  On the plane, if you walk in a straight line forever, you never come back.  On a sphere, if you walk in a straight line you arrive back where you started in a finite time.  On a plane, parallel lines never meet.  On a sphere, parallel lines do meet (think of the meridian/longitude lines on earth meeting at both poles).  

Three-dimensional curved space is harder to visualize, and we’re not going to try.  It’s enough to accept that a creature trapped on a two dimensional surface who believes it is flat will be surprised at some of the phenomena of curved 2-D space described above.  If you can tentatively accept that you are a creature trapped in a curved 3-D space that you think is flat, you can anticipate that some surprising things will be observed if that space is actually curved.  

The main counter-intuitive result of the fact that we live in a curved 3-D space is that we have mistaken concepts of what “at rest” and “accelerating” mean.  We think we’re at rest when we’re standing, sitting, or lying down on something connected to the surface of the earth.  According to Einstein, this is not a rest state, it’s an accelerated state.  According to GR, the only time you’ve not been accelerating in your life is when you’ve been freely “falling” in earth’s gravity.  Probably the longest freefall you’ve experienced is the time between stepping off the high dive and hitting the water in the pool.

How can anyone say lying in bed is acceleration?  In classic Einstein style, we use thought experiments.  Imagine you’re in a closed box that appears to be an elevator car.  If this box was placed out in space an infinite distance away from any mass, you’d float freely, with no particular tendency to go to one side or another.  If you held a ball and then released it, the ball wouldn’t move.  Now imagine some force grabs one side of the elevator car and starts pulling at it, accelerating at 9.8 m/s^2.  Immediately, you would find one of the sides of the car rushing towards you.  As you hit the side, it would immediately seem to become the “floor”, and the side the force was pulling on would seem to become the “ceiling”.   If you regained your footing, it would feel just like standing up on earth.  If you held a ball in your hand, your hand would impart an acceleration to the ball which comes from the acceleration of the floor traveling through your body.  If you let go of the ball, it would stop accelerating and inertia would cause it to travel in the same direction at a constant speed.  Since the elevator car is accelerating, it would soon catch up to the ball and the “floor” would hit it.  Unless you knew what was happening, you would interpret this as the ball “falling” “down”.  This is how acceleration and gravity are the same thing, or at least indistinguishable.  

Back to you lying in bed, the matter comprising the earth and your house and bed are constantly accelerating and pushing on you.  This explains the force you feel the bed pressing against you, just like the ball experienced the force of your hand pressing against it in the elevator car.  If an empty shaft to the center of the earth suddenly appeared under your bed, you’d no longer feel the force of anything pressing against you.  Very relaxing and comfortable!  You might think that you’d start “falling” but this is only because you have a habit of comparing your motion to things on the surface of earth.  Relative to the curved space you inhabit, you’d finally be at rest (or in inertial motion, actually, so at rest in some suitable inertial reference frame), while objects on or in the earth accelerate away from you.

That’s really all there is.  It’s taken me quite a long time to become even remotely comfortable with the explanation above.   But if you want to understand how curved space explains gravity, changing your habitual notions of “at rest” and “accelerating” is the only way.

As remarkable as the Theory of General Relativity is, many mysteries about the nature of gravity remain.  Matter following geodesics through curved space makes some sense, and allows you to keep your intuitive idea of space as a set of locations that may or may not contain matter.  But the other half of the theory, where matter causes space to curve, doesn’t really have an intuitive explanation behind it.  What is matter and what is space?  How exactly does matter cause space to curve?  It may well be that space is just another mathematical fiction like Newton’s force.  It may be that there are particles called gravitons that transmit the force of gravity, and their discovery may make curved spacetime unnecessary.  We also haven’t united General Relativity and Quantum Mechanics into a theory of Quantum Gravity, so there will doubtless be many surprises when that happens.

My Summary of Special Relativity

As you may know, Einstein’s theory of relativity is actually two related theories.  The Theory of Special Relativity, published in 1905, describes the nature of space and time for the special case of inertial reference frames (coordinate systems that aren’t accelerating relative to each other and have no gravity).  The Theory of General Relativity, published ten agonizing years later, covers the more general case of accelerating reference frames, which Einstein asserted are the same as gravity.  The math involved in general relativity is much more advanced, but special relativity is where most of the counter-intuitive weirdness comes from.  This essay covers special relativity (SR).

A reference frame is a construct we use to describe a set of locations in the universe that are rigidly connected so they move together at the same speed.  Such constructs are useful because we can draw imaginary coordinate axes on them and thereby apply all the tools of mathematics to study events that happen in the frame.  Obviously, not everything in the universe is rigidly connected to everything else, and things move at different speeds.  Therefore, we’ll need many different reference frames to accurately describe the universe.  

Imagine you’re in the back of a pickup truck driving at a constant speed down a straight road, and your friend is standing on the sidewalk as you drive by.  As you and your friend wave to each other, you are in inertial reference frames that are moving relative to each other at a constant velocity, based on the speed and direction of the truck relative to the sidewalk.  When the relative velocity is constant, we say the motion of the reference frames is inertial (since the law of inertia says objects will travel in this way if no forces act on them).

It’s been know for hundreds of years that this kind of relative motion exposed deep laws of physics.  For example, if you reached out your hand and gave a high five to your friend on the sidewalk, both of you would feel the slap with equal force.  You could say your friend on the sidewalk is at rest while you and the truck are moving, but this is an arbitrary designation.  This can only be said if you affix the coordinate axes to the sidewalk.  If you fix the coordinate axes to the truck, then it’s you who’s at rest and your friend on the sidewalk who is moving relative to the coordinate axes or reference frame.  This thought experiment points out that the notion of absolute rest and motion that Newton championed is not as useful as the notion of relative rest and motion.  After all, both truck and sidewalk are on the rotating and orbiting Earth, which is in the solar system which orbits the galaxy, etc., but none of these other motions make any difference in how the participants in these experiments perceive each other.  

This simplest form of relativity is known as “Galilean relativity”, because Galileo Galilei was the first to state it clearly.  His most famous postulate of relativity is that if you were inside a windowless cabin on a ship on a clam lake, there is no experiment you could do that would tell you if the ship was at rest (relative to the lake) or moving in a straight line at a constant speed.  As soon as the ship accelerated or turned, you could feel it immediately.  We’ll come back to that when we talk about gravity, but for inertial reference frames, inertial (constant-speed straight-line) motion can’t be detected.

With Galilean relativity, velocities add up in simple ways.  This is immediately intuitive.  Imagine you’re in the back of the pickup truck with a paintball gun, preparing to prank your friend on the sidewalk.  If the truck is going 20 mph and the paintball gun shoots at 30 mph, then if your aim is true, your friend on the sidewalk will feel as if the paintball has hit him at 50 mph, because that’s its speed relative to the sidewalk.  The velocity of the paintball from the gun is added to the velocity of the truck and gun.  

This intuitive picture seemed to work, and Newton’s laws of mechanics are based on it.  The triumph of the theory of special relativity is to recognize that this isn’t quite right; it’s only an approximation.  At low speeds, it’s a very good approximation, so how was the inaccuracy discovered?  The answer involves the speed of light.

By the end of the 19th century, the speed of light was known to be finite and had been measured to some degree of accuracy.  The physicists of the day assumed that everything, including light, obeyed the principle of Galilean relativity.  Given this assumption, light had to be moving relative to something, and they gave this something the name “luminiferous ether”.  The ether was taken to be the substance which filled and defined a “universal rest” reference frame, relative to which everything else could be measured.  

Many experiments were done to try to measure the speed of the earth relative to the ether.  The first experiment that was done well enough to be trusted by the bulk of the physics community was by Albert Michelson and Edward Morely.  The Michelson-Morley experiment used an interferometer, the same instrument used in the recent LIGO apparatus that detected gravitational waves.  An interferometer uses light sent down two perpendicular arms that reflect it back to the right angle vertex.  This is generally done with a partially reflective mirror oriented at 45 degrees to a light source.  The mirror “splits” the light so that some is reflected and goes down one arm and some is transmitted and goes down the other arm.  When these partial beams are reflected back to the mirror, the two beams are partially reflected and partially transmitted again.  This combines light from both arms of the interferometer and sends the combined signal to a detector.  See this animation on Wikipedia (just the left hand side of the diagram for now).  

The lengths of the arms can be calibrated so the light that travelled on the two arms is in anti-phase when it reaches the detector.  This means all the crests of one light wave line up exactly with the troughs of the other light wave so they completely cancel each other out and the detector sees nothing.  After such a calibration, if a subsequent change in experimental conditions causes the light to take longer to travel one arm versus the other, then the light won’t be perfectly in anti-phase anymore, and the destructive interference will be incomplete.  Then the detector sees a signal, and the amount of signal can be used to measure the difference in time taken by the light that traveled the different arms.  

Michelson and Morley expected to be able to see a difference in the time taken by the two light beams when they changed the orientation of their interferometer.  In a sense, they were searching for the ether, the reference of absolute rest.  They expected that when one arm was oriented parallel to the direction of Earth’s motion relative to the ether, the path light would have to travel on this arm would be slightly longer than the light that took the path perpendicular to the apparatus’ motion relative to the ether.  See this analysis to understand why, but it’s just basic Euclidean geometry, algebra, and the assumption of Galilean relativity.  Or just look at the right side of the Wikipedia animation to get an intuitive feel for why the travel times were expected be different, resulting in a phase shift of the light beams.

The Michelson-Morley experiment is said to be the most famous and consequential “failed” experiment of all time.  Of course “failed” is the wrong word.  The experiment produced a “negative” result, meaning it didn’t show the expected signal from the expected path length differences of the two arms of the interferometer.  This negative result was a major anomaly demanding an explanation, since it contradicted the prevailing theory of a stationary ether and Galilean relativity.  And as all good anomalies do, this one spawned a scientific revolution.

If there truly was no way to measure motion relative to a fixed ether, this had immediate logical and mathematical consequences.  It was not Einstein, but several other physicists of the late 19th century who worked out these implications.  Hendrik Lorentz got a lot of the credit, and the transformations used in special relativity still bear his name.  Einstein’s major contribution was to show that the Lorentz transformations relating the velocities of relatively moving reference frames were a consequence of only two principles, and didn’t require the existence of an ether or a frame of absolute rest.  The two principles from which SR is derived are:

1. The laws of physics have the same form in all frames of reference.
2. The speed of light is the same for all observers in all frames of reference.

The first principle is an axiom, taken as given without proof.  Most people don’t have a lot of trouble accepting it, but the second principle is a bit more counter-intuitive.  It basically says that Galilean relativity is false.  It doesn’t have to be taken on faith, since that’s what you get if you believe the results of the Michelson-Morley experiment are correct.  They observed that light travelled the same speed no matter how they oriented their apparatus relative to the orbital motion of the earth.  People didn't notice the departure from Galilean relativity until the late 19th century because it only becomes significant for things traveling near the speed of light.  The value of the speed of light, c (approximately 3 x 10^8 m/s or 186,000 mi/s), also falls out of Maxwell’s Equations of electromagnetism, which led Maxwell to propose that light was an electromagnetic wave.

The Lorentz transformation is simply a set of formulas that allow you to convert distances and time in one reference frame to these same quantities with respect to another reference frame that’s moving inertially relative to the first.  When making up such formulas, it’s convenient to pick our coordinate systems so that the relative motion only occurs in the x-axis direction.  In that case, Galilean relativity has a simple transformation:

where x, y, and z are the coordinates in the “first” reference frame (which we’re choosing to call “at rest”), the primed coordinates are the “moving” reference frame, and v is the relative speed of the reference frames.  This is the transformation used in the example above to determine that the speed of the pickup truck and the paintball should be added together to determine what speed your friend on the sidewalk perceives (the minus sign could be a plus sign, depending on how you choose your coordinates).  

The Lorentz transformation is a little more complicated:

 where

The gamma factor is simpler than it looks.  If your speed is much less than c (the speed of light), then the second term under the radical () is very close to zero.  When that happens, gamma is very close to 1, and the Lorentz transformation approximately reduces to the Galilean transformation.  (This is why Galilean relativity seemed right for so long).  On the other hand, if your speed is very close to c, then the value of gamma skyrockets.  Here are some sample values:

The most surprising thing about the Lorentz transformation is that the x coordinate isn’t the only one affected by relative motion in the x direction.  The time coordinate is also affected.  This makes a little more sense when you realize the whole purpose of creating the transformation in the first place was to make the speed of light constant in all reference frames.  Since speed is distance/time, of course time must be affected in order to keep the distance/time ratio constant.

This mixing of space and time is what gives SR some of its truly weird and counterintuitive properties.  I don’t know if people can get used to it, but I haven’t yet.  If you plot the x and t coordinates (omitting y and z since they’re not interesting and we run out of dimensions to draw them in) along with the primed x and t coordinates, we get this remarkable plot:

Notice how the primed axes no longer appear to be at right angles to each other.  However, this is only because we’ve drawn the graph from the perspective of the unprimed coordinates.  Unexpectedly, if we adopted the other perspective, the primed axes would look “normal” and the unprimed axes would appear in exactly the same positions as the primed axes shown above.  This illustrates the counterintuitive reciprocal nature of the time and length dilation effects.  

For any “event” that occurs on this “spacetime” graph, the two different coordinate systems will ascribe different relative amounts of space vs. time to the event.  The rectangle and parallelogram on the diagram are meant to represent “dropping a perpendicular” to each coordinate axis in order to measure an event’s position and time.  Notice the different amounts of time and space ascribed by the two coordinate systems to the event at the upper right corner of the rectangle and parallelogram (compared to another imagined event at the shared origin).

The only things that are unchanged in both coordinate systems are the 45 degree x = t and x = -t lines.  These represent light rays travelling in the positive and negative x directions.  (The units have been chosen so that the speed of light is equal to 1).  Again, this shouldn’t be surprising since the whole purpose of all of this is to construct an analytic framework where the speed of light is constant in all inertial reference frames.  

Newtonian physics and Galilean relativity don’t rotate space and time into each other like this.  Newton never had the need to add a time axis to his motion diagrams because he assumed there was one absolute reference of time, and that it was possible to consider “all points in space at a given time”.  This is simply not possible when it’s a different time at different points in space that are moving relative to one another.  In SR, clock readings can only be made locally.  We normally make assumptions about what distant clocks read, but these assumptions are invalid.  As long as you insist on thinking of space and time separately, as if there existed “all of space at a given time” then SR will seem bizarre and paradoxical.  Well, there is no unique time for all of space.  Doesn’t exist.  You were misinformed.  Get over it.

The easiest way to start to accept that there is no absolute universal time is to understand the relativity of simultaneity.  That is, whether or not two events are simultaneous depends on how you’re moving relative to the events.  All other times reduce to this, since you’re always comparing one event of interest to another event (the tick of a clock).    If you believe the tilted coordinate axes described above, then this animation from Wikipedia should convince you that events can be simultaneous in one reference frame, non-simultaneous in another, and non-simultaneous in the opposite order in yet a third reference frame (the white bar represents the progression of time in the reference frames).  If you aren’t on board with the tilted axes, then try this video.  

How can events occur in different orders depending on who’s watching?  Doesn’t that potentially mix up the order of cause and effect?  No, because the order of events can only change for events with spacelike separation.   To understand what this means, you need to imagine that every event has a “light cone” that precedes and follows it.  The surface of the cone represents the possible paths light rays can take.  As shown below, this is a spacetime diagram where units are again chosen so the speed of light is 1 (it plots at 45 degrees), the vertical axis is time, and only two spatial dimensions are shown.

 

If one event is caused by another, then the effect event must be inside the light cone of the cause event, since some some signal or piece of matter must pass between them for them to be causally related, and nothing can travel faster than the speed of light.  Events with spacelike separation do not have light cones that overlap, so they are said to be causally disconnected.  For causally connected events, all observers will agree on the order of events, so causality is saved.  The diagram below shows several light cones that can be imagined to represent causally connected and causally disconnected events.  Only a timelike path can connect causally related events.

The ultimate effect of the Lorentz transformation is that relatively moving observers disagree about both time and distance.  This leads to the fact that moving observers experience both time dilation and length contraction.  Time dilation as derived from the Lorentz transformation is expressed with this formula relating the difference in time (𝝙t) between two events in primed and unprimed coordinate systems:

In this formula, the closer v gets to c, the smaller the quantity under the radical becomes, and the larger the primed 𝝙t becomes compared to the unprimed 𝝙t.  This means if you’re moving relative to a clock, the clock will appear to be running slowly, since your 𝝙t is larger than the 𝝙t in the clock’s reference frame.  However, in the clock’s reference frame, it appears to be running normally.   This result has been verified by flying atomic clocks around on planes and then later comparing them to ground-based clocks.  Taking this to the extreme, if you could actually travel at the speed of light, a clock at rest would appear to be standing still.  

The Lorentz transformation can be used to derive the following formula for length contraction of an object as seen from a reference frame moving relative to that object:

where L0 is the length of the object in the reference frame where it is at rest, v is the relative speed of the two reference frames, and 𝛾(v) is shown in expanded form on the right.  This formula shows that if the relative motion is much slower than the speed of light, the length is nearly the same in both reference frames.  As the relative speed approaches c, the length of the object as seen from the moving reference frame approaches zero.

Is there anything that stays the same in every inertial reference frame?  As a matter of fact, there is.  If you combine the space and time separation (deltas) between two events using the formula below, you get the quantity on the left hand side, which is called the “spacetime interval”:

The wonderful thing about the spacetime interval is that it’s invariant, meaning it won’t change when you view the two events from a different reference frame that’s moving inertially with respect to the first.  The two reference frames may see different amounts of space vs. time, but the spacetime interval will always be the same.

An even more remarkable takeaway from this formula is that c is essentially used as a conversion factor between time and space.  The remarkable thing is not that electromagnetic radiation is observed to move at this speed, but that such a fixed conversion factor exists at all.  

This conversion factor between time and space also leads to the last thing I’ll say in this essay on special relativity.  The fact is, everything, including you, is always travelling through spacetime at the speed of light.  If you are sitting still in your reference frame, then all of your spacetime travel is along your time axis, and you’re travelling on that axis at the speed of light, as the spacetime interval formula indicates.  As soon as you start travelling in space, then some of your velocity is used up by this travel, and your progress through time must be less than c (as seen from your former rest frame).  This is one way to look at time dilation (your clock will be seen as running slow by an observer you left behind).

What does this say about light itself?  The time dilation formula:

isn’t very helpful.  If you imagine a photon that left a distant star a billion years ago and entered your eye just now, then in your unprimed coordinates, 𝝙t is one billion years.  The reference frame of the photon is moving at c, so the denominator in the formula is zero.  This prevents us from calculating 𝝙t'.  No matter what 𝝙t is (no matter how far away the star is), the result is the same.

This says that light is expending all of its velocity on travel through space, so there’s no velocity left for travel through time.  This seems to indicate that time does not pass at all for a photon.  You could try to say that 𝝙t' for the photon is infinity, or zero, but neither one really works.  There is an incredible variety of answers to this question on the Internet, and most of them are pretty incoherent.  If you ever get a chance to talk to a photon and ask it what happened on its trip, please let me know.

9/11 - Negotiation = Trump

When I was growing up in the 60's and 70's, airport security was nonexistent by today's standards, and planes getting hijacked was a thing that happened from time to time, like school shootings today.  In those days, some of the hijackings were political, but many were just done for money.  Commandeering a plane was just a way to acquire a bunch of hostages whose lives could be traded for cash.  This rarely worked out for the hijackers, whose plane would often be taken away from them by force, even if innocent lives were inevitably lost along the way.

These hijack-for-profit extortionists would kill hostages if necessary to gain tactical advantage, but murder didn't seem to be their primary goal.  So I thought the authorities should just pay the ransom to ensure the safe release of all the innocents.  It had to be explained to me why this seemingly obvious strategy was a naive idea.  The way it went was that if we give in to their demands, this will only encourage other groups to copy the same strategy, resulting in even more hijackings in the future.

Even this meager reasoning has been lost in today's rhetoric on terrorism.  Leaders from all countries and of all political persuasions just flatly state: "we don't negotiate with terrorists".  Why is this policy never examined?  In the hostages-for-cash scenario, there is a certain grim logic to the reasoning, but does the same calculus necessarily apply to today's political terrorism?

You might think the same logic applies universally, but a deeper analysis shows otherwise.  True, if political groups got what they wanted, their tactics might become more popular in the future.  But what do they want?  Would the demands of all similar groups be insatiable, growing without bound until the whole world was engulfed in perpetual terror?

If you only consume mainstream news, you wouldn't even know they had any demands, let alone what they are.  Terrorist groups like Al Qaeda, ISIS and the like are depicted as mindless sub-humans, bent on a completely irrational path of destroying us Westerners, who are just minding our own business when they strike out at us.  Within this narrative, the only acceptable response is to annihilate them with superior violence.

This characterization of today's terrorists is false.  They are human beings, trying to build a society just like us.  They have very specific demands of us, many of which are quite reasonable.  Meeting their demands would inevitably involve a reduction in western violence in their lands, which would cool the anger that's directed back at us.  Therefore, negotiation and willingness to meet some of their demands would result in a de-esalation of violence.

How did I come by such a contrary viewpoint?  Is it hopelessly naive?  A turning point for me was reading the book Imperial Hubris.  Unless you read a source like this, you and I can't have a rational discussion about things, because you're wallowing in ignorance.  Did you know that Al Qaeda articulated a very short, very specific list of demands, years before 9/11?  Why was this not widely reported, especially after 9/11 resulted from our failure to meet their demands?  Because the unconditional, unexamined "we don't negotiate with terrorists" mantra makes such thoughts unthinkable, let alone admissible in public debate.

If we at least entertain the possibility that people acting with violence are rational human beings who are simply out of options, then ending the cycle of violence becomes possible.  Why is this unthinkable?  The United States acts with violence, yet most Americans consider this rational, just and necessary action.  Why isn't the "enemy" given the same benefit of the doubt?  Answer: because they've been dehumanized so we don't need to consider their concerns.

What are the demands Al Qaeda made prior to 9/11?  Things like 1) Stop militarily occupying Muslim holy lands, 2) Stop propping up corrupt dictators in the Arab world, and 3) Stop blind support of Israel and dehumanization of the Palestinians.  Are such demands really beyond the pale, not even worthy of discussion?  To the US political and media elite, they apparently are, but what do you think?  Read Imperial Hubris and decide for yourself how bad the bad guys really are.

How does Donald Trump fit into this story?  I'll argue that someone like him is inevitable when a refusal to negotiate with others is the universal stance on both sides of the political divide in America.  Through eight years of Bush and eight years of Obama, the narrative pushed by both parties and their allied media has been the dehumanization of Muslim enemies, and a refusal to even admit they could have legitimate grievances with us.  When was the last time you saw a media interview of someone from a group branded as terrorist by the US?  Obama can talk about how we're at war with the Terrorists(TM), not Islam, but this doesn't help.  If Obama is such a nuanced, thoughtful statesman, why can't he admit the terrorists themselves are also human beings?

When superior violence is the only acceptable response to terrorism, escalation is inevitable, as the history of the War on Terror has shown.  If a genuinely peace-minded leader emerged and tried to negotiate with terrorists, the media would crucify him.  Opponents would call him a terrorist-lover, and the media would repeat this without challenge.  Authoritarianism through superior violence saturates our culture, from school to policing to war.  In such an atmosphere, one begins to wonder why it took so long for a Trump to appear.  And the media look around in horror, wondering how this could have happened, never considering their role in making it inevitable.

Did you know that Al Qaeda's threat, if their demands weren't met, was to draw us into an endless, global war that would bankrupt us?  That gives you a different take on 9/11 and the War on Terror, doesn't it?

How can we get out of this mess?  Start making negotiation a regular part of foreign policy discussions.  Remember when diplomacy used to be a respected role?  Today, diplomats only role is to convey threats of violence in euphemistic language.  The media can change this by simply asking questions.  Why has the so-called terrorist group done this horrifying act?  What do they hope to gain?  Why do they hate us?  Are their demands reasonable or not?  These questions have answers, but we never hear them because journalists and public officials never ask.  You can start by asking them yourself.

My Summary of Quantum Mechanics

As some of you know, I have an interest in physics.  I’ve been reading “popular” books for many years, marveling at the strangeness of our universe.  Eventually I became frustrated with popular accounts, where authors interpret the math and experimental results for us.  It seemed to me that many authors were in love with complexity and paradox.  They said the math was just that way, indeed the universe was just that way.  I doubt the universe is fundamentally paradoxical, and I decided I could no longer take their word for it.  I decided to learn physics for real, with the math.  

As an electrical engineering major, I took a full series of calculus and a couple of physics courses.  I thought I could rely on this background and just jump into a mathematical treatment of quantum mechanics, which seemed the richest source of paradox.  I shortly had to admit that after 30 years, calculus can be utterly and completely forgotten.  Plus, I had never completed linear algebra.  Undaunted, I started a thorough math review that included calculus, some pre-calc concepts, and linear algebra.  This was done mostly on Coursera and Khan Academy.  

After this prep, I still could not penetrate the two QM books I had in my possession, and I became depressed.  Then it occurred to me to ask the Internet if there was a better book.  The Internet delivered, in the form of a recommendation for David J. Griffiths’ book Introduction to Quantum Mechanics.  This book made all the difference for me.  It’s crystal clear, and builds up the concepts in a logical way that other books don’t for some reason.  He even includes an appendix that teaches you the specialized linear algebra concepts you need to know.  I read the first three chapters quite closely, but mostly skipped the problems and skimmed many of the later chapters that went over complex calculations that didn’t interest me.

So, what does the theory of Quantum Mechanics tell us?  What does the math look like?  Other books had given me the impression that QM was based on linear algebra, but this isn’t really true.  It’s based firmly on calculus, like other parts of physics.  In this respect, I think it’s misleading to say that QM is somehow “discrete”.  It’s true that things like the energy levels in atoms are calculated as a set of discrete values, and yes, linear algebra is used in these calculations, but these solutions come ultimately from equations of good old fashioned continuous variables.  This is especially true of Schrodinger’s Wave Mechanics formulation, but seems equally true in Heisenberg’s Matrix Mechanics formulation, where matrix elements are computed as Fourier coefficients of functions of continuous variables.

The foundation of quantum (wave) mechanics is the wave function, denoted by the Greek letter psi.  The wave function is a function of four continuous variables (time and three spatial variables), and represents the state of the system being examined.  Thus, each system has a different wave function, just as each system has a different equation of motion in classical mechanics.  In both schemes, the time evolution of the system’s state function is governed by a differential equation.  In classical mechanics, it’s Newton’s laws, i.e., F = ma (which is a differential equation since a is the second derivative of position).  In QM, the time evolution of the system’s state is governed by the Schrodinger Equation.  This differential equation is complicated enough if you pretend space has only one dimension (x), so that’s how it’s normally introduced.

Here i is the imaginary unit, h is Plank's constant, m is the (non-relativistic) mass of the system (i.e., particle), Psi is the wave function, V is the potential affecting the particle, and H is the Hamiltonian. a representation of the total energy of the system.

A “solution” of a differential equation is any function that varies in the required way that the differential equation is satisfied.  Now, partial differential equations are notoriously hard to find solutions for, and the time-dependent Schrodinger equation above contains differentials of Psi with respect to both position and time.  So as a practical matter, the equation is separated into a time-dependent part and position-dependent part.  This results in a simpler equation called the time-independent Schrodinger Equation, whose solutions are the so-called “stationary states” (because they don’t evolve in time).  

It turns out that the solutions to the general (time-dependent) Schrodinger Equation can always be expressed as linear combinations of the stationary states.  It’s these linear combinations that account for one of the oft-cited mysterious features of QM, superposition.  When the state of the system is a combination of sub-states, people say that the sub-states are superposed, i.e., simultaneously true to some fractional degree.  

When a system’s state is measured, only one of the sub-states is found, but more on this later.  In this way, measurements become one of the other mysterious features of QM.  What people say is that the system’s state (i.e, wave function) was in a superposition (e.g, linear combination of sub-states) before measurement, and that the act of measurement “collapses” the wave function to only one of the sub-states.  Mathematically, this happens because the result of a measurement is calculated by multiplying a particular matrix by a vector representation of the state, Psi.  So in this case, the elements of the vector are actually functions (the stationary state solutions to the time-independent Schrodinger Equation).  So this is where the linear algebra comes in, but the vector space is an infinite-dimensional space of orthogonal wave functions .  The functions and matrices are rather special, so it always turns out that the wave functions are eigenvectors of the measurement matrix.  An eigenvector of a matrix is a vector that comes out merely scaled (rather than, say, rotated or otherwise distorted) when multiplied by the matrix.  The scaling factor is called the eigenvalue, and in general is different for each eigenvector and matrix.  Each type of measurement, e.g., position, momentum, or energy, is represented by a different matrix, usually called an operator.

That’s the gist of it.  Doing physics involves defining a system with some initial conditions, solving the Schrodinger Equation, and then computing the different eigenvalues that may be returned by a measurement.  However, the theory cannot say which eigenfunction/eigenvalue pair may be selected by a given measurement.  It could be any of the component wave functions in the system’s superposed state.  The theory can say only the probability of selecting a certain component out of the superposition, which equals the square of the coefficient each component has in relation to the other components in the superposition.  For this statistical interpretation to make sense, the wave function must satisfy the property that the sum of all probabilities is 1, i.e.,

This requirement not only means that all acceptable wave functions must be “square integrable”, but also that they typically must be “normalized”, which means multiplying then by some factor that makes the integral above equal 1, rather than whatever it equaled when you chose the wave function.  This strikes me as a massive hack, albeit a hack that works.  Wave functions are guessed and normalized (i.e., fudged) so that they fit into the statistical interpretation of measurements of the wave function state.  The normalization requirement also rests on the assumption that the probabilities sum to 1 because “the particle must be found somewhere”, (or found with some energy, if that’s what’s being measured).  All of this is not derived from some physical principle or logical necessity.  It’s all just cobbled together into a system that happens to be useful for measuring the things we know how to measure.

This statistical interpretation of the measurement operation is perhaps the most frustrating part of QM for me and many others.  It certainly seems to be the jumping off point for how to interpret the theory, which is my main interest.  What struck me in my study is how ad-hoc the whole scheme is, and how far divorced from physical reality the math is.  In classical mechanics, a particle “really has” the properties represented by the quantities in the equations, and if you make a measurement, you get the instantaneous value of one of the variables.  I believe, along with Einstein, that a proper atomic theory should be the same way.

The statistical interpretation of QM is also the source of the famous “uncertainty principle”.  Many authors speak, wrongly, as if the uncertainty principle applies to a single particle, saying that the more precisely a particle’s position is known, the less precisely its momentum can be known.  (There are other pairs of complementary properties also).  This is not at all what the uncertainty principle says.  Being part of the statistical interpretation, the uncertainty principle states that the product of the standard deviations of two complementary observables has a lower limit:

That’s standard deviations, as in the statistical properties of repeated measurements.  Thus, the principle says nothing about the “actual” properties of a single particle, but only about the statistical character of repeated experiments done on multiple particles that start in the same state.  

Now it isn’t quite fair to compare quantum measurements with classical measurements.  For classical measurements, you can generally devise a measurement apparatus that doesn’t significantly disrupt the phenomenon being measured.  Not so for quantum phenomena at the atomic level.  You can’t just “look at” an electron to see where it is.  Shining even a single photon onto the electron will invariably alter the electron’s state in a large way.  But this is the only way we know how to do measurements.    

I believe the limitation of statistical predictions is a shortcoming of QM theory and our disruptive experimental techniques.  It’s hard to imagine other experimental techniques, but I believe there is some underlying reality that’s described deterministically by a theory beyond quantum mechanics.  Many others follow Bohr in insisting that the theory is complete and that our universe “really is” statistical in nature, composed of superpositions, etc. I don’t see any basis for such an interpretation.  In my view, QM is simply a limited calculation tool.  People make all sorts of assumptions about how particles exist in certain states and how wave functions collapse, when in fact these are just assumptions, not facts.  Many equations and experimental results are wildly over-interpreted beyond what they actually reveal.  We have no idea yet what “really exists” at the atomic level.  We have some operational theories that give good predictions for some experiments, but I don’t think we can say what nature really “is”.  

My future studies of QM will delve into what exactly is a “measurement”.  How does the apparatus work?  Why are they so destructive?  Some people believe that the wave function collapse is not a real phenomena, but simply an apparent one that is a result of the system under study becoming entangled with the measurement apparatus so that the system’s wave function experiences “decoherence”.  When I find out what that means, I’ll let you know,