## Pillars of Science: Summary and Questions

I've now completed my Pillars of Science series.  My goal was to analyze why Science is  such an amazingly effective method for discovering new truths about the world.  Here are the 6 "Pillars" I identified.  Of course, Science is a multifaceted word: it can refer to a method, a set of theories, or a community.  Understanding how Science works really requires thinking about all 3 together.

Intro:

A. How do we test scientific ideas?

B. What kinds of ideas can be tested scientifically?

C. Who can test them effectively?

Having laid this preparatory groundwork, in the next few weeks I'd like to get to a more exciting and controversial topic: I plan to discuss Christianity specifically in the context of each of these 6 Pillars to see how well it holds up.  (But before I get to that, I plan to post a bit about whether there are any other evidence-based ways of looking at the world, besides Science.)

You see, in this blog I am taking seriously the "What about Science?" objection to Christianity.  Many people think that the basic principles of Science are somehow refute or undercut religious views.  These are supposedly based on something called "faith" which is diametrically opposed to "evidence".  While everyone knows that some scientists are religious, many people think this is only possible because of "compartmentalized thinking" in which the two different approaches to life are somehow sealed off in different compartments so that the "evidence" compartment isn't allowed to explode the "faith" compartment.

Now those of us who practice the spiritual discipline of Undivided Looking obviously approve of UN-compartmentalized thinking, in which we think of reality as a whole, without making special exemptions for parts of life we don't want to subject to critical scrutiny.  Somewhat paradoxically, this does not require us to disapprove of compartmentalized thinking.  In certain respects Science itself is based on compartmentalized thinking (see Pillar III).

And we couldn't stop doing it even if we tried, because our brains are wired for compartmentalized thinking.  (Especially the male brain, which is more likely to delegate tasks to particular regions of the brain, whereas the female brain is more likely to think using connections between different parts of the brain.  See e.g. this study.)  But what we can and should do sometimes, is make a conscious effort to look at things together, rather than separately.

Since I'm going to be referring back to these six Pillars of Science, I'd like to ask for some reader feedback.  Do you think my discussion of these Pillars could be improved?  I'd like to solicit criticisms on any of the following issues, or anything else you can think of:

• Is there any practice which is important to Science which I have not included in the Pillars?  Or which I should have emphasized more?
• Is there anything which I've said is important for Science, which actually isn't?  Are there branches of Science which do without any of these things?
• My perspective is that of a physicist who works on fundamental issues.  But there's lots of other scientific fields: Biology, Geology, Chemistry, etc.  Do you think someone from these fields might have prioritized different aspects of scientific practice than I did?
Posted in Scientific Method | 38 Comments

## Coordinates don't matter

In my last post about spacetime, I explained how the geometry of spacetime is determined at each spacetime point by a set of 10 numbers.  These 10 numbers are packaged together into a $4 \times 4$ matrix called the metric, which is written as $g_{ab}$.  The subscripts $a$ and $b$ stand in for any of the 4 coordinate directions (in a 4-dimensional spacetime).  Since the metric is symmetric, i.e. $g_{ab} = g_{ba}$, there are 10 possible numbers in this matrix.  The value of these 10 numbers depends on your position and time,which makes them a field, specifically the gravitational field.

However, there is an important caveat in all this.  The coordinates which you use to describe a given spacetime are totally arbitrary.  For example, a flat 2-dimensional Euclidean plane can be described using Cartesian coordinates $-\infty < x < +\infty$ and $-\infty < y < +\infty$.  In this coordinate system, the distance-squared is given by the Pythagorean formula

which can be written in terms of the metric as

On the other hand, for applications involving rotations, it's often useful to use polar coordinates: $0 \le r < +\infty$ (the distance from the origin) and $0 \le \theta < 2\pi$ (the angle around the origin, measured in radians).  They're related to the original coordinate system by

In polar coordinates, the distance-squared is given by

where the extra $r^2$ factor comes in because circles that are a greater distance from the origin have a larger circumference, so there's more space as you move outwards.  This can be written in terms of the metric like this:

(Note: I've given these coordinate systems their traditional coordinate names to make them look more familiar, but this is actually just a redundancy to make it easier for humans to think about it.  I could have written the two coordinates as $(x^0, x^1)$—the superscript being a coordinate index, not an exponent—and then you could tell whether it was Cartesian or polar coordinates just by inspecting the formula for the metric.)

Now the point is, these two coordinate systems describe the same geometry in a different coordinate system.  If we were playing pool (or billiards) on a planar surface, and you wanted to describe how billiard balls bounce off of each other, you could equally well describe it using either coordinate system.  The physics would be the same.

Of course, the language you use to describe the system differs.  Suppose that I analyze a collision using Cartesian coordinates, while you use polar coordinates.  And suppose we had to communicate to each other what happened.  If you say to me, "The cue ball had a velocity in the $x^1$ direction", then I'll get confused because $x^1$ means something different to me than it does to you.  These kind of statements vary under a change of coordinate system, they are "relative" to your coordinate-perspective.  So if you want to communicate with me, you have to find a way to describe what's going on which does not refer to coordinates in any way.  For example, you could say "The cue ball hit the 3 ball, which knocked the 8 ball into a pocket."  Since the two balls and the pocket are unique physical objects, we can all agree on whether or not this happened, no matter what coordinate system we use.  These kind of statements are invariant under a change of coordinate system.  The goal of coordinate-invariant physics is to describe everything in this sort of way.

Here's another way in which coordinate systems can let you down: when you use polar coordinates, there are places where the coordinates go kind of funny.  For example, when you're going around the origin clockwise in the direction of increasing $\theta$, and you arrive at $\theta = 2\pi$, you immediately teleport back to $\theta = 0$ since you've come full circle.  Even stranger, space seems to come to an end at $r = 0$ (the origin) since there's no such thing as negative $r$.  And if you're sitting right at $r = 0$, the different values of $\theta$ all refer to the same point as each other.  However, in reality we know that nothing weird is happening to the geometry at any of these points, since nothing strange happens in Cartesian coordinates.  (A similar issue comes up in black hole physics.  The original set of coordinates found by Schwarzschild blow up at the event horizon, but actually nothing unusual happens there in classical general relativity.)

The upshot of all this for general relativity is the following: I told you above that you can describe general relativity using the metric $g_{ab}$, which involves 10 numbers at each point.  But this description actually has some redundancy in it, since there's infinitely many possible coordinates systems you could use (one for each way of labelling the points uniquely with four numbers), and the metric looks different in each one—it isn't an invariant object.

When a theory has redundancy like this, we say there is a gauge symmetry.  A regular symmetry says that two different states (i.e. configurations) of a system behave in the exact same way as each other.  A gauge symmetry is stronger than a regular symmetry: it says that the two configurations are actually the same physical state of affairs.  In general relativity, the choice of coordinates is a gauge-symmetry.  It is a mere human convention which doesn't correspond to any actual physical thing in Nature.

Of course, even if you aren't doing general relativity, you can still use whatever coordinate system you like!  Most games of billiards can be understood in the approximation where space is flat (unless you like to spice up your games with black holes and gravity waves, like the cool kids do!)  In flat space time, all coordinates are equal, but some are more equal than others.  Although nothing stops you from calculating in horrible coordinates, the laws of physics look especially simple in ordinary Minkowski coordinates, where the symmetries of spacetime look especially simple.  Since Newton's First Law of motion holds in these coordinates, we call it an inertial frame.  (Here I'm ignoring the downward pull of gravity, since in billards we're only interested in horizontal motions.)

However, if you're doing general relativity, then there's a property of spacetime which forces you to describe physics in a coordinate-invariant way; at least if you want the equations of the theory to look elegant and lovely instead of like horrendous cludge.  This property is called curvature—but we're out of time for today.

## God and Evil

Back in the comments section of my post on Giving Thanks, an old college friend and I are discussing the age-old problem of why God permits suffering and other evils.  This is a serious problem; in my view the Argument from Evil is the only really formidable positive argument for Atheism.  (By a positive argument for Atheism, I mean something that provides specific evidence against God's existence, rather than merely making the negative claim that there isn't enough evidence for Theism to believe it.  In order to show that Christianity is plausible, both claims must be addressed.)

The conundrum is famous: If God is the All-Knowing, then he knows what things are evil, if he is the All-Powerful, he should be able to prevent them, and if he is the All-Loving, then he will want to prevent evil.  So why is there evil?

The only way to solve the problem is to postulate the existence of some good thing which cannot exist unless evil either exists, or is at least possible.  (Common "defences" might refer to putative goods such as free will, the opportunity for humans to exercise virtues, the orderliness of the universe, an afterlife of a sort that depends on people having had certain experiences, etc.)  If the good is such that it is logically impossible to get it without (possibly) getting the evil too, then the defence would be successful, since when we say that God can do anything, we don't mean that he can or would create a logical contradiction.  (As C.S. Lewis says in The Problem of Pain, "Nonsense remains nonsense even when we talk it about God.")  I'm not going to attempt a detailed defence here, but I do want to make some general points about the Argument from Evil.

My first point is that God's omniscience actually makes the Argument from Evil weaker, not stronger.  The reason is that we humans are not omniscient.  If we are ignorant, there's no particular reason to assume that we know what is the morally best way to run a world.  Suppose that you wrote down a list of all the things you regard as good (happiness, knowledge, beauty, whatever).  Suppose you figured out a way to weight all of these factors numerically—of course, there's no way we could ever agree on how to do this, and I'm not convinced it even makes sense, but let's run with it—so that you could assert that some possible kind of universe (call it $U$) is optimum: the best possible.

[Note for experts: my kinds of universes $U$ here aren't exactly the same as the "possible worlds" discussed by analytic philosophers.  If the best possible kind of universe contains something like free will or nondeterminism, there will be multiple "possible worlds" $W_1, W_2 \ldots$ consistent with the same overall plan $U$ of the universe, some of which may be morally better or worse compared to the others.]

Now if God knows about even a single kind of goodness that we are ignorant of, or if he weights the various kinds of goodness differently than us in any way, then of course God will view some other kind of universe $U^\prime$ as best.  It seems infinitely unlikely that $U = U^\prime$ just by coincidence, so it seems to be almost certain that the universe will appear to us to contain evils that we can't explain.  One can argue about whether this is a sufficient explanation, but it's definitely something that has to be taken into account.  The idea that a superhuman entity which created the universe will see things exactly the way we do is absurd:

“For my thoughts are not your thoughts,
neither are your ways my ways,”
declares the Lord.
“As the heavens are higher than the earth,
so are my ways higher than your ways
and my thoughts than your thoughts.”
(Isaiah 55:8-9)

The second point I'd like to make, is that the Argument from Evil has emotional force as well as intellectual force.  Atheists tend to get annoyed when Theists suggest that Atheists don't believe in God because they resent him.  I've certainly seen plausible cases of this, but I don't want to speculate that all Atheists are this way, since I don't like making unfounded accusations about individual people's characters.  (Maybe that's why my Politics category only has one post in it so far.)

Nevertheless, leaving the Atheists aside for a moment, I think I can say from an examination of my own heart, and conversations with other people, that it's easy to carry an unconscious grudge against God for various real or imagined grievances in our lives, or the lives of those we care about.  Even if we have no grudge, there can be a deep sense of pain from all the kinds of grief that we don't understand.

So the Argument from Evil carries emotional force as well as intellectual force.  There's no necessary reason why an intellectually satisfying answer should be an emotionally satisfying answer, or vice versa.  One should bear this in mind when evaluating the intellectual arguments, since we may be asking from an argument something that no argument can do.

Finally, I believe that Christianity has resources for addressing the Argument from Evil which don't exist in generic-brand Theism, or indeed in any other religion.  It's much too simple to say that the existence of evil contradicts Christianity, when in fact the most basic doctrine of Christianity logically implies the existence of evil.

The basic doctrine is that 1) we human beings are wicked and deserve punishment, and that 2) in order to forgive us, God became an innocent human being and allowed himself to be tortured to death by us, and that 3) this act provides us with spiritual healing now, as well as physical immortality for all eternity.  Now regardless of whether you like this idea, even if you find it implausible or downright incomprehensible, you must admit that it's an idea about how God relates to evil, and uses it for the sake of good.  If there were no such thing as innocent suffering, Christianity wouldn't even be possible.  If Christianity is true, then God has arranged things so that the most important thing that ever happened was a horrible but redemptive evil.  All other evils, we view in the light of the Cross.

Posted in Theology | 27 Comments

## A Universe from Nothing?

Today I went to a talk by Lawrence Krauss entitled “A Universe from Nothing”, which had the following abstract:

The question, "Why is there something rather than nothing?" has been asked for millenia by people who speculate on the need for a creator of our Universe.  Today, exciting scientific advances provide new insight into this cosmological mystery: Not only can something arise from nothing, something will always arise from nothing.  Lawrence Krauss will present a mind-bending trip back to the beginning of the beginning and the end of the end, reviewing the remarkable developments in cosmology and particle physics over the past 20 years that have revolutionized our picture of the origin of the universe, and of its future, and which have literally revolutionized our understanding of both nothing, and something.  In the process, it has become clear that not only can our universe naturally arise from nothing, without supernatural shenanigans, but that it probably did.

In the first 45 minutes, he provided an animated and reasonably clear explanation of concordance cosmology, the current version of the Big Bang model, dating from the discovery in 1998 that the expansion of the universe is accelerating (rather than decelerating as one would expect from the attractive gravity of ordinary matter).  This is exciting but now well-established work, which I've heard about a hundred times before, but was probably new to many of the people in the audience.  It was peppered with occasional off-hand snears at Republicans, Theology, and Young Earth Creationism, but for the most part it was a pretty valiant stab at popularizing an important set of 20th century discoveries.

The real reason I was there, of course, was to listen to his claims in the last 15 minutes that modern cosmology somehow points to the nonexistence of a Creator.  His claim was that there is evidence that the universe came from "Nothing" according to physical processes, and this apparently is supposed to undermine the religious view that God created the world supernaturally.  There were so many things wrong with this part of his talk, both a physics and a philosophical perspective, that I'm not entirely sure where to begin.  But let's try anyway.

His Slam on Theology.  Krauss said that Theology wasn't based on empirical evidence, so therefore he didn't believe it.  That was it.  He didn't seem to take any particular theological ideas seriously enough to even try to define them, let alone refute them.  There was no indication that Religion had any other origin besides a bunch of clueless dudes sitting around asking "Why is there Something rather than Nothing?"  (In the case of Christianity, I thought it had more to do with a guy claiming to be God, doing miracles, and dozens of people saying that they saw him alive after he was killed.  But what do I know?)

But let's get back to cosmology, since that was the subject of his talk.  It used to be that Christians believed that the world was created a finite time ago, out of Nothing.  Although some of them, like St. Thomas Aquinas, said that God could have created a universe with an infinitely long past.  Atheists had (and have) a diversity of opinions, but most of them thought that things would make more sense if the universe were around forever, since then maybe you wouldn't have to explain where it came from.  Then Big Bang cosmology came along, and it now seems—provisionally speaking—like the Universe really did have a beginning.  Now some atheists think they can refute the Christian view that God created the Universe from Nothing by arguing that the world did emerge from Nothing.  The role-reversal here is a little strange.

The universe can only come from nothing if you define a certain kind of something as being "Nothing".  Duh, because any explanation by its very nature must explain one thing in terms of some other thing!  This other thing must be taken for granted for purposes of the explanation.  Now, Krauss actually referred to 3 different ideas which he called "Nothing #1, #2, and #3":

Nothing #1: an "empty" spacetime a.k.a. the vacuum.  In ordinary non-speculative quantum field theory (QFT), the "vacuum state" (the configuration of fields with the lowest energy) is actually filled with so-called virtual particles which can affect physics in various ways.  At least, that's what the popularized physics books say; if one actually studies quantum field theory rigorously, people tend to use somewhat different language since the notion of "virtual particle" can be difficult to define.  But let's spot him the terminology since he was talking to a popular audience.

Krauss claimed that if you start with an empty space which has no virtual particles in it, virtual particles will appear, and this is "something" coming from "nothing".  This is bosh, since strictly speaking, there's no such thing in QFT as a state with no virtual particles.  (If there were, it would be infinitely different from the vacuum state, and would therefore have an infinitely large energy.  That's not nothing at all!)  If anything can colloquially be called "Nothing" in QFT, it is the vacuum state.  But this state already has all those virtual particles in it.  And as time passes, this vacuum evolves to....wait for it....itself!  That's right, if you agree to call the vacuum state Nothing, then Nothing comes out of it.  (He seemed to think this story might change once you take gravity into account, due to negative energies, but I didn't really understand this suggestion so I won't comment on it.)

The QFT vacuum isn't nothing.  Of course, from a strict philosophical point of view, the vacuum state of QFT is not Nothing since it's filled with all those virtual particles, and even aside from that, there's the space and time geometry, which is not Nothing.  To fix this he started taking up a different kind of nothing:

Nothing #2: the absence of any space or time.  This actually connects to an interesting quantum gravity idea known as the "Hartle-Hawking state" or the "no-boundary boundary condition".   (Jim Hartle is on my floor at UCSB, by the way.)  The suggestion is that the laws of physics not only tell you how the universe at one time evolves to a later time, they also tell you what the initial state of the universe is.

In some sense, one can think of this state as emerging out of Nothing #2.  However, the sense in which this is true is subtle.  There's another sense in which the Hartle-Hawking state does not emerge from Nothing; rather it has existed for an infinite amount of time— the popular physics articles never mention this, for some reason!  This is an interesting and important idea, but I think it deserves to be in it's own post, after I've explained QFT better.  The important thing to know is the following:

The crucial physics here is totally speculative!  It was entirely based on speculative ideas about quantum gravity which anyone working in the field would admit are not proven.  This is because we currently have no experimentally testable theory of quantum gravity!  (Nor do we even know how to formulate a consistent theory of quantum gravity mathematically, except perhaps in some special situations that probably don't apply to the beginning of our universe)

I mentioned this in the Q&A afterwards.  My comment seemed to aggravate him a little, since he thought he'd been sufficiently clear about this.  But I discovered that at least one member of the audience was still unclear on which parts were speculative, and which weren't, at the end of the lecture.  In my experience, one has to be crystal clear about this sort of thing when speaking to a popular audience, or they tend to walk away thinking that "Science" has proven things when it hasn't.

Atheists such as Krauss scorn theology as being completely non-empirical.  They claim it is not based on evidence of any sort.  I find it extremely ironic when this sort of atheist thinks that speculative quantum gravity ideas are just the right thing to further bolster their atheism.  Suppose you think that Science is better than Religion because it is based on evidence, and suppose you also want to refute Religion by using Science.  Here's a little hint: consistency would suggest using a branch of Science that actually has some experimental data!

The universe has zero energy.  Krauss thinks that the universe coming out of Nothing has been made more plausible by cosmology.  To understand his terminology, you have to know that roughly speaking) a closed universe means that space at one time is finite in volume, and shaped kind of like a sphere, so that if you travel around the universe far enough you come back to where you started.  On the other hand, in a flat universe, space at one moment of time is shaped like ordinary Euclidean geometry, and is infinitely large.  Current observations indicate that the universe is flat.  As far as I could tell, Krauss' argument can be translated into these terms:

1. The total energy of a closed universe is zero.  (It's tricky to define energy in general relativity, but according to one commonly used definition, this is true.)
2. Conservation of energy suggests that if the universe came from Nothing, it should have zero energy.
3. If there was a period of extremely rapid expansion at the beginning of the universe (as evidence suggests there was—this is called inflation), then whether or not the universe started out closed, it should look flat today.
4. But the universe does look flat,
5. Therefore Science suggests that the universe was created out of Nothing,
6. Therefore there is no need for God.

Perhaps I'm missing some crucial steps in his argument.  But there seem to be several enormous leaps of logic in there.

The Hartle-Hawking state isn't Nothing either.  Strictly speaking, even the Hartle-Hawking idea doesn't strictly get the universe out of Nothing, since it says that the initial state of the universe depends on the laws of physics.  Now the laws of physics aren't nothing.  So if, for example, you are wondering if there is any role left for the Creator, then one might say he picked the laws of nature.

Now, there's all sorts of difficult philosophical issues involved in what's called the Cosmological Argument for the existence of God.  But it's hard to get into them with someone like Krauss who is so dismissive of Philosophy.  The trouble with people like that is that it isn't possible to just find things out using Science instead of Philosophy.  That's because you have to do Philosophy to know what is or is not implied by Science.  People who dismiss Philosophy still end up doing it; they just do it badly, without a critical examination of their premises.

Nothing #3: the string theory multiverse.  Krauss acknowledges that the laws of phyiscs themselves might seem to call  for an explanation.  Especially since the various constants of Nature seem to be "fine-tuned" to allow the existence of life (I'll go into this in much more depth later).  On the face of it, this seems to be at least some mild evidence for the existence of God, but Krauss would never admit such a thing.

He suggests that we can explain this fine-tuning if string theory turns out to be true.  That's because string theory has an enormous number of different possible configurations, that look like universes with different laws of physics.  Some people have suggested that if there's a gazillion different universes (known as the "multiverse"), each with its own laws of physics, that it's not surprising that one of those universes should support life.  Krauss admitted that there was some dispute as to whether this idea counts as "Science", what with it being totally speculative and arguably untestable.  But what I want to know is, why the \$@#& would we ever refer to an infinite number of universes, governed by the principles of string theory, as a Nothing?

I should say that this review is based entirely on Krauss' talk.  I have not read his book, but I have read this negative review by philosopher St. Feser.

Posted in Reviews | 2 Comments

## Pillar of Science VI: Community Examination

Scientific Results are Examined Collaboratively.

Scientists do not work alone, but in a particular kind of community.  The last stage of a research project is publishing and explaining the results.  Assuming these results get noticed, this begins the process of further review, critique, confirmation and rebuttal by other scientists.  No one person is smart enough to see things from all angles.  We need help from others to look in a clearer, less fragmented way.  Perhaps one could call this undivided looking?

Science is not just a set of facts, or an abstract procedure for testing ideas.  It is an ethical, truth-seeking community.  The love of truth is embodied in the alliance of particular, fallible humans, united by a common geeky interest in finding something out.  Together we create a public deposit of information which can be used to find new things out.

Because the community as a whole is truth-seeking, in the long run it reduces the need to trust the competency and ethics of the original researchers.  If someone fakes an experiment (or else just makes an innocent mistake), other people will be unable to replicate the result, and eventually the truth will come out.

Healthy scientific collaboration encourages reasonable dissent.   Otherwise group-think can insulate the community from effective criticism of accepted ideas.  Some people say that scientists should proportion their beliefs to the evidence.  However, there's also some value in diversity of opinion, because it permits subgroups to work on unpopular hypotheses.  I suppose things work best when the scientific community taken as a whole proportions its research work to the evidence.

One might argue that collaboration is not strictly necessary to Science.  Imagine a solo scientist doing careful experiments in secret, and drawing the correct conclusions from them.  (Even in this case the scientist would be drawing on public ideas which had gone before, "standing on the shoulders of giants", as the saying goes.)  But in practice, the benefits of discussion are so great that it's hard to imagine a successful modern scientist working completely alone.  Hence the symbiosis of Science with the Academy.

Individuals who think they can revolutionize Science all by themselves are almost always crackpots, the sort of crazy person I described one pillar ago.  If you want to see clearly, you have to expose yourself to the light.

Posted in Scientific Method | 1 Comment

## Geometry is a Field

In Time as the Fourth Dimension?, I explained how to calculate the distance (or duration) squared between any two points of spacetime, using a spin-off of the Pythagorean theorem:

Then I explained the Ten Symmetries of Spacetime, i.e. ways to shift or rotate the coordinate system $(t,\,x,\,y,\,z)$ that don't change the formula for $s^2$.

Well, it turns out that I lied.  The formula isn't actually true, except in the special case that there is nothing in the universe.  A significant reservation, I know.  Instead, what's true is that the geometry of space is a field, meaning that it varies from place to place, depending on where you are!  However, if you zoom in really close at any particular point, it looks similar to the formula I told you.

The field that says what geometry is like at any given place and time is called (brace yourself) the gravitational field.  In order to describe it, we use something called the metric, which indicates what the geometry of spacetime looks like at any given point.  The way this works is, suppose we have two points $p$ and $q$ which are very close to each other.  Suppose we want to know the distance between these points.

Since the points are really close to each other, we call the distance between them $ds$, where the $d$ is just a reminder that we're using Calculus to study infinitesimal quantities.  If you don't know Calculus, just pretend these are really small numbers.  We want to figure out what $ds$ is, if we know the infinitesimal coordinate differences $(dx,\, dy,\,dz,\,dt)$.  The way we do this is by generalizing the heck out of the Pythagorean theorem.  I'll write it down, and then explain what it means:

The right-hand side of the equation consists of every possible way of multiplying two of the coordinate distances $(dx,\, dy,\,dz,\,dt)$.  There are 4 different ways to pick the first $(dx,\, dy,\,dz,\,dt)$, and 4 different ways to pick the second, which gives $4 \times 4 = 16$ possible combinations in all.  However, multiplication is commutative so e.g. $dx\,dy = dy\,dx$.  So I added terms like that together; that's where the factor of 2 came from.  Taking that into account, there's 10 terms in all.

The funny $g$ things with subscripts are just functions of spacetime, i.e. they are just numbers that depend on where you are, i.e. they are fields.  In the special case where we pick these numbers to be $g_{xx} = g_{yy} = g_{zz} = +1,\,g_{tt} = -1$ and the rest zero, we get the geometry I told you about, which goes by the aliases "Minkowski space", "flat spacetime", and "Special Relativity".  In all other cases we have what is colloquially called "curved spacetime" which is the province of "General Relativity".

The formula above looks kind of ugly, but we can prettify it by choosing good notation.  We collectively refer to all ten of these gravitational fields as the metric, denoted $g_{ab}$, where subscripts like $a$ and $b$ can refer to any of the four coordinate labels.  (People often call these labels $(0,\,1,\,2,\,3)$ instead of $(x,\,y,\,z,\,t)$ to avoid confusion, since the metric itself says which of the coordinate directions behave more like space, and which behave more like time, and this can vary from place to place!)  Then we write the four coordinate differences $(dx,\, dy,\,dz,\,dt)$ collectively as $dx^a$, where the superscript says which of the four it is.  Finally, we make up a rule called the Einstein summation convention, that if we ever see the same letter as both a subscript and as a superscript, we add up all of the four possible ways for them to be the same (i.e. both 0, both 1, both 2, or both 3).  These are just changes in how we write things, not substantive changes, but they let us rewrite that long ugly equation like this:

There, isn't that much prettier?

Suppose we want to find the distance (or duration) between two points which are NOT infinitesimally close to each other.  In that case, we have to choose a path between the two points, since the amount of distance (or duration) depends on which path you choose, and in a curved spacetime there's not necessarily one "best" path.  This shouldn't seem that strange, since even in everyday life we know perfectly well that the distance between San Francisco and L.A. depends on which highway you take, and the distance between Tokyo and New York depends on which way around the globe you fly.  (It's totally intuitive for distances, but when the duration depends on the route you take through spacetime, people call it the Twin Paradox and act all shocked!)

So this is the first main idea of General Relativity: the geometry of spacetime is a field which varies from place to place.  This field affects matter by determining the paths that things take through space and time, but it also is affected by matter—we call this gravity.  The second main idea is that coordinates are an arbitrary choice; I'll tell you about this later.  The third main idea is the Einstein equation which says how matter affects the metric.  I haven't told you anything about this equation yet, but once I do, you would in principle be able to calculate everything about the gravitational field from that one equation.

There can also be distortions of the spacetime geometry which exist independently of matter.  These gravity waves are to gravity what light is to electromagnetism, ripples in the field which travel through empty space, and can be emitted and absorbed.  The propagation of these waves is also determined by the Einstein equation.  Since gravity comes from massive objects, gravity waves are emitted when extremely large masses oscillate, for example when two neutron stars orbit each other.  We know gravity waves are there, but we haven't detected them directly.  However, we hope to detect them soon with the LIGO experiment.

UPDATE: I realized that I never said how you would calculate the distance between two points, once you choose a path.  The answer is that you chop the path into lots of tiny little line segments, and find the length of each line segment using the metric.  Then you add them all up.  If you know Calculus, this can be done using an integral.

Posted in Physics | 2 Comments

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## Giving Thanks

Today is Thanksgiving Day (in the United States), a day set aside for us all to remember the things in life we are grateful for.   Fortunately, Nicole and I started the process of gratitude earlier this week when we finished writing and mailing our thank-you notes for the useful and beautiful presents we got by agreeing to spend the rest of our lives together.

All of us have been supported by other people in many ways, or we would not be alive.  All of us should be grateful more often for those things.  Those of us who believe in God have the privilege of also having someone to thank for the blessings of life that don't come from other human beings, such as the sun and moon, stars and trees, happy coincidences, good health and harvests, etc.  Even when the good things come from other people, we can still accept it as ultimately coming from the hand of God, who has, after all, provided those other people with the ability and conscience to help us.

But what about when bad things happen?  Is it consistent to attribute everything good that happens to God, but then turn around and say that God has no responsibility for any of the bad things that happen?  Should we blame God for the bad as we praise him for the good? If religious folk thank God even for the indirect results of God's providence, that are mediated through human choices, why should we not take the same attitude for bad things caused indirectly by God?

Some people say: God does not cause evil, he only permits it.  This idea can be comforting to those who have suffered greatly, because then they don't have to deal with resentment towards a God who inflicts suffering as well as joy.  Others may find this a pedantic distinction, saying that God is equally responsible for the evils he permits.

The Bible, on the other hand, doesn't seem to refrain from attributing sorrowful events to God:

When disaster comes to a city, has not the Lord caused it?  (Amos 3:6)

When the evil comes from other people, this is in one sense a violation of God's will, who has most definitely commanded us to love our neighbors (Lev. 19:18), strangers (Lev. 19:34), and enemies (Ex. 23:4-5, Prov. 24:17-18, 25:21), and who has set a day of judgement in which wrongdoers will be punished.  When a woman is raped, this horrible crime arises not because God approves of rape, but because God allows the will of wicked men to affect other people.

Nevertheless, God does allow it, and the Bible is not shy about describing such things as being (in another sense) God's decision and will.  When the righteous St. Job loses everything, including his children, to a combination of "natural" disasters and bandit attacks, what does he do?

At this, Job got up and tore his robe and shaved his head. Then he fell to the ground in worship and said:

“Naked I came from my mother’s womb,
and naked I will depart.
The Lord gave and the Lord has taken away;
may the name of the Lord be praised.”

In all this, Job did not sin by charging God with wrongdoing.  (Job 1:20-22)

Job attributes the disaster to God's "taking away", but he does not blame God by charging him with "wrongdoing".  What gives?  How is it possible for God to do something evil without being evil?  The key is what the patriarch St. Joseph says to his brothers, when he forgives them after they had sold him into slavery:

“Don’t be afraid.  Am I in the place of God?  You intended to harm me, but God intended it for good to accomplish what is now being done, the saving of many lives.” (Genesis 50:20)

One and the same act can be both evil and good, depending on whose intentions we consider.  The selling of Joseph into slavery is evil as done by his brothers, because they intended to harm him.  It is good as done by God, because God's intentions were different: God did it in order to save lives.  (I am not trying to make any comment about free will here; presumably if Joseph's brothers had freely chosen not to sell him into slavery, then God would also have chosen something different.)  Thus God can condemn what people do, while simultaneously using it for his good plan.

That must be why, after the Apostles were flogged for teaching about Jesus, they were

The apostles left the Sanhedrin, rejoicing because they had been counted worthy of suffering disgrace for the Name.  (Acts 5:41)

Why on earth did they take this attitude?  St. Paul explains it like this:

We also rejoice in our sufferings, because we know that suffering produces perseverance; perseverance, character; and character, hope.  And hope does not disappoint us, because God has poured out his love into our hearts by the Holy Spirit, whom he has given us.  (Romans 5:3-5)

And we know that in all things God works for the good of those who love him, who have been called according to his purpose.  (Romans 8:28)

The idea that God does not cause bad things to happen is a superficial teaching.  It evades the cross and forgets the gospel message that we are to rejoice and thank God for everything that happens to us.  That is why St. James tells us to

Consider it pure joy, my brothers, whenever you face trials of many kinds, because you know that the testing of your faith develops perseverance.  Perseverance must finish its work so that you may be mature and complete, not lacking anything.(James 1:2-4)

But perhaps James was just copying his brother's idea:

“Blessed are you when people insult you, persecute you and falsely say all kinds of evil against you because of me.  Rejoice and be glad, because great is your reward in heaven, for in the same way they persecuted the prophets who were before you.” (Matt. 5:11-12)

In conclusion, it doesn't make any sense to thank God for the good things in life and absolve him for the bad things.  No, we should also credit the bad things to God, and give thanks for them as well.  Not because the evil is imaginary, but because he intends to use it to build us up into more loving people, for the sake of the salvation of the world.

But perhaps your last year was actually quite pleasant, as mine was.  In that case, let's not forget to thank him for the obvious blessings as well.

Posted in Theology | 7 Comments

## Pillar of Science V: Ethical Integrity

Scientists must have Integrity.

Because Science involves an ethical principle, the love of truth, its practice cannot be unmoored from principles of morality.  A hypothesis can only be put to a fair test by a person who prefers knowing the truth even if it shows that their previous position was wrong (a corollary is that science becomes unreliable when there is political pressure to come to particular conclusions, such as the Lysenkoist biology mandated by Stalin, or the Deutschephysik of Nazi Germany).

This virtue is sometimes referred to as “objectivity”, but this word suggests a sort of dispassionate neutrality which is not actually characteristic of most real scientists—we actually tend to get rather excited about our work, or we wouldn't be doing it.  A better term for this virtue is humility: when doing research the scientist must take the posture towards the universe of a learner, rather than a teacher.  Unfortunately, some effective scientists are conceited and arrogant towards their peers, but when scientists take the same attitude towards Nature, they continue to defend ideas long after they become discredited, and become useless to Science.

It's also extremely common for working scientists to get mailings from laypersons who believe themselves to have revolutionized large areas of science, despite having imprecise, untested, and often meaningless ideas.  This psychotic disconnect from reality is nearly always accompanied by severe egoism, showing by contrast the way that humility characterizes true science.

Of course, humility does not involve taking the view that all knowledge is unreliable and tentative, since this would actually inhibit the discovery of truth!  (In modern times, revolutions in Science usually do not totally invalidate our previous understandings; instead the previous theories survive as approximations.)  The proper attitude of a learner is: “Test everything; hold onto what is good” (1 Thessalonians 5:21).

A second virtue of science is honesty.  Scientists must refrain from fudging results or misleading other scientists.  Honesty requires noting the factors weighing against a conclusion as well as those weighing for it.  They must also take precautions against bias, not in the sense of being unbiased (none of us are), but preventing that bias from contaminating their results.  Hence the need for experimentalists to do proper error analysis, use control groups, double-blind tests, etc.  Experiments that show the absence of an effect should be published as well as experiments that show the presence of an effect, even if such results are less likely to result in fame and respect.

I was going to include some juicy long excerpt from Feynman's famous commencement address on "Cargo Cult Science".  But too much of it was relevant to what I'm saying!  You should just go and read the whole thing.

## The Numinous

A few weeks ago I started to describe what holiness means, and someone requested that I go into more detail.

One way to approach this is through the concept of the numinous, described in the classic work The Idea of the Holy by the Blessed Rudolf Otto.  This book was a significant influence on St. Lewis, who discusses the numinous especially in his introduction to The Problem of Pain.  The concept of the numinous is difficult to explain because most of the language we use to describe it has come to mean other things.  In English, the words "awesome" and "awful" both used to mean the same thing: the feeling of dread, wonder, uncanniness, terror, or reverence one gets in the presence of something you believe to be eerie or supernatural.  As Lewis points out, we use the same word "afraid" when we say that someone is a jungle is "afraid of tigers" as that someone in a haunted house is "afraid of ghosts".  But in the first case, the fear is just for our own safety, whereas in the case of ghosts one is afraid of what the ghost IS, more than what it will do to you.

Please note, I am not claiming that ghosts exist, but rather using them as an example to make a point about human psychology.  Just as we have a sexual instinct which responds to sexual stimuli, so we have another instinct which responds when we believe we are encountering supernatural stimuli.  The hairs stand up on the back of our neck and we feel chilly.  In that sense it feels like fear, even though the experience may be pleasant or unpleasant, and we may or may not be concerned for our physical safety.  Atheists, pagans, and Christians all experience this feeling on certain occasions; the difference is how they interpret it.

In our own minds, we can feel numinous feelings without making any connection to ethical concepts; a pagan or a pantheist may feel that they are worshiping a Spirit which is beyond human notions of good or evil.  However, when ethical concepts do intrude, a special composite feeling arises.  In the case where the object is perceived as Numinous Evil, we call the feeling that arises in us Horror.  This feeling can be excited by natural objects which seem "eerie" such as corpses or creepy insects.  (Lewis claims that there is no survival advantage in this feeling, but it seems to me that avoiding diseased corpses and dangerous insects may well have evolutionary advantage.)  It can also be excited when we read or watch movies about vampires, werewolves, demons etc.  (This assumes that the movies treat the topic seriously, of course.  Monsters that think and act just like regular people are humorous, since we expected a numinous thrill and then it was a false alarm).

When the object is perceived as Numinous Good, this composite idea is nothing other than the Holy.  (Unfortunately, there's a lack of grammatical parallelism here, in that "Holy" refers to the Object about which we have numinous feelings, whereas "Horror" refers to the feelings themselves.)  The distinctive characteristic of holiness is that ethics itself becomes imbued with supernatural significance.  This experience is not always happy.  As the classic example, consider Isaiah chapter 6:

In the year that King Uzziah died, I saw the Lord seated on a throne, high and exalted, and the train of his robe filled the temple. Above him were seraphs [burning ones], each with six wings: With two wings they covered their faces, with two they covered their feet, and with two they were flying. And they were calling to one another:

“Holy, holy, holy is YHWH of hosts;
the whole earth is full of his glory.”

At the sound of their voices the doorposts and thresholds shook and the temple was filled with smoke.

“Woe to me!” I cried. “I am ruined! For I am a man of unclean lips, and I live among a people of unclean lips, and my eyes have seen the King, YHWH of hosts.”

Then one of the seraphs flew to me with a live coal in his hand, which he had taken with tongs from the altar. With it he touched my mouth and said, “See, this has touched your lips; your guilt is taken away and your sin atoned for.”

Then I heard the voice of the Lord saying, “Whom shall I send? And who will go for us?”  And I said, “Here am I. Send me!”

If you are a nonreligious person, I hope you tried to read that as you would some passage in a fantasy novel, with "suspension of disbelief".  Put aside how you feel about Christianity in general, and just ask how this passage makes you feel, as if it were a fictional work of art.

Doubtless Isaiah knew beforehand that he had ethical shortcomings; perhaps he lied, or berated someone.  But before, it was a matter of merely personal regret, excused by the fact that everyone does it.  In the presence of this astounding vision, his guilt becomes something completely different: a feeling of uncleanness and shrinking before a majestic purity, that even the angels had to hide their faces from.  It was like coming into a formal dinner party stinking, and wearing no clothes at all.

This is a numinous problem, not just an ethical problem.  So it needs a numinous solution.  The coal from the altar makes "atonement" for Isaiah's uncleanness.  That is, it allows Isaiah to become a participant in the numinous, in a way that covers up or removes his guilt.  Only then can St. Isaiah hear God's call to be a prophet, denouncing the sins of others.

It's a mistake to try to argue that Christianity is true before the audience knows what Christianity is.  Before people can understand Christianity, they have to understand the basic concepts in which it is expressed.  Without the concept of holiness, nothing we say about God deserving worship, or about Jesus dying on the cross for our sins, or about love requiring purity—none of it makes any sense at all!

## Firewalls

There's been a huge kerfuffle in the quantum gravity community since this summer, when some people here at UCSB published a paper arguing that (old enough) black holes may actually be surrounded by a wall of fire which burns people up when they cross the event horizon.  This is huge, because if it were true it would upset everything we thought we knew about black holes.

General relativity is our best theory of gravity to date, discovered by Einstein.  This is a  classical theory.  (In the secret code that we physicists use, classical is our code-word for "doesn't take into account quantum mechanics".  Don't tell anyone I told you.)

In my other posts on physics, I've been trying to explain the fundamentals of physics in the minimum number of blog posts.  This post is out of sequence, since I haven't described general relativity yet!  But I wanted to say something about exciting current events.

In classical general relativity, a black hole is a region of space where the gravity is so strong that not even light can escape.  They tend to form at the center of galaxies, and from the collapse of sufficiently large stars when they run out of fuel to hold them up.  A black hole has an event horizon, which is the surface beyond which if you fall in, you can't ever escape without travelling faster than light. The information of anything falling into the black hole is lost forever, at least in classical physics.

In the case of a non-rotating black hole, without anything falling into it, the event horizon is a perfect sphere.  (If the black hole is rotating, it bulges out at the equator.)  If you fall past the event horizon, you will inevitably fall towards the center, just as in ordinary places you inevitably move towards the future.

At the center is the singularity.  As you approach the singularity, you get stretched out infinitely in one direction of space, and squashed to zero size in the other two directions of space, and then at the singularity time comes to an end!  Actually, just before time comes to an end, we know that the theory is wrong, since things get compressed to such tiny distances that we really ought to take quantum mechanics into account.  Since we don't have a satisfactory theory of quantum gravity yet, we don't really know for sure what happens.

Now it's important to realize that the event horizon is not a physical object.  Nothing strange happens there.  It's just an imaginary line between the place where you can get out by accelerating really hard, and the place where you can never get out.  Someone falling into the black hole just sees a vacuum.  If the black hole was formed from the collapse of a star, the matter from the star quickly falls into the singularity and disappears.  The black hole is empty inside, except for the gravitational field itself.

We don't know how to describe full-blown quantum gravity, but we have something called semiclassical gravity which is supposed to work well when the gravitational effects of the quantum fields are small.  In semiclassical gravity, one finds that black holes slowly lose energy from thermal "Hawking" radiation.  This radiation looks exactly like the random "blackbody radiation" coming from an ordinary object when you heat it up. Here's the important fact: You can prove that the radiation is thermal (i.e. random) just using the fact that someone falling across the horizon sees a vacuum (i.e. empty space) there.

The Hawking radiation comes from just outside the event horizon.  It does not come from inside the black hole, so in Hawking's original calculation it doesn't carry any information out from the inside.   Nevertheless, for various reasons I can't go into right now, most black hole physicists have convinced themselves that the information eventually does come out.

As the black hole radiates into space, it slowly evaporates, and eventually probably disappears entirely (although knowing what happens at the very end requires full-blown quantum gravity).  If the outgoing Hawking radiation carries all the radiation out, then for a black hole at a late enough stage in its evaporation, the radiation must not be completely random, because it actually encodes all the information about what fell in.

The gist of what Almheiri, Marolf, Polchinski, and Sully argued, is that if we take both of these statements in bold seriously, then it follows that the black holes are NOT in the vacuum state from the perspective of someone who falls in.  Instead you would get incinerated by a "firewall" as you cross the horizon.  (It's not clear yet whether this is only for really old black holes, or if it applies to younger ones too.)  That's if we still believe there is an "inside" at all.  The argument shows that semiclassical gravity is completely wrong in situations where we would have expected it to work great.

If this is right, then it's devastating to the ideas of many of us who have been thinking about black holes for a long time.  As a reluctant convert to the idea that information is not lost, I'm wondering if I should reconsider.  At the end of this month, I'm going to Stanford for a weekend, since Lenny Susskind has invited a bunch of us to try to get this worked out.  Exciting times!

Posted in Physics | 6 Comments

## Pillar of Science IV: Precise Descriptions

Science involves Precise Description.

To be capable of being confirmed or ruled out at the high levels of reliability associated with Science, a hypothesis must be stated in a way which is precise enough to do definitive tests.  Even if a scientific hypothesis may not be experimentally testable at the present time, a precise formulation helps indicate ways that it could be tested in the future.  If experiment is capable of making everyone eventually agree on whether the idea worked or not, then the words it is expressed in shouldn't mean different things to different people.

Mathematical models of the physical world are the most precise form of description available, because they can describe complicated systems with perfect exactitude.  In theoretical physics, this kind of quantitative description is the usual way to make things precise.  We like to think about systems that are simple enough to describe mathematically (of course, this requires first making certain approximations).  In fact, fundamental physics is so mathematical that, even when there are no or few experiments, one can often make progress just by demanding that the model be logically consistent, and that it conform to known physical principles.  (Known, because they have been tested in other situations where we can do experiments.)  Mathematical consistency is nearly our only guide in speculative fields like my own (quantum gravity); however, it cannot completely substitute for observations, since no matter how consistent or beautiful your model is, Nature could always do something else when you finally are able to take a look.

So Math is great when you can get it.  Nevertheless, systems which are less regular, more complex, or less well-understood (such as biological life) cannot always be described mathematically, but may still be described through technical vocabulary that minimizes imprecision, without removing it altogether.  I'm not a biologist, so I'm probably not the best person to ask how this usually works, but I didn't want to give the impression that math is the only way to make ideas precise.  Even in physics, at one time it was possible to describe everything in words.  The great experimenter St. Faraday (whose work helped established the concept of the electromagnetic field), once wrote a letter to St. Maxwell (who wrote down the equations for electromagnetism) expressing surprise at Maxwell's need to translate everything into mathematical equations.  Yet no one could accuse Faraday's journals of being imprecise.

But not all concepts will do.  Ideas that are apprehended in words or images rich with heavy associations or mottled with variegated meanings—in short, using the common language of humanity—such ideas are excluded from Science.  Not because it is impossible to discuss and test these ideas; if that were true, then it would be impossible to think accurately at all about most matters of ordinary human concern.  Rather, it is because they involve elements of human and holistic judgement which are unsuitable for scientific inquiry.  The question “Is xenophobia a frequent cause of war?” could be given an informed and accurate answer by a historian, but it does not become a scientific question until the terms “xenophobia”, “frequent”, and “war” are given technical meanings sufficiently precise that a social scientist can do a statistical analysis.

## Medieval Bashing

Recently I ran across a pretty good explanation of the Higgs mechanism (hat tip Siris) by a certain Rob Knopp, which I thought I'd link to because of its connection with my previous post on fields.  When I first looked at his blog, it seemed like maybe I'd found a kindred spirit: someone who blogs on science while identifying as Christian.  Unfortunately, it turns out he actually denies almost all traditional Christian beliefs.  On this blog, religion won't mean something watered down until it makes few if any factual claims.  Instead I adhere to the red-hot supernaturalist "original brand" of Christianity that includes real miracles, a divine Incarnation, Atonement, inspired Scripture (including the Old Testament), the Second Coming, etc.  Just in case you were wondering.

[Jesus] lived an errant life, eschewing the temporal power that monarchs would later claim was justified by the "divine right of kings," even eschewing extreme temporal influence.  He preached the opposite of a lot of medieval social philosophy: the poor are not lesser humans and thus worthy of their lot, but if anything exalted. (Blessed are the poor, and all of that.)

Assuming that this was meant as a criticism of Medieval Europe (and not, say, the caste system of Medieval India), this is almost as far off base as if he said that Medievals had landed on the Moon.  The Medievals did not believe that the poor are "lesser humans and thus worthy of their lot".  It would be much more accurate to say that they believed that the poor were superior humans and that poverty is, not indeed strictly necessary for salvation, but highly desirable for anyone wanting to live more spiritually.  Which explains all those people who swore vows of poverty in order to live in monasteries.  It's almost as though they were familiar with the teachings of Jesus on the subject of poverty!

It's true that Medieval Europeans believed that the rule of earthly Kings was ordained by God.  But the claim that so-and-so was the rightful King of England originally had nothing whatsoever to do with spiritual superiority, any more than your claim to own your car, or the President's claim to be legitimately elected, implies any notion of being spiritually superior to other people.  Medieval Christians (like Ancient and Modern Christians) believed that God has ordained the existence of human governments to enforce justice, and that therefore (barring exceptional circumstances) it is our duty to obey whatever government one happens to live under.  There was indeed a much more extreme theory of the "divine right of kings" that basically said that the monarch could not be resisted under any circumstances whatsoever.  However, this theory was popular, not in Medieval times, but rather the Early Modern era (roughly the 16th-18th centuries).

As St. Chesterton pointed out, people are happy to accuse almost anything that seems old and bad as "Medieval" without checking to see what Medieval people actually thought and did.  In reality, prior to the Renaissance, the Medievals:

• invented Academia as we know it, and founded the first Universities,
• strongly believed that one could discover the nature of the Universe using logical reasoning based on appropriate authorities,
• held in high regard ancient pagan learning and culture,
• like all educated people after Aristotle, knew the Earth is spherical and that the universe is huge compared to the Earth,
• prohibited military attacks on civilians, and tried to restrict war to certain days,
• ended nearly all chattel slavery in Europe (but see below),
• developed the notion that government must respect certain human rights (in feudalism, serfs were tied to the land but had customary rights which the lord was required to respect),
• officially taught that witchcraft was impossible, and that the popular belief in witches was a superstition to be discouraged.

Yes, you heard that right.  They didn't burn witches.  They did burn heretics, but those were real whereas witches were a figment of the peasants' imagination.  Witch trials didn't become popular until the supposedly more enlightened Renaissance and Early Modern Era (mainly 1484-1750).  Even then they didn't burn witches, they hung them.  (What about all that stuff about dumping the witch in water to see if she floats?  You know, either she floats and is convicted, or she drowns and is posthumously acquitted.  Of course, no one would actually be so stupid as to devise a trial system that kills the innocent on purpose.  In reality, the witch-hunters would pull those who sank out of the water before they drowned.)

The early Medievals did occasionally use trial by ordeal, when the evidence of guilt was doubtful—for other crimes than witchcraft, which they didn't believe in, remember!  However, the trials actually appear to have been rigged to produce acquittals.  In any case, these mostly ended soon after 1215 when the Church refused to allow priests to cooperate.  In England this method was replaced with trial by jury.

The Modern Period also brought the racist version of the slave-trade into the world.  Unfortunately, because of failure to turn the other cheek when pagans or Muslims would capture Christians as slaves, some slavery of non-Christians was permitted.  Unfortunately, this meant that slave-traders were in existence when the New World was opened up...

The idea that the Medieval scholars believed the earth was flat is a lie invented in the 19th century by rationalists eager to find a historical precedent useful for mocking creationists, as documented by the historian St. Jeffrey Burton Russell here and here (I haven't read his book but I've seen him talk).  For more information about the actual Medieval worldview, you can't do better than reading St. Lewis' wonderful book The Discarded Image.  I should have put a gazillion more links in this post, but you all know how to use Wikipedia.

The idea of inevitable moral progress with time is much easier to believe if you only have a superficial notion of history.  I do think that we've made important progress in justice over the past 2 or 3 centuries, but notice that a lot of these involve undoing the moral mistakes of the Early Modern era.  Like all eras, the Medievals had many moral blind spots.  But then again, so do we.

## Fields

What is the world made out of?  In the most usual formulations of our current best theories of physics, the answer is fields.  What are those?

Well, if you know what a function is, you're already most of the way there.  A function, you will recall, is a gadget where, for any number you input, you can get a number out as an output.  We can write $f(x)$ where $x$ is the number you input, and $f(x)$ is the number you output.  The function $f$ itself is the rule for going from one to the other, e.g.  For example $f(x) = \sqrt{\sin x^2 + 1}$.

Now, nothing stops you from having a function that depends on multiple numbers as input; for example the function $f(x,\,y) = xy^2 + x^3y$ depends on two input variables, $x$, and $y$.  If there are $D$ input numbers, then the $D$-dimensional space of possible combinations of input numbers is called the domain of the function.

Also nothing stops you from having the output be a set of several numbers.  In this case we would need some sort of subscript $i$ to refer to the different possible output numbers.  For example, if we had a function with one input number $x$ and three output numbers $y$, then we could write $f_i(x)$, where $i$ takes the values 1, 2, or 3.  Then $f_i(x)$ would really be just a package of three different functions: $f_1(x)$, $f_2(x)$, and $f_3(x)$.  So if you specify the input $x$, you get three output numbers $(f_1, f_2, f_3)$.  If there are $T$ different output numbers, the $T$ dimensional space of possible outputs is called the target space.

Now a field is just a function whose domain is the points of spacetime.  For example, the air temperature in a room may vary from place to place, and it may also change with time.  So if you imagine checking all possible points of space in the room at all possible times, you could describe this with a temperature field $T(t, x, y, z)$.  However, the temperature field isn't a fundamental entity that exists on its own.  It subsists in a medium (air) and describes its motion.  When the air molecules are moving around quickly in a random way, we say it's hot, and when they start to move around slower, we say it's getting chilly.  An example of a field which actually is fundamental (as far as we know) would be the electromagnetic field.  This has 6 output numbers, since the electric field can point in any of the 3 spatial directions, and the magnetic field also has 3 numbers.

For a while in the 19th century, scientists were confused about this.  They thought that electromagnetic waves had to be some sort of excitation of some sort of stuff, which they called the aether.  That's because they were assuming (based on physical intuitions filtered through Newtonian mechanics) that matter is something solid and massy, which interacts by striking or making contact with other things.  The 20th century scientific advances partly came from realizing that its okay to describe things with abstract math.  Any kind of mathematical object you write down satisfying logically consistent equations is OK, as long as it matches experiment.  So electromagnetic waves don't have to be made out of anything.  They just are, and other things are (partly) made out of them.

In our current best theory of particle physics, the Standard Model, there are a few dozen different kinds fields, and all matter is explained as configurations of these fields.  I can't tell you exactly how many fields there are, because it depends on how you count them.  Not counting the gravitational field, there are 52 different output numbers corresponding to bosons, and 192 different output numbers corresponding to fermions (Don't worry about what these terms mean yet).  So you could say that there are 244 different fields in Nature, each with one output number.

That sounds awfully complicated.  But there's also a lot of symmetries in the Standard Model which relate these output numbers to each other.  This includes not only the Poincaré group of spacetime symmetries, but also various internal symmetries related to the dynamics of the strong, weak, and electromagnetic forces.  They are called internal because they don't move the points of spacetime around.  Instead they just mutate the different kinds of output numbers into each other.

So normally, particle physicists just package the output numbers into sets, such that the numbers in each set are related by the various kinds of symmetry.  (For example, the 6 different numbers of the electromagnetic field are related by rotations and Lorentz boosts.)  Each of these sets is called a field.  In future posts I'll give more details about the different kinds of fields.  As always, questions are welcome.

UPDATE: I forgot to include the 4 vector components of the spin-1 gauge bosons, so the numbers of degrees of freedom of the bosons were wrong before.  Note to Experts: These are the "off-shell" degrees of freedom before taking into consideration constraints or gauge symmetry.  Note to Non-Experts: the numbers in this post are just for flavor, in order to give you the sense that there are a LOT of different fields in Nature.  You won't need to understand how I got these numbers in order to enjoy future posts!

## Pillar of Science III: Approximate Models

Science requires Approximations.

Every kind of professional activity changes the way you think.  It rewires your brain so that even when you're off the job, things start looking a certain way.  For example, to a computer programmer everything looks like an algorithm.  To a teacher, everything is pedagogical.  As a physicist, what goes through my head every day is approximations.

Every time I think about a situation involving black holes, or prove a theorem, or do a calculation, I always have to keep in the back of my mind what kinds of physical effects I'm ignoring, not taking into account.  This habit has leaked out into my thinking about life in general.  Ideas don't have to be just true or false, instead they can be good approximations in some contexts, and bad approximations in other contexts.

(Partly related: before I started doing physics seriously, I think I had the idea in the back of my head that when I went to grad school, I'd learn how to calculate the really hard problems.  But it turns out there is no way to calculate the answers to the hard math problems.  There are only clever tricks for simplifying hard problems so that they become easy problems.  Frequently, the clever trick is finding some parameter that can be taken to be small, in order to justify some approximation.  The way this works is: first you figure out what happens if the parameter is zero, and then you calculate the tiny effects of it not quite being zero.)

It is impossible for any model of the universe to capture every feature of reality, or else it would be too complicated for human beings to analyze, or to compare to experiment, at the high level of precision demanded by science.  Consequently every scientific theory is applicable only to some limited range of phenomena.  In other words, it isolates some feature of reality which is as free as possible from contaminating influences, and which is simple enough to be either measured experimentally or calculated theoretically (ideally, one can do both, to compare the theory to the experiment).

Therefore, Science consists of a bunch of partly overlapping models which cover different patches of reality. Some of these patches are smaller, and cover very specific situations (e.g. Bernoulli's principle for fluid dynamics) and others cover a very broad range of situations (e.g. Quantum Electrodynamics, which covers everything related to electricity, chemistry, and light). None of these patches covers everything, and the two broadest patches, Quantum Field Theory and General Relativity, cannot yet be fully reconciled with one another.

One of the implications of this principle is that scientific revolutions seldom result in the complete discrediting of the old well-established theory. The reason is that if the first theory explained a significant patch of data, the new theory can only supercede the old one if it explains all the things the first one explained, and more. Usually this means that the old theory is a limiting situation or special case of the new theory. Thus the old theory is still valid, just in a smaller patch than the new theory. For example, Einstein's theory of General Relativity superceded Newton's theory of gravity, but it predicts nearly the same results as Newton in the special case that the objects being considered are travelling much slower than light, and their gravitational fields are not too strong.

Thus the empirical predictions of Newton's theory are still correct when applied in the proper domain.  However, the philosophical implications regarding the nature of space and time could hardly be more different in the two theories, because the Newtonian theory regards space and time as fixed, immutable, separate entities, while Einsteinian theory regards spacetime as a single contingent field capable of being affected by the flow of energy and momentum through the spacetime.

Philosophers who reason from scientific discoveries should take warning from this: although the empirical predictions of a theory usually survive revolution, the philosophical implications often do not.  Thus our current scientific views on such matters should be taken as somewhat provisional.  On the other hand, it would be even more foolhardy to try to discuss the philosophical nature of space, time, causality, etc. without taking into account the radical changes which Science has made to our naïve intuitions about these concepts.  Some improvement of our thinking is better than none.

Posted in Scientific Method | 6 Comments

## Does your vote make a difference?

Tomorrow is election day here in the US, so I'm going to have a post about elections.  It's strictly nonpartisan—people seeking bile can look elsewhere.

People often say things like "I'm not voting because it's unlikely that one vote could make a difference".  Sometimes they sound rather cynical about it, which I find a little odd, seeing as the whole point of democracy is that no individual gets to dictatorially make all the decisions for everyone.  But let's suppose they are just trying to be practical.  Is this a rational attitude to take?

There are many possible reasons to vote, even when your vote won't make a difference.   For example, it could be urged that voting fulfils a civic duty, that casting an informed vote forces you to become educated about the issues of the day, and to come to a decision about them.  It could also be urged that there is some political responsibility being exercised by the members of the majority even when the majority is by greater than one, or a value of protest in voting for the minority position.  Presumably people who "waste" their vote on third parties, or other hopeless causes, are primarily thinking about the communicative effects of voting.

But let's leave all this aside, and just think about the sort of power that comes from exercising a vote that changes the outcome.  This only happens when 1) the votes for two options are tied, so that you cast the deciding vote, or else 2) the votes for two options differ by one, so that you can accept that result or else cause a tie (the resolution of ties depends on the election system.  If ties are resolved randomly, the final result will change half the time).  For simplicity, let's assume the election has just two options: Yes and No.

How unlikely this is depends on the number of voters V, and the odds that the election will end up tied (within one vote).  Voters don't vote randomly, instead the more popular candidate will usually receive more votes. A certain fraction $x$ of the voters will choose to vote Yes, while $1-x$ will vote No.  Since we aren't certain what $x$ will be in advance, we have to model it by a probability distribution over possible values of $x$.  This is a function $p(x)$ describing the probability density of getting any particular outcome $x$.  (Here I'm assuming that $V$ is large so that $x$ can be approximated as a continous variable.  Note that because $p(.5)$ is a probability density rather than a probability, it can be greater than 1).

The odds of your vote making a difference can now be calculated to be:

This is a small number.  However, how important would it be if you got lucky and your vote did make a difference?  The results of the election presumably affect a large number of people, roughly comparable to the number $V$ of voters.  So the factors cancel out, and the expected effect of your vote does not depend on the size of V!  Instead it is proportional to $p(.5)$, which is equal to 100 times the odds that the vote lies within the critical percentile between 49.5% and 50.5%.

Another way of putting this, is if it would be worthwhile for everyone to miss their lunch to vote, so that everyone can get (what you believe is) a political benefit, then it is worthwhile for you to vote so that maybe everyone can get the political benefit.  Here I'm assuming you are voting for altruisic reasons, to benefit everyone.  The expected selfish benefits to you personally are indeed tiny.

It might still be rational to abstain from voting if 1) the odds of the election being close are small, or 2) the expected difference in benefit from the options is small.  These factors obviously depend on the specific election, but in general one expects that important issues are at stake.  So unless the outcome is a foregone conclusion, it is indeed rational to vote because of its deciding effects.

Does that mean that, if you're a resident of a non-Swing-State like California, you shouldn't vote, because (regardless of whether you support or oppose him) we all know that the state will go to Barack Obama?  If you said that, shame on you!  Go look at the local races instead of just thinking about the Presidency.  Although the number $V$ of voters affected by local politics is much less, the odds of you making a difference are proportionally greater.  The size cancels out, so you should regard local, state, and national races as being approximately equally important.  (Although I suppose one could argue that the President is twice as important, because he determines foreign policy as well as domestic policy.)

Posted in Politics | 3 Comments

## The Ten Symmetries of Spacetime

Previously, I described the main formula of Special Relativity:

This formula tells us the amount of distance squared between two points (if $s^2 > 0$) or the amount of duration squared (if $s^2 < 0$).  (By using some trigonometry we can also use this formula to figure out the size of angles, so this encodes everything about the geometry).  All the crazy time dilation and distance contraction effects you've probably heard about are encoded in this formula.

Today I want to talk about the symmetries of spacetime.  What I mean by a symmetry is this: a way to change the coordinates $(t,\,x,\,y,\,z)$ of spacetime in a way that leaves the laws of physics the same.  Now I haven't told you what the laws of physics are, but the important thing is that they depend on the geometry of spacetime.  So that means that we need to check in what ways we can change the coordinates of spacetime without changing the formula for $s^2$.

The first kind of symmetry is called a translation.  This consists of simply shifting the coordinate system e.g. one meter to the right, or one second to the future.  This doesn't affect the formula for $s^2$ since it only depends on the coordinate differences $\Delta t$, $\Delta x$ etc.  We can write a time translation like this:

i.e. the new time parameter $t^\prime$ equals the old one plus some number $a$.  Similarly, the three possible kinds of spatial translations are:

By choosing the numbers a, b, c, d, arbitrarily, one obtains a four dimensional space of possible translation symmetries.

The second kind of symmetry is more complicated, but you've certainly heard of it before—it's called a rotation.  If we have two spatial coordinates, then we can rotate them by some angle $\theta$ (measured in radians), which leaves all the distances the same.  The algebraic formula for a rotation looks like this:

That involves some trigonometry, but things look a bit simpler if we take the angle $\theta$ to be a really tiny parameter $\epsilon$, and just consider the resulting infinitesimal coordinate changes $\delta x \approx (x^\prime - x)$:

Translated into English, that says that if you rotate the y-axis of your coordinate chart a little bit towards the x-axis, you have to rotate the x-axis a little bit away from the y-axis (or vice versa if $\epsilon$ is negative).  I'm too lazy to draw this, but if for some reason you can't visualize it, a little bit of figeting with any rigid flat object should convince you.

Now actually we have three different spatial coordinates: x, y, and z.  That means that you can actually rotate in 3 different ways: along the x-y plane, the y-z plane, and the z-x plane.  Of course there are other angles you can rotate at as well, but they are all just combinations of those three; in other words the space of possible rotations is 3-dimensional.

But now, what about the time direction?  It would feel terribly lonely if it were left out, and in fact it is also possible to rotate spacetime about the t-x plane, the t-y plane, and the t-z plane.  However, remember how time is not quite the same as space?  Instead, it's just like space except for a funny minus sign.  So not surprisingly, the formula for a rotation also has a funny minus sign—or rather, a funny absence of a minus sign:

So if you rotate the t-axis towards the x-axis (which corresponds to changing your coordinate system so that you are travelling at a constant speed), then the x-axis has to rotate towards the t-axis (which means that your notion of simultaneity has to change as well).  If you know how to integrate this with calculus, you can get the effects of a finite "rotation" in space (called a Lorentz boost) through an "angle" $\chi$:

In the above, cosh and sinh are functions similar to cosine and sine but defined using hyperbolas instead of circles.

So this rotation has some wierd properties: It describes a crazy world (ours!) in which things rotate in hyperbolas instead of circles.  That's because of the minus sign in the formula for $s^2$ above, which makes it so the points of equal distance (or duration) correspond to hyperbolas instead of circles.  This has some additional consequences: 1) Because hyperbolas are infinitely long, the "hyperbolic angle" $\chi$ ranges from $-\infty$ to $+\infty$, unlike circular angles which come back to where you started after you rotate through $2\pi$ radians.  2) Because the two axes both move towards (or both move away) from each other, when you do a really big rotation it scrunches everything up towards $t = x$ or $t = -x$.  What this means is that when you accelerate objects more and more, they don't go arbitrarily fast.  Instead they just get closer and closer to the speed of light.

In conclusion, spacetime has 10 kinds of symmetry: 4 kinds of translations and 6 kinds of rotations.  The space of possible symmetries is 10 dimensional.  It is called the Poincaré group.

P.S. In this whole discussion I have ignored the possibility of reflection symmetries such as $t \to -t$ or $x \to -x$.  These are also symmetries of the formula for $s^2$, but they are discrete rather than continuous—there's no such thing as a "small" reflection the way you can have a small rotation.  Adding these in doesn't change the fact that the Poincare group is 10 dimensional.  However, these transformations are actually NOT symmetries of Nature.  They are violated by our theory of the weak force.  The only discrete symmetry like this which is preserved by the weak force is CPT: the combination of time reflection, space reflection, and switching matter and antimatter.

Posted in Physics | 2 Comments

## Pillar of Science II: Elegant Hypotheses

Scientific Theories must be Elegant.

Since there are always infinitely many different hypotheses which fit any set of data, there must be some prior beliefs which we use to decide between them.  Any hypothesis which has an excessive number of entities or postulates is unappealing, and gives rise to the suspicion that it works because of special pleading or force-fitting the data rather than because it has any deep connection with Nature.  So all else being equal, scientists prefer hypotheses which are simple, uniform, common-sensical and aesthetically pleasing.

At least part of this requirement is captured in the principle known as Occam's razor, which in the original form proposed by Occam translates to “Entities are not to be multiplied without necessity”.  Of course, one may be forced to postulate complexities if the data rules out any simpler hypothesis, but even here one must pick among the simplest of an infinite number of possible explanations for the same data.

This criterion of elegance is informed by previous scientific work as well as by a priori considerations.  It also varies from field to field: a particle physicist should be much more reluctant to postulate a new force of nature than a cellular biologist is to postulate a new kind of organelle.

Because many important scientific theories have greatly defied prior expectations, it is best not to turn these a priori expectations into hard and fast rules which would prevent too many hypotheses from being considered altogether.  Instead, scientists mainly use intuition and rules-of-thumb to judge which theories are worth considering.

There are many famous cases where the elegance of a new theory was used to predict confidently the results of an experiment.  Einstein once quipped about Planck that

...he did not really understand physics, during the eclipse of 1919 he stayed up all night to see if it would confirm the bending of light by the gravitational field [as predicted by Einstein].  If he had really understood the general theory of relativity, he would have gone to bed the way I did.

Nevertheless, ultimately the criterion of elegance is subordinate to observations.  It doesn't matter how beautiful or simple your theory is, if it gets the facts wrong.  To be sure, sometimes experiments turn out to be wrong too, especially when they go against fundamental principles of theory (like the recent supposedly faster-than-light neutrinos thing).  But if, in the long run, experimental observation can't correct our prejudices, then there's no point in doing science.   Nature may be beautiful but that doesn't mean that she (or her Creator) cares about our personal aesthetic of how things should be run.  In the greatest popularized physics lectures of all time, Feynman advises that:

Finally, there is this possibility: after I tell you something, you just can't believe it.  You can't accept it.  You don't like it.  A little screen comes down and you don't listen anymore.  I'm going to describe to you how Nature is—and if you don't like it, that's going to get in the way of your understanding it.  It's a problem that physicists have learned to deal with: They've learned to realize that whether they like a theory or they don't like a theory is not the essential question.  Rather, it is whether or not the theory gives predictions that agree with experiment.  It is not a question of whether a theory is philosophically delightful, or easy to understand, or perfectly reasonable from the perspective of common sense.  The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense.  And it agrees fully with experiment.  So I hope you can accept Nature as She is—absurd.

I'm going to have fun telling you about this absurdity, because I find it delightful.  Please don't turn yourself off because you can't believe Nature is so strange.  Just hear me all out, and I hope you will be as delighted as I am when we're through.

Excellent advice for anyone who wants to see the world scientifically.  Perhaps you can already see some implications for religious views, but we'll go into that some other time.

## All Saints Day Roundup

In honor of All Saints Day, here are some links to the saints on my blogroll:

This blog has its own canonization policy: every serious Christian, whom I refer to by name in the 3rd person, is a "Saint" (e.g. St. Faraday).  This policy is inspired by how the word "saints" was used in the early church to refer to ordinary Christians, e.g. St. Paul addresses one of his letters to "the saints in Ephesus", meaning every person in the congregation.  It emphasizes the fact that the Holy Spirit dwells inside every person who gives their life over to Jesus in order to become one of his Fathers's children.

The Hebrew word qadosh means something sacred which is set apart and dedicated to God's service, while the English word holy is related to whole or wholesome.  In its most proper sense, holiness is a property of God alone, and expresses that he is Good, not just in some conscientious ethical sense, but in the sense of a numinous, awe-inspiring Otherness which, for those fortunate enough to experience it, overpowers us with its majestic glory and weightiness.   The bodies of the "saints" are living Temples in which the Holy One dwells, and we become holy in a derivative sense, sanctified because of his presence inside of us.

Imagine a pond, which has some sort of flowers growing on its surface (a little like water lilies).  Most of these flowers float aimlessly on the surface, but some of them grow stems downwards in to the water.  This makes them rather awkwardly shaped at first, but when the stems reach the ground, they attach to the solid earth underneath.  From then on, the flowers share in the Solidity of the ground beneath.  They no longer drift with the surface currents, and they receive nutrients from below as well as above.  This is only an analogy, but perhaps it gives an idea of the kind of difference that holiness makes to a life.

When I call all Christians saints, this is to bring home the awareness of this astonishing fact.  It is not intended to deny that we all struggle in many ways with sin and bad habits, grieving his Spirit, and that we are therefore in constant need of forgiveness, from God and from one another.

Nor is it intended to deny that some people, because of their fellowship with Jesus, through suffering and joy, become especially holy in a way that serves as a special example of holiness to the rest of us.  I think of St. "Father John", the priest of Holy Trinity Orthodox church of Santa Fe, who cannot be in the same room with anyone without expressing deep love for them.

Nor do I mean to imply that only religious people can be ethical—if by ethics, one means a conscientious effort to be courageous, kind, honest, generous and self-controlled.  However, nonreligious people cannot be, and are not even trying to be, holy in the sense described above—unless indeed they have a relationship with God without knowing it.  (For we must never forget, that even before a person has a relationship with God, God is still having a relationship with them.  Like a host at a party, he provides them with food, drink, and entertainment, and if they happen to be ungrateful or mistreat the other guests, he takes it personally.)  For Christians, ethics comes out of holiness, because of God's love for us; it does not come out of conscientiousness.  That is the most important distinction between religious and nonreligious ethics.

## Time as the Fourth Dimension?

You've probably heard that time is the fourth dimension.  What does it mean?  It should seem rather fishy that time should be the same sort of thing as a spatial dimension.  We all know that you can only go in one direction in time—towards the future!  Time is measured in seconds, space in meters, etc.  It turns out though, that time can be thought of as the same sort of thing as space, but not exactly: there's just one tiny change that makes everything turn out different.

Let's start with two dimensional boring old Euclidean geometry of the sort one learns about in high school. It's called two dimensional because you can specify the location of a  point using two numbers $(x,\,y)$.  Now the most important equation—from which everything else about geometry follows—is the Pythagorean theorem.  This allows you to compute the distance $r$ between any two points $(x_1,\,y_1)$ and $(x_2,\,y_2)$.  If we define $\Delta x = x_2 - x_1$ and $\Delta y = y_2 - y_1,$ then we can think of $\Delta x$ and $\Delta y$ as the sides of a right-angled triangle.  Then the Pythagorean theorem says that then the distance is

Now it turns out that the exact same formula works in three dimensional Euclidean space as well; you just have to add in the $z$ coordinate:

Now our world has only three dimensions of space, so far as we know (though that hasn't stopped physicists from exploring ideas such as string theory where there are more dimensions of space).  Nevertheless, nothing stops us from imagining that there are more spatial directions, say four.  (If you haven't read the classic Flatland, go do it now.  Geometry as social satire!)  In that case, one would simply give the extra dimension a new name (say $w$ since we're out of letters at the end of the alphabet) and then write down

Voilá!  Four dimensional space has arrived!

What has not yet arrived is spacetime.  Each of these dimensions are all exactly the same (it doesn't matter which one we call "first", "second", "third", or "fourth" because of rotational symmetry).  There is no notion of past or future.  To get a time coordinate, we have to do something to make it special.  And what we do is very simple, we just put in a minus sign:

In deference to custom, I've renamed $r$ as the separation $s$, but that's just a change of notation.  What you should be noticing instead is how similar this equation looks to the last one.  In fact, just as all of Euclidean geometry essentially follows from the Pythagorean theorem, so all of what's called Special Relativity, or the geometry of Minkowski space $(x,\,y,\,z,\,t)$ essentially follows from this one equation.  (Minkowski was the name of Einstein's math teacher, who first pointed out this way of understanding relativity.)

Now, what is the effect of this minus sign?  It turns out that it changes things rather a lot.  In Euclidean geometry, the thing inside of the square-root is always positive.  But because of the minus sign, the $s^2$ of two different points can be either positive, negative, or zero:

• If $s^2 > 0$, then its square root represents the distance between the two points, just like in Euclidean geometry.  In this case we say that the two points are "spacelike" separated.
• If $s^2 < 0$, then the thing under the square-root is negative, so when we take the square-root we get an imaginary answer.  Fortunately, that's okay!  We just flip the sign of the thing under the square root, and intepret it as a duration instead of a distance.  In this case the two points are "timelike" separated.
• If $s^2 = 0$, then there is neither distance nor time between the two points.  In this case, we say that the two points are "lightlike" separated.  That's because rays of light (in a vacuum) travel along paths whose points are lightlike separated—in a sense, because they are travelling equally in space and time, but they experience no time or distance as they travel.

I should say that I'm using units here where the speed of light (normally called $c$) is equal to one.  That means that e.g. if we measure time in seconds, we have to measure space in light-seconds (the distance light travels in one second).  Otherwise, the beautiful equations would get all cluttered up with $c$'s flying all over the place.

We're used to dividing up time into three parts relative to ourselves: past, present, and future. The present is just an infinitesimal sliver, so in a sense this division is into two parts: points to the past have $\Delta t < 0$ compared to you, while points to the future have $\Delta t > 0$ compared to you.

However, special relativity tells us you have to chop up spacetime in a more complicated way.  Bearing in mind that you each live in a particular place as well as a particular time, you can chop up spacetime into three different regions.  The future is points that are timelike separated to you and have $\Delta t > 0$; these are the points of spacetime that you can affect.  The past is points that are timelike but have $\Delta t < 0$; these are the points that can affect you.  Then there is elsewhere, the points that are spacelike separated.  These points can neither affect, nor be affected, by each other.  The three regions are separated by the "light cone", which consists of the points that you could send a lightray to (or from).  I'm too lazy to draw a picture right now, but you can see a pretty good explanation here.