## Trip to Europe

I'm in the first week of a 5 week trip to Europe.  I'm in England and will be visiting France, England again, Germany, and Poland (where I'll be attending the GR20 conference in Warsaw).

Expect posting to be light to nonexistent.  However, if you leave comments, I'll try to eventually respond.

## True Justice

After writing about the death penalty recently, I was reflecting about the real meaning of Justice.  It's tempting to think that Justice refers to the thing which happens (or should have happened) in Law Courts.  And of course we hope that the laws and the "justice system" will work out in a way which is actually just.  However, there is a sense in which the justice system is a million miles away from true Justice, if we define Justice as harmonious reciprocal relationships.

Even if the justice system worked perfectly on its own terms, it would be a mistake to think that this is Justice.  The fact that crimes are committed (or else people are suing each other in civil courts) means that the harmonious relationships in society have already been disrupted.  Our Law Courts are, at best, a means for correcting injustice, and even then they can only do so in limited respects:  judges can restore property and restrain criminals, but they cannot change people's hearts to love each other again.

We call a hospital part of the "health-care system" not because lying in a hospital bed is Health, but because it is something we use to remedy sickness.  The best sort of Health is not needing to go to the hospital in the first place.

In the fourth year of King Darius, the word of the Lord came to Zechariah on the fourth day of the ninth month, the month of Kislev.  The people of Bethel had sent Sharezer and Regem-Melek, together with their men, to entreat the Lord by asking the priests of the house of the Lord Almighty and the prophets, “Should I mourn and fast in the fifth month, as I have done for so many years?”  (Zechariah 7:1-3, NIV)

Some context is important here.  The Jews had formerly been captured and exiled to Babylon, as a divine punishment for their sins.  Jerusalem and its Temple had been destroyed, and the fast in question commemorated that.

But now the Persians are in charge, and they have authorized the City and Temple to be rebuilt.  So the Bethelites have a natural question.  Do we have to still keep fasting or not?  The fast has become part of their religious practices, and they want to know whether it still applies to them.  What will Zechariah tell them?

Religious people naturally trend into thinking of religion as a certain set of rules which have to be kept, as if it were a secular legal code and they just have to stay on the right side of the law.  They want to know which way God wants things to be—but in fact either Yes or No would be misleading, because God wants a different sort of thing entirely:

Then the word of the Lord Almighty came to me:  “Ask all the people of the land and the priests, ‘When you fasted and mourned in the fifth and seventh months for the past seventy years, was it really for me that you fasted? And when you were eating and drinking, were you not just feasting for yourselves? (7:4-6)

Stop asking whether you should fast or feast—it's the wrong question.  Instead ask why you were fasting, and why you were feasting.  Was it really for God, or was it just to mourn your own sorrows and celebrate yourself?

And the word of the Lord came again to Zechariah: “This is what the Lord Almighty said: ‘Administer true justice; show mercy and compassion to one another.  Do not oppress the widow or the fatherless, the foreigner or the poor. In your hearts do not think evil of each other.’ ” (7:8-10)

The Lord replaces the people's question with a different command—do justice, resuce the oppressed.  THIS is the point of all of the religious rules, not which days are appropriate for fasting.  This is reiterated later:

“These are the things you are to do: Speak the truth to each other, and render true and sound judgment in your courts; do not plot evil against each other, and do not love to swear falsely. I hate all this,” declares the Lord.  (8:15-17)

There is indeed a role for Law Courts in this notion of Justice.  Zechariah was speaking to a broken society which had lost its bearings, which needed legal stability and fair dealing in order for any reconstruction to occur.  But the requirement of Justice goes deeper than just institutions.  The Just person is not just characterized by legal justice but by honesty and integrity in all of his dealings.

The Law Courts are a means and not an end.  What end it is a means towards may be seen in this beautiful passage:

This is what the Lord Almighty says: “Once again men and women of ripe old age will sit in the streets of Jerusalem, each of them with cane in hand because of their age.  The city streets will be filled with boys and girls playing there.”

This is what the Lord Almighty says: “It may seem marvelous to the remnant of this people at that time, but will it seem marvelous to me?” declares the Lord Almighty.  (8:4-6)

We have now nearly ascended the treacherous craggy slopes of Mount Justice.  Peering into the misty summit, upon which the Earthly Paradise is located, what do we see?  Children playing games with each other!  And sentimental elders looking on and reminiscing.

We do not see here the perfect restoration of body at the Resurrection, but we see the highest vision of Justice between humans beings which any society here and now can attain.  Doubtless the children sometimes accuse each other of cheating.  But the ideal of neighborliness is there, which is indeed the point of the command to Love your Neighbor.  This is Justice.

There is also a harmonious relation of the entire people to God:

This is what the Lord Almighty says: “I will save my people from the countries of the east and the west.  I will bring them back to live in Jerusalem; they will be my people, and I will be faithful and righteous to them as their God.”  (8:7-8)

This is Justice too.  The establishment of a truly just earthly society (harmony between human beings) requires also a correct relation to the God who works justice and righteousness in the earth:

This is what the Lord Almighty says: “You who now hear these words spoken by the prophets who were there when the foundations were laid for the house of the Lord Almighty, let your hands be strong so that the Temple may be built.  Before that time there were no wages for man or beast.  No one could go about his buisness safely because of his enemy, for I had turned every man against his neighbor.  But now I will not deal with the remnant of this people as I did in the past,” declares the Lord Almighty.

“The seed will grow well, the vine will yield its fruit, the ground will produce its crops, and the heavens will drop their dew. I will give all these things as an inheritance to the remnant of this people.”  (8:9-12)

God, humans, animals, the environment; all harmoniously related.  This is Justice.

Once the Temple is established (not just as a building but in our hearts) then there is a bond between neighbors which allows children to play safely in the streets.  Humans and animals can be fed for their work, because they are treated fairly.  Commerce is possible because people don't need to be afraid of aggressors (this is why the Law Courts aren't optional).  Responsible cultivation of Nature is possible because the Temple trains us that things which belong to God are sacred.

This is what the Lord Almighty says: “The fasts of the fourth, fifth, seventh and tenth months will become joyful and glad occasions and happy festivals for Judah. Therefore love truth and peace.”

This is what the Lord Almighty says: “Many peoples and the inhabitants of many cities will yet come, and the inhabitants of one city will go to another and say, ‘Let us go at once to entreat the Lord and seek the Lord Almighty. I myself am going.’ And many peoples and powerful nations will come to Jerusalem to seek the Lord Almighty and to entreat him.”

This is what the Lord Almighty says: “In those days ten people from all languages and nations will take firm hold of one Jew by the hem of his robe and say, ‘Let us go with you, because we have heard that God is with you’.”  (8:18-23)

Real Justice is attractive, and causes celebration and emulation.  It is no longer a question of rules, but of God's promises.  Whether or not you abstain from anything else, abstain from injustice.  Days for producing Justice are always festivals.  Therefore, rejoice always whatever you do.

## The Connection

Suppose we have a field $\Phi$ in a curved spacetime, and we want to know how fast it is changing as you move in some direction in space or time.  Because there is more than one possible direction to move in, we have to select a vector $\delta x^a$ which tells us which direction in the coordinate space $x^a$ to move in (remember, $x^a$ stands for a list of all 4 spacetime coordinates.)  Then we can calculate it by taking a partial derivative.  If your calculus is rusty, the partial derivative $\partial_a$ is defined by:

In other words, we compare the value of $\Phi$ at two different points ($x^a$ and $x^a + \epsilon\,\delta x^a$).  As $\epsilon$ gets smaller, these two points get closer and closer together, so the values of $\Phi$ typically get more and more similar, but because we divide by $\epsilon$ we end up with a nonzero answer in the limit.  I've written $\partial_a$ instead of $(\partial / \partial x^a)$ because I'm lazy.

That was the formula for the partial derivative in a particular direction $\delta x^a$ (which is itself a list of 4 numbers).  If we want to have a list of all 4 possible partial derivatives at each point, we can just write $\partial_a \Phi$ without the $\delta x^a$.  This is the partial derivative covector, where a covector is a thing which eats a vector (like $v^a$) and spits out a number.  That's almost the same thing as a vector, but not quite, which is why its index is downstairs instead of upstairs.  (You can convert between covectors and vectors by using the metric, e.g. $\partial_b \Phi = g_{ab} \partial^a \Phi$, where as usual we sum over all 4 possible values of the index.)

Now, $\Phi$ was a scalar field, meaning that it didn't have any indices attached to it.  What if we tried to do the same trick with some vector field $v^a$ (or a covector $v_a$)?  Well, nothing stops us from taking the partial derivative of a vector in the exact way:

Unfortunately, this turns out to be a stupid thing to do.  The problem is that (before we take the limit) it involves comparing two vectors at different points.  But in a curved spacetime, it doesn't make sense to talk about the same direction at different points, because coordinates are arbitrary.  There's no particular sense in comparing the "t" component of a vector at a point $x_1$ with the "t" component of a vector at another point $x_2$, because the definition of "t" is arbitrary.  If you change the coordinate system at $x_2$ but not $x_1$ you'll get confused.

In a curved spacetime, you can only compare vectors at different points if you select a specific path to go between the two points.  You can then drag (or if you prefer, parallel transport) the vector along this path, but if you choose a different path you might get a different answer.

Well here, because the points are really close, there's an obvious path to pick.  Since spacetime looks flat when you zoom up really close, you can just parallel transport along the very short straight line connecting the two points.  This allows you to relate the coordinate system at the starting point $x_1$ to the destination point $x_2$.  Thus, when we take the derivative, we want to compare $v^a(x_1)$ not to the same coordinate component of $v^a(x_2)$, but to the parallel translated component of the vector.  When we do this, we get the covariant derivative, defined as follows: $\nabla_a$:

Well, that's not very useful until I tell you what capital gamma means.  It's called the Christoffel symbol or the connection, and it tells us how to parallel transport vectors by an infinitesimal amount.  Basically if you take a vector pointing in the $c$ direction and drag it a little bit in the $a$ direction, then $\Gamma^{b}_{ac}$ says how much your vector ends up shifting in the $b$ direction, relative to your system of coordinates.  It turns out that the bottom two indices are symmetric: $\Gamma^{b}_{ac} = \Gamma^{b}_{ca}$.

Similarly, if you want to define the covariant derivative of a covector, you just have to attach the indices a little bit differently:

The minus sign comes in because covectors are the opposite of vectors, so they need to do behave oppositely under a coordinate change.  Or, if you have a complicated tensor with multiple upstairs or downstairs indices, you have to have a separate correction term involving $\Gamma$ for each of the indices.  How tedious!  But, in the case of a scalar field $\Phi$, we get off scot free: the covariant and partial derivative are just the same.

If your spacetime is flat and you use Minkowski coordinates, then $\Gamma = 0$.  But even in flat spacetime you can have $\Gamma \ne 0$ if you use a weird coordinate system, like polar coordinates.

All of this is a little bit circular so far, since I haven't actually told you how to calculate $\Gamma^{b}_{ac}$ yet.  It's just some thing with the right number of indices to do what it does.  In fact, you could choose to think of the connection $\Gamma^{b}_{ac}$ as a fundamental field in its own right, in which case there would be no need to define it in terms of anything else.  But that is NOT what people normally do in general relativity.  Instead they define the connection in terms of the metric $g_{ab}$, because it turns out there is a slick way to do it.

We want to find a way to use the metric to compare things at two different points.  In other words, the metric is a sort of standard measuring stick we want to use to see how other things change.  But obviously the metric cannot change relative to itself.  (If you define a yard as the length of a yardstick, then other things can change in size, but the stick will always be 1 yard by definition.)  Therefore, the covariant derivative of the metric itself is zero: $\nabla_c g_{ab} = 0$.  But if we write out the correction terms we get:

We can use this equation to solve for $\Gamma$ in terms of the metric.  To do this, we just switch around the roles of the $a$, $b$, and $c$ indices to get

and

By adding up two of these equations and subtracting the other, and dividing by two, one can prove that

We can then define $\Gamma^{d}_{ab}$ directly as

To do that, we had to introduce something called the inverse metric $g^{ab}$.  You get this by writing the metric $g_{ab}$ out as a matrix and inverting it.  (Technically we write $g_{ab} g^{bc} = \delta^c_a$ where $\delta^c_a$ is a very boring tensor which is always 1 if $a$ and $c$ are the same index, and 0 if they are different.)

So then, the connection (which allows us to transport vectors from place to place) can be written in terms of the first derivative of the metric.  We'll need to take a second derivative of the metric to get the curvature $R^{a}_{bcd}$, but that will be the subject of another post.

## Thoughts about the Death Penalty

In the previous post I mentioned that I support the death penalty, without providing any details.  Commenter Darryl asks:

Aron, (and other commenters, too!) I wonder if I might probe your notions on the death penalty. Given that in the US alone at least 120 (a fairly conservative estimate depending on which sources one consults) persons on death row have been exonorated by DNA evidence, what are your thoughts on this quote I came across the other day: "So, as long as the death penalty is in place, you are pretty much guaranteed to occasionally execute an innocent person."

I agree with the quote at the end of Darryl's comment.  However, I would add that "So long as imprisonment is in place, you are pretty much guaranteed to occasionally imprison an innocent person," and "So long as fines are in place..." etc.  Any human system of justice, no matter how careful, risks punishing the innocent.  And the more cautious the system is to protect the innocent, the more often the guilty will remain unpunished.

In this respect, the only thing that is different about the death penalty is that once you do it, it cannot be reversed.  However, this is a relative rather than an absolute distinction.  If you sentence someone to 25 years and then the person is exonerated 20 years later, you can give back the 5 years but you cannot give back the 20 years.  (In politically unstable countries, the reversibility of imprisonment may even be an argument for the death penalty, if one is worried e.g. that the genocidal dictator's old faction will retake power and set him free.)

I also do not think that the death penalty is really even an order of magnitude worse than life imprisonment.  If I were unjustly accused of a crime, I would prefer A) 50% chance of being released to rejoin my family and my career; 50% chance executed, over B) 100% chance of life imprisonment w/o possibility of parole.  Remember all those Revolutionary War slogans about liberty being more important than life!—do we still believe that?

Obviously, whenever an innocent person is executed this is a travesty of justice, a very bad outcome.  We live in a modern, first-world nation in which long-term humane imprisonment is possible.  Therefore, if there is no particular good to be had from executions rather than imprisonment, then it is our moral duty to abolish the death penalty.  But is that really true?

The trouble with most debates about the death penalty is that the begin in the middle (quibbling about statistics) rather than at the beginning, which is to ask why one might have the death penalty at all.  Until you first ask what the purpose of the death penalty is, you cannot know whether the trade-off is worth it.

Here are some possible motivations for punishing the guilty:

1. compensation of victims
2. prevention of future wrongdoing
3. deterrence
4. rehabilitation of the offender
5. retribution (i.e. because they deserve it)

In our system of justice, we make a clear distinction between civil proceedings (which exist to compensate the wronged) and criminal proceedings (which exist to vindicate the state's abstract interest in justice).  I think that this is a good distinction, and that therefore #1 is irrelevant to the current question.

With respect to #2 and #3, life imprisonment and executions are probably about equally good.  In principle, if execution is worse, people ought to be motivated more strongly to avoid it.  But I'm not sure it works out that way in practice, given criminal psychology, and the remoteness of the punishment from the crime.  (However, there is an important exception in the case of inmates who murder prison guards or other prisoners—in this case locking them up obviously doesn't prevent the crime, and if they already have a life sentence, the death penalty seems necessary as escalation.)

With respect to #4, the death penalty may well be better than life imprisonment.  True, life imprisonment provides more time to repent.  But the death penalty provides better circumstances.  If the prospect of one's own immanent death—due to one's own crimes, no less—does not cause someone to reconsider their way of life, it is hard to see what would.  Here the nonreligious and Christian might part ways.  The former is likely to feel that it doesn't matter much if a person repents, if you have to kill them to get it.  While for those like myself who believe in an eternal divine judgment following death, it is a matter of the greatest importance.

But I think the core issue at stake is actually #5: does anyone deserve to die?  Is it possible for someone (such as the mass murderers described in my last post) to be so polluted by guilt, that it is a travesty of justice for them NOT to be executed?  I think so, and it is for this reason that I support the death penalty.

For example, a while back there were two Judges in Pennsylvania who accepted bribes from a privately-run juvenile jail, in order to send the kids that appeared before them to jail, regardless of the nature of the circumstances.  These men betrayed their position of power and deliberately perverted justice in a way likely to corrupt and destroy the innocent.  Words cannot express how reprehensible their crime was.  Actions are better.  They ought not to have been allowed to go on breathing.  (Instead they received 28 and 17.5 years.)

I said earlier that the death penalty didn't seem that much worse to me practically than life imprisonment, which is permanent exile, slavery, and confinement all wrapped up in one package.  But there is one other respect in which "death is different", namely that it is psychologically horrible (and fascinating).

Ironically, I think the controversy over the death penalty itself illustrates this psychological response.  Passions are stirred; zealous anti-death advocates hasten to show that the person was convicted unjustly.  Those 120 people who were exonerated due to DNA tests were probably lucky to be on death row—if not, their cases would probably have been neglected, and they would still be in jail.

Death is numinous.  In that post I wrote concerning the concept of atonement for guilt:

This is a numinous problem, not just an ethical problem.  So it needs a numinous solution.

As the Bible says in a chilling passage:

“Whoever sheds man’s blood,
By man his blood shall be shed;
For in the image of God

Unless there is such a thing as real guilt, there is also no such thing as real forgiveness.  Thank God that we have the "sprinkled blood that speaks a better word than the blood of Abel" (Heb. 12:24), which is sufficient to atone for all of our sins!

Now, how much evidence should we have before we execute someone?  I can see nothing wrong with the standard implicitly required by the U.S. Constitution, namely beyond a reasonable doubt.  If it is unreasonable to doubt whether the person did the crime, then it is unreasonable to take into account the possibility of their innocence when sentencing them.

In this country, no one is executed unless either (a) 12 citizens unanimously agree that they are guilty beyond a reasonable doubt, or (b) the accused waives his right to a jury trial.  After that there are usually 5 different levels of appeals courts empowered to inspect the case for procedural flaws.

It may well be that juries fail to apply beyond a reasonable doubt to the cases that come before them, and as a result some innocent people are found guilty.  But this has nothing per se to do with the death penalty.  If you are not found guilty beyond a reasonable doubt, the correct judgment is an acquittal—not life imprisonment.  Not even a $20 criminal fine can be imposed in this case. If our current justice procedure is unreliable, this is a problem for all cases, not just capital ones. If there are people on death row whose cases were never proven beyond a reasonable doubt, they should be released! That is not a good reason to commute the sentences of those criminals who were proven guilty beyond a reasonable doubt. Posted in Ethics, Politics | 17 Comments ## Atrocities and the Media If you're wondering about the lack of posts recently: for the last 2½ weeks I've been travelling on a research trip to Boston and Canada (to Waterloo, Montreal, and Vancouver). The sashimi in Vancouver is out of this world, by the way. Mostly I've been cramming my head with physics, but I did spare some time to read reports about the horrible news regarding the Boston marathon bombing, which killed 3 spectators and injured many more. This was after I had already left, so don't press me for eyewitness accounts of traumatized civilians. I also read the news regarding the lack of news covering Kermit Gosnell's "abortion" "clinic", in which poorly trained and unlicensed "doctors" murdered hundreds of babies (i.e. viable infants outside of their mothers) and allowed women to die due to malpractice and unsanitary conditions, which the Pennsylvania government received complaints about for decades but failed to stop. This, apparently, is not news so far as the mainstream national media is concerned, but due to the untiring efforts of St. Mollie at GetReligion, at least the media's failure to cover it is now news. Apparently the journalists are worried that anti-abortion* activists will make hay out of this story, and so therefore advocacy of "women's rights" must, paradoxically, include suppressing stories in which—even from the pro-abortion* viewpoint—the rights of women to safe health care were most certainly violated. One has to tread cautiously when making accusations of media bias, since it's so easy to be biased oneself in making those accusations. As a moderate Republican who lives and works among academics in California, I would normally feel rather embarrassed to complain about this, since people would look at me as though I were one of those conservatives, who thinks that Fox News is a straight news source but everyone else is commie pinkos. (Even though my political opinions are not the same as those around me, I worry sometimes about whether they have been influenced by the desire to look reasonable to other people when I explain them. Compared to the historical range of opinion, any given time in America there is only a narrow range of acceptable opinions, and this range is even narrower in academia. There's much less incredulity when my position is some sort of libertarianish compromise...) Nevertheless, the fact that Fox News is tilted to the right doesn't change the fact that the mainstream newsmedia is clearly left-leaning on certain particular cultural issues, as I think GetReligion has abundantly demonstrated. I think the media tries to be fair when covering presidential election politics, but it wears its opinions on its sleeve when covering hotbutton social issues. One also has to be careful in defining what one means by bias. There is nothing inherently wrong with: 1) media reporting issues which are more likely to be of interest to their constituents (e.g. spending more time on one party's primary election), or 2) straight-up advocacy or editorializing for ones preferred position. These should be distinguished from 1') concealing relevant facts which ought to be of interest in order to give a partisan slant on reality, and 2') advocacy pieces which pretend to be straight news coverage, but are not. In this case the news media has been caught out in a clear example of (1'). The sooner they correct their mistake, the quicker they can get to a world where conservative complaints of media bias really are crankish. In the meantime, let us pray for those who died in these atrocities, for their families and friends, and for the twisted souls of the those who murdered them—that they would receive their rightful justice, and that God would grant them repentance, "so that they might be judged according to men in regard to the body, but live according to God in regard to the spirit" (1 Peter 4:6). (*) Footnote: I refuse to use the terms pro-life and pro-choice on this blog, since I regard both of them as mealy-mouthed euphemisms. There was once a time when real men and women were willing to state their opinions plainly, even if it meant they had to be "anti-" something, or actually mention the name of the thing they are supporting/opposing. (I can respect the term "pro-life" when it is used as a catch-all term for "anti-abortion, anti-euthanasia, and anti-death-penalty", since in this case an abstraction is useful. Although I do support the death penalty myself.) The term "pro-choice" is even more Orwellian, since it completely sweeps under the rug the nature of the choice being made, using a term so vague that it could just as well apply to e.g. school vouchers. Every decision to do anything can be characterized as a choice; the term avoids grappling with the fact that this choice involves killing one's own flesh and blood. The usual argument against the term "pro-abortion" is that the proponents do not advocate that all women terminate their pregnancies. But that's silly, because almost nobody wants that. Imagine if people who supported the death penalty refused the label because "we don't believe that everyone should be executed, so we prefer the term "pro-justice". Clarity is sacrificed in order to gain cheap points. Posted in Politics | 4 Comments ## Random Stuff Random links I've run across in the past couple months: • Math and physics notes by Bob Geroch, about Gödel's theorem, topology, quantum field theory and other things. • An homage to Catalan independence. When I was in Barcelona I saw the Sagrada Familia cathedral designed by St. Gaudí, which they've been constructing since 1882. They plan to finish around 2028. I also saw some of his weird apartment buildings. • Theodore Dalrymple, a writer for the New English Review, who reminds me of St. Chesterton, albeit with only about a tenth the wit and widsom. I mean this as an outrageous compliment, you should seriously read his essays. Posted in Links | 7 Comments ## Chastity: not just for religious folk In this time and place, the secular world does not place a high value on chastity, which seems to be regarded as some sort of eccentric religious virtue. So long as one avoids disease, and harassment in the workplace, and uses contraception to avoid "unwanted pregnancy", the only touchstone is consent. (Since paedophilia is still taboo, "consent" is interpreted using a legal fiction that says that those below e.g. 18 cannot consent.) Married types are often faithful to their spouses, but the unmarried do what they like. It is true that there are some distinctively religious (as well as some distinctively Christian) reasons to be chaste. But it is silly to think these are the only reasons. Traditional sexual morality is not founded exclusively on mystical doctrines about God. It is also based on practical experience, concerning some basic biological and psychological facts about human beings. Facts which are going to operate regardless of your opinion about any transcendental realities. That is why most traditional cultures have discouraged fornication, even those which have quite different doctrines about the gods and the meaning of earthly pleasures. It is not because they listened to their priests, but because they listened to their grandmothers. I will however have to assume one mystical dogma in this post, namely that there is such a thing as ethics, which commands do not harm your fellow human being, and that this rule is obligatory. Furthermore, that ones fellow human beings includes ONESELF, so that the ethical person is first and foremost committed to human flourishing in the one example of humanity which they actually have control over. Therefore, you cannot excuse a self-destructive behavior by saying "it doesn't hurt anyone else", because if you really love what is good, you will want to see it produced in yourself as well as others. Besides, by destroying yourself you lose your ability to help other people, and cause anyone who loves you to suffer. By the same token, it is insufficient if the other person consents, because people sometimes consent to things which are harmful to them. You have to actually decide whether the act of sex would be beneficial or destructive to the other person. Next comes the biology and psychology. The first fact is that sexual passions cause an intense pleasure, comparable in intensity to addictive drugs. There is nothing inherently wrong in this, but it does mean that we are unlikely to be very sober-minded when making choices about it. Therefore, whatever principles ethics may suggest, they ought to be enforced by clear and explicit social rules in order to avoid ambiguous situations. The second fact is that we are a species which pair-bonds through sex, conditioned by chemicals such as oxytocin. Therefore, the usual consequence of sex is to create a strong emotional attachment between the two participants. It may be possible to avoid this by deliberately shielding oneself with emotional barriers, but this is unreliable. And what will happen if one person pair-bonds while the other person keeps their reserve? Worse still, emotional distancing seems likely to lead to a sexuality based not on love but on a sort of contempt for ones own body, or for other people. It's worth noticing in this connection, that lust is not always "love", even if by "love" we only mean romantic affection. Like our sense of humor, lust can even relish cruelty, to oneself or others. There is a reason why certain words for this act have become curse words. For these reasons, the ethical person will refrain from casual sex, and will only make love in the context of a genuine relationship based on mutual affection and friendship. Furthermore, these relationships had better be exclusive if they are to be stable. Jealousy is another of those awkward facts about human nature which must be taken into account. It's easy for people (especially immature people who are unaware of their own limitations) to decide that they are sophisticated and mature enough to handle an "open relationship", but the notorious instability of such relationships proves otherwise. Besides, although jealousy is a vice in most contexts, I do not think it is wrong is this context. At least, "free love"—the vision of an idealized (almost communist) human nature in which all can be shared equally among close friends, but no one is entitled to private relationships unsuitable for sharing with others—does not at all appeal to me. Finally, if these relationships are to result in long-term happiness then there needs to be an explicit understanding that the relationship is intended to be permanent. The alternative is to pair-bond with someone (or more likely, a series of someones), under conditions in which you can reasonably expect that you will eventually be strangers, or even enemies, to that person. I did not date in college, but I was friends with several couples who dated and eventually broke up. When people are emotionally involved, but not emotionally committed, it is a trainwreck waiting to happen. From an objective outside viewpoint, it seems like the heartbreak outweighs the joy. Modern dating practices are bad enough in this respect, but adding sexual bonding to the mix makes it much worse. The other awkward little fact about sex is that it makes babies. As commonly practiced, contraception is not 100% successful. Even secular-liberal sexual morality recognizes the problematic nature of an "unwanted pregnancy". Any time a man has sex with a fertile woman whom he is unwilling to marry, he risks making her choose between (a) having an abortion, or (b) having a child grow up with an absent father (and possibly a reluctant mother too). I believe that (a) is immoral. But rather than get sidetracked about whether there are "secular" reasons for this, I will simply point out that in any case it is a difficult and likely traumatic decision for the woman to make. Another one of those awkward facts about human biology is that a pregnant woman's body is full of hormones trying to get her to feel a strong emotional bond to the unborn homo sapiens growing inside of her. And as for (b), an ethical person should realize that the interests of children in having loving parents is a thousand times more important than their own interest in sexual or romantic thrills. On the other hand, if in romance you are seeking the deepest interests of the other person, then this interest, like the interest of the child, is best supported by marriage. A gentleman does not use a lady and then discard her when she becomes inconvenient. Nor does he abandon his own flesh and blood. This post is not intended to say anything against love. Chastity is love: it wills the good of its beloved, by controlling sensual desires, for the sake of the beloved's completeness and integrity. Posted in Ethics | 20 Comments ## Unacceptable Many wonderful things happened at my church today, but as we all know the First Rule of Blogging is that one should always focus on the negatives. Admittedly this Rule is in direct conflict with the Christian rule, "Do not let any unwholesome talk come out of your mouths, but only what is helpful for building others up according to their needs, that it may benefit those who listen." (Ephesians 4:29) Perhaps some Christian bloggers think that the Rule for what goes into their keyboard, is different from the Rule for what comes out of their mouths. Still, there is a time and a place for criticism in "building others up"; I pray that the Lord would give me the right spirit and tone. What bothered me didn't have anything to do with the church service itself; it was a flyer for another event. It was a Methodist organized "Good Friday breakfast", with some speaker talking, and the fateful line was something like the following: Individual tickets—$35.  Sponsor a table of eight: Bronze Circle: $250, Silver Circle,$500, Gold Circle, $1000. There would be nothing particularly remarkable about seeing this sentence in an advertisement for a political rally, a theatre meet-and-greet, or a fundraiser for some other type of secular non-profit corporation. Yet I couldn't help but feel like there was a contrast between the point of the event—which one presumes has something to do with Christ's Crucifixion—and the means chosen to finance the event and make it available to the public. Perhaps, before mediating on Jesus' Passion, the organizers of this event should meditate on the following Bible passages: In the following directives I have no praise for you, for your meetings do more harm than good. In the first place, I hear that when you come together as a church, there are divisions among you, and to some extent I believe it. No doubt there have to be differences among you to show which of you have God’s approval. When you come together, it is not the Lord’s Supper you eat, for as you eat, each of you goes ahead without waiting for anybody else. One remains hungry, another gets drunk. Don’t you have homes to eat and drink in? Or do you despise the church of God and humiliate those who have nothing? What shall I say to you? Shall I praise you for this? Certainly not! (1 Corinthians 11:17-22) While this breakfast probably does not include a specific Communion ceremony celebrating the Lord's Supper, its purpose is still centered around remembering the same event. We do not come to the Cross as "Bronze" or "Gold" circle members, we come as sinners saved by grace. Nor is anyone excluded from the Cross because they cannot afford to pay$35 for a meal.  As it is written:

Since you call on a Father who judges each man’s work impartially, live your lives as strangers here in reverent fear.  For you know that it was not with perishable things such as silver or gold that you were redeemed from the empty way of life handed down to you from your forefathers, but with the precious blood of Christ, a lamb without blemish or defect.  (1 Peter 1:17-19)

God will judge us all impartially, so we ought to be afraid to draw wicked distinctions between those who have the silver and gold, and those who do not.  The "reverent fear" has to do with the fact that God is zealous for his holy Name and will not leave people unpunished who discriminate in this way.

If the organizers of this event suddenly decide to fear God and respect the poor, there is an easy solution.  Try the phrase "suggested donation", and make explicit the fact that those who cannot pay are still allowed to attend.  Remove the silliness about different levels of prestige associated with different contribution levels.  And then maybe you will understand what happened at the Cross better.

My brothers, as believers in our glorious Lord Jesus Christ, don’t show favoritism.   Suppose a man comes into your meeting wearing a gold ring and fine clothes, and a poor man in shabby clothes also comes in.   If you show special attention to the man wearing fine clothes and say, “Here’s a good seat for you,” but say to the poor man, “You stand there” or “Sit on the floor by my feet,”have you not discriminated among yourselves and become judges with evil thoughts?  (James 2:1-4)

This passage of Scripture was instrumental in the formation of the Free Methodist denomination.  The other Methodists were literally charging people money in order to sit in reserved pews.  I saw some of these myself when I visited an old Anglican church in Williamsburg, Virginia (a sort of colonial "living history" tourist attraction), which has been in continuous operation since 1711.  It's rather charming to sit in the Pews reserved for "George Washington" and "Thomas Jefferson", but less charming to think of poor people being unable to sit down in the church because of their lack of funds.

That's what makes Free Methodists "free"—you're free to sit wherever you want, although it was also associated with their political activism to end slavery, unfortunately still necessary.  "Freely you have received, freely give" (Matthew 10:8).  Who will put the Free back into these Methodists?

(If you're curious, the "method" part of Methodism was basically small group Bible studies, with literacy training for poor workers who needed it.  This was also politically subversive back in the day, since workers who could read demanded better treatment from their employers...)

Now the Church is the manifestation of the Kingdom of God here on earth.  Its King is Jesus Christ, and we do the things that we do in order to please him, not the world.  A lot of the things that Christians bicker about, Jesus might not care one way or another.  "Man, who appointed me a judge or an arbiter between you?" (Luke 12:14).  But if there's one thing that made Jesus flip out, it was contaminating God's holy place with commercialism:

When it was almost time for the Jewish Passover, Jesus went up to Jerusalem.  In the temple courts he found men selling cattle, sheep and doves, and others sitting at tables exchanging money.   So he made a whip out of cords, and drove all from the temple area, both sheep and cattle; he scattered the coins of the money changers and overturned their tables.  To those who sold doves he said, “Get these out of here!  How dare you turn my Father’s house into a market!”   His disciples remembered that it is written: “Zeal for your house will consume me.” (John 2:13-17)

When I see things like the flyer I mentioned, or church bazaars, I wonder what Jesus would do today.  Possibly something that would get him arrested.  (According to the synoptic Gospels, he was indeed arrested just a few days after making a similar scene.)  Presumably if I were holier—if I had more zeal for God's house—I would also fly into a rage and turn over tables, instead of simply noticing the inconsistency and calmly writing a blog post about it.

The bottom line is this.  The Church belongs to Jesus, and we are not at liberty to run it like a business.  Not even like a non-profit business.  True, a shrewd Christian leader with business experience might well be able to extract valuable life lessons from how the business world works.  But this cannot include the lesson that status and privilege is distributed on the basis of money.  The Church is an anticipation of the New Jerusalem, the home of righteousness, in which the only kind of riches that counts is being rich towards God.

Update: upon returning to church the next week, I found that I was mistaken about the Good Friday breakfast being sponsored by a Methodist organization.  It was actually being sponsored by the YMCA, the speaker being the CEO of some Christian organization.  The event was clearly explicitly Christian, so my criticisms still apply, except for the remark about putting the "free" back into the Free Methodists.

Posted in Theology | 2 Comments

## The Curvature Tensor

One of the things I've been trying to do on this blog is to explain Einstein's theories of relativity.  Here are my previous posts on this subject:

The first two have to do with Einstein's first theory of Special Relativity, in which spacetime is taken to be fixed.  The third post describes what fields are.  The last four describe his second theory, General Relativity, in which the geometry of spacetime is itself a field: the gravitational field.

The gravitational field is also called the metric $g_{ab}$, where $a$ and $b$ are indices selected from the four spacetime dimensions $(0, 1, 2, 3)$.  However, the exact form of the metric depends on your choice of coordinates, and it is always possible to pick coordinates so that the metric at any point looks just like the metric of Special Relativity.  For this reason, in order to discuss curved spacetimes (those that differ from the flat spacetime of Special Relativity, e.g. the gravitational field of the Sun or the Earth) we need to compare the metric at different points.

The basic idea is this: Suppose we are sitting at one particular point $X$ of spacetime, and we imagine we have an infinitesimal arrow whose base sits at the point $X$ and which points (ever so infinitesimally) away from $X$ in some direction.  We call such a thing a vector and write it as $v^a$, where $a = (0, 1, 2, 3)$.  This is actually a set of four numbers: $v^0$ would be a number saying how far $v$ points in the 0-direction, $v^1$ would be a number saying how far $v$ points in the 1-direction, and so on.

Now, suppose we take this vector on a vecation tour through spacetime.  Perhaps it would be simpler just to think about space for a minute.  Imagine vectors lying on a 2-dimensional sphere.  (Warning: when an ordinary person uses the word sphere, they usually mean to include the interior, but when a math or physics person says sphere, they only mean the surface!)  Imagine for example, that the vector $v^a$ lies on the surface of a Earth, and pretend that the vertical up-down direction doesn't exist.  So on a random point on the Earth's surface, $v^a$ can point north, south, east, west, or in-between, but it can't tilt up or down.

Imagine that $v^a$ starts on the North Pole, pointing towards Hawaii.  Suppose we slide the vector down in the direction it is pointing, until it hits the Equator.  The vector now points South.  Now let's drag the vector 1/4 of the way around the Equator, without rotating it.  The vector still points South.  Finally we drag it back to the North Pole.  It has now been rotated 90 degrees from its original position!  This is true even though on each step of the journey, we were careful not to rotate it as it travelled.  (Dragging vectors without rotating them is called parallel transport, by the way.)

This happens because the geometry of the surface of the Earth is intrinsically curved, i.e. the geometry of the sphere is not flat.  This must be distinguished from extrinsic curvature, which refers to something of lower dimension being bent within a higher-dimensional space.  The surface of the Earth is extrinsically curved in our ordinary 3-dimensional space, but that's not what were talking about.  We're talking here about the geometry of the sphere, quite apart from whether it is or is not embedded in some higher dimensional space.

The distinction is quite important in the case of our 4-dimensional spacetime, because (as far as we know) it is not embedded in any kind of higher-dimensional space.  When we say that spacetime is curved, we do not mean that spacetime is sitting in some kind of  "hyperspace" in which it is bent into a funny shape.  We mean that if you move vectors around in spacetime in a loop (i.e. a path that starts and ends at the same spacetime point $X$), it may come back rotated compared to its original position.  That is what curvature means.

I should note that it's okay to drag these vectors either forwards or backwards in time, or along spacelike directions.  That's because these are imaginary vectors serving as a visualization aid to probe the geometry of the spacetime.  They aren't tangible physical objects which have to travel slower than light, and towards the future.

Now, ever since Archimedes, math people have liked to study things by breaking them up into tiny infinitesimal pieces.  So we want to think about what happens if you drag a vector around an infinitesimal loop.  To define this, we imagine we have a vector $v^a$, which we drag around in a tiny parallelogram whose sides are given by vectors $x^a$ and $y^a$ pointing away from $X$.  When we drag $v^a$ along this parallelogram, if there is curvature it can come back rotated as an ever-so-slightly different vector $w^a$.  To keep track of this we write:

where $R^a_{bcd}$ is a gadget with four indices known as the Riemann curvature tensor'', which keeps track of the amount of curvature at any given point $X$$\epsilon$ is an infinitesimally tiny parameter which keeps track of how big the sides of the unit parallelogram are, for vectors of some unit length.

What's a tensor?  The metric $g_{ab}$, vectors such as $v^a$, and $R^a_{bcd}$ are all examples of tensors.  Tensors are similar to vectors, except that they are allowed to point in any number of directions.  A tensor is a kind of field which is allowed to depend on two things: 1) which point you are at in space and time, and 2) zero or more spacetime indices written as subscripts or superscripts $a, b, c ...$.  These indicate the total number of vectors it takes as inputs or outputs.  Note that the Riemann curvature has 3 of its indices downstairs and 1 upstairs.  That notation tells us that it eats 3 vectors as inputs and spits out 1 vector as an output.

In any equation involving tensors, each index letter is repeated, either 1) once in each term of the equation, always upstairs or always downstairs, or 2) twice in the same term of the equation, once upstairs and once downstairs.  In case (1) we interpret the tensor equation as being true for any possible choice of index, as long as it is the same for all terms, on both sides of the equals sign.  In case (2), we consider all 4 possible choices for the index and add them together (the Einstein summation convention).  These rules prevent us from doing nonsensical things like e.g. trying to add scalars and vectors together.

Tensors are not themselves coordinate-invariant, but when you change your system of coordinates, the value of the tensors changes in a particularly simple way.  This makes them useful when trying to describe physics in a coordinate-invariant way.  So long as you follow the rules in the previous paragraph, a tensor equation is a coordinate-invariant idea, i.e. if it is true in one coordinate system it is true in all of them.  That's because if you change your coordinates, both the left-hand-side and the right-hand-side of the equation change in the same way, so it doesn't matter.

The last thing I need to say here is that the Riemann curvature tensor $R^a_{bcd}$ is not a new field additional to the metric tensor $g_{ab}$.  If you know what the metric is, you can work out the Riemann tensor.  At any given point, $R^a_{bcd}$ depends on the metric $g_{ab}$, its first derivative $(\partial / \partial x^c) g_{ab}$ and its second derivative $(\partial / \partial x^d) (\partial / \partial x^c) g_{ab}$.  But the formula looks complicated, so I'll spare you for the time being, until I can think of a simple way to justify it.

Posted in Physics | 2 Comments