Monthly Archives: February 2013

Unacceptable

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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.

It turns out that there is an explicit rule about this in the New Testament:

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.

The Curvature Tensor

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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:

Time as the Fourth Dimension?
The Ten Symmetries of Spacetime
Fields
Geometry is a Field
Coordinates don’t matter
All points look the same

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: $$w^a = v^a + \epsilon^2 R^a_{bcd}\,v^b x^c y^d,$$ 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.

Wisdom Break

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Gentle Readers,

I know it’s been more than two weeks since I made any new top-level articles, but I hope to resume doing it soon.  This was because: i) I got very involved arguing with people on the comment sections of my blog, with consequent fatigue, and ii) I’m in the process of having my wisdom teeth removed (3 on Monday and 1 in a few hours).  I’m not in a whole lot of pain, but any encouragement is still welcome.

Nevertheless, sometime soon I hope to resume where I left off, trying to talk about general relativity in an accessible way. It’s been a long time since I’ve had a physics post, and I’m beginning to miss them.

I’m also going to continue talking about theology, but in a somewhat less argumentative way.  As St. Lewis says in Reflections on the Psalms, “A man can’t be always defending the truth; there must be a time to feed on it.”  So for the next little while, I’ll be focussing more on explaining what I believe than why I believe it (though of course the two things are connected!).  Hopefully this will also result in those posts being more accessible to ordinary readers: one of the hazards of debate is that one loses track of any audience besides the people one is arguing with.

So in a few days, Lord willing, I will return to this endeavor with more wisdom, if fewer teeth.  May those still reading this blog be blessed, may those who have given up on it be blessed, and may those who never started reading it also be blessed.  Amen.