Saturday, July 28, 2012
After venting my frustration about black holes yesterday, I engaged in an epic internet search for anything that could help me understand. Eventually, to my great surprise, I actually found what I was looking for. Bottom line: things can come out of singularities in the same way they come in. When people say that things can't escape black holes what they mean is that every time-like geodesic that goes inside the event horizon has one end at the singularity. But which end? The answer is that mathematically, it can be either the beginning or the end, just as I thought. The difference is that if things are going into the singularity we call it a black hole. If things are coming out of the singularity, we call it a white hole. A black hole and a white hole can exist at the same place. they are essentially the same thing except that one is operating in reverse.
Here is how I discovered this: I started by reading up about the event horizon, the boundary of no return on a black hole. I found this picture on wikipedia:
The message is that nothing can escape black holes because the forward looking light cone is tilted toward the center. But I also found this document, which makes it clear that the event horizon depends on the coordinate system (which we can associate with a specific observer). So, for example, a moving observer might observe the event horizon at a different place than a static observer. In a way, the event horizon isn't real.
I also found this chart which emphasizes the fact that to an outside observer, something falling into a black hole will appear to get stuck at the event horizon forever:
Then I noticed a small note at the end of the wikipedia article. Namely, that in some coordinate systems the light cones don't get tilted at the event horizon. In particular, in Kruskal-Szekeres coordinates light-like vectors are always straight lines. This coordinate system also reveals that the space in and around a black hole has some interesting geometry that can be divided into four sections:
In this image, the grey areas represent the singularity, the big X is the event horizon and the blue lines represent points at the same distance. Each blue line is like a spherical shell. There are two exterior regions (right and left) and two interior regions (top and bottom, minus the grey).
It turns out that mathematically, the top part of the interior represents world lines that originate outside the black hole and end at the singularity. The bottom part is a mirror image that represents world lines that originate on the singularity and end up outside the black hole. This has become known as a white hole. I had been wondering about the existence of this phenomenon for a long time and it turns out that is is right there on wikipedia, if only I had known what to look for. They don't teach this in your standard class on general relativity, but it is openly available on the internet.
The catch with white holes is that, just as how a person falling into a black hole will appear to an outside observer to stall forever on the event horizon, something coming out of a white hole will appear to have been at the event horizon infinitely long in the past. This implies that the white hole has been around forever. Since people don't think the universe has been around forever, they may not actually exist. I suppose a white hole could also be created by some quantum mechanical process that is the equivalent of how a black hole evaporates.
In any case, the universe still makes sense in reverse and the only way we can tell backwards from forwards is by referring to entropy.
Friday, July 27, 2012
Yesterday was my birthday and one of my former students sent me a link to an interesting video about black holes. Unfortunately, all it did was open a hornet's nest of questions about space and time that I have been suppressing for years. So I am going to present my priors and questions and if anyone is feeling really kind they can help me figure out the answers.
First, some background. I tend to view space-time as an unchanging, always present, 4-dimensional object. Our moving three dimensional view is a byproduct of the nature of our consciousness. Our perception of time corresponds to the local direction of entropy. So let me say a bit about that.
Entropy can be understood as a measure of statistical states. A system in a more unlikely state has lower entropy. I like to imagine a pool table with a bunch of balls bouncing around. If I take a photograph of the table it is unlikely that the balls will happen to all be crammed up in one corner. So this is a state of low entropy. If they did happen to be crammed in one corner and I had to guess where they would be in a few minutes, my guess would be that they would be less crammed (higher entropy). This is the law that entropy increases.
A long time ago I was interested in this process and I read a book by one of the founders of the field of statistical mechanics. I think it was Boltzman, but I can't remember. Anyway, my notion at the time was that if I had to guess where the balls had been a few minutes before my photo, I would also guess that they would be less crammed. The direction of time has nothing to do with it. Boltzman (or whoever it was) suggested that, in fact, time is defined by the direction of entropy and not vice versa. That is, if we take a video of the table, but we don't know which end of the video is the beginning and which is the end, by convention we take the part where the balls are all crammed into a corner and call that the beginning.
So what does this have to do with black holes? The above analysis is based on the idea that the laws of physics are the same backwards and forwards in time. The only difference between backward and forward is entropy, and that association is purely conventional. But how does this apply to black holes? Supposedly, things can fall into black holes but they can't come out. That means there would be a difference between forwards and backwards in time other than entropy.
Actually, things are a little more complicated than that. Like the video says, when something approaches a black hole, to an outside observer they never actually fall in. They just slow down and fade away at an infinitely slow pace when they get near what is called the event horizon. Theoretically, this process could be reversed if the right kind of energy were emitted just beyond the event horizon. That is, we could play the video in reverse and it would make just as much physics sense. We would just assign the video a direction based on entropy as in the case of the billiard balls.
But the video goes on to talk about what crossing the event horizon looks like from the perspective of a person falling in. Interestingly, they aren't immediately torn apart or anything, and they can't sense themselves slowing down. They just pass right through and their field of vision starts to narrow like they've got tunnel vision.
But here is my problem. On the inside of a black hole, is it true that falling in further is an irreversible process?
I can't comprehend how that is possible. And don't just tell me that you would have to go faster than the speed of light to get out. I have heard that already. What I don't get is that presumably the path of a falling object follows some geodesic through space-time. If the object is falling toward the center of the black hole, then the geodesic extends from some place that is further form the center to some place that is closer. But if there is a geodesic there, why couldn't something follow it in reverse?
If a person falling into a black hole took a movie of his fall, and we played it in reverse, could we tell the direction of time without relying on entropy?
This is one of those questions that the vast majority of physicists can't answer (at least in my experience of asking them while a physics graduate student). But perhaps I am just being lazy. If no one can help me I will be forced to write out the solutions to a simplified black hole, find the geodesics myself and see if that answers the question. But it is my birthday (well, yesterday it was) . . . so if you know the answer, please save me the trouble.
Update: Maybe White Holes have something to do with it.
Tuesday, July 24, 2012
After my last post, ROC of the Island pointed out the connection between the friend problem and the Monty Hall problem. Monty Hall is a famous probability question that has stumped more than one clever person. It comes from the game show “Let’s Make a Deal”.
At a certain point in the show, the participant has to choose among three closed doors. A sports car lies behind one door, and the other two hide goats. After the participant chooses a door, the host reveals a goat from one of the two remaining doors (since there are two goats, at least one of the remaining doors must be a goat). Then the participant chooses between sticking with their original door or switching. Should they switch?
Many people come to the conclusion that it doesn’t matter. It seems like a 50/50 choice. But it isn’t. If you switch you have a 66% chance of getting the car and if you don’t switch you only have a 33% chance.
Treedrep makes an interesting point in the comments that deserves some clarification. My quiz example is a bit different from the Monty Hall problem because you don't have the option of switching. So when the instructor knowingly reveals a false option form the remaining choices, your probability of getting the right answer doesn't change. However, as Treedrep notes, the likelihood of getting the right answer if you swtich does change to 37.5%.
To me, the reason the Monty Hall problem is interesting is that only one of the probabilities changes.
Monday, July 23, 2012
Here is an interesting statistics puzzle from the blog econlog:
Prove that for a given random person, a random friend of theirs will be more popular than they are.
The author of the blog, Arnold Kling, objects, stating that all we know is that the two people are friends. Since we have the same information about both people we can't come to different conclusions about them. I responded with some examples, including this one:
"Just for fun lets try a different configuration, where 6 people have 5, 4, 3, 3, 2, and 1 friends. In this case if we select a random person, then a random friend, there is a 6% chance that they will be equally popular, 29% chance that the friend will be less popular, and a 65% chance that the friend will be more popular."
If you are interested, you can draw out all of the friend relationships and test it yourself. But the principle behind this is rather interesting. Isn't Kling right that we know just as much about both people?
Not really. The problem is that we selected the two people in different ways. The first person was selected at random, and the second person was selected from a set of friends. And popular people are over-represented in friend sets. Still, it is interesting that we come to different conclusions so I would like to present a slightly modified version.
Suppose a random person selects a random NBA player and compares height. I would guess that the NBA player is taller because tall people are over-represented in the NBA. The key to understanding the friend bit is to realize that the set of friends is kind of like the NBA. The first selection is totally random so we the expected number of friends is average. But the second choice is from a restricted set (like the NBA). A popular person has more relationships, so their number is tossed into the pot more times and are more likely to be chosen. So even though both people are friends with each other, we actually know more about the second person because of the set they were chosen from.
Saturday, July 21, 2012
Apparently North Korea may be ready to begin liberalizing its economy. According to the article, one indicator that the country is an oppressive dictatorship is that 200,000 out of its population of 24 million people live in prison camps. So what does it say about the US when we have about 2.5 million people living behind bars? Since the North Korean numbers can't be verified, the US has the highest documented incarceration rate in the world (about six times the incarceration rate of Canada).
But, that isn't what I wanted to write about today. The other day I found an interesting discussion of workplace coercion on the blog Bleeding Heart Libertarians. This is a continuation of a debate that has been going on in the Libertarian blogosphere about the extent to which employers should be regulated in their treatment of employees.
The author of the post draws our attention to the fact that employers have power over employees because there is an unequal distribution of capital. Employees wouldn't be subject to coercion if they could employ themselves. But the distribution of capital is protected by the state. That is, employees can't rise up and take over a business because the state has a monopoly on the use of force and they prevent that sort of thing.
So in a way, the state is responsible for the current distribution of capital. And if those with capital can only maintain this level of inequality due to the use of state force, it doesn't seem unfair for the state to limit how they can use that inequality to economically coerce their employees (e.g., employers shouldn't be able to demand sex).
Another really interesting point made by the author: he states that if an employer forces an employee to do something because it is necessary to compete in the marketplace, he doesn't classify that as coercion. It is just a survival necessity.
I really like this last point, but it leads to issues. As I mentioned in my last post on military discipline, different people have different ideas of what is necessary in order to achieve their objectives. One employer may think it is necessary to regulate bathroom breaks while another is willing to let their employees work from home. I don't think these decisions are always based on necessity. Baseline standards, traditional methods, and individual preferences come into play as well.
But ultimately, the author argues thus: employer coercion is only possible due to state protection of property. Therefore, the state should place limits on how employers can use their state protected power.
There is one major problem with this that the author didn't really discuss. State limits on coercion may be effectively impossible, or even counterproductive in the face of economic inequality. How could this be? Can't the state simply state that sexual harassment, for example, is illegal?
The problem is that employers compete for employees with a mix of wages and working standards. If the state forcefully improves working conditions in one area, employers may simply reduce wages or conditions in another area. An employer that requires his employees to commit sex acts may have to pay a wage premium over other employers in the industry. If it is made illegal, wages will go down. If there is a minimum wage that prevents wages from decreasing, the regulation may make it so the employer doesn't want to hire anyone at all.
We can decide that the state is justified in regulating workplace coercion, but this doesn't answer the question of whether such regulation actually improves the lot of workers. I am not saying that we shouldn't regulate, just that we have to be cognizant of the unintended consequences. The kind of deal that employees get may be predetermined by the level of economic inequality. Trying to regulate coercion may be akin to playing whack-a-mole with working conditions.
Sunday, July 15, 2012
"Veteran commanders feared that too much emphasis on formal discipline risked weakening the reliance on personal commitment, bravery, and unit pride that had repeatedly brought victory to the IDF."
- Quote about the Israeli military from the website country-data.com.
I just completed my first week of the Judge Advocate Staff Officer Course. Maxwell AFB is a training base and while JASOC is a fairly disciplined environment, all around me are new officers being marched about in parade formations and being subjected to all kinds of minor hazing.
I always assumed that learning discipline and attention to details regarding rules and standards was an integral part of any military. But apparently I was wrong. The quote above about the Israeli military was pretty eye opening. It kind of reminds me of Pericles' funeral oration where he says (if I remember correctly) that Athenian soldiers are superior because they fight for something they believe in rather than out of fear. Still, formal discipline is a major part of the American military tradition and I don't think it is going anywhere.
My perception is that military commanders in the US see regular discipline as good training for combat bravery. The interesting thing is that the Israeli's see them as conflicting values. It is amazing that different traditions can be based on such contradictory beliefs. And both militaries are pretty effective.
So are there really any downsides to discipline? In addition to thinking that formal discipline is a good thing, I believe most American officers think that pretty much anyone worth having in a military can learn to conform. If they don't want to live a military life, they shouldn't join the military. Also, there are a fair number of people (especially in the military) who prefer a very structured existence.
A story told by one of our instructors concerned a Sergeant who came into her office and suggested that one of her subordinates (a new officer) needed to improve her officership. The new officer allowed enlisted personnel to address her without the proper courtesies. This offended the Sergeant and he thought it sent a bad message to the young airmen. The instructors intended message was that we need to demand respect because people notice. What I took away is that some people feel a need for social order and they want to enforce it even if they aren't at the top of that order (a Sergeant is an enlisted rank, and thus lower than even the newest officer). If discipline were relaxed, these people would fee less pride in their association and they form a pretty solid core of the military. So one function of our tradition of discipline is to uphold a sense of esprit de corps among those people who prefer structure.
I don't believe that discipline detracts from personal bravery, as the Israeli's fear, unless the institutional culture is based on a different set of attitudes. However, I do think that some people don't respond well to this kind of environment. Most of them do stay out of the military. I think that most creative minds don't like as much rigidity as the military demands, so the culture excludes or interferes with innovation. Does the military actually need creative people in its ranks? I am not sure. Maybe it can make do with creative contractors. Maybe people who like structure can be tactically creative.
Sunday, July 8, 2012
I am now at Maxwell AFB, Alabama for a few months of AF training. On the way we stopped in Memphis and saw the Rock and Soul Museum among other things.
The interesting thing about the museum is how it highlighted the effect of farmers being displaced by farm machinery after WWII. Millions of people moved to the cities, and they brought their music with them. Memphis was one location that became the new home for a lot of former sharecroppers. They had strong traditions in blues, gospel and country music.
In my last pot I mentioned how the transition away from an agricultural society led to higher rates of high school graduation. It also led to a revolution in American culture during the 1960's. Music was a big part of it.
One thing that is interesting about the new music was that it was often directed toward teens. I think that as long as teenagers were working with adults in the fields they were never able to develop their own culture. As soon as they were all plopped together in high schools and subjected to an artificial discipline, they went crazy trying to rebel against their parents.
I don't really think we have fully adapted to being a non-agrarian society. Think about it. We were mostly farmers for thousands of years. I don't think people realize just how recently it was that all of this changed (about 50-60 years ago). It is hard to overestimate how big of an impact this has had on history.
Tuesday, July 3, 2012
I have been contemplating a model of education where students are required to attend school, but not required to attend any particular class. That is, school should still serve the babysitting function and provide opportunities for education but we shouldn't make learning mandatory.
One implication of this model is that we would have to change our concept of high school graduation. Just attending school for 12 years would be pretty meaningless in terms of signaling educational attainment if you haven't actually gone to any classes. We would probably need a series of exams that students could take to prove competency in a particular subject (if they wanted to).
In this scenario, we might have a lot more students who don't live up to current educational standards. For example, they may never learn algebra. I don't think society is ok with this. Should we be ok with it?
A number of people have made the argument that one of the big drivers of economic growth in the US has been the increasing number of people who have graduated from high school (and college). Tyler Cowen makes this argument in The Great Stagnation. But what is it about high school that has been so important for economic growth?
This interesting blog post discusses the fact that after the 1952 birth cohort (who would have graduated around the year 1970) the high school graduation rate leveled off (hence the great stagnation). Interestingly, this article describes another phenomenon that leveled off in about 1970: agricultural employment. Perhaps high school graduation rates did not drive economic growth. They were a result of changes in the economy. I don't think that fewer people worked on farms because more people were going to school. I think more people went to school because they were no longer needed on farms.
As agriculture grew more efficient, fewer people were needed to supply our demand for food and so kids were sent to school. Also, since there weren't as many farm jobs available people sought other kinds of employment, including employment in jobs that require a certain level of education. I think that the lack of agricultural employment created demand for education just as it made room for kids to spend more time at school.
Note that Mississippi was the last state to enact compulsory education laws in 1917. But it still took more than 50 years before high school education was the norm. Again, I think the graduation rates were driven by changes in the economy.
I that the nation benefited a lot from increased educational opportunity. There were probably a lot of really bright kids working on farms in the early 20th century who could have been more productive in other sectors. However, I don't think that the benefit came so much from having everyone graduate from high school so much as it came from giving everyone the opportunity to do so. When everyone has educational opportunities (as opposed to the expectation that they stay home and work on the farm) bright kids are identified and steered toward higher educational attainment. But the majority of high school graduates still end up in jobs that don't really require them to know algebra.
So my argument is that as long as we have a system that can give all people equal educational opportunities we don't really actually need them to have equal educational attainment. If we let kids play when they are young and eventually decide for themselves what they want to learn the economy will not collapse. The needs to the economy will still dictate what jobs are available and wages will be set according to the balance of supply and demand. But trying to force every student to learn before they are ready is a Sisyphean task.
I grew up thinking that I lived in a democracy. It is one of the first things I learned in school. But I didn't question the fact that my school itself wasn't democratic. I had virtually no say in how the place was run. Our nation may be governed by the masses, but the local institutions that we work and live in on a daily basis are usually not.
In high school I had a big debate in my government class about whether an Israeli kibbutz can be considered a communist organization. My classmates thought that since they typically sell their produce on the market, they can't be considered communist. I argued that since within the kibbutz they own things in common they should be considered communist.
My point then was that we should be able to distinguish between different levels of organization. But it never occurred to me to apply the same logic to my own situation.
So today I read a long blog post discussing whether we should adopt laws that regulate the workplace. The conclusion is that if we don't regulate employers, workers will be subject to significant loss of freedom. Common examples of workplace restrictions include regulation of what you wear at work, when you can take lunch or bathroom breaks, what you can say, who you can date...you get the idea. Employers may even threaten to fire a woman who refuses to provide sexual favors.
One response to the issue of workplace freedom is that people are free to leave. Since they choose to work at a given place, they should either agree to the restrictions on their freedom or find another job. In a way this is kind of like saying that people who live in a country ruled by a dictator should either put up with the tyranny or move to another country (assuming they are free to do so). The trouble is that it isn't always easy to leave a job (or a country), and we may not have any realistic options that offer more freedom.
Perhaps there is another way. Perhaps it is possible for us to try and push for change from within. Or maybe not. Is it realistic for a student to try and make their school more democratic, or for a soldier to try and change the way the military operates?
I am not ready to say that limiting freedom of contract is the right way to go. We run into what I call the "sweatshop problem." If we prevent people from operating sweatshops, those people who be employed in them will probably be forced into some even more degrading kind of work.
What I am ready to say is that I personally need to be more conscious of what freedoms I am willing to give up in order to work somewhere. I have mostly chosen to work in highly structured environments (military, public school) where I had to give up significant freedoms that directly impacted my happiness and job satisfaction. I should be more careful about choosing where I work.
I also need to try and push for more freedom in whatever organization I am in. This doesn't necessarily mean rejecting authority outright. To a large extent it may involve trying to develop trust. If your boss trusts you, they are less likely to try and micromanage. But it also involves speaking your mind when the time is right. First and foremost we need to realize that our freedom is important and it isn't won easily either on a national scale or within more local institutions. Start small. What is the best way to develop freedom in your immediate environment?