This week's physics puzzler.

This question is prompted by a passage from Brian Greene's Fabric of the Cosmos. Greene is discussing the fact that in our usual experience entropy is proportional to volume (other things being equal) whereas for black holes the entropy is proportional to the area of the horizon. On page 479 of chapter 16 he says
Were you to double the radius of a black hole, its volume would increase by a factor of 8 while its surface area would only increase by a factor of 4.
This is misleading. His radius, area, volume relations are only valid for flat space, in curved space the ratios are different. The radius circumference relations for a circle on the surface of a two sphere illustrate the point.

So here's the question, in two parts:

a) What are the radius, area, volume relations (for a surface r=constant) in a Schwarzschild space-time?

b) Do the notions of radius and volume even make sense for a black hole?

Comments

  1. Anonymous5:41 PM

    I don't remember enough GR to calculate the 'volume' relations for Schwarzschild spacetime, but for (b) I think it's fair to say that familiar notions of radius and volume are pretty much kaput for a black hole.

    We can obviously see the event horizon, and thus a familiar surface area, but when we go inside there our radial coordinate becomes timelike and conversely.

    So for a BH radius, I think people just talk about the pseudo-radius that would correspond to the observed surface area were it embedded in a flat spacetime.

    BH "volume" is more confusing. To give a familiar measurement of volume, you need to convert the within-horizon radial measurement into a timelike measurement for outside guys. Googling this a neat answer comes up:
    BH_volume = surface area * lifetime
    http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970808.html

    ReplyDelete
  2. Anonymous6:56 PM

    Hartle calculates length, area, volume and four-volume on ppgs. 146-148 of his book Gravity for general diagonal metrics (including Schwarzschild spacetime), so that solves (a). The calculation should break down at the horizon for a BH, on account of the coordinate singularity. Inside the BH, all calculations collide with the singularity at the "center."

    ReplyDelete

Post a Comment

Popular posts from this blog

Anti-Libertarian: re-post

The most pointed critique of social security and liberalism in general is the libertarian critique. The problem with libertarians, for a liberal, is that we have too much in common. We both believe in individual rights, tolerance of individual differences, and dislike government prescription of religion. The basic difference, it seems to me, is the different answers we give to Cain's (with thanks to the Captain - my original version had the corpse asking the question!) famous question: "Am I my brother's keeper?" That's not really my basic bitch against the libertarians though. My real complaint is the same as my complaint against most religion - its premise is a fraud. For those who can't stand to wait for the punchline, I believe that trying to implement libertarian principles leads to tyranny or social disintegration. Demonstrating that takes some historical (and pre-historical) context. For all but the last 15,000 or so of the 100,000 years the human spec
Read more

Uneasy Lies The Head

Uneasy lies the head that wears a crown ........... Shakespeare Henry IV, Part II. So it is with us and our slightly distant primate cousins , so it seems. A 9-year study of wild baboons in Kenya by Princeton University and the Institute of Primate Research may turn conventional assumptions of alphas males upside down. The study found that alpha males have high testosterone levels, which allows them to dominate other baboons, get access to more food, and attract better mates. This is expected. Surprisingly, though, the alpha males also have high levels of stress hormone glucocorticoid, which scientists measured from their feces samples. In fact, the stress of alpha baboons was on par with low-ranking baboons. The convention wisdom is that access to food and females provides life security and thus less stress for dominant males. Conversely, weak males who are constantly threatened by starvation and physical violence from stronger males are thought to be more stressed. The st
Read more

We Call it Soccer

The US played its best, and maybe only good game of World Cup, but lost thanks to brilliant goalkeeping by the Swedish keeper and the bizarre method soccer uses to settle ties when three US stars could not put the ball on frame from the 12 yard spot.  At the end it came down to a blocked shot that spun in by maybe a couple of centimeters. Well, that's football - too much like life.   Which brings me to a modest proposal.  Replace the thirty minute overtimes with as many 7-player 15 minute periods as needed to settle it.  Scoring should be much easier with fewer players.
Read more