Some Notes on Black Hole Thermodynamics

I noticed, while reading The Black Hole War, by Leonard Susskind, that he presents Jacob Bekenstein's simple argument about Black Hole Entropy. On page 150 he lays out the basic idea in numbers, but encourages the interested reader to work it out with equations. Astonishingly, this very easy to do.

Prof. Bekenstein argued that, to add one (e) bit to the black hole, one k of entropy, one should drop in a photon with a wavelength just the right size to resolve the black hole, but not smaller, which would provide more detail than needed. In other words, hit the hole with a photon who's wavelength is comparable to it's radius, just right sized to “see” the Black Hole.

From there the argument proceeds with a very few lines to result that the surface area increases by a constant amount, proportional to the square of the Planck Length.

Here

So adding one bit increases the surface area by a constant amount, which is on the order of a square Planck Length and therefore the Black Hole Entropy is roughly it's surface area in square Planck Lengths.

All this depends on the relationship between the radius and the mass of the Black Hole,

,and without this relationship the argument does not work.

Then on page 169 Prof. Susskind makes the offhand comment that Bekenstein had actually calculated the black hole temperature without noticing, since the temperature is found from the amount of energy needed to add one bit of information (using the conversion factor k). In other words, from the above,

,which compares nicely with Hawking's result

All this was a real eye opener for pod pod and well worth the price of Prof. Susskind's book.

Pod pod also found some online lectures by Prof. Don Marolf which address this topic from a more technical point of view. The lectures can be found, along with a lot of other great stuff, including, a self contained introduction to Quantum Field Theory by Prof. Tony Zee, calculation of the temperature of an observers cosmological horizon in the case of de Sitter Space by Prof. Maulik Parikh, an introduction to Particle Physics by Prof. Alan Martin, and Cosmology talks by Prof. George Ellis, all at the African Summer Theory Institute. Oh yeah, plus a bunch of String Theory stuff too! The site is actually pretty cool.

As usual, pod pod invites reader comments on this post and especially if you think I am all wet. Thanks in advance for your scathing critiques.

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Some Notes on Misner Thorne and Wheeler, Chapters 14 and 27

I was motivated to examine MTW 14.7, the subject of my previous post, because I wanted to understand the calculation of the Einstein Tensor in MTW Box 14.5. Box 14.5 presents a calculation of the curvature components for the Friedmann metric based on the Cartan Structure Equations. This is used in Chapter 27 to write the Friedmann Equations for an idealized homogenious isotropic cosmological model, aka the Friedmann Lemaitre Robertson Walker model. What follows are just some small details concerning the last bit of Box 14.5 where the contraction is taken in order to form the Einstein Tensor. I do not address the Cartan Equations at this time because cosmology beckons. Plus those calculations are tedious but also presented in pretty good detail in MTW. Anyhow, you have to contract to get the Friedmann equations so that is the topic of this post.

Plug into Einstein Field Equations...

OK. So that was like totally easy. Now these two Friedmann equations can be combined to get a relationship between the acceleration in the scale factor, the density and pressure, and the cosmological constant.

Components of the Einstein Curvature Tensor

Misner, Thorne, and Wheeler give explicit expressions for the components of the Einstein Curvature Tensor. Pod Pod devotes this post to a detailed explanation of these formulas.