Re: Research Project

[George Jones <gjones@uvi.edu>]

In article <10011231538.ZM14660@unity.ncsu.edu> Christopher Thom
Eling wrote:

>        Hello. I'm wondering if anyone has any ideas for a research project in
>General Relativity suitable for an undergraduate. I have a professor I'm going
>to work with, but we are having trouble coming up with something good.

First, a few questions, then a few suggestions.

Is this for this semester, or next?

Is the course a course on general relativity, or a project course?

What is your background in general relativity?

A very interesting, newly published book is "Exploring Black Holes:
Introduction to General Relativity," by E. F. Taylor and J. A. Wheeler.
This book is written at the sophomore level, i.e., it uses calculus, but
does not use differential geometry, and contains many sections that
might be used as projects. Examples include, pages B-25 to B-26 about
what it's like inside a Schwarzschild black hole, and chapter D about
Einstein's rings. It might be fun to create images of Einstein's rings
on a computer.

Matthew Benacquista gives some good ideas about orbits in article
<3A1E84B3.B7852268@imt.net>. Something that fairly straightforward,
but not completely trivial, is to plot photon paths on a computer
screen as the computer solves the differential equations of motion.
Today's computers are fast enough that an intuitive Euler's method
(i.e., f'(x) = df/dx gives f(x+h) ~ f(x) +f'(x)*h for small h, and
iterate to build up the solution) can be used to solve the differential
equations. Page 4-9 of the above book gives a couple od more subtle
points that have to be considered when programming this.

I have implemented photon path programmes both on a computer using
Borland C++, and on an inexpensive programmable TI-83 graphing
calculator. A downside of using Borland C is that (as far as I know,
but I would love to be corrected) pixel graphics programmes have to be
run as DOS applications. Microsoft Fortran does not have this
disadvantage, and also allows one the ability to point-and-click the
starting point. I assume Micosoft C++ has similar facilities.

Overflow error causes interesting spiral-graph images to sometimes
be generated on both the calculator and computer.

Implementing the programme would also give you the (simple) skill
of mapping coordintes to pixel coordinate, which is useful for
dispaying images in general (not just In GR).

Once implemented, the programme could be used to investigate
both the photon sphere and event horizon of the black hole.

A theoretically more challenging extension is to use freely-falling
frames, as well as Schwarzchild coordinates.

Another possible project is black hole thermodynamics. If the
Hawking result that black hole temperature is inversely
proportional to mass is taken as given, or derived using
hand-waving methods as in "A First Course in General Relativity,"
by B. F. Schutz, then numerous interesting results can be reproduced.

For examples, consider the followind for a Schwarzschild black hole.

M ~ R and T ~ (R)^(-1). The thermodynamic relation dE = T dS (and
E = M) then give that S ~ R^2, i.e., the entropy of a black hole
is proportional to its surface area. Historically, the reverse of
this argument was thought to imply that black holes could not
have entropy proportional to their area.

Also, the Stephan-Boltzmann law that for blackbody radiation,
dE/dT ~ Area * T^(4), when combined with the above and integrated,
gives that the evaporation time for a black hole is proportional
to the cube of its mass. (Use dE/dt = - dM/dT, Area~R^2~M^(-2), and
T^4~M^(-4), and separate variables.)

I have taught of these thermodynamic ideas to students at 2
universities in sophomore thermodynamics classes, and they usually
go over well. You could look in Wald's "General Relativity" for
extensions (for example, rotating black holes).

Another possible project is the changing role that the
cosmological constant(?) has played in cosmology,
incuding how the latest empirical data leads to accelerating
universes and quintessence. I haven't looked into this,
so I can't give you any references.

IF you want more details, you can e-mail me.

Regards,
George