Let us consider the work done in moving an object of mass [tex]m[/tex] from the surface of an object of mass [tex]M[/tex] and radius [tex]R[/tex] out to infinity. Recall that work [tex]W[/tex] is given by the product of the force [tex]F [/tex] and distance [tex]r[/tex]. If the force is constant we can use the formula

[tex]W = F r. [/tex]

However in this case the force is a function of distance and we need to integrate to get the correct result,

[tex]W = \int_a^b F(r) dr. [/tex]

Newton’s law of gravitation, gives the force between two bodies separated by a distance [tex]r[/tex] as

[tex] F(r) = - \frac{G m M} {r^2} ,[/tex] where [tex]G[/tex] is the gravitational constant.

Substituting this into our formula and integrating gives,

[tex]W =\int_{R}^{\infty} - \frac{G m M} {r^2} dr [/tex]

[tex] = \left[ \frac{G m M}{r} \right]_R^{\infty}[/tex]

[tex] = \lim_{r \rightarrow \infty} \frac{G m M} { r} - \frac{G m M} {R} [/tex]

[tex] = - \frac{G m M} R [/tex]

The work done is given by the difference between the final kinetic energy [tex]KE_f[/tex] and the initial kinetic energy [tex]KE_i[/tex]. At infinity the object has zero velocity and consequently [tex]KE_f =0[/tex]. The initial kinetic energy of the object is given by

[tex] KE_i = \frac{1}{2} m v_e^2,[/tex]
where [tex]v_e[/tex] is the initial velocity (escape velocity).

[tex]W = KE_{f}-KE_{i} = 0 - \frac{1}{2} m v_e^2 [/tex]
so that

[tex] - \frac{1}{2} m v_e^2 = - \frac{G m M}{R} [/tex]
after some rearrangement, we find

[tex]v_e = \sqrt{\frac{2 G M} {R}}.[/tex]

Notice that the escape velocity [tex]v_e[/tex] is dependant only on the mass of the body [tex]M[/tex].

We can use this classical (i.e. non-relativistic) result to find the Schwarzschild radius of a black hole. To do this we set [tex]v_e = c [/tex] in the previous formula, where [tex]c[/tex] is the speed of light.

Some algebraic rearrangement yields,

[tex] R = \frac{2 G M} {c^2} [/tex]

2005 is the World Year of Physics. This year marks the 100th anniversary of Einstein’s miraculous year   in which he published three revolutionary papers:

• March 1905 Explanation of the photoelectric effect.
• May 1905 Explanation of Brownian motion.
• June 1905 The theory of special relativity.

Three of his 1905 papers can be found at the Chronology of Milestone Events in Particle Physics. These papers are

• Concerning an Heuristic Point of View Toward the Emission and Transformation of Light. (Introduction of light quanta).
• On the Electrodynamics of Moving Bodies. (The great relativity paper.)
• Does the Inertia of a Body Depend on its Energy Content ? (Septemebr 1905, E = mc² paper).

Einstein’s paper on Brownian motion is available here.

As part of the Einstein centenary I’ve decided that it’s time to brush up on some general relativity (GR). A few years back I was going to do the GR course in the local physics department. When I started my degree, there were no physics prerequisites and there was no way in hell I was going to do any physics labs, I’m a theorist not an experimentalist (I did a huge amount of lab work in a previous life). So I happily went along, until 4th year when they changed the rules, so a few of my fellow students and I were left up the creek without a quantum mechanical analogue of a paddle , we needed physics prereqs, even though the course was given by mathematicians and there were no formal prerequisites the year before. Actually the informal prerequisite used to be “sufficient mathematical maturity”, which meant having done such a shitload of maths that you could probably handle anything they threw at you.

Anyway, enough griping. I never got any formal training in relativity but that doessn’t matter, I think 5 years of research has given me enough mathematical maturity. When I started my PhD my supervisor gave me a rather large pile of notes that took me more than a year to work through, and that was only to get up to speed with the topic.

Fortunately I’ve been doing my PhD part-time (until recently I was working full time) so I’ve had more than enough time to read all the related literature (and it is voluminous, considering this particular field only begain around 1990). I really should be working on something related to my PhD but I’d rather fiddle with the equations a bit more than write them up and get them published. I have two papers sitting here that are almost ready to submit to the journals. I need to trim them a bit before submitting, I think I’ve put in too many details. I’ve taken an interlude in my PhD studies recently and have been reading on whatever topic takes my fancy: differential geometry, cubic equations, relativity and a fair bit of cosmology. Lately I’ve been fascinated by the accelerated expansion of the universe, but I’ll save my musings for another post.

I’ve got three wonderful books on relativity sitting on my desk right now and a bunch of reprints, preprints and other stuff all over the place.

Now I just need to find a few spare hours, a place to put my whiteboard and some peace and quiet.