${\mathrm{ds}}^2={\mathrm{dx}}^2+{\mathrm{dy}}^2+{\mathrm{dz}}^2-{\mathrm{dt}}^2$

the metric for flat space (no gravity). This gives us a nice simple expression for the distance between points in Minkowski space i.e.

$\Delta s^2=(x_1-x_2{)}^2+(y_1-y_2{)}^2+(z_1-z_2{)}^2-(t_1-t_2{)}^2$

(We use this metric in flat space because light cone (e.g. $x=ct$) are invariant under Lorentz transformations, the transformtions between inertial frames of reference that leave the speed of light invariant; and ${\mathrm{ds}}^2=0$ along light cones is hence invariant.)

Other metrics are more complicated, they may depend on position and hence we have to integrate along a path to determine its length.

The geodesic postulate states that the path followed by particles within a metric will minimise the length of the path i.e. ${\int }_A^B\mathrm{ds}$. To actually calculate the geodesic equations that determine these paths requires something called calculus of variations, but we don't need to actually calculate them at the moment.

We do need one more fact though. We know that a postulate of relativity is that no particles can travel faster than the speed of light; we also know that the only ones that do travel at the speed of light are massless e.g. photons. With this in mind, we can define the separation between two events in space time (i.e. two 4D co-ordinates as above) in three ways using ${\mathrm{ds}}^2$:

a) The two events are timelike separated if ${\mathrm{ds}}^2<0$;

b) The two events are spacelike separated if ${\mathrm{ds}}^2>0$;

c) The two events are lightlike separated if ${\mathrm{ds}}^2=0$.

If two events are timelike separated in flat space, then this basically means that there exists a frame of reference in which they occur at the same point in space but at different times (don't worry about why this is for now; if you don't like this then we can prove it but we have to do into algebraic detail which I think isn't useful at the moment.) Indeed, this time separation is the least time separation between the two events in any frame (to show this, think about the invariance of the interval $\Delta s^2$.) We call this separation the proper time (you probably know this as the time measured in the frame in which an event occurs; this is just the same thing.)

We have a second postulate, the clock postulate, which says that any clock following a path through space time (called a world line) measures the proper time for that path. We have that

$\tau ={\int }_A^B\mathrm{dt}$

where $d\tau =\mathrm{ds}$ when ${\mathrm{ds}}^2$ is positive (i.e. when two close by points are timelike separated, $\mathrm{ds}$gives the infinitessimal increment in proper time, $d\tau$.

One final word of warning - you may notice some books which have ${\mathrm{ds}}^2$ having the opposite sign. This is just a different convention; you have to just invert the definitions of timelike and spacelike.

I realise may be very hard to understand as i've skimmed over some issues, let me know how you get on with it. We'll need the idea of proper time for the next bit.

In a bit more detail, Lorentz transformations preserve the Minkowski distance above i.e. $\Delta s^2$; and hence this preserves the e.g. timelike nature of the two events. So it definitely possible for e.g. two timelike events to be separated only be space, as this changes the sign of $\delta s^2$. However it is non-trivial to show that a frame of reference exists such that two events are at the same point in space, and this is where the algebra comes in.

OK, will post the next bit of the explanation in a bit...

The link between these curved spaces and gravity is provided by the Einstein field equations. The Schwarzchild metric is the (unique) solution to the Einstein equations that is spherically symmetric (that is, the behaviour of the metric only essentially varies with respoect to the radial distance $r$) and satisfies the 'vacuum' Einstein equations for $r>0$.

We find that the above definition eventually gives us

${\mathrm{ds}}^2=-(1-r_s/r){\mathrm{dt}}^2+{\mathrm{dr}}^2/(1-r_s/r)+r^2(d{\theta }^2+\sin {\theta }^{2}d{\phi }^2)$

(the calculation of this is a) very long and b) involves tensors, which I definitely do not want to introduce here!)

It appears that we have a singularity in our metric at $r=r_s$. However we can show that this singularity is simply a co-ordinate singularly, one that has manifested itself due to a 'bad choice of co-ordinates'. Indeed, we find the curvature tensor is perpendicular to $r^{-6}$, so the singularity is not very singular. (Sorry I'm not elaborating on this but it would be very lengthy indeed to fully introduce tensors!)

What this tells us is that essentially that the Schwarzchild radius is not special, and that in some sense it is no different from any other radial distance.

OK, think it's probably best to let you read this before carrying on. The next bit will unfortunately have to make more assumptions that I can't really prove quickly, relating to the dynamics of falling objects in this metric. From this we can get a first glimpse of this property of objects never seeming to fall into black holes.

BTW, please feel free to ask questions - you are right Colin, this is very interesting, and questions are more than welcome.

We assume that our object is on a radial path i.e. $d\tau =d\phi =0$. We suppose $\tau$ is the proper time of the particle and $t$ the time measured by an observer at infinity.

It is possible to deduce that

${\displaystyle {(\mathrm{dr}/d\tau )}^2=E^2-(1-\frac{r_s}{r})}$

where $E$ is the energy constant, and also that

${\displaystyle \frac{\mathrm{dt}}{d\tau }(1-\frac{r_s}{r})=E}$

(These equations are essentially a form of the geodesic equations. They are derived from looking at the Lagrangian of the system and looking for constants of the motion. If this means nothing to you then don't worry, if it does I could explain how to get the above equations).

We obtain from the first equation that

${\displaystyle -d\tau =\mathrm{dr}/\sqrt{E^2-(1-\frac{r_s}{r})}}$

If $E$ is alrge enough (andsince this corresponds to energy, we just let our particle have enough energy as we like) then we can integrate the above to show that $\tau$ is finite at $r=r_s$ (I think you need a sinh squared substitution to show this.)

However,

${\displaystyle \mathrm{dt}=\frac{E}{\frac{1-r_s}{r}}\frac{\mathrm{dr}}{\sqrt{E^2-(1-\frac{r_s}{r}}}}$

${\displaystyle \Rightarrow \mathrm{dt}\approx \frac{-\mathrm{dr}{r}_s}{r-r_s}\text{to first order}}$

${\displaystyle \Rightarrow t~-r_s\log \frac{r-r_s}{r_s}}$

i.e. $t$ is infinite at $r$ tends to the Schwarzchild radius.

I'll post some more when I get time- sorry about the long delay between posts! If you want to read a book on this stuff, I can recommend two texts:

Introducing Einstein's Relativity by Ray D'Inverno;

Relativity: Special, General and Cosmological by Wolfgang Rindler.