In Newtonian mechanics, when a particle is moving along a path in space, the history of the particle is described in terms of absolute time: 
.
In general relativity, the same particle path is described by a world line in 4-dimensional spacetime. The world line consists of successive events in the particles history, specified by the respective points in spacetime. Each point 
, 
consists of a time coordinate 
and 3 coordinates in space 
.
According to Einsteins principle of general covariance, it is possible - and sometimes nescessary - to formulate the physical laws in equations, which takes the same form whatever coordinates 
we choose. This step from a description in inertial frames to a general covariant description also means, that we cannot use t as an absolute parameter any more. In relativity, t is just another coordinate ( 
), and we introduce the more natural parameter 
- the
proper time in our description of the world lines: 
.
The proper time is the time read by "a little physicist" with a callibrated
standard watch travelling with the particle. The proper time is independent of any coordinate system, and thus an invariant quantity.
But how do we use this to specify world lines in spacetime?
In Euclidian 3-dimensional space, any curve can be parametrised by using the arc length s. Between neighbouring points and 
, i=1..3; the arc length 
is given by Pythagoras' theorem in Cartesian 
coordinates:
or in polar 
coordinates:
The separation 
is independent of the choice of coordinate system and
is of course always positive. The point can be described in an arbitrary coordinate grid. If the original Cartesian coordinates (x,y,z) are functions of the new coordinates:\
and the coordinate differences are expressed by:
we can reformulate the expression for by substituting
and 
. We find:\
In the last expression the convenient Einstein summation convention is introduced, it says: sum over each repeated index (eg. i and k). Further more we have defined the metric tensor 
which describes the coordinate system and the curvature of space.
The metric tensor can also be written in matrix-form:\
Only 6 of these 9 functions 
are independent, because
. If we use the rectangular Cartesian coordinates,
takes the simple diagonal form:
.
Now we will look at the metrics and metric tensors of some simple surfaces, and the curvature of the surfaces.
Surfaces
The metrics discussed in the previous section, can be used to define a surface (the separation defines the allowed lines to move in). In table 1, the metrics of some simple surfaces are listed.
Table 1: The plane in Cartesian and polar coordinates and the sphere in polar coordinates.
Figure 1: Simple surfaces. A plane and a sphere corresponding
to table 1.
The (infinite) plane is of course a flat surface. But the sphere curves into the third dimension. The sphere is normally illustrated as a curved 2D surface mapped into three dimensions. This process is called
embedding. We can imagine a sphere of a given radius existing on it's own (where flat creatures could walk around), but because we live in 3D space, it is easier to think of a sphere embedded in three dimensions. We will now investigate the concept of curvature a little further because it is essential in the study of Black Hole geometry
Curvature
With the analytical expression - the metric defining a surface - we can use Gauss' curvature formula (see [2] p. 67) to calculate the curvature K at any point of the surface. We will write it in the restricted case where the metric tensors are in diagonal form, and the metric is orthogonal (all grid lines for the two coordinates 
always cross at right angles). The formula includes differentiation of
and 
in each direction:
Applied to the the three surfaces in table 1, we get K=0 for the plane in both Cartesian and polar coordinates (as we should). The spherical case, where 
and
gives 
.
The curvature of a sphere is thus 
. The curvature goes to infinity when 
and the sphere looks like flat space (K=0) when 
.
Gauss' curvature K is an invariant geometrical property of any surface, because it does not depend on the choice of coordinate system (as seen with the case with the plane). The curvature of a surface can change from one point to another. The analytical formula is quite complicated already in two dimensions. Actually this is the only independent function out of 16 in the curvature tensor. In n dimensions the so called Riemann curvature tensor 
has 
components, but
because of many symmetry relations (e.g. 
) and the special symmetries of the object (e.g. spherical symmetry) most of the components are identical, and the tensor reduces in complexity. For the n=2 case, there was only one ( 
-15) function (the Gauss formula). For n=3 we will later find an expression for the Schwarzschild metric.
When we work with curved space with dimensions higher than 2, we cannot rely on our geometrical intuition.
One example is negative curvature. What does that mean?
In table 2, three qualitative measurements of the curvature of surfaces with different curvature are listed:
Table 2: The facts from Euclidian geometry will not hold when space curves.
After this short space-travel intermezzo, we will return to the 4D space-time.
Until now it has been pure geometry and mathematics. The physical universe is described by a 3 dimensional space and a time dimension. This 4-D quantity is called Minkowski spacetime. The separation in 4-D spacetime is described by a metric in very much the same way as space alone. We cannot use Pythagoras' theorem directly, because time has to be treated with the opposite sign:
Now there's three different cases:
The separation of two events are said to be timelike. This is the case of material bodies.
The separation is spacelike and cannot lie on any worldline (also called absolute separated).
The separation is lightlike. This describes the worldline of a light ray, and is
called a null separation.
In 2 dimensional space, we can draw a lightcone into the future (the radius increases as time goes by). This represents light propagating in circles. In real 3-D space, light (eg. from a star) propagates in all directions defining a sphere with growing radius as time passes.
We are now ready to deal with the metrics describing Black Holes. In the next section, we will take a short look back in history...
