General Relativity

The basic idea of general relativity (GR) is, that the spacetime is a four dimensional topological manifold M with a differentiable structure. To exclude mathematical details such as changing the manifold by e.g. including certain subsets in the family of open sets, one demands that the manifold has a extremal topology. (I won't get into details here)
One basic feature of GR is, that one does not necessarily want the manifold to be embedded in a higher dimensional space. Therefore one seeks to express all quantities of interest in terms of features of the spacetime itself. This is done by defining the tangentvectorspace in a point

. I.e. a vector is an equivalence class of curves through p in M. This defines a vectorbundle on M, the tangentbundle

. The tangentspace has a dualspace (that is the space of linear operators on

), called the cotangentspace

. This space is also a vectorspace which gives one another bundle, the cotangentbundle

. With these vectorspaces (bundles) one can define tensors (tensorfields) in p (on M). A tensor of type

is a multilinear mapping that assigns to p vectors and q covectors a number:

One of the assumptions of GR is that all physical fields are described by tensorfields and the describing field equations are tensor equations and therefore independent of the co-ordinate patch used. For example one assumes (for good reasons), that the gravity field should be associated with the curvature of the spacetime and that the gravitational field is described by a nondegenerate symmetric tensorfield g of type (2,0). Therefore it assigns to two arbitrary vectors a number as follows

In abstract tensorial notation g may be written as

. Since g describes the distance of two points, that are infinitesimal far apart in terms of their co-ordinates, one often writes

In GR one deals with a so called "lorentzian metric". This means that there exists co-ordinates, such that g has the following form in a point:

Using the metric one can identify the tangent with the cotangent space in a point: i.e.

with

can be viewed as an element of

.
A classification of vectors can also be done:
If

then v called

like.
This leads to the classification of curves, hyperplanes, etc. into spacelike (timelike, null) curves [hyperplanes] if the tangentvector [normalvector] is spacelike (timelike, null) [timelike (spacelike, null)] throughout the curve [hyperplane].
Since points on the spacetime correspond to events in the "real world" one needs to compare the values of tensorfields in different points of the manifold. So given the value of a tensorfield in two points p, q (that means one has got two tensors one in p and the other one in q) one can compare these two tensors by paralleltransporting, lets say, the tensor in q to p along a curve

. The notion "parallel transport" is defined via the so called connexion (covariant derivative).
Let

denote the tangentvector along

, than one says a vector v is paralleltransported if the covariant derivative along a curve vanishes:

By imposing certain properties (that I will not quote here) one can extend the action of

to all types of tensors.
This derivative operator is unique if one imposes the following:

If one computes condition ii) in a co-ordinate basis one gets a formula for the

The extension of the first of Newton's axioms to GR is, that a freely falling particle follows a geodesic of the spacetime. A geodesic is a curve, with a tangentvector that is paralleltransported.
Let

denote the geodesic then its tangentvector

obeys

then, inserting this into equation (1) one gets the geodesic equation in a co-ordinate basis:

Using this equation one can determine the physical meaning of the

:
Set

and

. It follows that

So the

determine the acceleration of the particle moving on the geodesic.
As in classical mechanics it is possible to derive the geodesic equation via a variation principle:
A geodesic is the curve that extremizes the functional

Remark:
For a lorentzian manifold the extremum is a maximum.

The notion of paralleltransport lets one define the curvature of the manifold. The idea is to look at the paralleltransport of vectors along closed curves as can be seen in picture 1

Picture 1: On the left hand side a flat surface is shown whereas on the right a curved surface is depicted

The curvature is proportional to the change of the vector per area enclosed by the curve. As mentioned before this quantity should be a tensorfield, the Riemann tensor. It is defined by:

The Riemann tensor governs the difference in acceleration of two neighbouring free falling particles, the so called tidal forces. These forces also exist in Newtonian theory. There the trace over the equations for the tidal forces gives the field equations for the gravitational field. Doing the analogue in GR gives the Einstein Field Equations (EFE):

Where

is the Einstein tensor,

the Ricci tensor and

the Ricci scalar or curvature scalar.

, is the universal gravitational constant and

is the stress-energy tensor.
One can regard the

as being dimensionless whereas the co-ordinates

have a dimension of

. Then it is clear that the

have a dimension of

since they are derivatives of the metric with respect to the co-ordinates. Therefore R has dimensions

and for

follows a dimension of

.
The stress-energy tensor is a symmetric tensor and of the form

Where

is the energy density (i.e. in electrodynamics the energy density of the electromagnetic field),

is the energyflux density (i.e. in ED the Poynting vector) and

is the stress tensor (i.e. in ED the Maxwell-stress-tensor, a generalisation of the notion of pressure)
* It is encouraging if one is interested in quantization of gravity, that the EFE can be obtained by a variational principle (which means that it is possible to give a Lagrangian/Hamiltonian formulation of GR). The action for the spacetime is:

The first term is the Einstein-Hilbert action for the Metric g, the second one is the Gibbons-Hawking surfaceterm, which is often ignored and the third term is the action for all the fields that live in the spacetime (i.e. Yang-Mills fields, electromagnetic fields, ...).
The equations of motion for the above action are fieldequations for the matterfields (coupled to gravity) as well as the EFE with a stress energy tensor of the form