General Relativity

1 Introduction

Einstein's General Relativity is a theory of gravity which is based on the Equivance Principle that asserts that a uniform gravitational field (next to an infinite flat massive body) is indistinguishable from a uniform accelleration.

$\nabla \cdot \mathbf {A} =-4\pi G\rho$

2 Coordinate Systems

We will use the Einstein Notation. Consider Cartesian space in $\mathbb {R} ^{2}$, covered by Cartisian coordinates $(x^{1},x^{2})\,$. Pythagorous can be used to give the squared differential displacement:

$ds^{2}=(dx^{1})^{2}+(dx^{2})^{2}\,$

Now consider a general curvilinear coordinate system (CS) $(y^{1},y^{2}\,)$. Since the lines of the CS are neither parallel nor perpendicular, we form the same quantity:

$ds^{2}=g_{11}(dy^{1})^{2}+2g_{12}dx^{1}dx^{2}+g_{22}(dy^{2})^{2}\,$

In general the metric tensor $\mathbf {g} \,$ is a function of position. The terms $g_{11}\,$ and $g_{22}\,$ describe the relative spread of the axes, while $g_{12}\,$ describes how the axes are not perpendicular.

A particular geometry is called Flat if it can be described by a Euclidian geometry. In general, curved manifolds (such as the surface of a sphere) require a more complex metric.

Example: Metric Tensor in Polar Coordinates

Consider polar coordinates in Euclidean space:

$x=r\cos(\theta ),\;y=r\sin(\theta )$

The differential distance is given by:

$\mathrm {d} s^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}=\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}\,$

which shows the components of the metric tensor are:

$g_{rr}=1,\;g_{\theta \theta }=r^{2},\;g_{r\theta }=0$

Transformation

Now consider a general transformation between two general CS $x,$ and $y,$, and a scalar field $\phi ,$. The scalar field does not depend on the CS used, thus:

$\phi (y)=\phi (x);$.

Now,

Contravariant vector transformation:

$V^{n}(y)={\frac {\partial y^{n}}{\partial x^{m}}}V^{m}(x)$

Covariant vector transformation:

$V_{n}(y)={\frac {\partial x^{m}}{\partial y^{n}}}V_{m}(x)$

Derivative of a Tensor

Scalar

Consider a scalar field $\phi \,$. Clearly the value of the field at any point P is independent of the CS used to label P. If the field is constant on one CS, will be constant in any CS. Note that:

${\frac {\partial \phi }{\partial x^{m}}}$

is a vector.

Vector

Now consider a constant vector field $\mathbf {V}$ in flat space, in the Cartesian CS. The covariant components of $\mathbf {V}$ are also constant - that is, the projections on the axes:

${\frac {\partial V_{m}}{\partial x^{n}}}=0$.

If the vector field is now labelled by a general curvilinear CS, we see that the components of the vectors now change over position, even if the vector is constant, because the CS is not "constant":

${\frac {\partial V_{m}}{\partial x^{n}}}=0$.

Moreover:

${\frac {\partial V_{m}(x)}{\partial x^{n}}}=0\Rightarrow {\frac {\partial V_{m}(y)}{\partial y^{n}}}=0$

which states that vectors do not transform as tensors under a CS change.

Covariant Derivative

We would like to find a new definition of a derivative that maps a vector to a vector, and a tensor to a tensor. "Covariant" in this context means the quantity is independent of the CS chosen. Suppose there exists:

$T_{mn}(x)={\frac {\partial V_{m}}{\partial x^{n}(x)}}$

So:

$T_{mn}(y)={\frac {\partial x^{r}}{\partial y^{m}}}{\frac {\partial x^{s}}{\partial y^{n}}}T_{sr}(x)={\frac {\partial x^{r}}{\partial y^{m}}}{\frac {\partial x^{s}}{\partial y^{n}}}{\frac {\partial V_{r}(x)}{\partial x^{s}}}$

Now compare with (expanding):

${\frac {\partial V_{m}(y)}{\partial y^{n}}}={\frac {\partial }{\partial y^{n}}}{\frac {\partial x^{r}}{\partial y^{m}}}V_{r}(x)={\frac {\partial x^{r}}{\partial y^{m}}}{\frac {\partial V_{r}(x)}{\partial y^{n}}}+{\frac {\partial }{\partial y^{n}}}{\frac {\partial x^{r}}{\partial y^{m}}}V_{r}(x)$

The first term is equal to $T_{mn}(y)\,$, but we have a second term, which we denote by $\Gamma ^{r}{}_{mn}\,$

Hence in gernal we need to differentiate in a way to cancel the second term to get back a tensor:

$\nabla _{p}T_{mn}={\frac {\partial T_{mn}}{\partial y^{p}}}+\Gamma ^{r}{}_{pm}T_{rn}+\Gamma ^{r}{}_{pm}T_{mr}$

The Christoffel Symbols can be found from the metric tensor:

$\Gamma ^{a}{}_{bc}(y)={\frac {1}{2}}g^{ad}\left({\frac {\partial g_{dc}}{\partial y^{b}}}+{\frac {\partial g_{db}}{\partial y^{c}}}-{\frac {\partial g_{bc}}{\partial y^{d}}}\right)$

Parallel Transport and Curvature

Space is called "flat" (like Cartesian space) when it is possible to choose a CS such that the covariant derivavtive of the metric tensor is everywhere zero:

$\nabla _{r}g^{mn}=0$ that is the metric tensor reduces to the delta function: $g_{mn}=\delta _{mn}\,$

A space may be flat in some areas, but there may be curvature at other points. It is always possible to choose a CS such that at a given point, the derivavtive of the metric is zero - that is in, a small enough neighbourhood of any point of a continuous, differentiable space looks flat.

Consider a vector V moving from point A to B over a curve, whilst remaining parallel to itself. This is notion of parallel transport, and can be extended to general nonflat spaces. The covariant derivavtive:

$\nabla _{s}V^{m}={\frac {dV^{m}}{ds}}+\Gamma ^{m}{}_{np}V^{n}{\frac {dx^{p}}{ds}}$

If a vector is parallel transported such that the covariant derivative is always zero, the vector follows a geodesic; geodesics are in some sense the "straightest" lines that exist in the manifold. For example: all straight lines in flat 2D space are geodesics; on the surface of a sphere, all great circles are geodesics.

The geodesic condition is:

${\frac {d^{2}x^{m}}{ds^{2}}}+\Gamma ^{m}{}_{np}{\frac {dx^{n}}{ds}}{\frac {dx^{p}}{ds}}=0$