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.

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

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:

$d{s}^2=(d{x}^1{)}^2+(d{x}^2{)}^2$

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:

$d{s}^2=g_{11}(d{y}^1{)}^2+2g_{12}d{x}^{1}d{x}^2+g_{22}(d{y}^2{)}^2$

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 $\varphi ,$. The scalar field does not depend on the CS used, thus:

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

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 $\varphi$. 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:

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

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 ${\mathrm{\Gamma }}^r{}_{mn}$

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

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

The Christoffel Symbols can be found from the metric tensor:

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

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:

${\mathrm{\nabla }}_r{g}^{mn}=0$ 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:

${\mathrm{\nabla }}_s{V}^m=\frac{d{V}^m}{ds}+{\mathrm{\Gamma }}^m{}_{np}{V}^n\frac{d{x}^p}{ds}$

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:

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