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	Einstein's [http://en.wikipedia.org/wiki/General_relativity General Relativity] is a theory of gravity which is based on the [http://en.wikipedia.org/wiki/Equivalence_principle Equivance Principle] that asserts that a uniform gravitational field (next to an infinite flat massive body) is indistinguishable from a uniform accelleration.		Einstein's [http://en.wikipedia.org/wiki/General_relativity General Relativity] is a theory of gravity which is based on the [http://en.wikipedia.org/wiki/Equivalence_principle Equivance Principle] that asserts that a uniform gravitational field (next to an infinite flat massive body) is indistinguishable from a uniform accelleration.

	:<math>\nabla \cdot \mathbf{A} = - 4 \pi G\rho</math>		:<math>\nabla \cdot \mathbf{A} = - 4 \pi G\rho</math>

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== 1 Introduction ==

Einstein's [http://en.wikipedia.org/wiki/General_relativity General Relativity] is a theory of gravity which is based on the [http://en.wikipedia.org/wiki/Equivalence_principle Equivance Principle] that asserts that a uniform gravitational field (next to an infinite flat massive body) is indistinguishable from a uniform accelleration.

:<math>\nabla \cdot \mathbf{A} = - 4 \pi G\rho</math>

== 2 Coordinate Systems ==

We will use the [http://en.wikipedia.org/wiki/Einstein_notation Einstein Notation]. Consider Cartesian space in <math>\mathbb{R}^2</math>, covered by Cartisian coordinates <math>(x^1, x^2)\,</math>. Pythagorous can be used to give the squared differential displacement:

:<math>ds^2 = (dx^1)^2 + (dx^2)^2\,</math>

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

:<math>ds^2=g_{11}(dy^1)^2 + 2g_{12} dx^1 dx^2 + g_{22}(dy^2)^2\,</math>

In general the metric tensor <math>\mathbf{g}\,</math> is a function of position. The terms <math>g_{11}\,</math> and <math>g_{22}\,</math> describe the relative spread of the axes, while <math>g_{12}\,</math> 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:

:<math>x = r \cos (\theta),\;y = r \sin (\theta)</math>

The differential distance is given by:

:<math>\mathrm{d}s^2 = \mathrm{d}x^2 + \mathrm{d}y^2 = \mathrm{d}r^2 + r^2 \mathrm{d}\theta^2\,</math>

which shows the components of the metric tensor are:

:<math>g_{rr} = 1,\;g_{\theta\theta} = r^2,\; g_{r\theta} = 0</math>

==Transformation==

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

:<math>\phi(y) = \phi(x);</math>.

Now,

Contravariant vector transformation:
:<math>V^n(y) = \frac{\partial y^n}{\partial x^m} V^m(x)</math>

Covariant vector transformation:
:<math>V_n(y) = \frac{\partial x^m}{\partial y^n} V_m(x)</math>

==Derivative of a Tensor==

===Scalar===

Consider a scalar field <math>\phi\,</math>. 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:

:<math>\frac{\partial \phi}{\partial x^m}</math>

is a vector.

===Vector===

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

:<math>\frac{\partial V_m}{\partial x^n} = 0</math>.

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":

:<math>\frac{\partial V_m}{\partial x^n} = 0</math>.

Moreover:

:<math>\frac{\partial V_m(x)}{\partial x^n} = 0 \Rightarrow \frac{\partial V_m(y)}{\partial y^n} = 0 </math>

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:

:<math>T_{mn}(x) = \frac{\partial V_m}{\partial x^n(x)}</math>

So:

:<math>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}</math>

Now compare with (expanding):

:<math>\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)</math>

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

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

:<math>\nabla_p T_{mn} = \frac{\partial T_{mn}}{\partial y^p} + \Gamma^r{}_{pm} T_{rn} + \Gamma^r{}_{pm} T_{mr}</math>

The Christoffel Symbols can be found from the metric tensor:

:<math>\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 )</math>

===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:

:<math>\nabla_r g^{mn} = 0</math> that is the metric tensor reduces to the delta function: <math>g_{mn} = \delta_{mn}\,</math>

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:

:<math>\nabla_s V^m = \frac{dV^m}{ds} + \Gamma^m{}_{np} V^n \frac{dx^p}{ds}</math>

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:

:<math>\frac{d^2x^m}{ds^2} + \Gamma^m{}_{np} \frac{dx^n}{ds} \frac{dx^p}{ds} = 0</math>

General Relativity - Revision history

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