User:Thepigdog/General Relativity - 02 - Contra variant tensors

Special relativity is OK in un-accelerated reference frames. But it does not apply in accelerated reference frames. And we saw that there are difficulties with the way gravity effects light that indicate the curved nature of space time.

We need to consider in general how to convert between reference frames. This theory is called tensor calculus.

In general tensor calculus applies in N dimensional space. The formulas apply in N dimensional space, but it is hard to imagine N dimensional space. By setting up the theory, we do not actually have to. The mathematics can do the work, and we can think in whatever spaces we are comfortable with.

Converting between co-ordinate systems

Suppose there are two co-ordinate systems x and y. Then each point in space may be identified in either co-ordinate system. Then y is a function of x.

$x=x(y)$

Or in components,

$x^{\mu }=x^{\mu }(y)$

Now consider a small displacement vector $dx$.

$dx=dx(y)$

Or,

$dx^{\mu }=dx\mu (y)$

Then calculus tells us that,

$dx^{\mu }=\sum _{\nu }{\frac {\partial x^{\mu }}{\partial y^{\nu }}}dy^{\nu }$

The displacement vector $dx$ is a direction in the co-ordinate system. Now suppose that there was a force being applied in that direction, or the wind was blowing in that direction. All these vectors are called contra-variant vectors and they transform in the same way as a displacement vector.

So suppose there is contra variant vector V (a vector that represents a magnitude and direction at a point in space). Its co-ordinates in the x system are represented by, $V_{x}^{\mu }$ and in the y system by, $V_{y}^{\nu }$ and,

$V_{x}^{\mu }=\sum _{\nu }{\frac {\partial x^{\mu }}{\partial y^{\nu }}}V_{y}^{\nu }$

Interaction of multiple vectors

Suppose there now two contra variant vectors A and B interacting together to form a matrix where the components are multiplied.

$A_{x}^{\mu }B_{x}^{\nu }=\sum _{\alpha ,\beta }{\frac {\partial x^{\mu }}{\partial y^{\alpha }}}{\frac {\partial x^{\nu }}{\partial y^{\beta }}}A_{y}^{\alpha }B_{y}^{\beta }$

Then the product is called a contra variant tensor of rank 2, represented by,

$T_{x}^{\mu \nu }=A_{x}^{\mu }B_{x}^{\nu }$

and,

$T_{x}^{\mu \nu }=\sum _{\alpha ,\beta }{\frac {\partial x^{\mu }}{\partial y^{\alpha }}}{\frac {\partial x^{\nu }}{\partial y^{\beta }}}T_{y}^{\alpha \beta }$

Einstein summation convention

The Einstein summation convention is to drop the the summation symbol when there is a repeated index that appears where one term is a multiplier, and one term is a divisor. Later this definition will be extended to co-variant vectors.

So the above tensor transformation may be written,

$T_{x}^{\mu \nu }={\frac {\partial x^{\mu }}{\partial y^{\alpha }}}{\frac {\partial x^{\nu }}{\partial y^{\beta }}}T_{y}^{\alpha \beta }$

This notation works because the distributive law allows us to move all the summation to the start. For example,

$(\sum _{i}m_{i})(\sum _{j}n_{j})=\sum _{i,j}m_{i}n_{i}$