Transformation Rules

For our math below, we are using the apostrophe (‘) to indicate the new version of a variable. We transform a and we get a’.

T a = a'

Contravariant Vector

v'^{\alpha} = v^\alpha \dfrac {\partial x'^\alpha } {\partial x^\alpha}

Covariant Vector

v'_{\beta} = v_\beta \dfrac {\partial x_\beta } {\partial x'_\beta}

Tensor

The text markup that we use below will not let us use Greek letters so we used a,b,c,d,e.

T'^{abc}_{de} = T^{abc}_{de} \dfrac {\partial x^a}{ \partial x'^a} \dfrac {\partial x^b}{ \partial x'^b} \dfrac {\partial x^c}{ \partial x'^c} \dfrac {\partial x'^d}{ \partial x^d} \dfrac {\partial x'^e}{ \partial x'^e}

We would like you to see an example where a martric operates on a vector to make a new vector. This will seem like a detour but we will end up with something that matches to what we have above.

\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} = \begin{bmatrix} v'_1 \\ v'_2 \\ v'_3 \end{bmatrix}

We can use the above matrix math to get equations for the three v prime variables:

v'_1 = a_{11}v_1 + a_{12}v_2 + a_{13}v_3

v'_2 = a_{21}v_1 + a_{22}v_2 + a_{23}v_3

v'_3 = a_{31}v_1 + a_{32}v_2 + a_{33}v_3

Scrutinize those three equations. Each new component accepts contributions from all three both components.

We can say that each component is linearly dependent on the set of old components.

Appendix A

We anticipate some confusion because the idea of there being rules suggests we will be telling you what is allowed and what is not allowed (and hopefully also be telling you why). What we have above is not a list of rules.

We can make some guesses based on some of the math work we’ve seen. If they tell us something is true, then anything that would make it false has to be disallowed, and we would expect the axioms (rules) to enforce the disallowal.

Appendix B

  • ab = c
  • I=df
  • I is an identity element

Iab= Ic = c

I=df

(df)ab = Ic

dfab = Ic

dafb = Ic

(da)(fb) = Ic

We can’t do the above unless we can move a,b,d,f around in a term and that requires Associativity.

(da)(fb) = c

Appendix C

\displaystyle v = \sum_i v^i e_i = \sum_j v'^j e'_j

Appendix D

When we chose a new basis it must be a linear combination of the old basis.

Appendix E

Really work if you and your Transformations that were simple stories of changing inches to centimeters for rotating a vector by an angle. Apologies if this was boring; we have transformations coming that involve somebody flying by the physics story in a spaceship at half the speed of light.

Appendix R

What the Red Queen says:

When we told you that the new components are all linearly dependent on the old components, that wasn’t a gift. That was a limitation. If you want a relationship between old and new that isn’t a set of linear equation, you can’T HAVE IT!