Spacetime and Geometry: Outstanding Issues

This page is just a list of unfinished business, which I may come back to later.

Exercise 3.01 Consequences of metric compatibility

I could not do the second part of this (proving the covariant derivative of LC tensor vanished).

Exercise 2.08 Exterior derivative and modified Leibnitz rule

I could not prove the modified Leibnitz rule for the exterior derivative. But I got very close. I wish I could prove it.

Exercise 2.06 Helix

In "Note c) The polar tangent vector" we discover that all is not well with the tangent vector the polar coordinates. It would be nice to resolve this.

Commentary 2.9 Differential forms

I proved that the wedge product of an n-dimensional 2-form and 1-form is completely antisymmetric therefore a 3-form. It would be nice to explore an n-dimensional 2-form and 2-form.

It would be even better to prove, for a p-form A and q-form B if (A∧B) was always (p+q)-form..

Commentary 2.7 Causality

In Part 4. A Diversion, I speculate on the trajectory of a photon or massive particle in the cylindrical spacetime, where travel back in time is possible. Sadly I was unable to plot it.

Rules for tensors, matrices and indices

In this tome I left three issues

Prove that S^μρ_σ from (4.5) is a well-defined tensor and that it would not be well defined if we contracted over two upper or two lower indices.
Prove that covariant coordinates vary the same way as the basis vectors (5a22).
Fill in more jargon on tensors