Journal: Feb 16, 2009

Bryan Newbold, bnewbold@mit.edu
http://web.mit.edu/bnewbold/thesis/

This past week we've been covering generic operators in 6.945, which is great because it's a set of questions I'm just getting around to. What are the important data-types/-structures/objects in general relativity, what are the important operators and procedures, and what are their arities and applicabilities? Here are some potential object types (I'm using "object" in a general sense, not with the usual CS implications):

manifold (arbitrary dimension)
vector
tensor
vector and tensor fields
metric (a rank-2 tensor)
1-form (with implied metric)
chart, atlas
coordinate system
parameterized curve
coordinate basis
intervals
Hamiltonian, Lagrangian
geodesic

Some basic operators:

norm/inner-product
contraction
tensor raising/lowering (w.r.t. a metric)
differential form
symmetrize/anti-symmetrize

Some tests:

Lorentzian? (of transformations)
Poincare? (Lorentzian plus translation)
timelike?, spacelike?, lightlike? (of intervals and curves)
zero-curvature? (of manifolds)
rank? (of tensors)
dimension? (of manifolds)

Some more specific operators:

Riemann curvature

And some tools:

evolve
find-geodesic
find-interval

Turns out I wasn't using the latest version of scmutils and that's why a bunch of calculus functionality was missing... whoops!

Time is tick ticking away but I feel that things are starting to fall together, doing problem sets for my GR class is super valuable, I wish i'd done more practice exercises before to test myself. But overall I feel pretty mathematically prepared.
Over the weekend I went back and tried to review some topics from topology like Baire dimension theory, embedding, and the fundamental group. I never really covered these topics when I studies topology before, so I wasn't able to gleam any insight.

(previous entry) - (next entry)