Notes

Notes: Generalized harmonic form of Einstein’s equations from a gauge-fixed action

The generalized harmonic formulation can be derived by adding a gauge-fixing term to the Einstein-Hilbert action

Notes: A family of ramp functions and the Beta function

Sometimes in a numerical method, you need to be able to continuously turn a calculation on or off in space or time.

(Notes) When is a metric conformal to Ricci-flat?

Integrability conditions when trying to solve for a conformal factor

Note on simple(r) equations for Einstein-dilaton-Gauss-Bonnet and dynamical Chern-Simons theories

Here to save you some algebra and column-inches

Note on commutation coefficients in two ways

An identity between vector commutation coefficients and coframe connection coefficients

Note on exterior, interior, and Lie derivative superalgebra

These three objects form a superalgebra! Whoa!

Note on Lie derivatives and divergences

When can you integrate-by-parts with Lie derivatives?

Note on generating a divergence identity

Special thanks to Ben Mares for coming up with this identity.

Note on a (dimension-dependent) Weyl identity

There is a nice 4-dimensional Weyl identity that I can never seem to remember off the top of my head; so I decided I need to write this note so I don’t have ...

Notes on the Weyl, E/B, and 3+1 decompositions of Riemann, and curvature invariants

The Weyl decomposition, further splitting into E/B, and computing in terms of 3+1 objects

Notes on the pullback connection

In an effort to keep myself organized, I decided I should type up some notes I have laying around.