### Math 741. Topics in Partial Differential Equations: Linear Stability of Black Holes

- Spring 22 -
Hans Lindblad

The lectures are MW 1.30-2.45 in Krieger 411.

The course starts with a general introduction to Einstein's equations, in particular the exact black holes
solutions, and the wave equation nature of Einstein's equations,
that can be found in the text books:

Carroll * An introduction to general relativity: Spactime and Geometry*

Hawkinng and Ellis *The large scale structure of space-time *

Ohanian and Ruffini *Gravitation and Spacetime *

Wald *General Relativity *

*Physics Unsimplified GR Lectures*

After that we will go into material from recent research papers about linear stability of black holes, by Hintz-Vasy or Fang in the case of positive cosmological constant and Holzegel-Dafermos-Rodnianski, Johnson, Hung and Haffner-Hintz-Vasy et. al. in the case of vanishing cosmological constant, to the extent that time permits.
Some background in Geometry and PDE would be helpful, and
for the later parts also Functional Analysis and Microlocal Analysis.

V. Schlue *Linear Waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes*

M. Dafermos, I. Rodnianski: *Lectures on black holes and linear waves*

M. Dafermos, I. Rodnianski: *The wave equation on Schwarzschild-de Sitter
spacetimes*

The preliminary plan is:

Einstein's equations and the Geometry of Spacetime.

The Exact Solutions to Einstein's vacuum equations.

The Cauchy Problem in General Relativity.

The Linearized Einstein equations around the exact solutions.

The wave equation on the background of the exact solutions.

Linear stability of Black Holes with positive cosmological constant.

Linear stability of Black Holes with vanishing cosmological constant.

My Lecture notes will be available on blackboard

Some interesting articles about black holes in the news:

*Are we living in a baby universe that looks like a black hole to outsiders?*