### The Lagrangian Method

### 1 Black Hole

- Location of apparent horizon as
determined by the residual of the apparent horizon equation
using Eddington-
Finkelstein coordinates. The expected plot for r=2 for M = 1 is found
.
- Location of Apparent Horizon
as determined by the general apparent horizon equation in Scwarzschild
coordinates.
- Case1: A test of the general
equation for 1 black hole reducing to the residual case.
- Case2: General
solution of apparent horizon location for 1 black hole.

### 2 Black Holes

- Location of the Apparent Horizon for two black holes at the
critical separation. The critical separation for one marginally trapped
surface encludes
both holes for the general apparent horizon equation in Schwarzschild
coordinates.
- Case3: Location of Apparent
Horizon for 2 bhs for the reduced case.
- Case4: Location of
apparent horizon for 2 black holess for the general case.