Making sense of formulae in this slide show

Taking latin superscripts and subscripts to range over spatial dimensions (1, 2, 3), and greek scripts to range over space and time dimensions (0, 1, 2, 3), this equation can be derived from ds² = g_αβdx^αdx^β where β_a = g_0a and α = √(β^aβ_a − (g₀₀)²). The g_ab’s provide a metric on a time-like 3D slice thru the 4D Lorentzian manifold. Greek letters used as real field variables and integer scripts are not to be confused.

The exogenous field variables α and β provide coordinate conditions.

On slide 4 there are the equations:
∂_tg_ab = −2αK_ab + ∇_aβ_b + ∇_bβ_a
∂_tK_ab = −∇_a∇_bα + α(R_ab + K_abK − 2K_acK^c_b) + β^c∇_cK_ab + K_ca∇_bβ^c + K_cb∇_aβ^c
I assume that latin indexes are raised with the 3D metric g and that ∇ is the covariant derivative on g. K_ab is a symmetric tensor that defines the extrinsic curvature of the 3D space as embedded in the 4D spacetime. Perhaps these were derived from Einstein’s empty space equations G_αβ = 0. These equations are in the conventional form of a Cauchy initial value problem.

Since these slides predict an extraordinary phenomenon based on Einstein’s equations—the expulsion of a black hole from a galaxy—they deserve extraordinary confirmation. Perhaps a straight forward program to produce a small numerical 4D patch of empty space time from the equations, followed by a calculation of G_αβ therein.

ds² = g_αβdx^αdx^β = g_abdx^adx^b + 2g_0bdx⁰dx^b + g₀₀dx⁰dx⁰ = g_abdx^adx^b + 2g_0bdt dx^b + g₀₀dt²
I presume that the somewhat mysterious definition of α makes things simpler somewhere.