ds

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 ds^{2} = g_{αβ}dx^{α}dx^{β}
where β_{a} = g_{0a} and
α = √(β^{a}β_{a} − (g_{00})^{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:

∂_{t}g_{ab} = −2αK_{ab} + ∇_{a}β_{b} + ∇_{b}β_{a}

∂_{t}K_{ab} = −∇_{a}∇_{b}α + α(R_{ab} + K_{ab}K − 2K_{ac}K^{c}_{b})
+ β^{c}∇_{c}K_{ab}
+ K_{ca}∇_{b}β^{c}
+ K_{cb}∇_{a}β^{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.

Contact

On deriving the first equation:

ds^{2} = g_{αβ}dx^{α}dx^{β}
= g_{ab}dx^{a}dx^{b}
+ 2g_{0b}dx^{0}dx^{b}
+ g_{00}dx^{0}dx^{0}
= g_{ab}dx^{a}dx^{b}
+ 2g_{0b}dt dx^{b}
+ g_{00}dt^{2}

I presume that the somewhat mysterious definition of α makes things simpler somewhere.