I can help. All of this stuff is coded cosmology kind of talk.
ds2 = dT2 - a(T)2dr2,
where geometrized units of c = 1 are used and dr2 is the interval square of the three spatial dimensions, namely, + dx2 + dy2 + dz2.
It only means that we looking at length, width, height, and time and taking a measurement basically using the Pythagorean approach of adding the squares of each term (l,w,h,t) and taking the square root to find a real distance of separation in space between two points. All I did was take a sphere - ignore the interior - and ask myself: How would someone evaluate the growth of a sphere over time if all they had was the surface area of the sphere? That is, with no radius. That led me to equation 3. Then, for the same sphere I used the standard radial equation for surface area (Eq. 4) and laid the equations side by side. I found two distinct - but inextricably related - timepieces. I saw that the clocks were oriented orthogonally, meaning I was probably looking at the SINGLE Minkowski spacetime term we call time - the 4th dimension - broken out into two distinct time zones.
The clocks were ticking off time in step with one another but with very different values. For every movement of the "surface" clock I saw the squared exponent move on the "radial" clock. After investigating their scale factors in Eq. 7, I said "That is strange. The only variables in Eq.7 are a) the opening surface area size Sp of the sphere, and b) the two clocks."
What if I naively inserted today's Hubble value on the radial clock - 13.7 billion years - to describe a vast universe and then inserted only 6000 years on the "surface" clock? What would Sp look like? Well, it turned out to be about the size of a grapefruit.
The bonuses were that 1) the metric remained Friedmann in style and so standard cosmology was not upset, and general relativity was upheld, 2) if I made the surface clock precisely the cosmic microwave background expansion and temperature, everything fit like a glove, 3) the Hubble expansion was vast.
And 4) a point I had missed in 2008 in the original paper --> if the Hubble frame is an accelerated frame (which it very much seemed to be, remember the squared exponent), and the CMB frame was the rest frame of the universe, then it may be very well impossible to rule out a 6000 year old universe by simply saying "Today's cosmology does not allow for a system that young". On the contrary, everything I was finding said "Wrong. It could very well be that young and I have just used today's modern cosmology to demonstrate it".