Thanks for your comment. It will probably take me months to understand Terry Tao’s derivation. I remember that I marked it as interesting when I first saw it but I could never find the time to study it carefully. Also, I consider E=mc2 to be a definition and I’m not sure if one should or could derive definitions. But I may be wrong. Since you mention a geometric interpretation for kinetic and potential energies I thought I comment on that. My understanding is that physicists call Kepler’s rule “conservation of energy” and they write it as (for unit mass)

So they labeled R2/T2 = kinetic energy and 1/R= potential energy (for unit mass). Since this is Kepler’s rule, it has a simple geometric interpretation. Given two orbits R_0 and R I have this geometry:

AE = R_0
BE = w_0
AD = R
DC = w

Then by Kepler’s rule

R^2 and R_0^2 are the areas swept by the radii. Please let me know what you think.

I would like to comment on a modification of first figure [circle] in item #2 Acceleration.

Perhaps you could draw the figure from this description.
Let c be the radius of the crcle.
From orthogonal X and Y axes, with the origin at the center of the circle, draw the chord.
Do this only for the usual Quadrant I [ , ], although this could be done for the other three quadrants.
Reflect the right triangle so that a square is formed.

Compare this to the Gini coefficient diagram.
GOTO the Terence Tao Einstein’s derivation of E=mc^2 discussion.

From my perspective, this may bring economics into relativity:
From the wiki page:
a - the shaded segment labeled gini index may be O(v^3) and
b - the remaining portion of the lower right triangle may be E’ or perhaps bonding energy [?];
From the constructed diagram:
c - may correspond to the upper triangle in Quadrant I,
d - the lower triangle in Quadrant I may correspond to the kinetic energy,
e - the entire square may correspond to potential energy.

Insight - maybe?
Rigor - no.
Geometry in GR - perhaps? - when letting m=1, v=c, a=c^2.

The chord may be the relative velocity of orthogonal particle movement or c*sqrt(2).
The diameter is 2c which may be the relative velocity of two particles traveling 180 degrees from each other, although the vector sum of energy is zero.