Jack SarfattiThe equation in question is (in index-free short hand)
DU/ds = dU/ds + {LC}UU
This is a tensor equation- a geometric object, and the choices of local coordinate patches of differential geometry are irrelevant.
We are interested in this book in the heuristic physical meaning of the equations not in the excess baggage of formalism that only obscures the essential physics leading many physicists astray into purely mathematical dead ends perhaps important to pure mathematics but not to physics. The Cornell philosophy of Hans Bethe, Ed Salpeter, Phil Morrison, Tommy Gold and Richard Feynman himself was to get the most physics with the least possible mathematics. This is in accord with Einstein's remark that any intelligent fool can make the subject more complicated than it need be. Indeed, this is the trend we see in modern theoretical physics today.
DU/ds is what accelerometers measure locally on the test particle being observed.
{LC}UU is what accelerometers measure locally on the detector observer of the test particle.
dU/ds is the apparent or kinematical 4-acceleration of the test particle relative to the detector.
The test particle and the detector are nearly coincident, i.e. their actual space-time separation must be small compared to the local radii of curvature of 4D spacetime for the equation not to break down. For example on Earth surface that curvature radius is about 10^13 cm, so this is not a problem for local experiments.
The Levi Civita connection {LC} in Einstein's GR physically describes the fictitious inertial pseudo forces that appear to act on test particles. These inertial forces are caused by real forces on the local noninertial frames lnifs measuring the motion of the test particle.
Newton's 2nd Law is for rest mass m constant
F = mDV/ds = real 4-force
V = 4-velocity of test particle
ds = proper time differential
DV/ds = dV/ds - {LC}VV = proper 4-acceleration. it is a gct group tensor
In a local inertial frame lif {LC} = 0
This is Einstein's equivalence principle
In a local non inertial frame lnif
{LC} =/= 0
- m{LC}VV = all the inertial fictitious pseudo forces that seem to act on the test particle from the POV of the properly accelerating lnif detector observer, but don't.
Note the g00 in the denominator that is zero at horizons.
Note also that the quantum Unruh effect in which vacuum zero point virtual photons are transformed into real black body radiation photons is proportional to the local tensor proper acceleration DU/ds of the detector accelerometer.