Actually, ##t## doesn't matter either - you do not need to measure it. All that matters is the value of ##\phi## at the perihelion of consecutive orbits.
Not consistently.
The shell theorem essentially relies on being able to pick a potential that is zero at infinity. Unfortunately, this is incompatible with an infinite constant density (as you have shown here). Considering the Newtonian gravitational potential such that
$$
\nabla^2 \phi = 4\pi...
So the first thing you should ask yourself is: "Does the expression make sense at all?"
What does
$$
\int T^{\mu\nu} \sqrt{-g}\, d^4x
$$
actually mean? In essence, you are adding together tensor components from different parts of your spacetime and this generally holds no particular meaning. The...
The fan accelerates the air from essentially standstill to some velocity. The sail reflects the air from that velocity to some velocity in the back direction. What is the net momentum change of the air? Is whatever velocity the air had between fan and sail even relevant?
Newton’s third law does not apply to the latter part. The forces you are talking about do not form a third law pair.
The third law partner of the force of the air on the sail is the force from the sail on the air.
Apart from the original introduction by Pauli (which was needed for momentum/energy conservation as mentioned, but at least as importantly for angular momentum conservation), there are also many arguments in hindsight. For example, without the neutrino the electron could not interact at all via...