When taught how airplanes fly, we are probably always given the Newtonian mechanism: that four forces act on an airplane; lift and weight on the vertical axis, and thrust and drag on the horizontal axis. Since gravity is a force in Newtonian mechanics, this seems to predict that passengers on an airplane in flight should feel both their weight due to gravity and additional force due to lift at all times, not just when taking off. However, this is not what happens; passengers in an airplane coasting at a constant velocity feel their normal weight due entirely to lift, as implied by general relativity. Is this intuition correct? What I'm really asking is, do general relativity and Newtonian mechanics predict different vertical forces on airplanes?
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$\begingroup$ This question seems to be ill-posed- but just draw a free-body diagram of a passenger in a plane and you will get the right answer. $\endgroup$– niels nielsenCommented Nov 17, 2021 at 7:13
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$\begingroup$ That should also depend on approximations that you made ... What will be your gravitational potential of earth etc... $\endgroup$– VidCommented Nov 17, 2021 at 7:40
2 Answers
Since gravity is a force in Newtonian mechanics, this seems to predict that passengers on an airplane in flight should feel both their weight due to gravity and additional force due to lift at all times, not just when taking off.
Whether you are coasting in an airplane or on the ground, if you have 0 net acceleration, the force you feel is the normal force of the airplane or the ground pushing you upwards. Passengers do not feel lift directly. The airplane feels lift and the passengers feel the seats or floor beneath them. Any difference they feel from their normal weight will be accompanied by a net vertical acceleration. $N = M(g+a)$: normal force provides net upward acceleration "a" after opposing acceleration due to gravity "g".
In the Newtonian description there are two opposing forces on the overall plane: lift and gravity. In the GR description the plane would travel along a falling geodesic due to gravity, but lift is a force that deviates the plane from its geodesic. I would say the descriptions are different so in a sense the described forces are different, but the predictions are (to very good approximation) the same in terms of what passengers feel and in terms of accelerations.
Newtonian mechanics emerges from General Relativity mathematically, .
When Euclidean coordinate lengths are replaced by the metric lengths of a curved geometry within Newton’s second law of motion, the metric form of the second law can be shown to be identical to the geodesic equation of motion of general relativity. The metric coefficients are contained in the metric lengths and satisfy the field equations of general relativity. and energies.
So the question to ask is if the velocities and masses of aircrafts moving in the atmosphere and in the earth's field are such as to need the corrections to the vertical forces from special and general relativity.
These corrections are necessary for the GPS system to work at the needed accuracy, but not large enough to be felt for passengers in aircrafts.
