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I'm looking for some simple paper/textbook explaining the fact that time dilation in the Schwarzschild's metric can be translated into the classic equation of motion in a gravitational field. That is:

Least action (Classical Mechanics) $\leftrightarrow$ maximal proper time d$\tau$ (General Relativity).

I'd rather avoid explanations with tensors (upper and lower indices) because I soon get lost with that notation. A simple (high school/undergraduate level) 1 (space)+1 (time) toy model could do to guide me to a deeper understanding and towards more general cases.

The closer thing I've found so far is Feynman's Lectures 42-8 - Motion in curved space-time but that is not as direct as I hoped... and much details are not explicitly stated

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    – Qmechanic
    Nov 23 '21 at 13:50
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As a guide to deeper understanding:

In classical mechanics we have that theory of motion can be expressed in terms of energy exchange.

As we know, newtonian gravity is a conservative force, hence we can define a potential energy. As we know, it is customary to take infinite radial distance (to the source of gravity) as the zero point of potential energy.

At every radial distance $r$ to the source of gravity the value of local potential energy is obtained by evaluating how much kinetic energy a test object acquires in the process of free falling from infinity to radial distance $r$.

As we know: Hamilton's stationary action is stated in terms of kinetic energy and potential energy. When the Lagrangian $(E_k-E_p)$ is inserted in the Euler-Lagrange equation the resulting differential equation expresses the constraint that in the case of motion along the true trajectory the rate of change of kinetic energy matches the rate of change of potential energy.

For further discussion of how Hamilton's stationary action gives the true trajectory see my answer to a question titled: 'Motivation for Lagrangian formalism'


With the above in place I turn to the river model of spacetime.

2006 article by Andrew J. S. Hamilton, and Jason P. Lisle: The river model of black holes

Given a metric it is possible to assign a velocity vector at each radial distance to a source of gravity.

Of course, as a matter of principle there is no such thing as assigning a velocity vector to spacetime. However, we can arbitrarily choose to assign a value of zero velocity at infinity, and from there work towards the source of gravity. The velocity assigned at each distance $r$ to the source of gravity is then the velocity that a test mass would acquire when released to free fall from infinity. So: while the choice of zero point is arbitrary: once a zero point is set then at every other distance the $r$ to the source of gravity the corresponding velocity value is determined.

As we know, traveling more spatial distance corresponds to a smaller amount of proper time elapsing.

The Pound-Rebka experiment in terms of the River Model:
At the base of the tower the spacetime is flowing at a larger velocity than at the top. We can define a difference is distance traveled in terms of difference in rate of flow of the river of space time. (Note especially: the evaluation is not directly in terms of flow-of-spacetime, it is difference in rate of flow.)

Compared to the object at the top the object at the base is traveling a longer spatial distance, hence for the object at the base a smaller amount of proper time elapses than for the object at the top. (Again, the evaluation is evaluating a difference. No self-contradiction is introduced by evaluating difference in distance traveled.)


For non-relativistic velocity the contribution of spatial curvature can be treated as negligable.

For the time curvature aspect: difference in amount of proper time that elapses and difference in gravitational potential are correlated.

The river model of spacetime allows evaluation of gravitational potential in terms of velocity with respect to the river of spacetime. Difference of velocity has a corresponding difference of kinetic energy.


In terms of relativistic physics:
Free fall from point A to point B is the trajectory of maximal proper time; any trajectory that is not inertial motion covers a longer spatial distance, hence a smaller amount of proper time elapses.

In terms of Hamilton's stationary action:
Along any trajectory that is not the true trajectory the rate of change of kinetic energy does not match the rate of change of potential energy.


So: my recommendation for background on the connection between kinetic/potential energy, and amount of proper time that elapses, is the article about the River Model of gravitational interaction.




[Later addition]
The correlation between Hamilton's stationary action and maximal proper time is via energy considerations.

That is, there is no direct connection.

There is a fundamental distinction. Hamilton's stationary action is not about satisfying an extremum condition. Hamilton's stationary action is about comparing rate of change of kinetic energy to rate of change of potential energy. There is a transition where the evaluation flips from looking for a minimum to looking for a maximum.

In the case of kinetic energy the exponentiation is to the power of 2; kinetic energy is proportional to the square of velocity. A very common power for the potential energy is minus 1; in the case of an inverse square force law the potential energy is to the power of minus 1.

For all functions for potential energy with the power smaller than 2 the true trajectory corresponds to a minimum of Hamilton's action

For all functions for potential energy with the power larger than 2 the true trajectory corresponds to a maximum of Hamilton's action.

The case where the potential increases with the square of the displacement (Hooke's law) is the in-between case between the above two. It's the point where there is a transition; the cusp where the evaluation flips from looking for a minimum of the action to looking for a maximum of the action.

What that shows is: the extremum is irrevant. The only thing that matters is the stationary criterium. The following cannot be emphasized enough: the name stationary action is important; very much not a technicality. The name 'least action' is in contradiction with the nature of Hamilton's action.


On the other hand: the amount of proper time of motion along a geodesic being maximal, is a true maximum Much like zero Kelvin being a true lowest possible value, the maximal amount of proper time is a true maximum.


So Hamilton's stationary action and maximal proper time cannot be brought in direct correlation. There is a fundamental distinction;

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