Maximizing proper time with parabolic trajectory in uniform gravitational field

Question

In Feynman Lectures, Vol II, Chapter 42, he states,

"In a uniform gravitational field the trajectory with the maximum proper time for a fixed elapsed time is a parabola."

How can I prove this?

Answer 1

The uniform gravitational field is oriented in the $-z$-direction, with acceleration $g$. Clock A is at rest at $z=0$ and reads time $t_A$. Clock B is allowed to move in three-dimensional space $(x, y, z)$. We look for the trajectory of clock $B$ that maximizes the proper time, $t_B$, experienced by clock B.

Let $T$ denote the time read by a clock at rest at height $z$. Then in relation to clock A at rest at $z=0$,

\begin{equation} dT \approx \left(1+\frac{gz}{c^2}\right)dt_A \end{equation}

At height $z$, clock B moves with velocity $v$ so

\begin{align} dt_B &= \sqrt{1-v^2/c^2}\,dT \\ &\approx \left(1-\frac{v^2}{2c^2}\right)dT \\ &= \left(1-\frac{v^2}{2c^2}\right)\left(1 + \frac{gz}{c^2}\right)dt_A \\ &\approx \left(1-\frac{v^2}{2c^2}+\frac{gz}{c^2}\right)dt_A \end{align}

and the proper time elapsed on clock B can be computed with the functional:

\begin{equation} \Delta t_B[z, \dot{x}, \dot{y}, \dot{z}] = \int \left(1-\frac{v^2}{2c^2}+\frac{gz}{c^2}\right)\,dt_A \end{equation}

The extremum of this functional is found with the Euler-Lagrange equation

\begin{equation} \frac{\partial L}{\partial q_i} - \frac{d}{dt}\left[\frac{\partial L}{\partial \dot{q}_i}\right] = 0 \end{equation}

where $L = 1-\frac{1}{2c^2}(\dot{x}^2+\dot{y}^2+\dot{z}^2)+\frac{gz}{c^2}$ and $q_i\in\{x,y,z\}$. This yields the equations of motion

\begin{cases} \ddot{x} = 0\\ \ddot{y} = 0\\ \ddot{z} = -g\\ \end{cases}

which are the equations for a parabola in the $z$-direction and uniform motion in $x$ and $y$, as we'd expect from Newtonian gravity in a uniform gravitational field. We know these equations extremize the functional, but we haven't shown they maximize the proper time. It is straightforward to see this trajectory indeed maximizes the proper time if we expand the integrand of the functional about this trajectory. Alternatively, there is a nice argument in this answer.

Note: throughout the problem we work in the Newtonian limit - weak gravitational fields and low velocities. Allowing strong fields and large velocities requires a full general relativistic treatment, in which the trajectory which maximizes proper time is parabolic at leading order, but will include higher order corrections.

Answer 2

A calculation of this result is given in "Box 1" on p.151 of:

"Action: Forcing Energy to Predict Motion," Dwight E. Neuenschwander, Edwin F. Taylor, and Slavomir Tuleja, The Physics Teacher, Vol. 44, March 2006, pages 146-152. https://doi.org/10.1119/1.2173320

Abstract:

In this paper we use scalar energy, rather than vector force and momentum, to predict how a particle will move. The result is a quantity called action. Action and its relatives undergird Newton's laws and transcend them, also predicting motion in the quantum world and in the curved spacetime of general relativity. An example exhibits action in action.

Edwin Taylor (a co-author of the article referenced above) has made available of a copy of the article at his website https://www.eftaylor.com/leastaction.html#forcingenergy