Question

Free particles travel on paths that \"extremize\" the proper time. In flat Minkowski spacetime there is only one world line that \"extremizes\" the proper time between two points and this extremum is always a maximum. In curved spacetime, particles still travel on world lines that \"extremize\" the proper time, but the \"extrema\" can be either local maxima or minima. This problem focuses on world lines in the \"weak field\" metric around a source of curvature, say Earth of mass $M_e$, with gravitational potential $\Phi(r)/c^2 = -GM_e/(rc^2) << 1$. Compute the following proper times for each of the following observer's world lines to lowest order in $v^2/c^2$ and $\Phi/c^2$ in an inertial frame at which Earth is at rest: (a) A particle in circular orbit about Earth between two spacetime events A and B that represent the particle starting then returning to the same spatial position in the orbit separated by one orbital period $P$ in time. Express your answer in terms of $P$ and $M_e$ and any necessary constants. (b) A particle held at fixed position during the aforementioned period $P$. Express your answer in terms of $P$, $M_e$, and any necessary constants. (c) Is the world line of (a) that of a free particle? Is the world line of (b) that of a free particle? Explain how the ratio of the two proper times is consistent with the concept discussed in the beginning of this problem.

          Free particles travel on paths that \"extremize\" the proper time. In flat
Minkowski spacetime there is only one world line that \"extremizes\" the
proper time between two points and this extremum is always a maximum.
In curved spacetime, particles still travel on world lines that \"extremize\"
the proper time, but the \"extrema\" can be either local maxima or minima.
This problem focuses on world lines in the \"weak field\" metric around a
source of curvature, say Earth of mass $M_e$, with gravitational potential
$\Phi(r)/c^2 = -GM_e/(rc^2) << 1$.
Compute the following proper times for each of the following observer's
world lines to lowest order in $v^2/c^2$ and $\Phi/c^2$ in an inertial frame at which
Earth is at rest:
(a) A particle in circular orbit about Earth between two spacetime events
A and B that represent the particle starting then returning to the same
spatial position in the orbit separated by one orbital period $P$ in time.
Express your answer in terms of $P$ and $M_e$ and any necessary constants.
(b) A particle held at fixed position during the aforementioned period
$P$. Express your answer in terms of $P$, $M_e$, and any necessary constants.
(c) Is the world line of (a) that of a free particle? Is the world line of
(b) that of a free particle? Explain how the ratio of the two proper times is
consistent with the concept discussed in the beginning of this problem.

Free particles travel on paths that ëxtremizeẗhe proper time. In flat
Minkowski spacetime there is only one world line that ëxtremizesẗhe
proper time between two points and this extremum is always a maximum.
In curved spacetime, particles still travel on world lines that ëxtremizeẗhe proper time, but the ëxtremac̈an be either local maxima or minima.
This problem focuses on world lines in the ẅeak fieldm̈etric around a
source of curvature, say Earth of mass Me, with gravitational potential
Φ(r)/c^2 = -GMe/(rc^2) << 1.
Compute the following proper times for each of the following observer's
world lines to lowest order in v^2/c^2 and Φ/c^2 in an inertial frame at which
Earth is at rest:
(a) A particle in circular orbit about Earth between two spacetime events
A and B that represent the particle starting then returning to the same
spatial position in the orbit separated by one orbital period P in time.
Express your answer in terms of P and Me and any necessary constants.
(b) A particle held at fixed position during the aforementioned period
P. Express your answer in terms of P, Me, and any necessary constants.
(c) Is the world line of (a) that of a free particle? Is the world line of
(b) that of a free particle? Explain how the ratio of the two proper times is
consistent with the concept discussed in the beginning of this problem.

Added by Caitlin B.

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