Two static Newtonian potentials are proposed for a laboratory region

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Redshift, local inertial frames and the failure of flat spacetime

Problem

Two static Newtonian potentials are proposed for a laboratory region, with

x

horizontal,

y

upward,

g_\star=9.81\ \mathrm{m\,s^{-2}}

and

k=1.00\times10^{-6}\ \mathrm{s^{-2}}

\begin{aligned} \Phi_1 &= g_\star y, \\ \Phi_2 &= g_\star y+\tfrac12k\,(x^2-y^2). \end{aligned}

((a)) Find the tidal field

\mathcal{E}_{ij}=\partial_i\partial_j\Phi

of each. ((b)) Decide, for each, whether the field can be drawn on flat spacetime with accelerated static observers, and give the reason. ((c)) In the field that cannot, two free particles are released at rest

1.00\ \mathrm{m}

apart along

x

. Find the magnitude of their relative acceleration, and how much their separation has shrunk after

10.0\ \mathrm{s}

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