Consider a toy universe with one time dimension and two space dimensions

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · The equivalence principle and gravity as geometry

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

Problem

Consider a toy universe with one time dimension and two space dimensions, coordinates

(t,x,y)

and metric signature

(-,+,+)

, so

n=3

. ((a)) Repeat the order-by-order counting of coordinate freedoms against metric derivatives used for four-dimensional spacetime, this time for

n=3

. At a point

P

, taken as the origin, expand the coordinate change as

x'^\mu = x^\mu + a^\mu{}_\nu x^\nu + \tfrac12 b^\mu{}_{\nu\rho}x^\nu x^\rho + \tfrac16 c^\mu{}_{\nu\rho\sigma}x^\nu x^\rho x^\sigma + \dots

with

b^\mu{}_{\nu\rho}

symmetric in

\nu\rho

and

c^\mu{}_{\nu\rho\sigma}

symmetric in

\nu\rho\sigma

. Count the independent parameters in

a^\mu{}_\nu

b^\mu{}_{\nu\rho}

and

c^\mu{}_{\nu\rho\sigma}

, and the independent components of

g_{\mu\nu}(P)

\partial_\alpha g_{\mu\nu}(P)

and

\partial_\alpha\partial_\beta g_{\mu\nu}(P)

, for

n=3

. Hence state how many second derivatives of the metric at

P

survive every choice of coordinates. ((b)) In this toy universe a weak, static field has potential

\Phi(y) = g_\star y

for height

y \ge 0

, so that

g_{00} = -(1+2\Phi/c^2)

and, for slow motion,

d\tau/dt \approx 1 + \Phi/c^2 - (\dot x^2+\dot y^2)/2c^2

. Starting from

\delta\int d\tau = 0

with the end events fixed, derive the equations of motion for

x(t)

and

y(t)

. A ball is launched from

y=0

with speed

v_0 = 20.0\ \mathrm{m\,s^{-1}}

30.0^\circ

above the horizontal, with

g_\star = 9.80\ \mathrm{m\,s^{-2}}

. Find the horizontal range at which it returns to

y=0

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