(a) A surveying drone hovers at the point m in a horizontal plane, so the plane polar…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Tensors in flat spacetime: index notation and covariant field equations

Problem

(a) A surveying drone hovers at the point

(x,y) = (6, 8)

m in a horizontal plane, so the plane polar radius is

r = 10

m. Using

r = \sqrt{x^2+y^2}

and

\phi = \arctan(y/x)

, write down the four Jacobian entries

\partial r/\partial x

\partial r/\partial y

\partial\phi/\partial x

\partial\phi/\partial y

at this point, then find the polar components

(dr, d\phi)

produced by the small Cartesian displacement

dx^\mu = (0, 0.020\ \mathrm{m}, 0, 0)

. (b) A four-vector has Cartesian components

A^\mu = (5, 4, 0, 3)

m (with

A^0 = c\Delta t

). Lower the index using

\eta_{\mu\nu} = \mathrm{diag}(-1,1,1,1)

to get

A_\mu

, then contract to find

A^\mu A_\mu

. State what type of object the contraction produces and what kind of interval (timelike, spacelike or null) it describes. (c) A two-index array has components

Q^{01} = 6

and

Q^{10} = 2

(all other components zero). Find

Q^{(01)}

and

Q^{[01]}

, and state how many independent components a general two-index tensor in four dimensions has in its symmetric part and in its antisymmetric part. (d) At an event, the fields are

\vec E = (3.0\times10^5, 0, 0)

V m

^{-1}

and

\vec B = (0, 0, 1.0\times10^{-3})

T. Using

F_{0i} = -E_i/c

and

F_{ij} = \epsilon_{ijk}B_k

, give the components

F_{01}

and

F_{12}

(take

c = 3.0\times10^8

m s

^{-1}

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