Download Sample Questions Q.1 : Consider two inertial reference frames S

Sample Questions
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Q.1 : Consider two inertial reference frames S and S , where S is moving with constant speed
v along the positive x-direction relative to S. Now imagine an observer in S sees a (electrically
charged) particle of rest mass m and charge q, start from rest at the origin of S and accelerate
uniformly with acceleration a in the positive y-direction. Find the velocity vector and acceleration
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vector of the same particle as seen from S . Find the rate of change of momentum (vector) of the
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particle in both S and S .
Q.2 It is given that the particle in reference frame S in Q.1 is accelerating because there is a
uniform electric field in the positive y direction (and no magnetic field). Find the magnitude of that
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electric field. What fields do an observer in S see ? Calculate all the components of both electric
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and magnetic fields as seen from S . Explicitly verify both the relations,
d
~ + ~v × B)
~
p~ = q(E
dt
in reference frame S and
d 0
~ 0 + ~v 0 × B
~ 0)
~ = q(E
0p
dt
0
in reference frame S .
Q.3 Show that the energy momentum tensor T 0µ transforms as a contra-variant four-vector using
the formula of T 0µ in terms of the electric and magnetic fields only . That means knowing how
the electric and magnetic fields transform under Lorentz transformation and knowing the relation
between T 0µ in terms of electric and magnetic fields show that T 0µ is a four-vector.
Q.4 Find ds2 in terms of appropriate parametric variables (such as θ, φ) on the following two
dimensional surfaces.
(a) Torus of smaller radius a and bigger radius b.
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(b) Ellipsoid of revolution obtained by rotating the ellipse xa2 + yb2 = 1 about the x-axis.
Q.5 : Consider a two dimensional surface described by the metric :
ds2 = (du2 + u2 dv 2 ) Cosh(2v) + 2u Sinh(2v) dudv
here Cosh(x) = (ex + e−x )/2 and Sinh(x) = (ex − e−x )/2.
Find the metric tensor gµν (u, v) and g µν (u, v) (choose x1 = u and x2 = v).
Find the connection coefficients Γµνσ (u, v).
From this find the equation of the geodesic.
Find the components of the Riemann curvature tensor.
What can you say about the nature of this surface ?