PHYS 5770 Grav Theory Spring 2014. Problem Set 9. Due Tue 8 Apr
1. Tidal forces falling into a Schwarzschild black hole
In the Schwarzschild or Gullstrand-Painlev orthonormal tetrad, or indeed in any orthonore
mal tetrad of the Schwarzschild geometry wher
PHYS 5770 Gravitational Theory Spring 2014. Problem Set 3. Due Tue 4 Feb
1. Equation of motion
The Euler-Lagrange equations of motion for a particle whose Lagrangian is L(x , dx /d)
are (this is equation (4.5) of the notes),
L
d
L
= .
/d)
d (dx
x
(1.1)
I
PHYS 5770 Gravitational Theory Spring 2014. Problem Set 4. Due Tue 11 Feb
1. Earth metric
The metric just above the surface of the Earth is well-approximated by
ds2 = (1 + 2)dt2 + (1 2)dr 2 + r 2 (d2 + sin2 d2 ) ,
(1.1)
where
GM
r
is the familiar Newtonia
PHYS 5770 Gravitational Theory Spring 2014. Problem Set 6. Due Tue 25 Feb
1. Geodesics in the Schwarzschild geometry
The Schwarzschild metric is
ds2 = B(r) dt2 +
1
dr 2 + r 2 (d2 + sin2 d2 ) ,
B(r)
(1.1)
where B(r) is the horizon function
B(r) = 1
2M
.
r
PHYS 5770 Gravitational Theory Spring 2014. Problem Set 2. Due Tue 28 Jan
1. The Rules of 4D Perspective
(a) Celestial ellipsoid
In terms of the photon energy-momentum 4-vector pk in an unprimed frame, what is the
photon energy momentum 4-vector pk in a p
PHYS 5770 Gravitational Theory Spring 2014. Problem Set 5. Due Tue 18 Feb
1. Equations of motion in weak gravity
Consider the Newtonian metric
ds2 = (1 + 2)dt2 + (1 2)(dx2 + dy 2 + dz 2 ) ,
(1.1)
where (x, y, z) is the familiar Newtonian gravitational pot
PHYS 5770 Grav Theory Spring 2014. Problem Set 7. Due Tue 4 Mar
1. Gravitational lensing
In a previous problem you found that, in the weak eld limit, light passing a spherical mass
M at impact parameter y is deected by angle
=
4GM
.
yc2
(1.1)
(a) Lensing
PHYS 5770 Grav Theory Spring 2014. Problem Set 11. Due Th 24 Apr
1. Geodetic and frame-dragging precession of gyroscopes
The purpose of Gravity Probe B was to measure the predicted general relativistic precession
of a gyroscope in the gravitational eld of
PHYS 5770 Grav Theory Spring 2014. Problem Set 10. Due Tue 15 Apr
1. Constant density star
Shortly after communicating to Einstein his celebrated solution, Schwarzschild (1916) sent
Einstein a second letter describing the solution for a constant density s
PHYS 5770 Grav Theory Spring 2014. Problem Set 8. Due Th 20 Mar
1. Geodesics in the FLRW geometry
The Friedmann-Lema
tre-Robertson-Walker metric of cosmology is
ds2 = dt2 + a(t)2 dx2 +
sin2 (1/2 x )
d2 + sin2 d2
(1.1)
where is a constant, the curvature co
PHYS 5770 Gravitational Theory Spring 2014. Problem Set 1. Due Tue 21 Jan
1. Lorentz transformation
Relative to person A (unprimed frame), person B (primed frame) moves at velocity v along
the x-axis. Derive the form of the Lorentz transformation between