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ASSIGNMENTS Week 3 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof ASSIGNMENTS Week 3 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof. A. Achterberg, Dr. S. Larsen and Dr. F. Saueressig This exercise explicitly keeps track of the speed of light. Use c = 3 × 108 m/s for explicit compu- tations. For solving exercises 1 and 2 the following compilation of masses and distances may be used: Earth: 27 Mass M⊕ = 5.97 × 10 g 2 −3 GM⊕/c = 4.43 × 10 m 6 Equatorial radius R⊕ = 6.378 × 10 m Rotation period 8.62 × 104 s Sun: 33 Mass M⊙ = 1.99 × 10 g 2 3 GM⊙/c = 1.48 × 10 m 8 Radius R⊙ = 6.96 × 10 m Exercise 1: the global positioning system (GPS) (Hand-in assignment) The GPS system consists of 24 satellites which orbit the earth on 6 equally spaced orbital planes. Each satellite emits a signal encoding its time of emission te and the location of the satellite. An observer who receives the signal at time tr, that is an interval ∆t = tr − te later, knows that she is located somewhere on a sphere of radius c∆t centered on the satellite. Signals from two satellites narrow the location down to the intersection of two spheres. Since GPS receivers typically don’t include clocks with sufficient accuracy to precisely pinpoint the arrival time of the satellite signals, it requires the signals from four satellites to locate a position in four-dimensional spacetime. In order to facilitate the computation, assume that the GPS satellite is in a 12-hour circular equatorial orbit of radius Rs from the Earth’s center. a) For military applications, the GPS system should work with an accuracy of 2 m. Estimate the resulting required accuracy to which time signals and time differences must be known to be able to meet this requirement. b) Use the framework of Newtonian mechanics to compute the velocity vs and the radius of the satellite’s orbit Rs in a frame where the Earth is considered at rest. The mass of the Earth is given in the table above. c) Use the data on the orbit to compute the fractional correction needed to compensate for the special relativistic effect of time dilatation. This fractional correction is defined as ∆τ − ∆τ E S (1) ∆τS where ∆τE and ∆τS are time intervals measured by a clock on Earth and on board of the satellite, respectively. In order to compute the necessary corrections, it suffices to work to 2 2 first order in vs /c . d) Compute the fractional corrections necessary to compensate the general relativistic effect resulting from the clocks being positioned at different gravitational potentials. You may use the formula derived in the lecture ΦS − ΦE ∆τE =∆τS 1 − , (2) c2 where ΦS and ΦE are the Newtonian gravitational potentials felt by the satellite and on the surface of the Earth, respectively. e) Compare the size of the corrections from special relativity, part c), and general relativity, part d), to the accuracy requirements, part a), and comment on their relevance for the stability of the GPS system. Exercise 2: Gravitational redshift During the lecture we have seen that for a static weak gravitational field, the spacetime geometry is specified by the line element 2 2 Φ(~x) 2 2 Φ(~x) 2 2 2 2 ds = 1+ (cdt) − 1 − (dx + dy + dz ) . (3) c2 c2 Here Φ(~x) is the Newtonian gravitational potential at ~x. Recall that the line element holds to first order in c−2 only. Thus terms containing higher inverse powers of c can consistently be dropped throughout the computation. a) Based on the line element (3), derive the formula for the gravitational redshift GM ω∞ = 1 − ω∗ . (4) Rc2 Here ω∗ is the frequency of a photon emitted at the surface of a star with radius R and mass M, while ω∞ is the frequency of the photon detected by an observer situated far away from the star (Robs → ∞). Use that the line element is actually independent of t, so that the coordinate time t is universal. b) Estimate the fractional change in the frequency, (ω∗ − ω∞)/ω∗, for a photon emitted by a 3 white dwarf star with mass M = M⊙ and R = 10 km. 2 Exercise 3: Accelerated reference frames Imagine a spaceship of length h accelerating uniformly with acceleration g. The special relativistic effects encountered on the spaceship may be analyzed within the following framework: a) Transform the line element of special relativity from the usual (t,x,y,z) coordinates to new coordinates (t′,x′,y′, z′) related by c x′ gt′ c x′ gt′ c2 t = + sinh , x = c + cosh − , y = y′ , z = z′ . (5) g c c g c c g ′ gt ≪ b) By expanding the right-hand side of the transformation for c 1, show that the spaceship at rest in the primed frame appears to be moving with acceleration g in the unprimed frame. c) Consider clocks carried by the spaceship, i.e., clocks that are at rest in the primed frame. Show that a clock at rest at x′ = h runs faster than a clock at rest at x′ = 0 by a factor (1 + gh/c2). d) Let the bottom of the ship be at x′ = 0 and its top at x′ = h. (The spaceship extends also in the y′ and z′ direction, but this is not relevant for the problem at hand.) Use the line element derived in part a) to show that the hight of the spaceship remains constant in time, i.e., the ship moves rigidly. 3.
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