Consider the following line element, ds^2 = −dt^2 + a^2 (t) (dx^2 + dy^2 ) + b^2 (t) dz^2 , where a(t) and b(t) are distinct functions. State whether or not this line element obeys the Cosmological Principle, if applied to describe the universe on large scales. Justify your answer. Consider the following line element,ds² = −dt² + a² (t) (dx² + dy²) + b² (t) dz²,where a(t) and b(t) are distinct functions. State whether or not this line element obeys theCosmological Principle, if applied to describe the universe on large scales. Justify your answer.

Answered: Consider the following line element,…

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Suppose a hypothetical universe is expanding (at some moment in time) at a rate of H. At this same moment the density of this Universe is ρ. (a) Confirm for yourself that this is a closed universe, given the values below. (b) Determine, and enter below, the spatial radius of curvature for this closed universe (at the same moment in time at which the values above are given). Values: H = 56 km s-1 Mpc-1 ρ = 4.9x10-25 kg m-3 Give your answer for (b) in units of Mpc, to the nearest integer (not in scientific notation - e.g., 1234).
Please answer all three parts! Thank you. Stanford has a linear particle accelerator (SLAC) which is 3 km long that produces electrons with a total energy of 50 GeV. These electrons lead exciting (albeit brief) lives, zooming along the accelerator before slamming into a target to produce other high-energy particles. a. Consider the viewpoint of one of the electrons. From the electron’s point of view, how long is the accelerator? Note that it is possible to answer this question without calculating the electron’s velocity. b. Let’s figure out how fast the electrons are traveling. Start by solving for β = u/c in terms of 1/γ following the trick we used in class. Use the binomial expansion if that is helpful. At what speed does a 50 GeV electron travel? c. The Large Hadron Collider (LHC) at CERN presently accelerates protons to a total energy of 6.5 TeV. Imagine a pulse of light, a 50 GeV electron, and a 6.5 TeV proton race each other along a 3 km distance. The light pulse will surely win…
Consider the following line element, ds² = -dt² + a² (t) (dx² + dy²) + b²(t) dz², where a(t) and b(t) are distinct functions. State whether or not this line element obeys the Cosmological Principle, if applied to describe the universe on large scales. Justify your answer.
5.8 Consider an expanding, positively curved universe containing only a cosmological constant (20 = Q.0 > 1). Show that such a universe %3D underwent a "Big Bounce" at a scale factor 1/2 ()". Apounce = (5.120) and that the scale factor as a function of time is
An accretion disc may form around a black hole. This is a thin disc of orbiting matter spanning radii r = Rin to Rout around the black hole. We assume that Rout » Rin and so we make the simplifying approximation that Rout → +∞o. The disc radiates according to the following equation 3 GM D(r, 0) = 1 CM (1-[B]"). 4 3 Here, r and are the usual polar coordinates with the origin at the centre of the disc. G is the gravitational constant, M is the mass of the black hole, Rin is the disc inner radius, M is the accretion rate - all these are constants. (a) Integrate D(r, 0) over the surface of the disc to find the total radiation output of the disc. (b) Find the total radiation in the case of Rin = 6GM/c².
Name: Hubble Distances Redshift z parameter The relativistic redshift is parametrized by z and given by Δ In terms of the scale factor, 2= X do - de de 1+z= ao a (2) Problem 01. Find the redshift z for a Hydrogen spectral line originally at 656 nm which has been observed at a wavelength of 1.64 μm. Astro 001 Fall 2022 Problem 02. How much smaller was the universe when this light was emitted? U₁ = DHO Using the redshift to measure the velocity, we find D~ (1) 0.1 Hubble's Law Hubble's Law states that the recession velocity of a redshifted galaxy is given by the product of the distance and the Hubble constant. (3) ZC Ho where c = 3 x 108 m/s and Ho = 2.3 x 10-18 s in standard units. The standard measurement of the Hubble constant is Ho = 71 (km/s)/Mpc. Problem 03. What is the distance in Mpc and ly to the galaxy measured in problem 01? 1 pc = 3.26 ly.
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Suppose that the universe were full of spherical objects, each of mass m and radius r, with the objects distributed uniformly throughout the universe as in the previous problem. (Assume nonrelativistic objects.) Given the density of these spherical objects (as you would have found in the previous problem), how far would you be able to see in meters, on average, before your line of sight intersected one of them? Values (note, different from the above problem): m = 3 kg r = 0.03 m Answer must be in scientific notation and include zero decimal places (1 sig fig).
The geometry of spacetime in the Universe on large scales is determined by the mean energy density of the matter in the Universe, ρ. The critical density of the Universe is denoted by ρ0 and can be used to define the parameter Ω0 = ρ/ρ0. Describe the geometry of space when: (i) Ω0 < 1; (ii) Ω0 = 1; (iii) Ω0 > 1. Explain how measurements of the angular sizes of the hot- and cold-spots in the CMB projected on the sky can inform us about the geometry of spacetime in our Universe. What do measurements of these angular sizes by the WMAP and PLANCK satellites tell us about the value of Ω0?
What magnetic field B is needed to keep 943-GeV protons revolving in a circle of radius 1.0 km ? Use the relativistic mass. The proton's "rest mass" is 0.938 GeV/c². (1 GeV = 10⁹ eV) [Hint: In relativity, mrelv²/r = quB in still valid in a magnetic field, mrel = ym.] Express your answer to two significant figures and include the appropriate units B = μA Value Units ?