Modern Physics
3rd Edition
ISBN: 9781111794378
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
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Question
Chapter 16, Problem 4P
(a)
To determine
Show that if Hubble’s constant is truly constant in time.
(b)
To determine
Find H as a function of time for the case of Einstein-de Sitter Universe.
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(a) Let L be the diameter of our galaxy.Suppose that a person in a spaceship of massm wants to travel across the galaxy at constantspeed, taking proper time τ. Find the kineticenergy of the spaceship. (b) Your friend is impa-tient, and wants to make the voyage in an hour.For L = 105 light years, estimate the energy inunits of megatons of TNT (1 megaton=4×109 J).
Consider the energy-momentum tensor
Tμv = (p+p) uu+P9μ
applied to the matter/energy distribution in the universe on large scales, and assume an equation
of state of the form p = wp, with w a constant. Determine the type of matter/energy dominating
the universe if the energy-momentum tensor is traceless, that is, T"μ = 0.
Consider a cosmological spacetime in which the line element is given by
ds? = a²(t)(-dt + dr² + dy² + dz²),
where a(t) > 0 is the scale factor. Two light rays tangent to l = (1,1,0, 0)
and l = (1,0, 1,0) are received at time t =
u" = (a-'(to), 0, 0, 0). Compute the observed angle between the correspond-
ing images.
to by someone with 4-velocity
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- The energy density ϵ in radiation is related to its temperature by ϵ=αT4. Compute the temperature when the Universe was 0.1 second old, using the Friedmann equation and its radiation-dominated solution a(t)∝t1/2.arrow_forwardName: 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.arrow_forwardThe Friedmann equation in a matter-dominated universe with curvature is given by 87G -pR² – k , 3 where R is the scale factor, p is the matter densi, and k is a positive constant. Demonstrate that the parametric solution 4G po 4тG Po R(0) (1 – cos 0) 3 k and t( (e – sin 0) 3 k3/2 solves this equation, where 0 is a variable that runs from 0 to 27 and the present-day scale factor is set to Ro = 1. %3Darrow_forward
- Some of the familiar hydrogen lines appear in the spectrum of quasar 3C9, but they are shifted so far toward the red that their wavelengths are observed to be 3.0 times as long as those observed for hydrogen atoms at rest in the laboratory. (a) Show that the classical Doppler equation gives a relative velocity of recession greater than c for this situation. (b) Assuming that the relative motion of 3C9 and Earth is due entirely to the cosmological expansion of the universe, find the recession speed that is predicted by the relativistic Doppler equation.arrow_forward(d) In a homogeneous and isotropic universe we can represent all distances s by a time- independent comoving distance x multiplied by a time-dependent scale factor a(t), such that s = xa(t). Show that this naturally leads to Hubble's law and hence obtain an expres- sion for Hubble's "constant" in terms of a(t) and its derivatives. Explain the limitations of Hubble's law in calculating distances for nearby galaxies with measured recession velocities.arrow_forwardGiven hubble expansion and the accelerating expansion of the universe, what equation would be used to determine the maximum go-return distance (the maximum comoving distance from a point of origin an entity can go from their point of origin and still be able to return in a finite period of time) given an object's velocity as a fraction of the speed of light (and the time it would take to get to any given comoving distance at some fraction of c)what would that distance be for the speeds of 0.8c 0.9c & 0.95c?The accelerating expansion of the universe also implies that the overall density of matter in the universe will decrease over time. what equation tells you how dense the volume of our go-return bubble would at a given future time compared to present date?arrow_forward
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