Difference between revisions of "Homogeneous Universe"
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| − | <p style="text-align: left;"> A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous. </p> | + | <p style="text-align: left;">A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.</p> |
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| − | <p style="text-align: left;"> | + | <p style="text-align: left;">Due to isotropy the distribution must coincide in any two points, because they can be transformed to each other by rotation by $180^\circ$ around the middle of the interval connecting them. The opposite is not true (see problem \ref{equ1}).[[#equ1]]</p> |
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Revision as of 19:41, 28 May 2012
Contents
Homogeneous and isotropic Universe, Hubble’s law
Problem 1.
Most cosmological models are based on the assumption that the Universe is spatially homogeneous and isotropic. Give examples to show that the two properties do not automatically follow one from the other.
A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
Problem 2.
Show that if some spatial distribution is everywhere isotropic then it is also homogeneous. Is the opposite true?.
Due to isotropy the distribution must coincide in any two points, because they can be transformed to each other by rotation by $180^\circ$ around the middle of the interval connecting them. The opposite is not true (see problem \ref{equ1}).#equ1
Problem 3.
What three-dimensional geometrical objects are both homogeneous and isotropic?
A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
Problem 4.
Why the notion of Big Bang regarding the early evolution of the Universe should not be treated too literally?
A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
Problem 5.
Show that the Hubble's law is invariant with respect to Galilean transformations.
A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
Problem 6.
Show that the Hubble's law represents the only form of expansion compatible with homogeneity and isotropy of the Universe.
A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
Problem 7.
Show that if expansion of the Universe obeys the Hubble's law then the initial homogeneity is conserved for all its subsequent evolution.
A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
Problem 8.
In the 1940-ties Bondi, Gold and Hoyle proposed a stationary model of the Universe basing on the generalized cosmological principle, according to which there is no privileged position either in space or in time. The model describes a Universe, in which all global properties and characteristics (density, Hubble parameter and others) remain constant in time. Estimate the rate of matter creation in this model.
A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
Problem 9.
Galaxies typically have peculiar (individual) velocities of the order of $V_p \approx 100~\mbox{km/s}.$ Estimate how distant a galaxy should be for its peculiar velocity to be negligible compared to the velocity of Hubble flow $V_H=H_{0}R$.
A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
Problem 10.
Estimate the age of the Universe basing on the observed value of the Hubble's constant (the Hubble time $t_H$).
A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
Problem 11.
Show that the model of the expanding Universe allows one to eliminate the Olbers' paradox.
A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
| Problem 1. A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous. |
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| A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous. |
