Difference between revisions of "Homogeneous Universe"
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= Homogeneous and isotropic Universe, Hubble’s law = | = Homogeneous and isotropic Universe, Hubble’s law = | ||
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<div id="02001"></div> | <div id="02001"></div> | ||
+ | === 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. | 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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</div> | </div> | ||
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=== Problem 2. === | === Problem 2. === | ||
− | + | 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 [[#02001 | problem #1]]). | |
− | 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 [[#02001 | problem]]). | + | |
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
<div class="NavHead">solution</div> | <div class="NavHead">solution</div> | ||
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=== Problem 3. === | === Problem 3. === | ||
There are three cases: three-dimensional plane (zero curvature), sphere (positive curvature) and hyperboloid (negative curvature). | There are three cases: three-dimensional plane (zero curvature), sphere (positive curvature) and hyperboloid (negative curvature). | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <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> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | |||
+ | <div id="02004"></div> | ||
+ | === Problem 4. === | ||
+ | The main qualitative difference from the usual explosion lies in the fact that the explosive charge is usually surrounded by atmospheric air. The expansion is then caused by the difference between the huge pressure of the gaseous products of the explosion and comparatively small pressure of the surrounding air. But when considering the expanding Universe, one assumes that the pressure (according to the cosmological principle) is uniformly distributed too. Therefore there are neither pressure gradients nor forces that could cause or even affect the expansion. The expansion of the Universe itself is the result of initial velocity distribution. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <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> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | |||
+ | <div id="02005"></div> | ||
+ | === Problem 5. === | ||
+ | Suppose the Hubble's law holds in the reference frame $K$, in which matter is at rest at the origin. Consider another frame $K'$, with origin at some point $A$, moving with velocity $\vec{v}_{A}$ relative to $K$. The quantities measured in the latter frame are denoted by primes. Then in $K'$ one has $\vec r' = \vec r - {\vec r_A}$ and | ||
+ | \[\vec v' = \vec v - {\vec v_A} | ||
+ | = H\vec r' = H\vec r - H{\vec r_A} | ||
+ | = H\left( {\vec r - {{\vec r}_A}} \right) = H\vec r'.\] | ||
+ | Therefore the distribution law for velocities in the new frame has the same form with the same value of Hubble's parameter as in the original one. | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
<div class="NavHead">solution</div> | <div class="NavHead">solution</div> |
Revision as of 21:14, 21 May 2012
Contents
Homogeneous and isotropic Universe, Hubble’s law
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.
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.
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 #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.
Problem 3.
There are three cases: three-dimensional plane (zero curvature), sphere (positive curvature) and hyperboloid (negative curvature).
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.
The main qualitative difference from the usual explosion lies in the fact that the explosive charge is usually surrounded by atmospheric air. The expansion is then caused by the difference between the huge pressure of the gaseous products of the explosion and comparatively small pressure of the surrounding air. But when considering the expanding Universe, one assumes that the pressure (according to the cosmological principle) is uniformly distributed too. Therefore there are neither pressure gradients nor forces that could cause or even affect the expansion. The expansion of the Universe itself is the result of initial velocity distribution.
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.
Suppose the Hubble's law holds in the reference frame $K$, in which matter is at rest at the origin. Consider another frame $K'$, with origin at some point $A$, moving with velocity $\vec{v}_{A}$ relative to $K$. The quantities measured in the latter frame are denoted by primes. Then in $K'$ one has $\vec r' = \vec r - {\vec r_A}$ and \[\vec v' = \vec v - {\vec v_A} = H\vec r' = H\vec r - H{\vec r_A} = H\left( {\vec r - {{\vec r}_A}} \right) = H\vec r'.\] Therefore the distribution law for velocities in the new frame has the same form with the same value of Hubble's parameter as in the original one.
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. |