Editing SSC/Dynamics/FreeFall (section)

==Relationship to Relativistic Cosmologies==
NOTE:  [http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner &amp; Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies.

Einstein's ''General Theory of Relativity'' generates the following "[http://en.wikipedia.org/wiki/Friedmann_equations#Equations Friedmann equations]" describing the (pressure-free) evolution of homogeneous, isotropic cosmologies in terms of the time-dependent scale length of the universe, <math>~\mathcal{R}(t)</math>:

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Evolution of Homogeneous, Isotropic Cosmologies
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\ddot{\mathcal{R}}}{\mathcal{R}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{4}{3}\pi G \rho + \frac{\Lambda c^2}{3} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~H^2 \equiv \biggl( \frac{\dot{\mathcal{R}}}{\mathcal{R}} \biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{Kc^2}{\mathcal{R}^2} + \frac{\Lambda c^2}{3}\, .</math>
  </td>
</tr>
</table>
</div>

</td></tr>
</table>

It is instructive to compare these dynamical relations with the analogous pair of equations that we [[#PressureFreeSummary|derived, above]], in the context of a Newtonian (and flat space) description of the free-fall collapse of a uniform-density sphere having a time-dependent radius, <math>~R(t)</math>:

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\ddot{R}}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
where, &nbsp; &nbsp; <math>~k(R_i,v_i)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
  </td>
</tr>
</table>
</div>

</td></tr>
</table>

If we set Einstein's cosmological constant, <math>~\Lambda</math>, to zero in the first set of equations, we see that the cosmological equations are mathematically identical to their Newtonian counterparts.  (Apparently, this mathematical overlap was first discussed in back-to-back papers in the ''Quarterly Journal of Mathematics Oxford'' by [http://adsabs.harvard.edu/abs/1934QJMat...5...64M Milne (1934)] and [http://adsabs.harvard.edu/abs/1934QJMat...5...73M McCrea &amp; Milne (1934)].)  This tight analogy can help us understand the physical origin of the various terms that appear in the set of relativistic equations, or ''vise versa''.

It is clear, for example, that the Newtonian equations that have been derived here in an effort to quantitatively describe the dynamical ''collapse'' of a uniform-density sphere could just as well be used to describe the dynamical ''expansion'' of a uniform-density sphere.  To achieve expansion (in the absence of a cosmological constant), however, the velocity <math>~v_i</math> that appears in our definition of the integration constant, <math>~k</math>, must be nonzero and positive.  Also, in order to successfully achieve an ''homogeneous'' expansion, the magnitude of <math>~v_i</math> that is assigned to fluid elements at different locations throughout the sphere must be linearly proportional to the radial coordinate of each fluid element.

Furthermore, it is clear that the "Hubble parameter," <math>~H</math>, is a (time-dependent) parameter giving the ratio of the instantaneous collapse/expansion velocity relative to the instantaneous scale length of the universe; and that the constant, <math>~K</math>, in part, relates to the "initial" collapse/expansion velocity imparted to the system.

<!--
As a final remark, it is easy to understand why analogies are drawn between the cosmological constant, <math>~\Lambda</math>, and pressure gradients in a gaseous sphere.  The standard version of the Euler equation that appears among the regular group of governing equations at the top of this page includes two "source terms":  One, describing the radial gradient of the gravitational potential that arises due to the "self-gravity" of the fluid sphere; this gives rise to the "<math>~\pi G \rho</math>" term in the our final set of dynamical equations and always acts to ''decelerate'' an expansion (or ''accelerate'' a collapse).  A second, describing the radial gradient of the fluid enthalpy.  In order to develop a Newtonian model of free-fall collapse, we explicitly set this second term to zero.  Had we not done this, a term analogous to Einstein's cosmological constant would have appeared in our final set of (Newtonian) dynamical equations.  While enthalpy gradients can, in principle, be either negative or positive, when they appear as a source term in the Euler equation in the context of astrophysical discussions, they usually appear in opposition to the gravitational acceleration; this allows for the construction of pressure-balanced, equilibrium configurations.  For similar reasons, in the pair of relativistic dynamical equations, the cosmological constant customarily appears with a sign in opposition to the "<math>~\pi G \rho</math>" term.  With this association in mind, it may be easier for the nonrelativist to appreciate [http://en.wikipedia.org/wiki/Equation_of_state_(cosmology) discussions of the equation of state of the universe]. 
-->