Annu. Rev. Astron. Astrophys. 1984. 22:
157-184
Copyright © 1984 by . All rights reserved |

**2.2. Basic Equations**

The standard results rest on three pillars: the conservation equations, the Raychaudhuri equation, and the energy conditions. The form of these equations depends on the nature of the matter in the Universe. It is assumed here that the usual description of matter on a cosmological scale as a "perfect fluid" is adequate at most times. Then (24, 25) the stress tensor of each component of matter or energy present takes the form

(4a) |

where *µ* is the (relativistic) energy density of the fluid
and *p* its
pressure. To complete the description of matter, we must give suitable
equations of state for the matter variables; in the hot early Universe,
the "standard" theory assumes (except during phases of pair
annihilation) that these are

(4b) |

describing a "Fermi gas" with temperature *T*
(107,
109).

THE CONSERVATION EQUATIONS The stress tensor
*T*_{ab} of matter that does not
interact with other matter or radiation obeys the equations
*T*^{db}_{; b} = 0.
If this matter is a perfect fluid (Equation 4a), the energy conservation
equation
*u*_{d}*T*^{db}_{; b} = 0 takes
the form

(5) |

where *t* is proper time measured along the fluid flow lines,
showing the
effect of expansion or compression of the matter on its energy density.
(The radius *R* is related to the volume *V* of a fluid
element by the relation
*V*
*R*^{3}.) The momentum conservation equation
*h*_{d}^{a}*T*^{db}_{; b}
= 0 is

(6) |

which shows how spatial pressure gradients generate acceleration (and thus observable gravitational redshifts).

THE RAYCHAUDHURI EQUATION Contracting the Ricci identity
*u*^{a}_{; bc} -
*u*^{a}_{; cb} =
*R*^{a}_{dbc}*u*^{d}
for *u*^{a} and using Einstein's field equations

(7) |

it follows that the volume behavior of the fluid is controlled by the Raychaudhuri equation (82, 24, 25, 45)

(8) |

where ^{2} is
the squared magnitude of the fluid vorticity and
^{2} the
squared magnitude of the shear (or rate of distortion). Because
*d*^{2}*R* / *dt*^{2}
is the curvature of the curve *R*(*t*), this equation shows
how rotation
tends to hold matter apart and distortion to make it collapse, with
(*µ* + 3*p*) the "active gravitational mass density" that
tends to cause collapse.

ENERGY CONDITIONS Various "energy conditions" have been proposed as physical restrictions on the matter (45); for present purposes, the major such conditions are

(9a) (9b) |

with Equation 9a needed for physical stability of matter (as follows from Equations 5 and 6) and Equation 9b implying that the gravitational effect of matter is attractive (see Equation 8). It is usually assumed that "reasonable" matter obeys these conditions.