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We shall now write down the basic equations for the radius of the Universe a(t) and the scalar field phi(t). All other fields and particles are neglected. We shall see that this is not just a simplification; it is also a sound approximation. The equations are

Equation 8.1 (8.1)

Equation 8.2 (8.2)


Equation 8.3 (8.3)

Equation (8.1) describes the behavior of the field in curved space; in this case, the curvature is that determined by the expansion. The characteristic combination adot / a = (1/a) da/dt = d ln a/dt = H is the well known Hubble constant. In the general case with V(phi), µ2 phi (which is dV / dphi for V = 1/2 µ2 phi2) is replaced by the derivative dV / dphi. Equations (8.2) and (8.3) are the Einstein equations, which can be obtained from the general equation

Equation 8.4 (8.4)

for the various components of Gik that do not vanish due to the symmetry of the problem (c ident 1 in (8.2) and (8.3)).

We shall apply the Einstein theory to Friedmann (FRW) universes. The last term in (8.3), i.e., the k-term - k = - 1 for a closed universe, k = 0 for a flat universe, and k = + 1 for a hyperbolic (open, infinite) universe - determines their spatial characteristics.

Since the Einstein theory includes the Newtonian theory, one can give a simple Newtonian interpretation of (8.2) and (8.3). In many books, it is explained that one can consider a comoving sphere of radius r(t) = xia(t), where xi = const << 1. A point on the surface of this sphere moves under the influence of the gravitational force produced by the mass M within the sphere of radius r, so that we have the equation of motion

Equation 8.5 (8.5)

and the equation of energy conservation

Equation 8.6 (8.6)

where Em is the kinetic energy per unit mass on the surface of the sphere (this specific kinetic energy is conserved). Comparing (8.6) with (8.3) we obtain

Equation 8.7 (8.7)

We see that a closed universe corresponds to negative Em, which explains why a closed universe collapses again after it expands. A flat universe corresponds to Em ident 0, i.e., parabolic motion, and the hyperbolic case corresponds to positive Em - infinite expansion. This is old stuff, enthusiastically explained in many books (for example, One, Two, Three, Infinity (Gamow 1971), or the famous The First Three Minutes (Weinberg 1976)). The next point is that the mass M is defined differently in (8.5) and (8.6).

Comparing them with their parent equations, we see that (8.5) is obtained from (8.2), so that the corresponding mass,

Equation 8.8 (8.8)

in contrast to

Equation 8.9 (8.9)

where V = 4pi r3/3 is the volume. Is this a contradiction? The law of the conservation of mechanical energy is a consequence of the Newtonian equation for the acceleration. Therefore, at first glance, it would seem that M5 neq M6 is impossible. However, the two equations in fact agree with one another beautifully, if we remember that the mass inside the comoving volume only remains constant in the unique case of zero pressure.

If the pressure is not equal to zero, the work done by the expanding sphere, pdv, must be taken into consideration. Let E denote the total energy within the sphere, E = Mc2. The first law of thermodynamics is

Equation 8.10 (8.10)

It turns out that (8.5) (with M5) and (8.6) (with M6) and (8.10) (written for M6) are consistent. (10) Each of these three basic equations can be thought of as resulting from the other two.

The difference between M5 and M6, and, in particular, the presence of the 3p term inside the parenthesis in (8.2) is inevitable.

As long as one assumed that the sky was filled with slowly moving stars, the average pressure could be neglected. The p = 0 equation of state is called the "dust" equation of state. In the radiation-dominated phase, p = epsilon/3, so that epsilon + 3p = 2epsilon, which makes a small quantitative change in the expansion law. This was duly accounted for in nucleosynthesis theory, but nobody much cared about the difference between M5 and M6.

The fact that p = -epsilon is possible with the scalar field is what drew attention to the subject. Of course, if p = -epsilon, epsilon + 3p = -2epsilon is negative, which means that gravitational attraction is replaced by gravitational repulsion!!

A particle at a given distance r from the origin is subject to a positive acceleration (relative to the origin) if the pressure within the sphere is large enough and negative. This is not a direct result of the pressure on the particles because (1) negative pressure is actually tension and (2) it is the difference in pressure, not the pressure itself, that gives rise to the accelerative force acting on a particle. Therefore, what (8.2) and (8.5) describe is the indirect influence of the pressure via the gravitational field created by the pressure - an effect due to Einstein's general relativity; this effect is totally unpredicted, even alien to the Newtonian theory of gravitation.

The gravitational repulsion due to negative pressure is what gives rise to expansion and the characteristic Hubble velocity distribution.

The first equation governing the evolution of the scalar field (8.1.) can be deduced from the energy conservation law, which, in turn, follows from the Einstein equations. Setting dE = -pdV and E = epsilonV, where V = const a3 and epsilon = 1/2 phidot2 + V(phi), one can obtain (8.1) after performing a few simple mathematical operations. This reflects a very important property of general relativity. The motion and transformation of matter and field must obey all the mechanical equations expressing energy and momentum conservation. (11)

If there is only one kind of gravitating substance, then this property of general relativity turns out to dictate the laws of motion. Along these lines, before World War II, Einstein and Infeld and Fock and Petrova independently showed that the laws describing the mechanical motion of a particle in curved space are a consequence of the geometry of space - time.

If there is only one field - the scalar field phi - then its equation of motion is also completely determined by the geometry of space-time. We shall study two cases in the next few sections: the Higgs V(phi) with one maximum and two minima and the linear equation corresponding to the quadratic potential V = 1/2 µ2 phi2.

10 To check this, take the time derivative of (8.5), accounting for the fact that M is not constant: dM / dt = -p(dV / dt) = -p . 4pi xi3 a2 (da/dt). Back.

11 The Maxwell equations, which dictate exact electric charge conservation, have a similar property. Back.

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