7. PROPERTIES OF THE SCALAR FIELD

It is a truism that once the genie has been let out of the bottle, he never wants to go back inside . . . .

The successful use of the Higgs scalar field aroused a new, long-lasting interest in scalar fields and their properties. The most important mechanical properties used are the values of the energy density, energy flux, momentum density, and stress in a given potential scalar field distribution. We have already written down the law that describes the evolution of the potential itself for the simplest case - flat, nonexpanding Minkowski space. In the next section, we shall rewrite it for an expanding universe with a "Friedman-Robertson-Walker" (FRW) metric. In keeping with the idea of a homogeneous, isotropic universe, we shall study the case of a scalar field that depends solely on the world time t, i.e., f(t). In accordance with the homogeneous nature of the problem, the field does not depend on the space coordinates. Since it is coordinate-independent, can be used to construct the time derivative d / dt = , but all the spatial components of the gradients ð / ðx = ð / ðy = ð / ðz = 0 are identically equal to zero, so that they do not define any preferred spatial direction. This is in accordance with the fact that the space is isotropic and expanding. It is clear from symmetry considerations that, in this case, the mechanical properties of the field are described by two quantities - the energy density and p, the isotropic pressure. ⁽⁸⁾

(7.1)

(7.2)

The first variant on the right-hand side of each equation is for the simple case - a linear field equation - while the second variant, after the "or," is written for a field with an arbitrary potential V() - for example, the Higgs potential (see the preceding section).

The first thing to realize is that the relation between the pressure and the energy density is not unique. It is impossible to construct a definite function p(). In particular, the energy density is always positive. This is clear in the linear case (the first variant) and this condition is imposed on V() in the second case (see the end of this section). The minimum, V_min() = 0, is a reasonable condition; if this condition were violated, it would give the ground state nonzero energy. However, the energy density > 0 can be distributed between the "kinetic energy" 1/2 ² and the potential energy differently in the excited state. There are two limiting cases: all of the energy is in the form of kinetic energy. In this case,

(7.3)

(7.4)

This is the stiffest possible equation of state. ⁽⁹⁾ On the other hand, if we choose the combination = const, i.e., 0 (or _min) and = 0, we then reach the paradoxical situation

(7.5)

We retain the notation p (pressure) even though, in this case, it is in fact isotropic tension along every spatial direction.

The fact that it is possible for the scalar field to produce tension (i.e., negative pressure) is what makes it the most important field in cosmology.

An electromagnetic field can produce tension along either the electric field or the magnetic field. In the Faraday concept, field lines are something like strained ropes. But this strain is more than compensated for by the repulsion between the field lines in the direction perpendicular to the field lines. A homogeneous electric or magnetic field would produce a highly anisotropic pattern of tension and pressure; it would make the expansion of the Universe anisotropic. A very tangled field (which is easy to obtain with a magnetic field, but difficult to obtain with an electric field) would produce an average positive pressure p = /3 after averaging the pressure and tension. The same equation of state is obtained for a gas of photons, neutrinos, or other noninteracting highly relativistic particles. These well-known facts are brought to the reader's attention for one purpose: to stress that the ability to obtain isotropic tension [or large (in terms of absolute value) negative quasihydrostatic pressure] is peculiar to the scalar field. This is the basis of the inflationary cosmological models and is, therefore, of the utmost importance for the remainder of the paper. The end of this section is written for the mathematically oriented reader. Others may go on to the next section without losing the thread of the argument.

It is appropriate here to give several more extremely general arguments that explain the negative pressure in the scalar field and, especially, the relation p = - (7.5) for

(7.6)

Condition (7.6) above means that , together with its derivative, is invariant under the Lorentz transformation: both the spatial part, grad₃ , and the four-vector grad , which consists of and grad₃ are equal to zero. This means that grad does not point in any particular direction in either space or time so that there is therefore no preferred "rest frame." The "rest frame" is defined by zero energy (flux and momentum). So (7.6) says, in effect, that all moving observers should be unable to detect an energy flux or a field momentum. How is this possible? If there is no flux in one frame and the density (at rest) is , a slowly moving (v/c = << 1) observer would then observe a flux v. Because of the equivalence principle, we can talk in terms of an energy density = c² and energy flux v. However, we did not take the pressure into account. The total energy flux consists of the mass flux v plus a second term, the work done by the pressure pv, so that j = ( + p)v. We magically see that p = - is precisely the condition for j = 0 in all moving frames of reference. The condition v << c is not binding: if j = 0 is valid for small v, then, by adding several small v, we obtain j = 0 for all finite v < c, provided, of course, that the condition p = - is satisfied. We shall finally give a formal proof that dates back to the early days of relativity: the local Minkowski metric is characterized by the diagonal metric tensor g₀₀ = 1, g₁₁ = g₂₂ = g₃₃ = - 1 (with all other g 0).

The fact that this tensor is invariant under the Lorentz transformation is the essence of Einstein's statement about space and time. The energy density and pressure are parts of the energy-momentum tensor T, which is also a second-rank tensor, like the metric tensor g. To be invariant under the Lorentz transformation, the T must be proportional to the g, which, in everyday language, reduces to (7.5).

This idea was used to introduce the cosmological constant, which is a way of expressing the assumption that the energy density of the vacuum is not zero.

This idea was alien to classical physics, in which the "vacuum is a place with nothing inside." However, in quantum physics, the vacuum is merely the lowest state (that with minimum energy density); however, the minimum of a function clearly does not necessarily occur at the zero of the function. A definite value of the minimum energy must be introduced so that the Lorentz invariance of the vacuum will not be destroyed; this leads to _v = -p_v = 0 (the subscript v stands for vacuum). From astronomical observations, we know that _v and p_v are extremely small (even values of zero are not excluded). A straight-forward evaluation of _v using quantum field theory yields _v ~ ± 10⁹³ erg cm^-3. This very precise cancellation requires a special explanation that has not yet been found - but this is another story.

We just noted the observational fact that _v and p_v are almost exactly equal to zero. Now, we turn to the scalar field. A region of space with a nonzero (or nonminimum) value of is not a vacuum! However, if all of the derivatives , grad₃ are equal to zero, this state then has the same symmetry as the vacuum, so that p = -; in this case, the ratio p / = -1, just as for a vacuum, but the absolute values of p and are quite different from p_v and _v. This is the end of our discussion of whether or not large negative pressure is possible in a scalar field.

We should also mention two other situations in the linear case. For simplicity's sake, we shall return to the case of nonexpanding Minkowski space. The equation for in the uniform case, in which it is independent of x, y, and z, is

(7.7)

The solution to (7.7) is

(7.8)

and we find that

(7.9a)

and

(7.9b)

The average pressure is zero on time intervals such that µ >> 1. The average field describes a collection of particles at rest, so there is, naturally, some energy, but the pressure is zero.

A chaotic dependence of on x, y, and z can describe scalar particles in thermal equilibrium. In this case, the average pressure varies from p << for thermal energies much less than the rest mass of the particles to p = /3 at high temperature, when we are dealing with an extreme relativistic gas of scalar particles.

The versatile nature of the scalar field is instructive. One can construct a building, a cathedral, or a disordered set out of the same bricks; the same is true of the scalar field and the corresponding particles. Different situations are characterized by different pressure values, ranging from p = - to p = +; the pressure can, in some cases, be anisotropic. The energy density is always positive, unlike the pressure. The last point is more linguistic than physical. In macroscopic physics (say, hydrodynamics or elasticity), the notion of pressure can only be used for relaxed systems, just like the notion of "equation of state", p = p() or p = p(, S) (where S is the entropy). The stress tensor in a nonrelaxed system is usually written as the sum of a pressure and a viscosity term. In the case of the scalar field and, in particular, the homogeneous field = (t), the system is obviously not relaxed, and not in thermodynamic equilibrium (one can obtain more entropy at a given energy density by introducing many incoherent waves). This means that our use of the word "pressure" is a linguistic oversimplification.

⁸ For the reader familiar with the interested in general relativity:

= T⁰₀, -p = T^y_y T^z_z, T⁰_x = T⁰_y = T⁰_z = 0,

T^y_x = T^z_y = T^x_z = 0.

The numerous zeros in the table of the T^k_i are due to the choice of coordinate system: the t = const hypersurface is chosen tangent to the = const hypersurface. Therefore, all spatial derivatives are zero at the point of observation, and there is no preferred direction. In a cosmological solution with uniform , the two hypersurfaces are not only tangent at one point but also identical throughout all space. This is not always possible. The best example is the static (time-independent) but strongly nonhomogeneous field between two charges at rest. The = const hypersurface is time-like; there is no tangent space-like t = const hypersurface at any point. In a coordinate system where everything is at rest the energy density = 1/2 (grad₃ )² + V(), and there is no momentum or energy flux; this is obvious in a static system. What is most important is that the stress is highly anisotropic. Taking the x-axis along the field gradient, ð / ðx = g, ð / ðy = ð / ðz - 0, we obtain compression along the x-axis (T^x_x = V - 1/2 g²) and tension along the y- and z-axes (T^y_y = T^z_z = V + 1/2 g²). The case of vanishing V() is similar to the Coulomb electrostatic field of an electric charge at rest, = e/r. But the sign of the pressure and tension are reversed for the scalar field relative to the electrostatic case: T^x_x = E² / 8, T^y_y = T^z_z = -E² / 8 (For an E along the x-axis, the electrostatic field depends on x alone). The change in sign of the stress tensor reflects the fundamental difference between the scalar and electrostatic fields: attraction of equal charges for the scalar field and repulsion of equal charges for the electrostatic field. Back.

⁹ This equation of state was introduced by the author in 1960 in a vain attempt to get rid of stellar collapse. The main idea which inspired me, but which was not stated in the paper, was wrong: even an incompressible fluid experiences gravitational collapse. But the results derived remain: p = = c² is the equation of state corresponding to a sound velocity equal to c, the speed of light. The equation of state for an extreme relativistic gas, p = /3, T = T = - 3p = 0, is not the limit. A ratio of pressure to energy density greater than 1/3 is possible: it can be as large as 1. Back.