This is the general case in which the energy-momentum tensor has no timelike or null eigenvector. There are no observed fields which have energy-momentum tensors of this form.

For type I, the weak energy condition will hold if /i ^ 0, /i+pa > 0 (a = 1,2,3). For type II it will hold if p1 > 0, pt Js 0, k Js 0, v = +1. These inequalities are very reasonable requirements and are satisfied by all experimentally detected fields. The condition will not hold for the physically unrealized types III and IV.

The condition will also holdfor the scalar field 0 postulated by Brans and Dicke and by Dicke (see Dicke (1964)). This field is required to be positive everywhere. It has an energy-momentum tensor of the form (3.6) where now m = 0. The energy-tensor of the other fields is <j> times what it would have been had the scalar field not existed.

The condition will not hold for the 'C'-field proposed by Hoyle and Narlikar (1963). This again is a scalar field with m zero, only this time the energy-momentum tensor has the opposite sign and so the energy density is negative. This allows the simultaneous creation of quanta of positive energy fields and of the negative energy C-field. This process occurs in the steady-state model of the universe suggested by Hoyle and Narlikar in which, as particles move apart due to the general expansion of the universe, new matter is continually being created to keep the average density constant. There is, however, a quantum mechanical difficulty associated with such a process. For even if the cross-section for the process were very small, the infinite phase space available to the positive and negative energy quanta would seem to result in an infinite number of such pairs being produced in a finite region of space-time.

Such a catastrophe could not occur if the weak energy condition held. If a slightly stronger condition holds then creation is impossible in the sense that space-time must remain empty if it is empty at one time and no matter comes in from infinity. Conversely, matter present at one time cannot disappear and so must be present at another time. The condition is

The dominant energy condition

For every timelike Wa, T^W^ Js 0, and TabWa is a non-spacelike vector.

This may be interpreted as saying that to any observer the local energy density appears non-negative and the local energy flow vector is non-spacelike. An equivalent statement is that in any orthonormal basis the energy dominates the other components of T^, i.e.

This holds for type I if ju > 0, — ¡jl < pa < /i (a = 1,2,3) and for type II if v = +1, k > 0, 0 ^ pt ^ k (i = 1,2). In other words, the dominant energy condition is the weak energy condition with the additional requirement that the pressure should not exceed the energy density. This holds for all known forms of matter and there is in fact good reason for believing that this should be the case in all situations. For the speed of sound waves travelling in the Ea direction is dpjd/i (adiabatic) times the speed of light. Thus dpjdju, must be less than or equal to one, as by postulate (a) in § 3.2 no signal can propagate faster than light. It follows that pa < fi, since, for every known form of matter, the pressures are small when the density is small. (Bludman and Ruderman (1968, 1970) have shown that there might be fields for which mass renormalization could lead to pressure being greater than the density. We feel, however, that this probably indicates a failure of renormalization theory rather than that such a situation would occur.) Now consider the situation depicted in figure 9 in which there is a C2

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