Knowing the conformal structure, one can choose a metric g which represents the conformal equivalence class of metrics and can evaluate the left-hand side of (3.3) for any test particle. Then the right-hand side of (3.3) determines (log i2); b up to the addition of a multiple of Ka§ab.

By considering another curve y'(t) whose tangent vector K'a is not parallel to Ka, one can find (log 0).b and so can determine Q everywhere up to a constant multiplying factor. This constant factor specifies one's units of measurement, and so can be chosen arbitrarily.

This is, of course, not the way one measures the conformal factor in practice; one makes use of the fact that there exist a large number of similar systems (such as the electronic states of atoms) whose internal motions define a number of events along the timelike curve which represents their position in space-time. The intervals between these events seem to be independent of their past history in the sense that the intervals measured by two nearby systems correspond. If one can effectively isolate them against external matter fields (so they must move on geodesic curves) and if one assumes their internal motion is independent of the curvature of space-time, then the only thing it can depend on is the metric. Thus the arc-length between two successive events on a curve must be the same for each pair of successive events on any such curve. If one takes this arc-length as one's unit of measurement, one can determine the conformal,factor at any point of space-

In fact it may not be possible to isolate a system from external matter fields. Thus for example in the Brans-Dicke theory there is a scalar field which is non-zero everywhere. However the conformal factor can still be determined by the requirement that the conservation equation J1"6.6 = 0 should hold. Thus knowledge of the energy-momentum tensor Tab determines the conformal factor.

3.3 Lagrangian formulation

The conditions (i) and (ii) of postulate (b) do not tell one how to construct the energy-momentum tensor for a given set of fields, or whether it is unique. In practice one relies heavily on one's intuitive knowledge of what energy and momentum are. However, there is a definite and unique formula for the energy-momentum tensor in the case that the equations of the fields can be derived from a Lagrangian.

Let L be the Lagrangian which is some scalar function of the fields x¥(i>a'"bc...ti> their first covariant derivatives, and the metric. One obtains the equations of the fields by requiring that the action time.

be stationary under variations of the fields in the interior of a compact four-dimensional region By a variation of the fields xF(1)a---(icdin 3) we mean a one-parameter family of fields *¥({)(u,r) where ue( — e,e) and re J', such that


where W({>a-bc m.a;c are the components of the covariant derivatives of But = (&%i)a-bc...ä)-,» thus the second term can be expressed as si -A,r»a-6e...-)

The first term in this expression can be written as f Qa,aAv= f Q"daa, J 3 J as where 0 is a vector whose components are

This integral is zero as condition (ii) is the statement that Al^ vanish at the boundary Thus in order that dl/du^Q should vanish for all variations on all volumes 3!, it is necessary and sufficient that the Euler-Lagrange equations,

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