Integrating by parts, dl_ du where = Va.ibVb. If this is zero for all K, it follows that

(p+p) = -Pib(gba+ VbV"), where fi = p{ 1+e) is the energy density and p = p2(de/dp) is the pressure. Thus the acceleration of the flow lines, is given by the pressure gradient orthogonal to the flow lines.

To obtain the energy-momentum tensor one varies the metric. The calculations may be simplified by noting that the conservation of the current may be expressed as

Given the flow lines, the conservation equations determine ja uniquely at each point on a flow line in terms of its initial value at some given point on the same flow line. Therefore — g)ja is unchanged when the metric is varied. But p2 = - g)ja (V - g)jb) su.

W> = (i°i6 —jCjc9ab) &9ab> T»" = (p(l+e)+p2^j

We shall call any matter whose energy-momentum tensor is of the above form (whether or not it is derived from a Lagrangian) a perfect fluid. From the energy and momentum conservation equations (3.1) applied to (3.8) one finds

These are the same as the equations derived from the Lagrangian. We shall call a perfect fluid isenlropic if the pressure p is a function of the energy density ¡i only. In this case one can introduce a conserved density p and an internal energy e and derive the equations and the energy-momentum tensor from a Lagrangian.

One may also give the fluid a conserved electric charge c (i.e. J°. „ = 0 where J = c V is the electric current). The Lagrangian for the charged fluid and the electromagnetic field is

so and thus

The last term gives the interaction between the fluid and the field. Then varying A, the flow lines and the metric respectively, one finds

Tab = (¡i+p) VaVb +pgab+(FacFbc- {g^F^F'*). 3.4 The field equations

So far, the metric g has not been specified. In the Special Theory of Relativity, which does not include gravitational effects, it is taken to be flat. One might think that one could include gravitation by keeping the metric flat and by introducing an extra field on space-time. However, experiments have shown that light rays travelling near the sun are deflected. Since light rays are null geodesies, this shows that the space-time metric cannot be flat or even conformal to a flat metric. One therefore has to give some prescription for the curvature of space-time. It turns out that this prescription can be chosen so as to reproduce the results of Newtonian gravitation theory in the limit of small slowly varying curvature. It is therefore not necessary to introduce an extra field to describe gravitation. This is not to say that there could not be an additional field that produced part of the gravitational effects. Such a scalar field has been suggested by Jordan (1966), and Brans and Dicke (see Dicke (1964)). However, as mentioned before, such an additional field could be regarded as simply another matter field and included in the total energy-momentum tensor. We therefore adopt the view that the gravitational field is represented by the space-time metric itself. The problem then becomes one of finding field equations to relate the metric to the distribution of matter.

These equations should be tensor equations involving the matter only through its energy-momentum tensor, i.e. should not distinguish between two different matter fields which have the same distribution of energy and momentum. This can be regarded as a generalization of the Newtonian principle that the active gravitational mass of a body (the mass producing a gravitational field) is equal to the passive gravitational mass (the mass acted on by the gravitational field). This has been verified experimentally by Kreuzer (1968).

To determine what the field equations should be, we shall consider the Newtonian limit. Since the Newtonian gravitational field equation does not involve time, the correspondence with Newtonian theory should be made in a metric which is static. By a static metric is meant a metric which admits a timelike Killing vector field K which is orthogonal to a family of spacelike surfaces. These surfaces may be regarded as surfaces of constant time and may be labelled by the parameter t. We define the unit timelike vector V as /-1K, where P = - KaKa. Then Va.b = - fa Vb, where = Va,b Vb = f~1f;bgab represents the departure from geodesity of the integral curves of V (which are of course also integral curves of K). Note that Va = 0.

These integral curves define the static frame of reference, that is to say, the space-time metric seems to be independent of time to a particle whose history is one of these curves. A particle released from rest and following a geodesic would appear to have an initial acceleration of — V with respect to the static frame. If / differs only slightly from unity the initial acceleration of a freely moving particle released from rest is approximately minus the gradient off. This suggests that one should regard /— 1 as the quantity analogous to the Newtonian gravitational potential.

One can derive an equation for this potential by considering the divergence of

« = (V"; b V»). a = F". b;aVb+ F". 6 V». a

= Rab V°Vb + (V°.a).bVt> + (VbVb)* = RabV°Vb.

But f".a= (f~1f-.bgab):a= -naf.^+f-'f-.tasr"

and /.o6 VaVb = -/;0 V-:b V» = -f^f.J,^, so one finds /. o6(^ + VaVb) = fRab VaVb.

The term on the left is the Laplacian of / with respect to the induced metric in the three-surface {i = constant}. If the metric is almost fiat, this will correspond to the Newtonian Laplacian of the potential. One would therefore obtain agreement with Newtonian theory in the limit of a weak field (i.e. when / ~ 1) if the term on the right is equal to 4nG times the matter density plus terms which are small in the weak field limit.

This will be the case if there is a relation of the form

where Kab is a tensorial function of the energy-momentum tensor and the metric, which is such that (4nG^R^ Va Vb is equal to the matter density plus terms which are small in the Newtonian limit. We shall for the moment assume a relation of this form.

Since Bab satisfies the contracted Bianchi identities Rmb. b = a, (3.11) implies Ka*ib = lK;b. (3.12)

This shows that the apparently natural equation Kab = knGTab cannot be correct, since (3.12) and the conservation equations Tb. b = 0 would imply T.a = 0. For a perfect fluid, for example, this would mean that /J>—3p was constant throughout space-time, which is clearly not satisfied by a general fluid.

In fact in general, the only first order identities satisfied by the energy-momentum tensor are the conservation equations. From this it follows that the only tensorial function Kab of the energy-momentum tensor and the metric which obeys the identities (3.12) for all energy-momentum tensors, is

where k and A are constants. The values of these constants can be determined from the Newtonian limit. Consider a perfect fluid with energy density ¡i and pressurep whose flow lines are the integral curves of the Killing vector (i.e. the fluid is at rest in the static frame). The energy-momentum tensor is given by (3.8). Putting this in (3.13) and (3.11), one finds f-.ab(9ab+ VaVb) =/(£*(/*+ 3?>)-A). (3.14)

In the Newtonian limit the pressure <p is normally very small compared to the energy density (We are using units in which the speed of light is unity. In units in which the speed of light is c, the expression fi+3p should be replaced by /i+3j>/c2.) One would therefore obtain approximate agreement with Newtonian theory if k = 8nG and if |A| is very small. We shall use units of mass in which G = 1. In these units, a mass of lO^gm corresponds to a length of 1cm. Sandage's (1961, 1968) observations of distant galaxies place limits on |A| of the order of 10-56cm-2; we shall normally take A to be zero, but shall bear in mind the possibility of other values.

One may then integrate (3.14) over a compact region^" of the three-surface {t = constant} and transform the left-hand side into an integral of the gradient of / over the bounding two-surface

J^/(47T(/t + 3p)) dcr = jj.^(g""4- VaVb) do-= [ f;a(9ab+VaVb)drb, where der is the volume element of the three-surface (t = constant} in the induced metric, and dr6 is the surface element of the two-surface in the three-surface. This gives the analogue of the Newtonian formula for the total mass contained within a two-surface. There are however two important differences from the Newtonian case:

(i) a factor / appears in the integral on the right-hand side. This means that matter placed in a region where /is considerably less than one (a large negative Newtonian potential) makes a smaller contribution to the total mass than does the same matter in a region where/is almost one (small negative Newtonian potential);

(ii) the pressure contributes to the total mass. This means that in some circumstances it can actually assist rather than prevent gravitational collapse.

are called the Einstein equations and are often written in the equivalent form (Rab - \Rgab)= Srr^. (3.15)

Since both sides are symmetric, these form a set of ten coupled nonlinear partial differential equations in the metric and its first and second derivatives. However the covariant divergence of each side vanishes identically, that is,

hold independent of the field equations. Thus the field equations really provide only six independent differential equations for the metric. This is in fact the correct number of equations to determine the spacetime, since four of the ten components of the metric can be given arbitrary values by use of the four degrees of freedom to make coordinate transformations. Another way of looking at this is that two metrics gj and g2 on a manifold ^ define the same space-time if there is a diffeomorphism 6 which takes gx into g2. Therefore the field equations should define the metric only up to an equivalence class under diffeomorphisms, and there are four degrees of freedom to make diffeomorphisms.

We shall consider the Cauchy problem for the Einstein equations in chapter 7, and shall show that, together with the equations for the matter fields, they are sufficient to determine the evolution of spacetime given suitable initial conditions, and that they satisfy the causality postulate (a).

The Einstein equations can be derived by requiring that the action

Js be stationary under variations ofgab, where L is the matter Lagrangian and A a suitable constant. For

The last term can be written g°»ARabdv = g°b((Ar<ab).^-(Ar\c).b)dv

Thus it may be transformed into an integral over the boundary dQ, which vanishes as AT^ vanishes on the boundary. Therefore

= f {^l((iR - A) <7a6 - Rab) + £7™} A^d«, (3.17)

and so iidl/du vanishes for all Agab, one obtains the Einstein equations on setting A = (167r)-1.

One might ask whether varying an action derived from some other scalar combination of the metric and curvature tensors might not give a reasonable alternative set of equations. However the curvature scalar is the only such scalar linear in second derivatives of the metric tensor;

so only in this case can one transform away a surface integral and be left with an equation involving only second derivatives of the metric.

If one tried any other scalar such as RabRab or RajxdRabcd one would obtain an equation involving fourth derivatives of the metric tensor.

This would seem objectionable, as all other equations of physics are first or second order. If the field equations were fourth order, it would be necessary to specify not only the initial values of the metric and its first derivatives, but also the second and third derivatives, in order to determine the evolution of the metric.

We shall assume the field equations do not involve derivatives of the metric higher than the second. If these field equations are derived from a Lagrangian, then the action must have the form (3.16). One could however obtain a system of equations other than the Einstein equations, if one restricted the form of the variations Agab for which the action was required to be stationary.

For example, one could restrict the metric to be conformal to a flat metric, i.e. assume „„

where rjab is a flat metric as in Special Relativity. Then

Ag-o6 = 2Q-iAQffa6 and the action will be stationary if

{(A(iR-A)gab-Rab) + Tab}AQgab = 0 for all AQ, that is if R + A~XT = 4A. From (2.30),

R = -6Q.-m]bcV<x = -en-m.^+izn-m.. a.dSr*, where | denotes covariant differentiation with respect to the flat metric ijab. If the metric is static, Q will be constant along the integral curves of the Killing vector K (it will be independent of the time t). The magnitude of K will be proportional to Q. Therefore f iab(gab+ V»V*)f-i = Cl.ab(gab+ VaVb) Q"1

Thus the Laplacian of/will be equal to — plus a term proportional to the square of the gradient of/. This last term may be neglected in a weak field. From the field equations, will be equal to

For a perfect fluid, T = ~/i + 3p. One will therefore get agreement with Newtonian theory if Ais small or zeroandvl-1 = — 247r.

This theory in which the metric is restricted to be conformally flat is known as the Nordstrom theory. It can be reformulated as a theory in which the metric is the flat metric rj and in which the gravitational interaction is represented by an additional scalar field As mentioned before, this sort of theory would be inconsistent with the observed deflection of light by massive objects, and it would not account for the measured advance of the perihelion of Mercury.

One could in fact obtain the observed deflection of light and the advance of the perihelion of Mercury if the metric was restricted to be of the form gab = W{Vab + WaWb), where Wa is an arbitrary one-form field. This would give the Newtonian limit in a static metric in which Wa was parallel to the timelike Killing vector. There could however also be other static metrics where Wa was not parallel to the Killing vector and these would not give the Newtonian limit. Further this restriction on the form of the metric seems rather artificial. It appears more natural not to restrict the metric, apart from requiring that it be Lorentzian.

We therefore adopt as our third postulate,

Postulate (c): Field equations Einstein's field equations (3.15) hold on

The predictions of these field equations agree, within the experimental errors, with the observations that have been made so far on the deflection of light and the advance of the perihelion of Mercury, though the question of whether there exists a long range scalar field which ought to be included in the energy-momentum tensor remains open at the present time.

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