B. The cosmological constant
Special relativity is very successful in laboratory physics. Thus one might guess any inertial observer would see the same vacuum. A freely moving inertial observer represents spacetime in the neighborhood by locally Minkowskian coordinates, with the metric tensor µ given in Eq. (14). A Lorentz transformation to an inertial observer with another velocity does not change this Minkowski form. The same must be true of the stress-energy tensor of the vacuum, if all observers see the same vacuum, so it has to be of the form
where is a constant, in a general coordinate labeling. On writing this contribution to the stress-energy tensor separately from all the rest, we bring the field equation (17) to
This is Einstein's (1917) revision of the field equation of general relativity, where is proportional to his cosmological constant ; his reason for writing down this equation is discussed in Sec. III.A. In many dark energy scenarios is a slowly varying function of time and its stress-energy tensor differs slightly from Eq. (19), so the observed properties of the vacuum do depend on the observer's velocity.
One sees from Eqs. (14), (18), and (19) that the new component in the stress-energy tensor looks like an ideal fluid with negative pressure
This fluid picture is of limited use, but the following properties are worth noting. 10
The stress-energy tensor of an ideal fluid with four-velocity uµ generalizes from Eq. (18) to Tµ = ( + p)uµ u - pgµ. The equations of fluid dynamics follow from the vanishing of the divergence of Tµ. Let us consider the simple case of locally Minkowskian coordinates, meaning free fall, and a fluid that is close to homogeneous. By the latter we mean the fluid velocity -- the space part of the four-velocity uµ -- and the density fluctuation from homogeneity may be treated in linear perturbation theory. Then the equations of energy and momentum conservation are
where cs2 = dp / d and the mean density and pressure are <> and <p>. These combine to
If cs2 is positive this is a wave equation, and cs is the speed of sound.
The first of Eqs. (22) is the local energy conservation law, as in Eq. (9). If p = - , the pdV work cancels the dV part: the work done to increase the volume cancels the effect of the increased volume. This has to be so for a Lorentz-invariant stress-energy tensor, of course, where all inertial observers see the same vacuum. Another way to see this is to note that the energy flux density in Eqs. (22) is (<> + <p>). This vanishes when p = - : the streaming velocity loses meaning. When cs2 is negative Eq. (23) says the fluid is unstable, in general. But when p = - the vanishing divergence of Tµ becomes the condition seen in Eq. (22) that = <> + is constant.
There are two measures of gravitational interactions with a fluid: the passive gravitational mass density determines how the fluid streaming velocity is affected by an applied gravitational field, and the active gravitational mass density determines the gravitational field produced by the fluid. When the fluid velocity is nonrelativistic the expression for the former in general relativity is + p, as one sees by writing out the covariant divergence of Tµ. This vanishes when p = - , consistent with the loss of meaning of the streaming velocity. The latter is + 3p, as one sees from Eq. (8). Thus a fluid with p = - / 3, if somehow kept homogeneous and static, would produce no gravitational field. 11 In the model in Eqs. (19) and (21) the active gravitational mass density is negative when is positive. When this positive dominates the stress-energy tensor is positive: the rate of expansion of the universe increases. In the language of Eq. (20), this cosmic repulsion is a gravitational effect of the negative active gravitational mass density, not a new force law.
The homogeneous active mass represented by changes the equation of relative motion of freely moving test particles in the nonrelativistic limit to
where is the relative gravitational acceleration produced by the distribution of ordinary matter. 12 For an illustration of the size of the last term consider its effect on our motion in a nearly circular orbit around the center of the Milky Way galaxy. The Solar System is moving at speed vc = 220 km s-1 at radius r = 8 kpc. The ratio of the acceleration g produced by to the total gravitational acceleration g = vc2 / r is
a small number. Since we are towards the edge of the luminous part of our galaxy, a search for the effect of on the internal dynamics of galaxies like the Milky Way does not look promising. The precision of celestial dynamics in the Solar System is much greater, but the effect of is very much smaller; for the orbit of the Earth, g / g ~ 10-22.
One can generalize Eq. (19) to a variable , by taking p to be negative but different from - . But if the dynamics were that of a fluid, with pressure a function of , stability would require cs2 = dp / d > 0, from Eq. (23), which seems quite contrived. A viable working model for a dynamical is the dark energy of a scalar field with self-interaction potential chosen to make the variation of the field energy acceptably slow, as discussed next.
10 These arguments have been familiar, in some circles, for a long time, though in our experience discussed more often in private than the literature. Early statements of elements are in Lemaître (1934) and McCrea (1951); see Kragh (1999, pp. 397-8) for a brief historical account. Back.
11 Lest we contribute to a wrong problem for the student we note that a fluid with p = - / 3 held in a container would have net positive gravitational mass, from the pressure in the container walls required for support against the negative pressure of the contents. We have finessed the walls by considering a homogeneous situation. We believe Whittaker (1935) gives the first derivation of the relativistic active gravitational mass density. Whittaker also presents an example of the general proposition that the active gravitational mass of an isolated stable object is the integral of the time-time part of the stress-energy tensor in the locally Minkowskian rest frame. Misner and Putman (1959) give the general demonstration. Back.
12 This assumes the particles are close enough for application of the ordinary operational definition of proper relative position. The parameters in the last term follow from Eqs. (8) and (21). Back.