Example: Minkowski space

2.4 Example: Minkowski space

In this section we continue our discussion of the prototype of an asymptotically flat space-time, namely Minkowski space-time. The motivation for doing so is partly to get more acquainted with the idea of conformal compactification and partly to show why this concept is the correct one for the description of radiation processes.

For the following discussion we refer to the rescaled metric

which is the metric of the Einstein cylinder. The standard conformal diagram for Minkowski space [125 ] is shown in Figure 2

Figure 2: The conformal diagram of Minkowski space.

Each point in the interior of the triangle corresponds to a 2-sphere. The long side of the triangle consists of all the points in the centre, $r = 0$ (i.e. $U = V$ ). The other two sides of the triangle correspond to null-infinity $ℐ ±$ . The points $i±$ are points in the centre with $U = V = ±π ∕2$ , while $i0$ is a point with $U = − π∕2,V = π ∕2$ . The lines meeting at $i0$ are lines of constant $t$ , while the lines emanating from $− i$ and converging into $+ i$ are lines of constant $r$ . In the four-dimensional space-time the lines of constant $t$ correspond to three-dimensional space-like hypersurfaces that are asymptotically Euclidean.

In the case of Minkowski space-time, the metric can be extended in a regular way to three points representing future and past time-like infinity and space-like infinity, but this is not generally so. Already in the case of the Schwarzschild metric, which is also an asymptotically flat metric, there are, strictly speaking, no such points because any attempt to extend the metric yields a singularity. However, it is common language to refer to this behaviour by saying that “the points $i0$ and/or $i±$ are singular”. The reason for this is related to the presence of mass. For any space-time that has a non-vanishing ADM mass, the point $i0$ is necessarily singular while the singularity of the time-like infinities is, in general, related to the fall-off properties of the energy-momentum tensor in time-like directions. In the case of the Schwarzschild solution (as in any stationary black hole solution), it is the presence of the static (stationary) black hole that is responsible for the singularity of $i±$ .

Figure 3: Spacelike hypersurfaces in the conformal picture.

Let us now assume that there is a particle that moves along the central world-line $r = 0$ , emitting radiation. For the sake of simplicity we assume that it emits electro-magnetic radiation, which travels along the outgoing null-cones to null-infinity. The null-cones are symbolized in Figure 2 by the straight lines going off the particle’s world-line at $45∘$ . We are now interested in the behaviour of the signal along various space-like hypersurfaces. In Figure 3 we show again the conformal diagram of Minkowski space. The generic features discussed below will be the same for any asymptotically flat space-time as long as we stay away from the corners of the diagram. The reason for choosing Minkowski space is simply one of convenience.

The vertical dashed line is the world-line of the particle that defines the time axis. We have displayed two asymptotically Euclidean space-like hypersurfaces intercepting the time axis at two different points and reaching out to space-like infinity. Furthermore, there are two hypersurfaces that intersect the time axis in the same two points as the asymptotically flat ones. They reach null-infinity, intersecting in a two-dimensional space-like surface. This geometric statement about the behaviour of the hypersurfaces in the unphysical space-time translates back to the physical space-time as a statement about asymptotic fall-off conditions of the induced (physical) metric on the hypersurfaces, namely that asymptotically the metric has constant negative curvature. This is, in particular, a property of the space-like hyperboloids in Minkowski space. Thus, such hypersurfaces are called hyperboloidal hypersurfaces. An important point to keep in mind is that the conformal space-time does not “stop” at $ℐ$ but that it can be extended smoothly beyond. The extension is not uniquely determined as we have already discussed in connection with the embedding of Minkowski space into the Einstein cylinder (cf. Figure 1 ). Thus, the extension plays no role in the concept of null-infinity but it can be very helpful for technical reasons, in particular when numerical issues are discussed.

We now imagine that the central particle radiates electro-magnetic waves of uniform frequency, i.e. proportional to $sin (ωτ)$ , where $τ$ is the particle’s proper time. This gives rise to a retarded electro-magnetic field on the entire space-time, which has the form $ϕ ∝ sin(ωu )$ , where $u$ is a retarded time coordinate on Minkowski space with $u = τ$ on the central world-line. We ignore the fall-off of the field because it is irrelevant for our present purposes. Let us now look at the waves on the various hypersurfaces.

In the physical space-time, the hypersurfaces extend to infinity, and we can follow the waves only up to an arbitrary but finite distance along the hypersurfaces. The end-points are indicated in Figure 3 as little crosses. The resulting waveforms are shown in Figure 4 .

Figure 4: Waveforms in physical space-time.

The first diagram shows the situation on the asymptotically Euclidean surfaces. These are surfaces of constant Minkowski time, which implies that the signal is again a pure sine wave. Note, however, that this is only true for these special hypersurfaces. Even in Minkowski space-time we could choose space-like hypersurfaces that are not surfaces of constant Minkowski time but which nonetheless are asymptotically Euclidean. On such surfaces the wave would look completely different.

On the hyperboloidal surfaces the waves seem to “flatten out”. The reason for the decrease in frequency is the fact that these surfaces tend to become more “characteristic” as they extend to infinity, thus approaching surfaces of constant phase of the retarded field.

The final diagram shows the signal obtained by an idealized observer who moves along the piece of $ℐ$ between the two intersection points with the hyperboloidal surfaces. The signal is recorded with respect to the retarded time $u$ which, in the present case, is a so-called Bondi parameter (see Section 4.3). Therefore, the observer measures a signal at a single frequency for a certain interval of this time parameter. A different Bondi time would result in a signal during a different time interval but with a single, appropriately scaled, frequency. Using an arbitrary time parameter would destroy the feature that only one frequency is present in the signal. This is, in fact, the only information that can be transmitted from the emitter to the receiver under the given circumstances.

Figure 5: Waveforms in conformal space-time.

The waveforms of the signal as they appear in the conformal space-time, i.e. with respect to a coordinate system that covers a neighbourhood of $ℐ$ , are shown in Figure 5 . In the specific case of Minkowski space-time, we use the coordinates $T$ and $R$ on the Einstein cylinder. The signal on the asymptotically Euclidean surfaces shows the “piling up” of the waves as they approach space-like infinity. The signal on the hyperboloidal surfaces looks very similar to the physical case. Since the field and the surfaces are both smooth across $ℐ$ , the signal can continue on across null-infinity without even noticing its presence. The points where $ℐ$ is crossed are indicated in the diagram as two little crosses. The values of the field at these points are the same as the boundary values of the signal in the third diagram. Here the signal in the same region of $ℐ$ as in Figure 4 is displayed, but with respect to the coordinate $U = arctan u$ which is not a Bondi parameter. Accordingly, we see that the wavelength of the signal is not constant.

What these diagrams teach us is the following: It has been convenient in relativity to decompose space-time into space and time by slicing it with a family of space-like hypersurfaces. In most of the work on existence theorems of the Einstein equations it has been convenient to choose them to be Cauchy surfaces and thus asymptotically Euclidean. Also, in most numerical treatments of Einstein’s equations the same method is used to evolve space-times from one space-like hypersurface to the next (see Section 4). Here the hypersurfaces used are finite because the numerical grids are necessarily finite. In the approaches based on the standard Einstein equations it makes no difference whether the grid is based on a finite portion of an asymptotically Euclidean or a hyperboloidal hypersurface. The fact that the space-time should be asymptotically flat has to be conveyed entirely by a suitable boundary condition, which has to be imposed at the boundary of the finite portion of the hypersurface (i.e. at the little crosses in Figure 3 ). However, this implies that the accuracy of the waveform templates obtained with such approaches depends to a large extent on the quality of that boundary condition. So far there exists no suitable boundary condition that would be physically reasonable and lead to stable codes.

In the conformal approach one has the option to “include infinity” by using the conformal field equations (see Section 3). Then the type of the space-like hypersurfaces becomes an issue. The diagrams show that the hyperboloidal surfaces are very well suited to deal with the radiation problems. They provide a foliation of the conformal space-time on which one can base the evolution with the conformal field equations. The solution obtained will be smooth near $ℐ$ and we “only” need to locate $ℐ$ on each hypersurface to read off the value of the radiation data (as indicated in the second diagram of Figure 5 ).