Unless the giddy heaven fall,
And earth some new convulsion tear,
And, us to join, the world should all
Be cramped into a planisphere.
As lines so loves oblique may well
Themselves in every angle greet;
But ours, so truly parallel,
Though infinite, can never meet.
Therefore the love which us doth bind,
But Fate so enviously debars,
Is the conjunction of the mind,
And opposition of the stars.
Andrew Marvell (1621-1678)
The empirical evidence for the existence of black holes – or at least something very much like them has become impressive, although it is arguably still largely circumstantial. Indeed, most relativity experts, while expressing high confidence (bordering on certainty) in the existence of black holes, nevertheless concede that since any electromagnetic signal reaching us must necessarily have originated outside any putative black holes, it may always be possible to imagine that they were produced by some mechanism just short of a black hole. Hence we may never acquire, by electromagnetic signals, definitive proof of the existence of black holes – other than by falling into one. (It’s conceivable that gravitational waves might provide some conclusive external evidence, but no such waves have yet been detected.)
Of course, there are undoubtedly bodies in the universe whose densities and gravitational intensities are extremely great, but it isn’t self-evident that general relativity remains valid in these extreme conditions. Ironically, considering that black holes have become one of the signature predictions of general relativity, the theory’s creator published arguments purporting to show that gravitational collapse of an object to within its Schwarzschild radius could not occur in nature. In a paper published in 1939, Einstein argued that if we consider progressively smaller and smaller stationary systems of particles revolving around each other under their mutual gravitational attraction, the particles would need to be moving at the speed of light before reaching the critical density. Similarly Karl Schwarzschild had computed the behavior of a hypothetical stationary star of uniform density, and found that the pressure must go to infinity as the star shrank toward the critical radius. In both cases the obvious conclusion is that there cannot be any stationary configurations of matter above the critical density. Some scholars have misinterpreted Einstein’s point, claiming that he was arguing against the existence of black holes within the context of general relativity. These scholars underestimate both Einstein’s intelligence and his radicalism. He could not have failed to understand that sub-light particles (or finite pressure in Schwarchild’s star) meant unstable collapse to a singular point of infinite density – at least if general relativity holds good. Indeed this was his point: general relativity must fail. Thus we are not surprised to find him writing in “The Meaning of Relativity”
For large densities of field and matter, the field equations and even the field variables which enter into them have no real significance. One may not therefore assume the validity of the equations for very high density of field and matter… The present relativistic theory of gravitation is based on a separation of the concepts of “gravitational field” and of “matter”. It may be plausible that the theory is for this reason inadequate for very high density of matter…
These reservations were not considered to be warranted by other scientists at the time, and even less so today, but perhaps they can serve to remind us not to be too dogmatic about the validity of our theories of physics, especially when extrapolated to very extreme conditions that have never been (and may never be) closely examined.
Furthermore, we should acknowledge that, even within the context of general relativity, the formal definition of a black hole may be impossible to satisfy. This is because, as discussed previously, a black hole is strictly defined as a region of spacetime that is not in the causal past of any point in the infinite future. Notice that this refers to the infinite future, because anything short of that could theoretically be circumvented by regions that are clearly not black holes. However, in some fairly plausible cosmological models the universe has no infinite future, because it re-collapses to a singularity in finite coordinate time. In such a universe (which, for all we know, could be our own), the boundary of any gravitationally collapsed region of spacetime would be contiguous with the boundary of the ultimate collapse, so it wouldn’t really be a separate black hole in the strict sense. As Wald says, "there appears to be no natural notion of a black hole in a closed Robertson-Walker universe which re-collapses to a final singularity", and further, "there seems to be no way to define a black hole in a closed universe, because it requires going to infinity, but there is no infinity in a closed universe."
It’s interesting that this is essentially the same objection that is often raised by people when they first hear about black holes, i.e., they reason that if it takes infinite coordinate time for any object to cross an event horizon, and if the universe is going to collapse in a finite coordinate time, then it’s clear that nothing can possess the properties of a true black hole in such a universe. Thus, in some fairly plausible cosmological models it's not strictly possible for a true black hole to exist. On the other hand, it is possible to have an approximate notion of a black hole in some isolated region of a closed universe, but of course many of the interesting transfinite issues raised by true (perhaps a better name would be "ideal") back holes are not strictly applicable to an "approximate" black hole.
Having said this, there is nothing to prevent us from considering an infinite open universe containing full-fledged black holes in all their transfinite glory. I use the word “transfinite” because ideal black holes involve singular boundaries at which the usual Schwarzschild coordinates for the external field of a gravitating body go to infinity - and back - as discussed in the previous section. There are actually two distinct kinds of "spacetime singularities" involved in an ideal black hole, one of which occurs at the center, r = 0, where the spacetime manifold actually does become unequivocally singular and the field equations are simply inapplicable (as if trying to divide a number by 0). It's unclear (to say the least) what this singularity actually means from a physical standpoint, but oddly enough the "other" kind of singularity involved in a black hole seems to shield us from having to face the breakdown of the field equations. This is because it seems (although it has not been proved) to be a characteristic of all realistic spacetime singularities in general relativity that they are invariably enclosed within an event horizon, which is a peculiar kind of singularity that constitutes a one-way boundary between the interior and exterior of a black hole. This is certainly the case with the standard black hole geometries based on the Schwarzschild and Kerr solutions. The proposition that it is true for all singularities is sometimes called the Cosmic Censorship Conjecture. Whether or not this conjecture is true, it's a remarkable fact that at least some (if not all) of the singular solutions of Einstein's field equations automatically enclose the singularity inside an event horizon, an amazing natural contrivance that effectively shields the universe from direct two-way exposure to any regions in which the metric of spacetime breaks down.
Perhaps because we don't really know what to make of the true singularity at r = 0, we tend to focus our attention on the behavior of physics near the event horizon, which, for a non-rotating black hole, resides at the radial location r = 2m, where the Schwarzschild coordinates become singular. Of course, a singularity in a coordinate system doesn't necessarily represent a pathology of the manifold. (Consider traveling due East at the North Pole). Nevertheless, the fact that no true black hole can exist in a finite universe shows that the coordinate singularity at r = 2m is not entirely inconsequential, because it does (or at least can) represent a unique boundary between fundamentally separate regions of spacetime, depending on the cosmology. To understand the nature of this boundary, it's useful to consider hovering near the event horizon of a black hole. The components of the curvature tensor at r = 2m are on the order of 1/m2, so the spacetime can theoretically be made arbitrarily "flat" (Lorentzian) at that radius by making m large enough. Thus, for an observer "hovering" at a value of r that exceeds 2m by some arbitrarily small fixed ratio, the downward acceleration required to resist the inward pull can be arbitrarily small for sufficiently large m. However, in order for the observer to be hovering close to 2m his frame must be tremendously "boosted" in the radial direction relative to an in-falling particle. This is best seen in terms of a spacetime diagram such as the one below, which show the future light cones of two events located on either side of a black hole's event horizon.
In this drawing r is the radial Schwarzschild coordinate and t' is an Eddington-Finkelstein mapping of the Schwarzschild time coordinate, i.e.,
The right-hand ray of the cone for the event located just inside the event horizon is tilted just slightly to the left of vertical, whereas the cone for the event just outside 2m is tilted just slightly to the right of vertical. The rate at which this "tilt" changes with r is what determines the curvature and acceleration, and for a sufficiently large black hole this rate can be made negligibly small. However, by making this rate small, we also make the outward ray more nearly "vertical" for a radial coordinate r that exceeds 2m by any given ratio greater than 1, which implies that the hovering observer's frame needs to be even more "boosted" relative to the local frame of an observer falling freely from infinity. The gravitational potential, which need not be changing very steeply at r = 2m, has nevertheless changed by a huge amount relative to infinity. We must be very deep in a potential hole in order for the light cones to be tilted that far, even though the rate at which the tilt has been increasing can be arbitrarily slow. This just means that for a super-massive black hole they started tilting at a great distance.
As can be seen in the diagram, relative to the frame of a particle falling in from infinity, a hovering observer must be moving outward at near light velocity. Consequently his axial distances are tremendously contracted, to the extent that, if the value of r is normalized to his frame of reference, he is actually a great distance (perhaps even light-years) from the r = 2m boundary, even though he is just 1 inch above r = 2m in terms of the Schwarzschild coordinate r. Also, the closer he tries to hover, the more radial boost he needs to hold that value of r, and the more contracted his radial distances become. Thus he is living in a thinner and thinner shell of r, but from his own perspective there's a world of room. Assuming he brought enough rocket fuel to accelerate himself up to this "hovering frame" at that radius 2m + r (or actually to slow himself down to a hovering frame), he would thereafter just need to resist the local acceleration of gravity to maintain that frame of reference.
Quantitatively, for an observer hovering at a small Schwarzschild distance r above the horizon of a black hole, the radial distance r' to the event horizon with respect to the observer's local coordinates would be
which approaches as r goes to zero. This shows that as the observer hovers closer to the horizon in terms of Schwarzschild coordinates, his "proper distance" remains relatively large until he is nearly at the horizon. Also, the derivative of r' with respect to r in this range is , which goes to infinity as r goes to zero. (These relations pertain to a truly static observer, so they don’t apply when the observer is moving from one radial position to another, unless he moves sufficiently slowly.)
Incidentally, it's amusing to note that if a hovering observer's radial distance contraction factor at r was 12m/r instead of the square root of that quantity, his scaled distance to the event horizon at a Schwarzschild distance of r would be r' = 2m + r. Thus when he is precisely at the event horizon his scaled distance from it would be 2m, and he wouldn’t achieve zero scaled distance from the event horizon until arriving at the origin r = 0 of the Schwarzschild coordinates. This may seem rather silly, but it’s actually quite similar to one of Einstein’s proposals for avoiding what he regarded as the unpleasant features of the Schwarzschild solution at r = 2m. He suggested replacing the radial coordinate r with = , and noted that the Schwarzschild solution expressed in terms of this coordinate behaves regularly for all values of . Whether or not there is any merit in this approach, it clearly shows how easily we can “eliminate” poles and singularities simply by applying coordinates that have canceling zeros or by restricting the domain of the variables. However, we shouldn’t assume that every arbitrary system of coordinates has physical significance.
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