Sharp Blue: making the complex simple

In this journal, you'll find my ephemeral thoughts. Perhaps I won't write anything profound, but I'll only post things that I think at least some of my audience will find interesting or amusing. Amongst the updates on my life, you will probably find short articles on science, technology, history, books and music. The things most worth reading are my articles "Cathedrals and Shuttles", "Flamininus at Corinth", "Enemies at the Gate", the series beginning with "The Economics of Space Transportation", the series beginning with "Maps of Physics", and "Living at the Fulcrum", the nearest thing I've written to a personal statement.

- Richard Baker

This will be a much more rambling post than is usual. On the Brin List, the conversation has turned to the fundamental nature of ethics, and from there, essentially, to whether "might makes right", or at least whether ideologies that endure are de facto the most effective and so most moral. During this discussion, one of the more erudite contributors said:

Historically, empires can last a long time. The eastern part of the Roman Empire, which was split by Constantine in the 300s, lasted roughly 1500 years, and was defeated by another empire. IIRC, the Chinese empire lasted about the same length until it was overtook by the Ghengas Kahn...who's rule ended up merging into that empire.

In reply I wrote quite a long, tangential message on the inaccuracies of these statements, and various other ideas that they brought to mind. I thought that perhaps some of my regular readers might like to read my reply too, even though it's not especially well structured and doesn't really present a proper argument. It is, however, the longest thing I've written for some months...

To begin with, Constantine reunified rather than splitting the administration of the Roman state. The history of the separation between West and East bears closer examination. Under the Republic, the Romans had a long history of the division of the supreme magistracy, first between two consuls and later into first an ad-hoc and later a formalised "triumvirate". This tendency briefly re-emerged during the second century with the co-imperium of Marcus Aurelius Antoninus and Lucius Aurelius Verus, which enabled the presence of emperors at several trouble-spots concurrently.

During the troubled third century this need for divided absolute authority became even more pressing and was formalised by the emperor Diocletian's institution of the "tetrarchy", in which there were two senior emperors ("Augusti") and two junior emperors ("Caesars"). It was Diocletian's intention that the Augusti should periodically abdicate in favour of their junior colleagues who would in turn appoint two new Caesars from the best men of the state. The succession of the emperors would thus be regularised, putting an end to the cycle of rebellion and civil war that had plagued the empire for fifty years. Unfortunately, it didn't work like that, as sons of the Augusti who had been passed over in favour of new, unrelated emperors, asserted their supposed hereditary rights, alternative centres of power crystallised and a new phase of civil wars began. The ultimate victor was Constantine, who became sole ruler of the Roman empire in 324.

Before Constantine, there had been many temporary Roman capitals - for many decades the capital had effectively not been Rome but wherever the emperor was. Under the tetrarchy, for example, the capitals of the Augusti had been Nicomedia in Asia Minor, Mediolanum in northern Italy, Sirmium in what's now Serbia and Augusta Treverorum (modern Trier). One of Constantine's several innovations was the establishment of a permanent new capital at Constantinople. Rather than this city being the capital of an "Eastern Roman Empire", it was the capital of the whole empire. Even during periods of division of the imperial authority, the empire itself was seen as a unitary whole and the usual procedure was for edicts to be issued in the name of all the current emperors and to be enforced across the Roman world.

It's commonly held that the final division of the Roman empire occurred in 395 at the death of Theodosius I, at which Honorius became emperor in the west and Arcadius in the East. From then until the extinction of the western dynasty in 476 there was always an emperor in Constantinople and another usually in Ravenna. However, even as these two centres of power solidified, the Roman world formally remained whole. The two emperors provided each other with military assistance even as late as a major joint naval expedition against the Vandals in 468. Even the man sometimes seen as the last fully legitimate western emperor, Julius Nepos, was appointed by the eastern emperor Leo I. Furthermore, following the overthrow of the last western emperor, Romulus Augustulus, many of the Germanic successor rulers claimed to be ruling not as independent kings but as representatives of the emperor at Constantinople.

As for when the Eastern remnant of the Roman empire fell, I think there were two very clear periods during which large swathes of territory were lost and the character of the empire deeply changed. The first was during the lightning conquests of the Muslim armies in the seventh century, which cut away from the empire the ancient Roman provinces of Syria, Palestine, Egypt and North Africa. Augustus might well have recognised the sixth century empire of Justinian as a successor, however much transformed by the passage of centuries, to his own; but the Byzantine empire of Heraclius and his successors was a different world. The second major collapse occurred with the defeat of Romanus Diogenes by the Seljuk Turkish sultan Alp Arslan at Manzikert in 1071. (The Seljuk sultanate was a successor to the Arab Caliphates that had inflicted the earlier defeats on the Byzantines.)

In any case, much of this is a distraction from the central questions: what endured for those 1500 or more years, and was it totalitarian. In my view the main continuity was that of the administrative bureaucracy created by the Romans, despite the changes at the highest levels of power, the shifts of culture and even the change of religion. During the first few centuries of the Empire, the military and civil leaders were essentially talented amateurs drawn from the senatorial class. A major development during the third century was the replacement of these aristocratic leaders by middle class, professional leaders, first in the military sphere under Gallienus and then in the civil administration under Diocletian and Constantine. Alongside this shift, the administrative bureaucracy expanded dramatically in size as the troubled empire sought to organise its still massive economic resources to meet its ever more desperate military needs. It's striking that the empire of the second century was run by an imperial staff of a few hundred bureaucrats but more striking that by the fourth century this had increased to tens of thousands.

It was this vast administrative machinery - and the parallel hierarchy of the Christian Church, with which it became increasingly entangled - that endured through so many changes of dynasty, provincial structure, prevailing religious orthodoxy and military organisation. Indeed, it even survived the collapse of Roman political authority in both East and West. The Germanic rulers of post-Roman Europe attempted to preserve the Roman administration and the Roman laws, but both fragmented and decayed during the first few centuries of the German states. Under Islam, however, the bureaucracy flourished, becoming the administration of the Ummayyad and Abbasid Caliphates. The civilisation of classical Islam fused the Arab religion with Roman administration and Persian elite culture.

(I think that this kind of continuity through administrative bureaucracy, or at least continuity of scribal and bureaucratic standards, pratice and culture, is typical of ancient civilisations, whether Roman, Egyptian, Mesopotamian or Chinese.)

As for totalitarianism, I think it's clear that it's a product of modern states. Even when the Roman rulers might have aspired to totalitarianism, such as during Diocletian's attempts to control the economy through edicts, or the exasperated attempts by Christian emperors to impose some kind of religious orthodoxy, the tools to do so - mass media, mass surveillance and so forth - simply were not available. Likewise, republicanism or democracy on scales larger than that of city-states are products of modern times. It's not clear to me that the endurance or otherwise of pre-modern empires has much to say about the prospects for democracy or dictatorship in the modern world.

I could say as much about China, but I'll spare you the details. However, it's incorrect both that Genghis Khan conquered China and that the empire the Mongols conquered had endured for 1500 years. Since the fall of the Tang dynasty in 907, China had been divided into a number of smaller states. During the period from 906 to 960, five dynasties rapidly succeeded one another in the north of China and the south was divided into ten or so small states. China was briefly reunified by the Song dynasty but by 1127 the northern part of the country had fallen under the rule of the non-Chinese Jin and Xia dynasties in the east and west respectively. These two northern dynasties were defeated by Genghis but the conqueror of China proper was his grandson Kublia, founder of the increasingly sinicised Yuan dynasty. The Mongols ruled China for a century until the Yuan were overthrown by the native Chinese Ming dynasty.

As with Rome, China passed through succeeding periods of political unity and disunity. Indeed, the normal state of affairs might have been a division into smaller states ruled by independent dynasties. From the first unification of China by the Qin dynasty in 221BC to the Mongol conquest in AD1271, China was only inarguably a single state from 221BC to AD220 under the Qin and Han, from 581 to 907 under the Sui and Tang and from 960 to 1127 under the Northern Song, or about 60% of that period. It was only during the Yuan, Ming and Qing that the idea of China as a coextensive political and cultural zone achieved an enduring reality. (Which is not to denigrate the earlier achievements of the Chinese. For example, at the time of the Mongol conquest the Song capital, Hangzhou, may have been the most populous, wealthy and sophisticated city in the world.)

I'll say even less about another civilisation that I know something about - ancient Egypt - but that one also wasn't a single "Egyptian Empire". Instead, four periods of unity (the Old Kingdom, Middle Kingdom, New Kingdom and Late Period) were separated by periods of political decentralisation or foreign domination, and I seem to recall counting something like fifteen distinct periods of ancient Egyptian imperialistic expansion. In this case too, the continuities across vast periods of time are not so much political as cultural and administrative.

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In the previous part of what has now become an ongoing series on the causal structure of various general relativistic spacetimes, I discussed the causal structure of the flat, Minkowski spacetime of special relativity, and of the Schwarzschild vacuum outside a spherical, uncharged, non-rotating star which collapses to form a black hole. In this part, I'd like to discuss the so-called Kruskal extension of the Schwarzschild vacuum. This is the general solution for a static, asymptotically flat vacuum (that is, a matter-free spacetime that looks like Minkowski spacetime when one is far away from the event horizon) containing a black hole. In other words, this time we'll consider a black hole that exists for all time, rather than one which forms from the collapse of a star.

As commenters have noted, the astrophysical black hole in the last article is not a time-symmetric solution as there's a star early in time and a black hole late in time. The time reverse of this solution is a "white hole", from which matter can emerge into the outside universe but into which no matter can fall. The Kruskal extension of the Schwarzschild metric is time symmetric and it contains both a black hole region and a white hole region (and is thus occasionally called a "grey hole"). The former contains a singularity that is in the future of some lightlike and timelike paths - those that enter the black hole - but in the past of none. Similarly, the latter contains a singularity that is in the past of some lightlike and timelike paths but in the future of none. More surprisingly, the full solution contains two asymptotically flat external regions, each of which is causally isolated from the other!

Figure 1 The causal structure of a maximally extended Schwarzschild spacetime

Even though these posts are about the causal structure of the spacetimes in question rather than their geometries, I feel that at this point I ought to say something more about coordinates I've used in this diagram. For observers at rest with respect to the hole and far away from it in one of the asymptotically flat regions, the t coordinate is just proper time as measured on a standard clock. Suppose this distant clock emits a regular "time signal": a flash of light to mark every second of its proper time. An observer elsewhere in the external vacuum who is at rest with respect to the distant clock will not in general receive one time signal flash per second of her proper time. If she's closer to the hole then she'll receive more than one flash per second as measured on her clock. (If she emits her own time signal flash once per second of her proper time, the distant clock will receive them less often than once per second of its proper time. This is the famous phenomenon of gravitational time dilation, which leads immediately to gravitational redshifts and blueshifts. Unlike the time dilation of special relativity its not symmetric with respect to the two clocks.) However, by adjusting the mechanism of her clock so it runs more slowly she can make it tick in synchrony with the time signals arriving from the asymptotically flat region. In this way - provided a timelike hypersurface is chosen as a "zero" of coordinate time - the t coordinate can be extended across the external region.

The r coordinate is much easier to grasp. It's a radial coordinate with respect to the hole that's chosen such that a sphere of constant r coordinate has an area of 4 pi r^2. (Every point in the Penrose diagram is such a sphere at a certain time.) Note, though, that this means that r is not a proper distance (that is, a distance measured by using measuring rods). As the spacetime is spherically symmetric it's easy enough to finish off our coordinate system by picking two angular coordinates, but these won't concern us here.

The event horizon of the hole is at an r coordinate of 2m (where m is the mass of the hole and I'm using coordinates in which the gravitational constant and the speed of light are both equal to 1). (This distance is called the Schwarzschild radius, but it's not, of course, a proper radius.) As I described in the last part, a distant observer watching an object fall into the black hole sees it fall ever more slowly towards the horizon (and at the same time it appears to get ever more redshifted and ever dimmer). The black hole's event horizon is thus at a t coordinate of +infinity. (Remember, though, that the falling object crosses the event horizon in a finite amount of its proper time.) In a similar way, the distant observer sees any objects that emerge from the white hole as having done so infinitely long ago in coordinate time. The white hole's event horizon is thus at a t coordinate of -infinity.

At some future time, I'd like to say something about the causal structure of rotating and/or electrically charged black holes, but in the next part of the series I'm going to focus on the causal structure of open and closed universes.

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A commenter writing about my earlier post, "Relativity, FTL and Causality" said:

As far as FTL being equivalent to time travel, the above explanation is correct. As far as FTL being impossible at present, it is not quite. Richard's explanation lacks a mention of black holes. An object (with non-zero mass), falling into a black hole from rest, will cross the event horizon AT THE SPEED OF LIGHT. Unfortunately we (the distant observers) would not be able to witness this event, since the falling observer would take forever (from our viewpoint) to reach the event horizon. In the falling observer's reference frame, however, he'd be flying at the speed of light all right! Moreover, he'd probably be able to time travel as well, as his speed continues to increase (to >c) inside the event horizon. Whether this would lead to any causality violations is unknown, since we can't see past the event horizon, and the unfortunate falling observer has a short time to live before he hits singularity, causality violations or not. However, from a purely theoretical point of view, we know that FTL travel is possible with black holes.

This comment about black holes is not true, at least in general relativity, the relativistic theory of gravity. It's correct that the radial velocity of an infalling object goes to zero at the event horizon when expressed in the coordinates of an inertial observer who is stationary with respect to the black hole and far away from it. In other words, the distant observer sees the falling object approach the horizon but never cross it. However, if a second observer falls in a windowless[1] spaceship she would notice nothing out of the ordinary, except for tidal effects, as she crosses the event horizon. If the tidal effects are sufficiently small (they can be made arbitrarily small by increasing the mass of the black hole) then she'd have no way of knowing whether she was falling into a black hole or just drifting through space. This is one aspect of a general feature of general relativistic spacetimes: as the region considered gets smaller and smaller it looks more and more like the spacetime of special relativity. Moreover, there's no frame in which her trajectory becomes spacelike so she can't be said to be travelling faster than light.

On the other hand, there's clearly something special about the event horizon of a black hole, and the singularity in its interior. What happens in a black hole spacetime is that the light cones near the black hole are "tilted" compared with what a naive observer at far away from the hole might expect. At the event horizon the tilting becomes so great that there are no future timelike or lightlike trajectories that can escape to infinity. The singularity is therefore not really at the "centre" of the black hole - although it looks that way in the coordinate system of the distant inertial observer - but rather in the future of all observers who fall across the horizon. And typically, and unfortunately for our daring black-hole explorer, not that far in the future.

One way to visualise this is through the use of Penrose diagrams. These are diagrams of the causal structure of a spacetime in which the points infinitely far away in space or time have been drawn at finite distances on the diagram through the use of a "conformal mapping" that leaves the causal structure intact. (The conformal mapping distorts the geometry of the spacetime to more clearly show its causal structure.) Rays of light in these diagrams still travel at 45 degrees to the vertical as in all of my Minkowski diagrams. The edges of the diagram are then the regions of timelike, spacelike and lightlike infinity - and singularities at a finite distance. This all sounds rather arcane, so let's look at an example:

Figure 1 The causal structure of Minkowski spacetime

On this diagram I've shown two possible paths for freely falling observers. (These paths are technically called "timelike geodesics".). Just as in my other diagrams of spacetime, at each event along the timelike path the path itself is in the light cone of the event, so all observers with mass travel slower than light. If we follow either of the two example paths - or indeed any timelike geodesic - forward in time we reach the region called "future timelike infinity". If we follow them backwards in time we reach "past timelike infinity". Similarly, light rays can shine from "past null infinity" and can extend in the future to "future null infinity". As we might expect, all the edges of this diagram are the appropriate kinds of infinity as Minkowski spacetime extends infinitely in space and time from any given event.

Now let's take a look at the causal structure of a spacetime that contains a spherical, uncharged, non-rotating star collapsing to form a black hole. (The spacetime for an eternally existing black hole has features I don't wish to discuss here.) In this case the spacetime outside the star isn't a Minkowski spacetime but a Schwarzschild spacetime. However, causally - and remember that Penrose diagrams show only the causal structure and not the geometry - the outside region is very much like Minkowski spacetime. The inside is very different though.

Figure 1 The causal structure of a physically realistic black hole spacetime. (The wiggles in the singularity are a notational convention and are not supposed to indicate any structure in the singularity itself.)

Once again, outgoing light rays originating from anywhere in the exterior vacuum region can reach future null infinity and (some) freely falling matter particles can reach future timelike infinity. But the diagram clearly shows that a light ray or matter particle from the region inside the event horizon of the black hole cannot reach future null or timelike infinity respectively. No matter in which direction or at what speed particles inside the horizon move, they can only ever reach the singularity. The singularity itself is a spacelike hypersurface on which all world-lines passing through the event horizon terminate. To escape from it, one would have to be able to travel faster than light. Otherwise it is, very literally, the end of time.

[1] If she were fortunate(?) enough to have windows to observe the outside world she'd see some strange things that I won't discuss here.

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