The most obvious candidate explanation of this sort is that something going wrong with the dynamical structure of the geometry of spacetime, which is to say, with the curvature of the spacetime. This suggestion is bolstered by the fact that local measures of curvature do in fact blow up as one approaches the singularity of a standard black hole or the Big Bang singularity. There is, however, one problem with this line of thought: no species of curvature pathology we know how to define is either necessary or sufficient for the existence of incomplete paths.
For a discussion of foundational problems attendant on attempts to define singularities based on curvature pathology, see Curiel ; for a recent survey of technical issues, see Joshi To make the notion of curvature pathology more precise, we will use the manifestly physical idea of tidal force.
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Tidal force is generated by the difference in intensity of the gravitational field at neighboring points of spacetime. For example, when you stand, your head is farther from the center of the Earth than your feet, so it feels a practically negligible smaller pull downward than your feet.
1. Spacetime Singularities
For a diagram illustrating the nature of tidal forces, see Figure 9 of the entry on Inertial Frames. Tidal forces are a physical manifestation of spacetime curvature, and one gets direct observational access to curvature by measuring the resultant relative difference in accelerations of neighboring test bodies. For our purposes, it is important that in regions of extreme curvature tidal forces can grow without bound.
It is perhaps surprising that the state of motion of an object as it traverses an incomplete path e. Whether an object is spinning or not, for example, or accelerating slightly in the direction of motion, may determine whether the object gets crushed to zero volume along such a path or whether it survives roughly intact all the way along it, as shown by examples offered by Ellis and Schmidt Indeed, the effect of the observer's state of motion on his or her experience of tidal forces can be even more pronounced than this. There are examples of spacetimes in which an observer cruising along a certain kind of path would experience unbounded tidal forces and so be torn apart, while another observer, in a certain technical sense approaching the same limiting point as the first observer, accelerating and decelerating in just the proper way, would experience a perfectly well-behaved tidal force, though she would approach as near as she likes to the other fellow who is in the midst of being ripped to shreds.
Things can get stranger still. There are examples of incomplete geodesics contained entirely within a well-defined, bounded region of a spacetime, each having as its limiting point an honest-to-goodness point of spacetime, such that an observer freely falling along such a path would be torn apart by unbounded tidal forces; it can easily be arranged in such cases, however, that a separate observer, who actually travels through the limiting point, will experience perfectly well-behaved tidal forces.
This example also provides a nice illustration of the inevitable difficulties attendant on attempts to localize singular structure in the senses discussed in section 1. It would seem, then, that curvature pathology as characterized based on the behavior of tidal forces is not in any physical sense a well-defined property of a region of spacetime simpliciter.
When we consider the physical manifestations of the curvature of spacetime, the motion of the device that we use to probe a region as well as the nature of the device becomes crucially important for the question of whether pathological behavior manifests itself. This fact raises questions about the nature of quantitative measures of properties of entities in general relativity, and what ought to count as observable, in the sense of reflecting the underlying physical structure of spacetime.
Because apparently pathological phenomena may occur or not depending on the types of measurements one is performing, it seems that purely geometrical pathology does not necessarily reflect anything about the state of spacetime itself, or at least not in any localizable way. What then does it reflect, if anything? Much work remains to be done by both physicists and by philosophers in this area, i.
See Bertotti , Bergmann , Rovelli , in Other Internet Resources, henceforth OIR , , Curiel and Manchak a for discussion of many different topics in this area, approached from several different perspectives. There is, however, one form of curvature pathology associated with a particular form of an apparently important class of singularities that recently has been clearly characterized and analyzed, that associated with so-called conformal singularities, also sometimes called isotropic singularities Goode and Wainright ; Newman a, b; Tod The curvature pathology of this class of singularities can be precisely pinpointed: it occurs solely in the conformal part of the curvature; thus, what is singular in one spacetime will not necessarily be so in a conformally equivalent spacetime.
These properties, along with the fact that the Big Bang singularity almost certainly seems to be of this form, make conformal singularities particularly important for the understanding and investigation of many issues of physical and philosophical interest in contemporary cosmology, as discussed below in section 7. The characterization of this kind of singularity has, so far, been confined to the context of cosmological models, including essentially all spacetimes whose matter content consists of homogeneous perfect fluids and a very wide class of spacetimes consisting of inhomogeneous fluids.
The dynamics of those cosmological models is largely governed by the behavior of the cosmological expansion factor, a measure of the relative sizes of local regions of space not spacetime at different cosmological times.
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If the universe's expansion stops, and the net gravitational effect on cosmological scales results in the universe's collapsing in on itself, this would be marked by a continual decrease in the expansion factor, eventuating in a Big Crunch singularity as the expansion factor asymptotically approached zero. The remaining dynamics of these cosmological models is encoded in the behavior of the Hubble parameter, a natural measure of the rate of change of the expansion factor.
A sudden singularity, then, is defined by the divergence of a time derivative of the expansion factor or the Hubble parameter, though the factor or parameter itself remains finite. In such cases, it may happen that the mass-density of the fluid itself, the expansion factor and its first derivative, and even the Hubble parameter and its first derivative, all remain finite: only the pressure and so the second derivative of the expansion factor diverges.
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Because the physical significance of quantities such as pressure is thought to be unambiguous, this feature of sudden singularities stands in marked contrast to the problems of physical interpretation that plague the standard type of singularity, discussed in section 1. Indeed, point particles passing through the sudden singularity would not even notice the pathology, as only tidal forces may diverge and not even all sudden singularities involve divergence of those : point particles, having no extension, cannot experience tidal force.
Although the discovery of sudden singularities has reinvigorated the study of singular spacetimes in the physics community Cotsakis , they remain so far almost entirely unexamined by the philosophy community. Nonetheless, they raise questions of manifest philosophical interest and import. The fact that they are such radically different structures from all other previously known kinds of singularity, for example, raises methodological questions about how to understand the meaning of terms in physical theories when those terms refer to structurally quite different but obviously still intimately related phenomena—the reasons for thinking of them as singularities are compelling, even though they violate essentially every standard condition known for characterizing singularities.
Another unusual kind of singularity characterized only recently characterized deserves mention here, because of its possible importance in cosmology. The physical processes that seem to eventuate in most known kinds of singular structure involve the unlimited clumping together of matter, as in collapse singularities associated with black holes, and the Big Bang and Big Crunch singularities of standard cosmological models.
A big rip , contrarily, occurs when the expansion of matter increasingly accelerates without bound in a finite amount of proper time Caldwell ; Caldwell et al. Again, standard concepts and arguments about singularities characterized as incomplete paths do not seem easily applicable here. Although big rips do have incomplete paths associated with them as well as curvature pathology, they are of such radically different kinds as to prima facie warrant separate analysis.
Recent work, codified by Harada et al. For homogeneous cosmological models filled with perfect fluids with a linear equation of state—the standard cosmological model—certain values of the barotropic index yield past, future, or past and future big rips that are such that every timelike geodesic runs into them, but every null geodesic avoids them. See note 7 for an explanation of the barotropic index. In other words, any body traveling more slowly than light will run into the singularity, but every light ray will escape to infinity.
This is not a situation that lends itself to easy and perspicuous physical interpretation. When considering the implications of spacetime singularities, it is important to note that we have good reasons to believe that the spacetime of our universe is singular. In the late s, Penrose, Geroch, and Hawking proved several singularity theorems, using path incompleteness as a criterion Penrose , ; Hawking , b, c, d; Geroch , , b, ; Hawking and Penrose These theorems showed that if certain physically reasonable premises were satisfied, then in certain circumstances singularities could not be avoided.
Notable among these conditions is the positive energy condition, which captures the idea that energy is never negative. These theorems indicate that our universe began with an initial singularity, the Big Bang, approximately 14 billion years ago. They also indicate that in certain circumstances discussed below collapsing matter will form a black hole with a central singularity. According to our best current cosmological theories, moreover, two of the likeliest scenarios for the end of the universe is either a global collapse of everything into a Big Crunch singularity, or the complete and utter diremption of everything, down to the smallest fundamental particles, in a Big Rip singularity.
See Joshi for a recent survey of singularities in general, and Berger for a recent survey of the different kinds of singularities that can occur in cosmological models. Should these results lead us to believe that singularities are real? Many physicists and philosophers resist this conclusion. Some argue that singularities are too repugnant to be real. Others argue that the singular behavior at the center of black holes and at the beginning and possibly the end of time indicates the limit of the domain of applicability of general relativity.
Some are inclined to take general relativity at its word, however, and simply accept its prediction of singularities as a surprising but perfectly consistent account of the possible features of the geometry of our world. See Curiel and Earman , for discussion and comparison of these opposing points of view. In this section, we review these and related problems and the possible responses to them. Let us summarize the results of section 1 : there is no commonly accepted, strict definition of singularity; there is no physically reasonable characterization of missing points; there is no necessary connection between singular structure, at least as characterized by the presence of incomplete paths, and the presence of curvature pathology; and there is no necessary connection between other kinds of physical pathology such as divergence of pressure and path incompleteness.
What conclusions should be drawn from this state of affairs? There seem to be two basic kinds of response, illustrated by the views of of Clarke and Earman on the one hand, and those of Geroch et al. The former holds that the mettle of physics and philosophy demands that we find a precise, rigorous and univocal definition of singularity. On this view, the host of philosophical and physical questions surrounding general relativity's prediction of singular structure would best be addressed with such a definition in hand, so as better to frame and answer these questions with precision, and thus perhaps find other, even better questions to pose and attempt to answer.
The latter view is perhaps best summarized by a remark of Geroch et al. In sum, the question becomes the following: is there a need for a single, blanket definition of singularity or does the urge for one betray only an old Aristotelian, essentialist prejudice? This question has obvious connections to the broader question of natural kinds in science. One sees debates similar to those canvassed above when one tries to find, for example, a strict definition of biological species.
Clearly, part of the motivation for searching for a single exceptionless definition is the impression that there is some real feature of the world or at least of our spacetime models that we can hope to capture precisely. Further, we might hope that our attempts to find a rigorous and exceptionless definition will help us to better understand the feature itself.
Even without an accepted, strict definition of singularity for relativistic spacetimes, the question can be posed: what would it mean to ascribe existence to singular structure under any of the available open possibilities? It is not far-fetched to think that answers to this question may bear on the larger question of the existence of spacetime points in general Curiel , ; Lam See the entries The Hole Argument and Absolute and Relational Theories of Space and Motion for discussions of the question of the existence of spacetime itself. It would be difficult to argue that an incomplete path in a maximal relativistic spacetime does not exist in at least some sense of the term.
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It is not hard to convince oneself, however, that the incompleteness of the path does not exist at any particular point of the spacetime in the same way, say, as this glass of beer exists at this point of spacetime. If there were a point on the manifold where the incompleteness of the path could be localized, surely that would be the point at which the incomplete path terminated. But if there were such a point, then the path could be extended by having it pass through that point.
It is perhaps this fact that lies behind much of the urgency surrounding the attempt to define singular structure as missing points. Aristotelian substantivalism here refers to the idea contained in Aristotle's contention that everything that exists is a substance and that all substances can be qualified by the Aristotelian categories, two of which are location in time and location in space.
limizardpi.tk Such a criterion, however, may be inappropriate for features and properties of spacetime itself. Several essential features of a relativistic spacetime, singular or not, cannot be localized in the way that an Aristotelian substantivalist would demand. For example, the Euclidean or non-Euclidean nature of a space is not something with a precise location. See Butterfield for discussion of these issues.
Likewise, various spacetime geometrical structures such as the metric, the affine structure, the topology, etc. Because of the way the issue of singular structure in relativistic spacetimes ramifies into almost every major open question in relativistic physics today, both physical and philosophical, it provides a peculiarly rich and attractive focus for these sorts of questions.
An interesting point of comparison, in this regard, would be the nature of singularities in other theories of gravity besides general relativity. Weatherall's characterization of singularities in geometrized Newtonian gravitational theory, therefore, and his proof that the theory accommodates their prediction, may serve as a possible testing ground for ideas and arguments on these issues. General relativity admits spacetimes exhibiting a vast and variegated menagerie of structures and behaviors, even over and above singularities, that most physicists and philosophers would consider, in some sense or other, not reasonable possibilities for physical manifestation in the actual world.
Manchak has argued that there cannot be purely empirical grounds for ruling out the seemingly unpalatable structures, for there always exist spacetimes that are, in a precise sense, observationally indistinguishable from our own Malament ; Manchak a that have essentially any set of properties one may stipulate.
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