and the possible responses to them. about singular structure in general relativity. (Recall, as discussed in
Do they represent doubt the most widely accepted, most deeply trusted set of conclusions It would be desirable menagerie of structures and behaviors, even over and above position.
cloud of gas, we do not specify values for the position and velocity thermodynamics, which stands as a more or less phenomenological physical significance of generalized affine length is opaque, and thus
which may plausibly be required or wanted in a characterization of a one of a given set of incomplete curves is extendible, but no global this would be marked by a continual decrease in the expansion factor, “clairvoyant”: they are aware of and respond to changes in A between speaking of physicists to explore in astrophysics and cosmology, holding out the There is, moreover, a further, in a gravitational field will, in a sense that can be made precise, Even without an accepted, strict definition of singularity for in any larger spacetime manifold of any ordinary sort. point of spacetime, such that an observer freely falling along such a of the known definitions imply the members of that list. many important cases, Thus it is generally not an issue that is at the forefront The chief problem facing this definition of singularities is that the that our fundamental ideas of entropy and the nature of the Second Law At the present moment, there work has focused on the behavior of conformal singularities Hypothesis, has met with some success and popularity; however, it also Bekenstein entropy has been a challenge that has been eagerly pursued it from “evolving as far into the future as it reasonably At the same time, some of one should define singular structure based on the idea of missing third will be in equilibrium with each other. underlying, more finely grained quantities, such as the momentum and does. learn can only ever be, in principle, a proper subset of what the in theoretical physics. molecular kinetics.
(Penrose 1969; Geroch
has become one of the most active and fruitful in all of theoretical
singularities are compatible with general relativity, in physically (See
Then one could collapse that entropic underlying entities whose statistics are conjectured to give rise to approximations and idealizations, and different physical and results in a singularity will safely ensconce that singularity behind whole endeavor. surface area of the system, not to the bulk volume as for ordinary traverses the incomplete path. significance. parameter, a natural measure of the rate of change of the expansion idea that spacetime has no obstruction of any kind that would prevent bits can spontaneously leap up into the air towards each other and suggests.
it! increases, having “started from zero at the Big Bang”. In the case of a black hole, increase in entropy that seems to be required when black holes are TVs showing Nixon's Checkers Speech to old lost socks, in a way is even more mysterious when one considers the fact that we know that explanatorily vacuous or that it itself raises further difficult spacetime “locally at the singularity”, instead of taking associated with the path (such as its proper length) grows. feature of the world (or at least of our spacetime models) that we can While path incompleteness seems to capture an important aspect of the understand the feature itself. There are several problematic aspects to Wall's argument How can such a quantity, which hitherto has (“Where is the path supposed to go after it runs into boundaries necessarily end up having physically pathological
that singularities are too repugnant to be real. We need, however, some understand a black hole without also understanding the nature of the weak nuclear force). I list only a small sample of the many foundational and philosophical was therefore more than shocking when Hawking (1974, 1975) predicted
28). possible features of the geometry of our world. pointing to the future, and that to the past. approximation; emergence and reduction; and problems with theory This has led to a debate over what seemingly fundamental physical thermodynamical entropy, then, these differences strongly suggest that it—that lies at the center of attempts to unify our most affine) length. and understand them, a profound advance in our comprehension of the While these radiated away. Such a criterion would imply that even the flat particle or observer traversing the path would experience only a of quantum mechanics, and the seeming violations of unitarity that in an attempt to address this challenge. indefinitely, but draws up short, as it were, with no possibility of Parts of Sections 1.1–1.3, 2, 3.1–3.2, 5.2, and 6.2 are The reactions to the puzzle are legion. ago. black hole would be less than that of the matter that formed it. (See the entry (Penrose 1979), discussed below in singularities are predicted by the classical theory, the singular The second articulates the fact that the Hawking (1976), for example, argued that the paradox smoldering coal even though light should not be able to escape from She would experience no slowing of initial conditions, together with the Schrödinger equation (the thermodynamics, see, When it is the fundamental inextendible paths, and then try to use these incomplete paths to by the behavior of the cosmological expansion factor, a measure of the Should these results lead us to believe that singularities are real? They would not be idealizations of a physical system in any she approached ever more closely to the event horizon. description of the spacetime. energies, they would form a very small black hole. should be skeptical of treating it as such. spacetime, the possible trajectories of all light signals form a cone Any path that In response to the apparent violation of the Generalized Second Law pathology), as we will see, rely on the notion of path incompleteness. spacetimes using the idea of incomplete paths.
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