Special Relativistic physics

Two features of special relativistic physics make it perhaps the most hospitable environment for determinism of any major theoretical context: the fact that no process or signal can travel faster than the speed of light, and the static, unchanging spacetime structure. The former feature, including a prohibition against tachyons (hypothetical particles travelling faster than light)[4]), rules out space invaders and other unbounded-velocity systems. The latter feature makes the space-time itself nice and stable and non-singular—unlike the dynamic space-time of General Relativity, as we shall see below. For source-free electromagnetic fields in special-relativistic space-time, a nice form of Laplacean determinism is provable. Unfortunately, interesting physics needs more than source-free electromagnetic fields. Earman (1986) ch. IV surveys in depth the pitfalls for determinism that arise once things are allowed to get more interesting (e.g. by the addition of particles interacting gravitationally).

4.3 General Relativity (GTR)

Defining an appropriate form of determinism for the context of general relativistic physics is extremely difficult, due to both foundational interpretive issues and the plethora of weirdly-shaped space-time models allowed by the theory's field equations. The simplest way of treating the issue of determinism in GTR would be to state flatly: determinism fails, frequently, and in some of the most interesting models. Here we will briefly describe some of the most important challenges that arise for determinism, directing the reader yet again to Earman (1986), and also Earman (1995) for more depth.

4.3.1 Determinism and manifold points

In GTR, we specify a model of the universe by giving a triple of three mathematical objects, <M,g,T>. M represents a continuous “manifold”: that means a sort of unstructured space (-time), made up of individual points and having smoothness or continuity, dimensionality (usually, 4-dimensional), and global topology, but no further structure. What is the further structure a space-time needs? Typically, at least, we expect the time-direction to be distinguished from space-directions; and we expect there to be well-defined distances between distinct points; and also a determinate geometry (making certain continuous paths in M be straight lines, etc.). All of this extra structure is coded into g, the metric field. So M and g together represent space-time. Trepresents the matter and energy content distributed around in space-time (if any, of course).
For mathematical reasons not relevant here, it turns out to be possible to take a given model spacetime and perform a mathematical operation called a “hole diffeomorphism” h* on it; the diffeomorphism's effect is to shift around the matter content T and the metric g relative to the continuous manifold M.[5] If the diffeomorphism is chosen appropriately, it can move around Tand g after a certain time t = 0, but leave everything alone before that time. Thus, the new model represents the matter content (now h* T) and the metric (h*g) as differently located relative to the points of M making up space-time. Yet, the new model is also a perfectly valid model of the theory. This looks on the face of it like a form of indeterminism: GTR's equations do not specify how things will be distributed in space-time in the future, even when the past before a given time tis held fixed. See Figure 5:
Special Relativistic physics Hole-diffeo-new

Special Relativistic physics Hole-diffeo-new

Figure 5: “Hole” diffeomorphism shifts contents of spacetime
Usually the shift is confined to a finite region called the hole (for historical reasons). Then it is easy to see that the state of the world at time t = 0 (and all the history that came before) does not suffice to fix whether the future will be that of our first model, or its shifted counterpart in which events inside the hole are different.
This is a form of indeterminism first highlighted by Earman and Norton (1987) as an interpretive philosophical difficulty for realism about GTR's description of the world, especially the point manifold M. They showed that realism about the manifold as a part of the furniture of the universe (which they called “manifold substantivalism”) commits us to an automatic indeterminism in GTR (as described above), and they argued that this is unacceptable. (See the hole argument and Hoefer (1996) for one response on behalf of the space-time realist, and discussion of other responses.) For now, we will simply note that this indeterminism, unlike most others we are discussing in this section, is empirically undetectable: our two models <M, g, T> and the shifted model <M, h*g,h*T> are empirically indistinguishable.

4.3.2 Singularities

The separation of space-time structures into manifold and metric (or connection) facilitates mathematical clarity in many ways, but also opens up Pandora's box when it comes to determinism. The indeterminism of the Earman and Norton hole argument is only the tip of the iceberg; singularities make up much of the rest of the berg. In general terms, a singularity can be thought of as a “place where things go bad” in one way or another in the space-time model. For example, near the center of a Schwarzschild black hole, curvature increases without bound, and at the center itself it is undefined, which means that Einstein's equations cannot be said to hold, which means (arguably) that this point does not exist as a part of the space-time at all! Some specific examples are clear, but giving a general definition of a singularity, like defining determinism itself in GTR, is a vexed issue (see Earman (1995) for an extended treatment; Callender and Hoefer (2001) gives a brief overview). We will not attempt here to catalog the various definitions and types of singularity.
Different types of singularity bring different types of threat to determinism. In the case of ordinary black holes, mentioned above, all is well outside the so- called “event horizon”, which is the spherical surface defining the black hole: once a body or light signal passes through the event horizon to the interior region of the black hole, it can never escape again. Generally, no violation of determinism looms outside the event horizon; but what about inside? Some black hole models have so-called “Cauchy horizons” inside the event horizon, i.e., surfaces beyond which determinism breaks down.
Another way for a model spacetime to be singular is to have points or regions go missing, in some cases by simple excision. Perhaps the most dramatic form of this involves taking a nice model with a space-like surface t = E (i.e., a well-defined part of the space-time that can be considered “the state state of the world at time E”), and cutting out and throwing away this surface and all points temporally later. The resulting spacetime satisfies Einstein's equations; but, unfortunately for any inhabitants, the universe comes to a sudden and unpredictable end at time E. This is too trivial a move to be considered a real threat to determinism in GTR; we can impose a reasonable requirement that space-time not “run out” in this way without some physical reason (the spacetime should be “maximally extended”). For discussion of precise versions of such a requirement, and whether they succeed in eliminating unwanted singularities, see Earman (1995, chapter 2).
The most problematic kinds of singularities, in terms of determinism, are naked singularities(singularities not hidden behind an event horizon). When a singularity forms from gravitational collapse, the usual model of such a process involves the formation of an event horizon (i.e. a black hole). A universe with an ordinary black hole has a singularity, but as noted above, (outside the event horizon at least) nothing unpredictable happens as a result. A naked singularity, by contrast, has no such protective barrier. In much the way that anything can disappear by falling into an excised-region singularity, or appear out of a white hole (white holes themselves are, in fact, technically naked singularities), there is the worry that anything at all could pop out of a naked singularity, without warning (hence, violating determinism en passant). While most white hole models have Cauchy surfaces and are thus arguably deterministic, other naked singularity models lack this property. Physicists disturbed by the unpredictable potentialities of such singularities have worked to try to prove various cosmic censorship hypotheses that show—under (hopefully) plausible physical assumptions—that such things do not arise by stellar collapse in GTR (and hence are not liable to come into existence in our world). To date no very general and convincing forms of the hypothesis have been proven, so the prospects for determinism in GTR as a mathematical theory do not look terribly good.

4.4 Quantum mechanics

As indicated above, QM is widely thought to be a strongly non-deterministic theory. Popular belief (even among most physicists) holds that phenomena such as radioactive decay, photon emission and absorption, and many others are such that only a probabilistic description of them can be given. The theory does not say what happens in a given case, but only says what the probabilities of various results are. So, for example, according to QM the fullest description possible of a radium atom (or a chunk of radium, for that matter), does not suffice to determine when a given atom will decay, nor how many atoms in the chunk will have decayed at any given time. The theory gives only the probabilities for a decay (or a number of decays) to happen within a given span of time. Einstein and others perhaps thought that this was a defect of the theory that should eventually be removed, by a supplemental hidden variable theory[6] that restores determinism; but subsequent work showed that no such hidden variables account could exist. At the microscopic level the world is ultimately mysterious and chancy.
So goes the story; but like much popular wisdom, it is partly mistaken and/or misleading. Ironically, quantum mechanics is one of the best prospects for a genuinely deterministic theory in modern times! Everything hinges on what interpretational and philosophical decisions one adopts. The fundamental law at the heart of non-relativistic QM is the Schrödinger equation. The evolution of a wavefunction describing a physical system under this equation is normally taken to be perfectly deterministic.[7] If one adopts an interpretation of QM according to which that's it—i.e., nothing ever interrupts Schrödinger evolution, and the wavefunctions governed by the equation tell the complete physical story—then quantum mechanics is a perfectly deterministic theory. There are several interpretations that physicists and philosophers have given of QM which go this way. (See the entry on quantum mechanics.)
More commonly—and this is part of the basis for the popular wisdom—physicists have resolved the quantum measurement problem by postulating that some process of “collapse of the wavefunction” occurs during measurements or observations that interrupts Schrödinger evolution. The collapse process is usually postulated to be indeterministic, with probabilities for various outcomes, via Born's rule, calculable on the basis of a system's wavefunction. The once-standard Copenhagen interpretation of QM posits such a collapse. It has the virtue of solving certain problems such as the infamous Schrödinger's cat paradox, but few philosophers or physicists can take it very seriously unless they are instrumentalists about the theory. The reason is simple: the collapse process is not physically well-defined, is characterised in terms of an anthropomorphic notion (measurement)and feels too ad hoc to be a fundamental part of nature's laws.[8]
In 1952 David Bohm created an alternative interpretation of non relativistic QM—perhaps better thought of as an alternative theory—that realizes Einstein's dream of a hidden variable theory, restoring determinism and definiteness to micro-reality. In Bohmian quantum mechanics, unlike other interpretations, it is postulated that all particles have, at all times, a definite position and velocity. In addition to the Schrödinger equation, Bohm posited a guidance equation that determines, on the basis of the system's wavefunction and particles' initial positions and velocities, what their future positions and velocities should be. As much as any classical theory of point particles moving under force fields, then, Bohm's theory is deterministic. Amazingly, he was also able to show that, as long as the statistical distribution of initial positions and velocities of particles are chosen so as to meet a “quantum equilibrium” condition, his theory is empirically equivalent to standard Copenhagen QM. In one sense this is a philosopher's nightmare: with genuine empirical equivalence as strong as Bohm obtained, it seems experimental evidence can never tell us which description of reality is correct. (Fortunately, we can safely assume thatneither is perfectly correct, and hope that our Final Theory has no such empirically equivalent rivals.) In other senses, the Bohm theory is a philosopher's dream come true, eliminating much (but not all) of the weirdness of standard QM and restoring determinism to the physics of atoms and photons. The interested reader can find out more from the link above, and references therein.
This small survey of determinism's status in some prominent physical theories, as indicated above, does not really tell us anything about whether determinism is true of our world. Instead, it raises a couple of further disturbing possibilities for the time when we do have the Final Theory before us (if such time ever comes): first, we may have difficulty establishing whether the Final Theory is deterministic or not—depending on whether the theory comes loaded with unsolved interpretational or mathematical puzzles. Second, we may have reason to worry that the Final Theory, if indeterministic, has an empirically equivalent yet deterministic rival (as illustrated by Bohmian quantum mechanics.)

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