In physics, the requirement that interactions in any spacetime region can influence the evolution of the system only at subsequent times; that is, past events are causes of future events, and future events can never be the causes of events in the past. Causality thus depends on time orientability, the possibility of distinguishing past from future. Not all spacetimes are orientable.
Causality and determinism
The laws of a deterministic theory (for example, classical mechanics) are such that the state of a closed system (for example, the positions and momenta of particles in the system) at one instant determines the state of that system at any future time. Deterministic causality does not necessarily imply practical predictability. It was long implicitly assumed that slight differences in initial conditions would not lead to rapid divergence of later behavior, so that predictability was a consequence of determinism. Behavior in which two particles starting at slightly different positions and velocities diverge rapidly is called chaotic. Such behavior is ubiquitous in nature, and can lead to the practical impossibility of prediction of future states despite the deterministic character of the physical laws. See also: Chaos
Quantum mechanics is deterministic in the sense that, given the state of a system at one instant, it is possible to calculate later states. However, the situation differs from that in classical mechanics in two fundamental respects. First, conjugate variables, for example, position x and momentum p, cannot be simultaneously determined with complete precision, the relation between their indeterminacies being Δx Δp ≥ℏ, where ℏ is Planck's constant divided by . Second, the state variable ψ gives only probabilities that a given eigenstate will be found after the performance of a measurement, and such probabilities are also all that is calculable about a later state by the deterministic prediction. Despite its probabilistic character, the quantum state still evolves deterministically. However, which eigenvalue (say, of position) will actually be found in a measurement is unpredictable. See also: Determinism; Eigenvalue (quantum mechanics); Nonrelativistic quantum theory; Quantum mechanics; Quantum theory of measurement; Uncertainty principle
Causal structure of spacetimes
Nonrelativistic mechanics assumes that causal action can be propagated instantaneously, and thus that an absolute simultaneity is definable. This is not true in special relativity. While the state of a system can still be understood in terms of the positions and momenta of its particles, time order, as well as temporal and spatial length, becomes relative to the observer's frame, and there is no possible choice of simultaneous events in the universe that is the same in all reference frames. Only space-time intervals in a fused “spacetime” are invariant with respect to choice of reference frame. The theory of special relativity thus rejects the possibility of instantaneous causal action. Instead, the existence of a maximum velocity of signal transmission determines which events can causally influence others and which cannot. See also: Spacetime
The investigation of a spacetime with regard to which events can causally influence (signal) other regions and which cannot is known as the study of the causal structure of the spacetime. Thus, in the Minkowski spacetime of special relativity, an event E can causally affect another if and only if there is a timelike curve which joins E and , and lies in the future of E. Such a curve is contained within the future light cone of E and connects E and . The light-cone surface is generated by null geodesics representing the velocity of light. One of the two halves of the light cone intersecting at the “present” is specified as the future, the other as the past. Paths lying within the light cone are timelike, those in the past region (or the present) being capable of influencing later events. Curves joining E with events outside the light cone are called spacelike. Their traversal requires velocities greater than that of light, and thus events which are spacelike-related cannot causally influence one another.
Closed causal curves
Different spacetimes, for example, those allowed by general relativity, are distinguished by their different causal structures. Because general relativity admits distinct spacetime metrics at adjacent points, the future directions of light cones can vary over short distances. (Light cones at adjacent points tilt with respect to one another.) Under these circumstances, it is possible for a continuous sequence of tilted cones to result in a timelike curve intersecting itself, producing a closed curve. An event on such a curve, however, both precedes and succeeds itself, and can be both its own cause and effect. Some solutions of the general relativistic field equations, for example, Gödel and Taub-Nut spacetimes, contain such closed causal curves.
However, many physicists hold that the existence of such curves is unphysical, and seek criteria to exclude them. Among the many such conditions that have been extensively discussed are the following. The causality condition excludes closed nonspacelike (that is, null or timelike) curves. Strong causality further excludes “almost closed” causal curves, wherein a nonspacelike curve returns more than once to the same infinitesimal neighborhood. Need for yet a further condition arises from the possibility that, in a quantum theory of gravity, the uncertainty principle would prevent the metric from having an exact value at every point, leading to the possibility that small variations in the metric would generate closed timelike curves. The condition of causal stability prevents such occurrences by defining a neighborhood of a point in which there are no closed timelike curves. Causal stability is the strongest condition excluding causal anomalies: violations of weaker conditions necessarily violate causal stability. If a spacetime is causally stable, the topology of its manifold follows from the causal structure, as do the differentiable and the conformal structures. Clearly a study of the causal structure gives deep insight into the characteristics of a spacetime.
In contrast, there are also many physicists who do not consider solutions of the equations of general relativity which contain closed causal curves to be unphysical, pointing out that, in the past, possibilities treated as unphysical have frequently turned out to be physically significant. For this reason (among others), they hold that solutions with closed causal curves should be taken seriously, and should be included among, for example, the possible histories of the universe in path-integral calculations in quantum theories of gravity.
Other causality violations
Violations of conventional causality could (hypothetically) arise in ways other than through closed causal curves. For example, the possibility has been considered that tachyons, faster-than-light particles, might exist. This is tantamount to the speculation that the past (or present) could be influenced by future events through the transmission of tachyons. Existence of such particles is generally rejected on both experimental and theoretical grounds. Causality violation would also result if there were more than one dimension of time, as there are of space. Such assignments are therefore usually excluded in theories of quantum gravity. See also: Quantum gravitation; Relativity; Tachyon