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Causality is the relationship between causes and effects.[1][2] It is considered to be fundamental to all natural science, especially physics. Causality is also a topic studied from the perspectives of philosophy and statistics.

Cause and effect in physics

In physics it is helpful to interpret certain terms of a physical theory as causes and other terms as effects. Thus, in classical (Newtonian) mechanics a cause may be represented by a force acting on a body, and an effect by the acceleration which follows as quantitatively explained by Newton's second law. For different physical theories the notions of cause and effect may be different. For instance, in the general theory of relativity, acceleration is not an effect (since it is not a generally relativistic vector); the general relativistic effects comparable to those of Newtonian mechanics are the deviations from geodesic motion in curved spacetime.[3] Also, the meaning of "uncaused motion" is dependent on the theory being employed: for Newton it is inertial motion (constant velocity with respect to an inertial frame of reference), in the general theory of relativity it is geodesic motion (to be compared with frictionless motion on the surface of a sphere at constant tangential velocity along a great circle). So what constitutes a "cause" and what constitutes an "effect" depends on the total system of explanation in which the putative causal sequence is embedded.

A formulation of physical laws in terms of cause and effect is useful for the purposes of explanation and prediction. For instance, in Newtonian mechanics, an observed acceleration can be explained by reference to an applied force. So Newton's second law can be used to predict the force necessary to realize a desired acceleration.

In classical physics, a cause should always precede its effect. In relativity theory the equivalent restriction limits causes to the back (past) light cone of the event to be explained (the "effect"), and any effect of a cause must lie in the cause's front (future) light cone. These restrictions are consistent with the grounded belief (or assumption) that causal influences cannot travel faster than the speed of light and/or backwards in time.

Another requirement, at least valid at the level of human experience, is that cause and effect be mediated across space and time (requirement of contiguity). This requirement has been very influential in the past, in the first place as a result of direct observation of causal processes (like pushing a cart), in the second place as a problematic aspect of Newton's theory of gravitation (attraction of the earth by the sun by means of action at a distance) replacing mechanistic proposals like Descartes' vortex theory; in the third place as an incentive to develop dynamic field theories (e.g., Maxwell's electrodynamics and Einstein's general theory of relativity) restoring contiguity in the transmission of influences in a more successful way than did Descartes' theory.

The empiricists' aversion to metaphysical explanations (like Descartes' vortex theory) lends heavy influence against the idea of the importance of causality. Causality has accordingly sometimes been downplayed (e.g., Newton's "Hypotheses non fingo"). According to Ernst Mach[4] the notion of force in Newton's second law was pleonastic, tautological and superfluous. Indeed it is possible to consider the Newtonian equations of motion of the gravitational interaction of two bodies,

\( m_1 \frac{d^2 {\mathbf r}_1 }{ dt^2} = -\frac{m_1 m_2 g ({\mathbf r}_1 - {\mathbf r}_2)}{ |{\mathbf r}_1 - {\mathbf r}_2|^3};\; m_2 \frac{d^2 {\mathbf r}_2 }{dt^2} = -\frac{m_1 m_2 g ({\mathbf r}_2 - {\mathbf r}_1) }{ |{\mathbf r}_2 - {\mathbf r}_1|^3}, \)

as two coupled equations describing the positions \( \scriptstyle {\mathbf r}_1(t) \) and \( \scriptstyle {\mathbf r}_2(t) \) of the two bodies, without interpreting the right hand sides of these equations as forces; the equations just describe a process of interaction, without any necessity to interpret one body as the cause of the motion of the other, and allow one to predict the states of the system at later (as well as earlier) times.

The ordinary situations in which humans singled out some factors in a physical interaction as being prior and therefore supplying the "because" of the interaction were often ones in which humans decided to bring about some state of affairs and directed their energies to producing that state of affairs—a process that took time to establish and left a new state of affairs that persisted beyond the time of activity of the actor. It would be difficult and pointless, however, to explain the motions of binary stars with respect to each other in that way.

The possibility of such a time-independent view is at the basis of the deductive-nomological (D-N) view of scientific explanation, considering an event to be explained if it can be subsumed under a scientific law. In the D-N view, a physical state is considered to be explained if, applying the (deterministic) law, it can be derived from given initial conditions. (Such initial conditions could include the momenta and distance from each other of binary stars at any given moment.) Such 'explanation by determinism' is sometimes referred to as causal determinism. A disadvantage of the D-N view is that causality and determinism are more or less identified. Thus, in classical physics, it was assumed that all events are caused by earlier ones according to the known laws of nature, culminating in Pierre-Simon Laplace's claim that if the current state of the world were known with precision, it could be computed for any time in the future or the past (see Laplace's demon). However, this is usually referred to as Laplace determinism (rather than `Laplace causality') because it hinges on determinism in mathematical models as dealt with in the mathematical Cauchy problem. Confusion of causality and determinism is particularly acute in quantum mechanics, this theory being acausal in the sense that it is unable in many cases to identify the causes of actually observed effects or to predict the effects of identical causes, but arguably deterministic in some interpretations (e.g. if the wave function is presumed not to actually collapse as in the many-worlds interpretation, or if its collapse is due to hidden variables, or simply redefining determinism as meaning that probabilities rather than specific effects are determined).

In modern physics, the notion of causality had to be clarified. The insights of the theory of special relativity confirmed the assumption of causality, but they made the meaning of the word "simultaneous" observer-dependent.[5] Consequently, the relativistic principle of causality says that the cause must precede its effect according to all inertial observers. This is equivalent to the statement that the cause and its effect are separated by a timelike interval, and the effect belongs to the future of its cause. If a timelike interval separates the two events, this means that a signal could be sent between them at less than the speed of light. On the other hand, if signals could move faster than the speed of light, this would violate causality because it would allow a signal to be sent across spacelike intervals, which means that at least to some inertial observers the signal would travel backward in time. For this reason, special relativity does not allow communication faster than the speed of light.

In the theory of general relativity, the concept of causality is generalized in the most straightforward way: the effect must belong to the future light cone of its cause, even if the spacetime is curved. New subtleties must be taken into account when we investigate causality in quantum mechanics and relativistic quantum field theory in particular. In quantum field theory, causality is closely related to the principle of locality. However, the principle of locality is disputed: whether it strictly holds depends on the interpretation of quantum mechanics chosen, especially for experiments involving quantum entanglement that satisfy Bell's Theorem.

Despite these subtleties, causality remains an important and valid concept in physical theories. For example, the notion that events can be ordered into causes and effects is necessary to prevent (or at least outline) causality paradoxes such as the grandfather paradox, which asks what happens if a time-traveler kills his own grandfather before he ever meets the time-traveler's grandmother. See also Chronology protection conjecture.

Distributed causality

Theories in physics like the Butterfly effect from chaos theory open up the possibility of a type of distributed parameter systems in causality. The butterfly effect theory proposes:

"Small variations of the initial condition of a nonlinear dynamical system may produce large variations in the long term behavior of the system."

This opens up the opportunity to understand a distributed causality.

A related way to interpret the Butterfly effect is to see it as highlighting the difference between the application of the notion of causality in physics and a more general use of causality as represented by Mackie's INUS conditions. In classical (Newtonian) physics, in general, only those conditions are (explicitly) taken into account, that are both necessary and sufficient. For instance, when a massive sphere is caused to roll down a slope starting from a point of unstable equilibrium, then its velocity is assumed to be caused by the force of gravity accelerating it; the small push that was needed to set it into motion is not explicitly dealt with as a cause. In order to be a physical cause there must be a certain proportionality with the ensuing effect. A distinction is drawn between triggering and causation of the ball's motion. By the same token the butterfly can be seen as triggering a tornado, its cause being assumed to be seated in the atmospherical energies already present beforehand, rather than in the movements of a butterfly.

Causal dynamical triangulation
Main article: Causal dynamical triangulation

Causal dynamical triangulation (abbreviated as "CDT") invented by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, and popularized by Fotini Markopoulou and Lee Smolin, is an approach to quantum gravity that like loop quantum gravity is background independent. This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. The Loops '05 conference, hosted by many loop quantum gravity theorists, included several presentations which discussed CDT in great depth, and revealed it to be a pivotal insight for theorists. It has sparked considerable interest as it appears to have a good semi-classical description. At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time. Using a structure called a simplex, it divides spacetime into tiny triangular sections. A simplex is the generalized form of a triangle, in various dimensions. A 3-simplex is usually called a tetrahedron, and the 4-simplex, which is the basic building block in this theory, is also known as the pentatope, or pentachoron. Each simplex is geometrically flat, but simplices can be "glued" together in a variety of ways to create curved spacetimes. Where previous attempts at triangulation of quantum spaces have produced jumbled universes with far too many dimensions, or minimal universes with too few, CDT avoids this problem by allowing only those configurations where cause precedes any event. In other words, the timelines of all joined edges of simplices must agree.

Thus, maybe, causality lies in the foundation of the spacetime geometry.

Causal sets
Main article: Causal sets

In causal set theory causality takes an even more prominent place. The basis for this approach to quantum gravity is in a theorem by David Malament. This theorem states that the causal structure of a spacetime suffices to reconstruct its conformal class. So knowing the conformal factor and the causal structure is enough to know the spacetime. Based on this Rafael Sorkin proposed the idea of Causal Set Theory. Causal Set Theory is a fundamentally discrete approach to quantum gravity. The causal structure of the spacetime is represented as a Poset, while the conformal factor can be reconstructed by identifying each poset element with a unit volume.

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Causality (physics)