Foundations of special relativity

The special theory of relativity has had a profound impact upon notions of time and space within the scientific and philosophic communities. This well-established model of local coordinate transformations in the universe is built upon two fundamental postulates:

1. • The principle of relativity: the laws of physics apply in all inertial reference systems;

2. • The universality of the speed of light: the speed of light in a vacuum is the same for all inertial observers, regardless of the motion of the source or observer.

The principle of relativity is not unique to the special theory of relativity; indeed it is assumed within Galilean relativity. However, if the equations of electrodynamics described by Maxwell's equations describe laws of nature, then the second postulate immediately follows from the first, since Maxwell's equations predict a universal speed of propagation of electromagnetic waves in a vacuum. The consequences of these postulates will be developed briefly.

Lorentz transformations

One of the most direct routes towards developing the transformations satisfying the postulates of special relativity involves examining the distance traveled by a propagating light pulse: (Δx)2 + (Δy)2 + (Δz)2 = (Δct)2.

Common cause (in)completeness and extendability

In principle, there are two ways to interpret Reichenbach's Common Cause Principle in general, each determined by how one views the status of the Principle with respect to the conditions of its validity: the Principle can be viewed as a falsifiable or a nonfalsifiable principle. In the falsificationist interpretation, the Common Cause Principle is a claim that can possibly and conclusively be shown not to hold for some empirically given events and their correlations; in the nonfalsificationaist interpretation, the Common Cause Principle cannot be falsified conclusively – whatever the actual circumstances. Is the Common Cause Principle falsifiable or nonfalsifiable?

We shall argue that the Common Cause Principle, as formulated in Chapter 2, is not falsifiable conclusively, but the argument cannot be trivial, since the Common Cause Principle is certainly not trivially nonfalsifiable: it is not true that every classical probability space (X, S, p) is provably common cause complete in the sense that for any A, B ϵ S that are correlated in p there exists a C ϵ S that is a (proper) common cause of the correlation between A and B. There exists common cause incomplete probability spaces (for instance the probability space described in Figure 3.1 is common cause incomplete). This makes the following definition nonempty:

Definition 3.1 The probability space (X, S, p)is common cause incomplete if there exist events A, B ϵ S such that Corrp(A, B) ＞ 0 but S does not contain a common cause of the correlation Corrp(A, B).

From special to general relativity

In general relativity, the “force of gravity”, which directly couples to a gravitating body through its mass, is replaced by relationships between geometric coordinates. This can be done because the inertial mass that relates the acceleration of a body from rest (a purely geometrical aspect) to the force through Newton's second law F = ma, is the same as the gravitational mass that couples the gravitational acceleration g to that body Fgravitation = mg. Newton tested this equivalence using various pendulums, and Eotvos [75] in 1889 verified the equivalence of inertial and gravitational mass to better than one part in 109. Gravity attracts different masses in a way that results in the differing masses having the same accelerations. This tenet embodies the equivalence principle, which will be discussed next. Since bodies of vastly differing constitutions and masses gravitate equivalently, one can then construct the trajectories of general gravitating masses in terms of geometric geodesies (special curves in the space-time), independent of the mass, charge, or internal structure of the gravitating body.

The principle of equivalence

The principle of equivalence forms the conceptual foundation of early formulations of the theory of general relativity. For present purposes, the principle of equivalence will be stated as follows: At every space-time point in an arbitrary gravitational field, it is possible to choose a locally inertial coordinate system such that (within a sufficiently small region of that point) the laws of nature take the same form as in an unaccelerated Minkowski coordinate system in the absence of gravity. Such an assertion inherently relates the inertial mass to the gravitational mass.

Causal closedness and common cause closedness

Assuming that Reichenbach's Common Cause Principle is valid, one is led to the question of whether our probabilistic theories predicting probabilistic correlations can be causally rich enough to also contain the causes of all the correlations they predict. The aim of this chapter is to formulate precisely and investigate this question.

According to the Common Cause Principle, causal richness of a theory T would manifest in T's being causally closed (complete) in the sense of being capable of explaining the correlations by containing a common cause of every correlation between causally independent events A, B. This feature of a theory is formulated in the next two definitions of causal closedness. In both definitions (X, S, p)is a probability space and Rind is a two-place causal independence relation that is assumed to have been defined between elements of S. We treat the relation Rind as a variable in the problem of causal closedness; hence, at this point we leave open what properties Rind should have to be acceptable as a causal independence relation – later we will return to the issue of how to specify it.

Definition 4.1 The probability space (X, S, p) is called causally closed with respect to Rind, if for every correlation Corrp(A, B) ＞ 0 with A ϵ S and B ϵ S such that Rind (A, B) holds, there exists a common cause C of some type in S.