Minkowski Space

The theme of this slender tome has its origins in the work of Hans Reichenbach, who suggested in 1925 that the ‘heuristic and inductive arguments’ that led to Einstein’s curved space-time description of gravity were unsatisfactory. But specific details of Reichenbach’s concerns aren’t immediately discernible within the concisely presented mathematical contents of this book. A more accessible perspective on the epistemology of relativity is Bas van Fraasen’s Introduction to the Philosophy of Time and Space.

A fundamental difficulty in the theory of time concerned the notion of ‘temporal order’. The challenge was to construct a model of spacetime that allowed for a physical basis for temporal relations. Reichenbach’s causal theory of time is a framework within which these matters were originally addressed. That, and other space-time theories (STTs), attempted to clarify the nature of space and time independently of special and general relativity. Such STTs are regarded as the pre-theories of specific space-time pictures of special and general relativity.

To develop relativity on an axiomatic basis, Reichenbach made use of three basic concepts; causation, coincidence and physical possibility. In terms of the these, he formulated the following definitions:

1. Event \(E_2\) is later than event \(E_1\) if, and only if, it is physically possible for there to be a chain of events \(s_1,s_2,\dots, s_k\) such that each event \(s_i\) is the cause of event \(s_{i+1}\), the event \(E_1\) coincides with \(s_1\), and \(E_2\) coincides with event \(s_k\).
2. \(E_1\) and \(E_2\) are indeterminate with respect to temporal order if, and only if, neither is later than the other.
3. An assignment, \(t\), of real numbers to events is a topologically admissible coordinatization if, and only if,

1. If \(E_1\) and \(E_2\) coincide, then \(t(E_1) = t(E_2)\)
2. If \(E_1\) is later than \(E_2\), then \(t(E_1) < t(E_2)\)

Moreover, if event \(E_1\) is the cause of event \(E_2\), then a small variation in \(E_1\) is associated with a small variation in \(E_2\) — but not conversely. Thus, \(E_2\) is later than \(E_1\) if it is caused by \(E_1\).

The purpose of Joachim Schröter’s book is to establish a mathematical model of spacetime that addresses matters such as temporal order and causation. It begins with five ‘axioms’ specifying Minkowski spacetime as differentiable manifold with a \(C^k\) atlas, \(A^s\), and a global chart whose image under a bijective mapping resides in \(\mathbb{R}^4\). The distinction here is that the elements of \(M^s\) are symbols for physical events, while \(\mathbb{R}^4\) contains the coordinates of those events. There is also the requirement for a \((0,2)\) tensor field (metric) \(g^s\) on \(M^s\) taking the form \[g^s(p)=dx_1^2+dx_2^2+dx_3^2-dx_4^2.\]

For this system to be connected to general relativity, \(g\) should be a solution of the Einstein field equations. And when the Einstein equations are expanded in terms of the coordinates of \(x=\varphi(p)\), the components of \(g\) are \(g^s_{\alpha\beta}\) with \(\alpha,\beta=1,2,3,4\). These enable the definition of one of the key concepts in this book: that of Minkowski chart, which is an arbitrary (non-global) chart \(\varphi’:M^s\to \mathbb{R}^4\) together with metric \(g^s = g^s_{\alpha\beta}dx^{\prime\alpha}\otimes dx^{\prime\beta}\).

The Lorentz transformation between charts \((M^s,\varphi^\prime)\) and \((M^s,\varphi^{\prime\prime})\) is the \(C^k\)-bijection \(\phi:\mathbb{R}^4\to\mathbb{R}^4\) such that \(\phi=\varphi^{\prime\prime}\circ\varphi^{\prime-1}\), which can be represented by a Lorentz matrix, classifiable as proper, orthochronous or antichronous (extensively discussed in chapters 1–4)

The next step is to show that the base set \(M^s\) has particular vector space and topological structures - each inherited from \(\mathbb{R}^4\) by means of the bijection \(\varphi’:M^s\to \mathbb{R}^4\). The vector space structure, of course, depends upon the chart chosen; but any two charts that are related by a Lorentz transformation result in same vector space structure on the base set. Regarding \(M^s\) as a vector space in this way means that it can be shown to be isomorphic to each of its tangent spaces. And since, on any tangent space, there exists an indefinite inner product, it becomes possible to define one directly on \(M^s\). The topology of \(M^s\), that comes from \(\mathbb{R}^4\), is the natural topology and is independent of \(\varphi\). \(M^s\) now has the dual existence of being a point-like set of physical events and a set of vectors, thereby making way for the definition of causal and chronological relations expressed in terms of future-pointing tangent vectors.

For this purpose, there is an introduction to the concept of Lorentz vector space, which is a real vector space of dimension \(n\) together with an inner product \(g:V\times V\to \mathbb{R}\) where \(V\) has basis \((e_1,e_2,\dots,e_n\) and \(g(e_\alpha,e_\beta)=g_{\alpha\beta}^s\). A Lorentz vector space \((V,g)\) is partitioned into three subsets by the following criteria: \(v\) is timelike if \(g(u,v)<0\), is lightlike if \(g(u,v)=0\) and is spacelike if \(g(u,v)>0\) or \(v=0\).

Vector \(v\) is causal if it isn’t spacelike. But the meaning of causality here differs from the usual meaning. Specifically, a causal vector isn’t the cause of something; instead, it determines a signal between events \(p_1,p_2\in M^s\). A signal from \(p_1\) to \(p_2\) may cause something at \(p_2\) — but not necessarily. In the relativistic sense, a causal vector relates to the propagation of a signal.

So, if \(C\) is the set of causal tangent vectors (timelike or lightlike) on a ‘Lorentzian manifold’, the subsets \(C^+,\, C^-\) consist of future-pointing and past-point tangent vectors respectively. If these subsets form a partition of \(C\), the manifold is time orientable. Following this definition, there are mathematical criteria for the classification of tangent vectors in terms of their membership if either \(C^+\) or \(C^-\).

At this stage in the book, \(M^s\) is set of vectors or a manifold consisting of a set of pointlike events. This allows for the introduction of two further structures. Firstly, the relation \(p_1\prec p_2\) for \(p_1,p_2\in M^s\) means that \(p_2-p_1\in C^+\) is future pointing, or \(p_2\) lies in the chronological future of \(p_1\). Then there is the causal relation \(p_1\leq p_2\), which means that \(p_2\) can be caused (or causally influenced) by \(p_1\). The fact that \(\leq\) is a partial ordering events in \(M^s\) is redolent of Reichenbach’s notion that events may be indeterminate with respect to temporal order. The relation \(\prec\) is, of course, transitive but not reflexive.

Having provided the foregoing foundations, discussion then centres upon the motion of point-like mass particles and light pulses in the in the context of special relativity. For this purpose, chapter 8 (Kinematics) develops the notions of worldlines, signals, observers, clocks, time dilation and length contraction. Also, based upon an observer-dependent notion of time and space, there are definitions of Newtonian velocity and acceleration and many more fascinating phenomena.

The only curious aspect of this book concerns the central concept (that of Lorentzian manifold), which is first mentioned on the opening page of chapter 1 and many times thereafter, but not formally defined until the very end of the book. This concept, which is central to spacetime theory, is ultimately defined as an \(n\)-dimensional semi-Riemannian Haussdorff manifold with metric signature \(s=n-2\).

And the overall conclusion is that, for each STT, the corresponding physical spacetime \(M\) can be described mathematically as 4-dimensional Lorentz \(C^k\) (\(k > 2\)) manifold whose points are represent point-like physical events. Moreover, the matters of temporal order and causation (that so bothered Reichenbach) have been mathematized well before the last chapter, which is the mathematical capstone examining further important ideas of relativistic theories.

Despite a few obvious typos, this excellent book has been produced to a high standard of accuracy and has been well-translated from German by Christian Pfeifer.

Peter Ruane’s teaching career involved the training of mathematics teachers (primary, secondary and high school). His postgraduate study concerned the application of differential geometry to matters of superconductivity, and he received the Seventh Annual Mathematical Gazette Writing Award in 2002.