User:Marc Goossens/Motivations of spacetime geometry

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THIS ARTICLE IS STILL IN SCAFFOLDING – PERSONAL DRAFT (intended for publication in 2008)

Motivations of spacetime geometry looks at the answers given by a multitude of scholars, notably philosophers, mathematicians and physicists to the question as to why space and time that 'surround us', exhibit certain characteristics, which can be described by mathematical models. Especially, the question is about the geometrical structure of these models, and how this may be made plausible, be explained and be better understood.

As this inquiry is about physical reality and the physical universe, purely mathematical geometry, of itself, is insufficient to provide an adequate answer. Nonetheless, the problem also touches upon deep geometrical topics too. By equipping the math with (formalized) physical assignments or 'interpretations' (as had been suggested by Einstein in 1924), physical axiomatics of space and spacetime tries to bridge this gap–with varying degrees of elegance, rigor and persuasiveness.

To achieve as this directly as possible, using only a minimum of intuitive primitive notions and without recourse to complex physical constructs (such as quantum mechanical atomic clocks) is in particular the object of physical Spacetime Theory or SST. This is prompted by the observation that space and time (possibly combined) provide the a priori backdrop to almost any other physical theory (including quantum mechanics). While this fundamental character of spacetime also intrigues philosophers, they typically tend to formulate more abstract epistemic questions, showing less interest in the actual physical motivation behind the mathematical geometry, or indeed which physical meaning and scope different such geometries possess to begin with.

Across all these fields of scholarship, the motivations for the Euclidean geometry of classical mechanics, the Minkowski geometry of special relativity as well as the pseudo-Riemannian (or Lorentzian) geometry of general relativity (even with some possible generalizations) are addressed. The various motivational approaches shed some light on the issue of foundations of spacetime.

The adopted classification of the various strains of explanations for the choice of a spacetime structure is due to Udo Schelb (Schelb 1997, pp. 49–59).

Deficits in the understanding of relativistic spacetime

The following account essentially follows Udo Schelb's extensive historical and systematic survey of the subject (Schelb 1997).

Main article: Foundations of spacetime

The classical Newtonian concepts of space, time and relativity may not be as self evident as they are often purported to be. The Minkowski geometry of special relativity is still less so, but the questions surrounding the rationale behind the choice of a mathematical spacetime model are the most striking in the field of general relativity.

Here, Einstein's famous proposal that a physical theory of space and time comprising a representation of gravity in geometric form, should be a continuous 'manifold' of dimension 4, equipped with a metric (symmetric) tensor field of Lorentz signature (see wikirefs) has withstood the test of time . Nonetheless this postulate, however successful, is not self evident, nor even well understood with regard to its physical motivation.

As is shown by the following survey and classification, numerous authors have made considerable efforts to arrive at a better understanding of the reasons or rationale behind the mathematical (and in particular: geometrical) structure of general relativity. The sheer number and variation of these motivations (sometimes even leading to modified geometry proposals mg wikiref) indicates the importance as well as the complexity of this problem, as well as the relevance of corresponding foundational research concerning this cornerstone of theoretical physics.

Classification of spacetime theories' rationale and scope

Classification of spacetime theories' rationale and scope

It is important to keep in mind that this classification is neither arbitrary nor absolute, and is intended primarily as a help for obtaining an overview on the subject matter.

Philosophical motivation

Whilst philosophers of science in general have formulated numerous questions about the geometry of space and time [1], they appear largely oblivious to the issue of physically motivated spacetime models with an empirically verifiable basis, that constitute the goal of SST. Instead, they debate whether any attributed spacetime geometry arises either as an a priori truth or as an empirical discovery, or whether it is merely a matter of convention (Malament 1986). They also wonder about whether spacetime is a substance, or about possible consequences of hypothetical violations of causality. The notable exception is Hans Reichenbach [2]. Already in 1924, he publishes a 150 page treatise on the axiomatics of spacetime, aiming to provide a systematic and physical basis (Reichenbach 1924). Still, this impressive endeavor is not fully consistent: Reichenbach employs concepts material intertial frames and rigid rods, thus falling short of his stated intention to erect a 'light geometry'. He also fails to clearly distinguish between mathematical and physical statements. Further, his usage of signal transmission times becomes unambiguous only in a physically unattainable limiting sense; as a result, his axiom IV, 1 becomes untestable. And the extension from special relativistic Minkowski spacetime onto general relativity is of a purely mathematical nature only (Schelb 1997, sect. 2.3.4, p. 73). Despite such shortcomings, not only the intent of his landmark approach but also several of Reichenbach's tools (real-points, signals and first-signals, pre-clocks and clocks, …) will prove to be of great value for inspiring later attempts (wiki ref meister-eps; sch-sch-SST).

Mathematical motivation

Mathematical geometers have devoted quite some attention to foundationally inspired 'theory of space', as epitomized by Hans Freudenthal's 1978 bundle (Freudenthal 1978). These studies touch upon some issues that are also relevant to axiomatic SST, for example regarding the Riemann-Helmholtz-Lie space problem [1].

Missed out on Lorentz signature! forthright

Mathematicians have also investigated the global structures (topology, orderings, …) of physical spacetime models (Pankai 1993), and devised axiomatic developments for these geometries. Still, their focus lies on characterizing and classifying geometries with the aid of a minimal set of postulates, primarily using group theoretical arguments of invariance under automorphisms. That their postulates should have a physical meaning and motivation is generally not their concern.

Nonetheless, developments mathematicians will deliver concepts and tools that can be applied fruitfully in SST axiomatics (like Hilbert's characterization of geometries by specifying incidence properties of congruences of curves) as well as in the resulting models of GRT (like Penrose's spinor and twistor methods):

In contrast, when it comes to axiomatics, most other mathematicians have focused on Minkowski spacetime:

  • Robb – Refining and extending his earlier A theory of Time and Space, A.A. Robb's 1936 Geometry of Space and Time (Robb 1936) builds up Minkowski space starting from a binary before-after (causal) relation in a point set. With over 400 pages presenting 30 (mainly mathematically inspired) postulates and proving 206 theorems virtually without any chapter structuring, it is not easily accessible. (see S sec 2.3.1, p 62-64)
    • Some physical inspiration can be drawn from Robb's arguments (presented in the introduction) that quantified measurements of length and time intervals are not elementary enough, and that time measurements (causal relationships between events first among them) hold precedence.
  • Carathéodory – Carathéodory's 1924 axiomatics of special relativity is a somewhat inhomogeneous mixture of more convincing and less well-founded elements. He intends to do away with measuring lengths and velocities, adopting temporal ordering and light propagation instead (Carathéodory 1924). He introduces 'light polygons', being a closed sequence of light signals along a set of material points at which mirrors are placed, and expresses postulates regarding these. The transformations taking light polygons to light polygons form a 15-parameter group. Reduction to Lorentz transformations is achieved by invoking the principle of special relativity.
    • Among the deficits of his proposal are Carathéodory's assumptions of 3-dimensional "space" to be endowed with certain topological properties, the possibility of measuring lengths after all, and some form of light carrying 'medium' (leading to the transformation group). (S 2.3.2, p 65)
  • Noll – Stemming from his work with Clifford Truesdell and others on continuum mechanics, Walter Noll also addressed what he calls 'Minkowskian chronometry'. First he starts purely mathematically, showing that a separation function defined on a set E and specified through its group of automorphisms, induces a pseudo-Euclidean geometry on E. Subsequently, he connects this with some more physical ideas like events, observers.and a (postulated, rather than constructed) signal relation between pairs of events (Noll 1964), (Matsko & Noll 1993).
  • Basri and Ax – From these authors stem two spacetime axiomatizations that are highly formalized and explicit in stating their primitives. Mathematically concise, they remain physically insufficient (S sect. 2.4.2 p 77-78). S. Basri has given a construction, primarily of special relativistic spacetime, cast in a strict language of formal logic, and following Reichenbach in setting out using particles, events, signals, first signals and observers. The way the notion of 'rigidity' is used makes it unsatisfactory from a physical point of view (Basri 1966). J. Ax follows more closely in the footstep of Robb, his work pointing the way towards the use of radar coordinates as exploited in STT (Ax 1978).
  • Suppes – In the context of his axiomatics of classical point mechanics and together with various co-authors, Patrick Suppes was also in search of a motivation for the Lorentz transformations. These authors readily adopt particle masses, forces and the speed-of-light constant c as undefined primitives, and postulate a hyperbolic quadratic 'relative distance' function (expressed in coordinates) between two spacetime points also without suggesting any kind of justification (Suppes 1959), (Suppes 1973)
  • Schutz – Along with Arthur G. Walker and George Szekeres, John W. Schutz has put forward a concise and well organized special relativistic spacetime axiomatics, with elements for an extension to general relativity. Some of the advanced geometrical axioms (e.g. using the projective notion of 'sprays'), though physical in intent, still lack a succinct physical meaning (Schutz 1973).

Intuitive motivations

Some authors straightforwardly present the mathematical structures as if they were physical space(time) 'itself', considering them self-evident, possibly on the basis of some terse reference to experiment. How patently obvious these structures really are, may be seen from the fact that different authors put forward different choices of geometry as compelling.

  • Weyl – Hermann Weyl takes as a starting point the 'observation' that spacetime forms a 4-dimensional continuum. Matter and gravitation govern a 'structural field' of this continuum… "by which it exerts a profound influence on physical occurrences; (…) and only by its effects on concrete phenomena in nature, can we recognize this structure" (Weyl 1931, p. 49) In his arguments, Weyl identifies physical elements directly with mathematical ones. The successes of general relativity explain why in order to come to 'a unified view of the world', a 'geometrization' of the whole of physics is called for (p. 51).
    • That according to general relativity, 'shifting a vector around in the world' may change its direction, but not its length, leads Weyl to propose an obvious generalization [2]. The resulting Weyl space structure includes entities that exhibit the mathematical behavior one expects also from an electromagnetic field, which Weyl hoped to unify with gravity in this way. The path of this 'accidental analogy' has nowadays been abandoned. Independently of this, Weyl geometries still constitute an important class of generalized spacetime models (see mg wikirefs).
  • Schrödinger – Like Weyl, Erwin Schrödinger has a generalization of Riemannian (Lorentzian) geometry in mind. Introducing the affine connection before the metric, there is no a priori ground for the connection to be symmetrical. Again, the hope to incorporate electromagnetism into the geometry of spacetime in this way, proved premature if not unfounded. Directly looking at the mathematics, and falling short of explicit physical arguments, Schrödinger's line of attack also remains implicit and intuitive. (See (Schrödinger 1987).)

Motivation through analogy

Another heuristic, non-axiomatic motivation for a particular choice of spacetime geometry is given by referring to certain analogies in other domains of physics, especially the microphysical structure of continua with spin, such as ferromagnets, liquid crystals or metallic lattices containing dislocations. Such considerations lead Hehl and co-authors to adopt a more general Riemann-Cartan geometry (Hehl & et al. 1989), (Hehl & et al. 1995).

Didactical "textbook" motivations

Adler Bazin Schiffer textbook S 2: finsler and weyl geometries are mentioned

Physical motivation

In his 1921 lecture on 'Geometry and Experience' (Einstein 1993), Einstein explicitly distinguishes between "practical" and "purely axiomatic geometry", the latter requiring notions like measuring rods, chains and clocks to be incorporated into "the formal system". Only the resulting "sum" (G) + (P) of geometry (G) and its physical principles (P) is subject to empirical verification.

  • Robertson – In his essay 'Geometry as a Branch of Physics' (Robertson 1964), H.P. Robertson has noted that the question regarding the geometrical properties of the universe around us is not a question to be answered merely by geometry as a branch of mathematics. The mathematical theory of Euclidean geometry, as a suite of axioms and theorems, is neither more nor less real with regard to our physical environment, than is some theory of (pseudo-)Riemannian geometry. There, internal consistency and deductive logical correctness are the sole criteria, and they are fulfilled by any of these theories.
    • Schelb goes on to state an assignment between 'the world of physical objects' and 'geometrical terms' (e.g. mathematical distance between mathematical points) as implied by Einstein, requires the mathematical theory to be expressed as a formal construct to begin with, and (in this case) its geometry (curvature, …) to be stipulated intrinsically, without any a priori reference to a possible embedding in a space of higher dimension (sect. 2.2, p 51). Otherwise, one would have to set up this 'surrounding geometry' on a physical footing to begin with. Still according to Robertson, the subsequent physical correspondence ought to be neutral, philosophically as well as 'physico-mathematically' (p 231, 236), and, for a theory as ground laying to physics in its entirety as space(time), ought to start from direct, 'pre-scientific concretion of our sense-data'[2]. (more in next section)

For more details, see e.g. (Schelb 1997, pp. 12–47). waar sloeg dit op terug?

Historical overwiew

(Schelb 1997, pp. 55–94).

Operational axiomatics and Spacetime Theory (STT)

or: physical axiomatics in the spirit of or leading up to STT

The Ludwig group: axiomatic theories of space and spacetime

A noteworthy series of publications have been broadly inspired by Günther Ludwig's 1974 initial question as to why physical space can be described by 3-dimensional Euclidean geometry (Ludwig 1974, chapter IV). This has been expanded upon by H.J. Schmidt (regarding Euclidean space) and D. Mayr (adding the time dimension).

The aim of these authors is to provide direct axiomatic bases for physical space and space-time as primary physical theories, so without the use of prior 'ancillary' theories. As such, they qualify as STT in the sense of J. Schröter.

The axiomatics relies to a large extent on the geometry classifications resulting from the solutions of Hilberts fifth problem and the Riemann-Helmholtz-Lie space problem by Yamabe, Freudenthal, Tits and others. (See Schelb sect 2.4 P 74-76 and MG Wikiref Ludwig STT and Yamabe.)

EPS axiomatics, precursors and successors

  • John L. Synge's chronometry – With his stated aim of "(setting) the theory (of general relativity) up afresh on a logical basis" (Synge 1964, p. 34), J.L. Synge heeds the criticism by 1946 Nobel laureate Percy W. Bridgman that, whilst Einstein had convincingly vanquished Newton's conception of spacetime using operational arguments, he (Einstein) had not succeeded in carrying this over also to general relativity (Bridgman 1949). Synge wants to provide such an operational basis (Synge 1960, p. 105).
    • To Synge, events are specified by 'four instrument readings' (coordinates), forming a 4-dimensional continuum. Neither 'time' nor 'space' are mentioned as constituents of the event (spacetime) manifold. Another primitive he uses is particle (material as well as photon), the 'history' of a particle forming a worldline of events. Yardsticks are banned; instead, according to the chronometric hypothesis (or axiom), each particle is equipped with a clock parameterization. Synge establishes no relationship between such a clock and the coordinate readings, but he does select standard clocks as those indicating 'proper time', stipulating that such a clock is realized by the frequency ticks of an atomic spectral line. His 'consistency hypothesis' takes is that this can be any such atomic spectral frequency. Thus, Synge adopts atomic clocks as a cornerstone of his chronometry, notwithstanding his own admission that "the modern mental picture of an atom and the radiation from it is so unclear" (Synge 1960, p. 106). Synge proceeds to add the Riemannian hypothesis, giving a coordinate expression for a Lorentzian metric as measuring time distance between neighboring events along a particle world line. His 'geodesic hypothesis' then states that free falling particle worldlines are precisely the timelike metric geodesics. Chronometry will continue to feature prominently in subsequent axiomatic physical spacetime motivations. However, later authors will seek to replace atomic clocks by light-based chronometry, which is more natural to the geometric setting, and avoids the need to appeal to 'higher' physical theories like quantum mechanics, for understanding the 'primary' concepts of space and time. In the nineties, work by J. Audretsch and C. Lämmerzahl reintroduces rudiments of quantum mechanics as a foundation for spacetime. (ref S p 90 - or see below)
  • Refinements and broadening by Castagnino, Enosh and Kovetz – Attempts to improve upon Synge's proposal have lead researches to consider more general geometries for general relativity, than the Lorentzian one. M. Castagnino replaces Synge's atomic clock with the light-clock of Marzke and Wheeler, leaving only paths of free particles and light pulses as constituents, forming a 'fundamental congruence'. Five 'hypotheses'[2] (axioms) establish its properties, leading to a conformal (light cone field) structure (Castagnino 1971).
    • As a geometrical construction, the Marzke-Wheeler clock rests on building a "physical parallelism" on the spacetime manifold, in a way that is strongly inspired by the Desargues construction from projective geometry, and for which Castagnino provides an exact analytical formulation. As Schelb notes though, the construction remains infinitesimal (by definition), so that this mathematical clock can not be regarded as truly operational in a physical sense (ref S p 83). Finally, Castagnino needs to enforce the reduction from a conformal to a Riemannian (Lorentzian) structure a bit artificial (hypotheses H5 and H6): as the author points out, dropping H6 leaves spacetime with the structure of a Weyl manifold. This was confirmed by Enosh and Kovetz (Enosh & Kovetz 1971), and later taken up by Perlick (see Perlick Weyl STT below).
  • Ehlers, Pirani and Schild's landmark spacetime axiomatics – In 1971, J. Ehlers, F. Pirani and A. Schild published the first paper with the EPS spacetime axiomatics, named after the authors, and which was to become the epitome of the search for a sound physical understanding of spacetime structure and geometry.
    • Main article: EPS spacetime

    As the groundwork for their proposal (ref), these authors adopt (world lines of) light rays and (world lines of) particles in a 4-dimensional space (manifold) of events. The 'continuity' of the manifold structure can motivated on the basis of light rays sent back and forth between particles. Coordinates are built by chronometry (as pioneered by Synge), employing a geodesic clock of Marzke-Wheeler or Kundt-Hoffmann type.

    If one takes it that at a given event, light rays determine local causality, in the sense that they distinguish (= topologically separate) 'future', 'past' and 'spatially out of reach' (at least infinitesimally, i.e. in tangent space, and according to the idea of special relativity); this gives rise to a conformal structure. Quite independently from this causal structuring using light, one may use the paths (world lines) of free falling particles to construct a projective structure on event space. Light rays are now conformal geodesics, free falling particles projective geodesics.

    A natural requirement of minimal compatibility between both these structures, again results in event space possessing a Weyl manifold structure. 'Enforcing' this Weyl structure to fold down even further to the intended Lorentz manifold of general relativity, finally requires a more complex (and non-local) postulate of compatibility.

    In conclusion, one may note the following remarks:

    • While EPS axiomatics 'addresses all the issues', not all aspects are treated with the same degree of completeness, detail and rigor—at least in the original paper. The event space's differentiable manifold structure, while not taken for granted, is only partially motivated. Neither is the concluding 'Riemannian hypothesis' formulated in axiomatic form; instead, only a qualitative geometrical indication is given of what possible arguments could look like.
    • At any rate, EPS has imparted a considerable boost to the search for a deeper understanding of the spacetime concept as postulated by Einstein for general relativity. Several enhancements have been published subsequently, also by the original authors; these have been bundled by Meister in his STT formulation of EPS (see 'Spacetime theoories' below).
    • EPS has also led to a revival of H. Weyl's 19XX proposal, albeit as a generalization of spacetime geometry, than as an attempt to unify gravity and electromagnetism, as he had originally intended.
    • EPS' use of a distinguished set of free falling particles (for generating the projective structure) has come under criticism. While Meister has argued that the argument is not circular, Schröter and Schelb have shown that this notion remains problematic, as two inequivalent such sets of particles may be found (ref); this has prompted them to propose an axiomatic STT that avoids this problem (see 'Spacetime theories' below.)
  • Perlick's Weyl space axiomatics – Taking EPS as a starting point and echoing the suggestions of Castagnino,Enosh and Kovetz, the 1983 master's thesis by Voker Perlick gives an elegant argument for considering Weyl geometry as a more general alternative to Lorentz geometry for general relativity (ref Perlick).

Spacetime Theories

  • Enhanced EPS
  • S² SST

See also

spacetime general relativity manifolds meister Schröter-Schelb foundations clocks in general relativity conformal manifold projective manifold geometrical constructions

Notes

  1. ^ See Schelb p 52-53, where he refers to (Grünbaum 1974), (Earman 1989), (Friedman 1983), (Sklar 1974).
  2. ^ For more on the space problem and the views of H. Weyl on general relativity, see also (Hawkins 1998), and G-structures.
  3. ^ This is precisely the ambition of SST as put forward by [[]Schröter]]; see MG wikiref.
  4. ^ These are themselves related to the solution of Hilbert's fifth problem by Hidehiko Yamabe.
  5. ^ On H. Reichenbach and his views on space and time, see also the Internet Encyclopaedia of Philosophy article by M. Murzi: http://www.iep.utm.edu/r/reichenb.htm#H2
  6. ^ Schelb qualifies some of Castagnino's axioms as ad hoc. For instance, the ones expressing the manifold structure, the hypercone signature, or the independence of clock time on the 'parallel ribbon' used (Schelb p 81).

References

  • Ax, James (1978). "The Elementary Foundations of Spacetime". Foundations of Physics. 8 (7–8): 507–546. doi:10.1007/BF00717578.
  • Basri, Saul A. (1966), A Deductive Theory of Space and Time, Amsterdam: North Holland
  • Bridgman, P.W. (1949), "Einstein's Theories and the Operational Point of View", in Schilpp, P.A. (ed.), Albert Einstein, Philospher-Scientist, LaSalle, Ill.: Open Court and Cambridge University Press, pp. 335–354
  • Carathéodory, C. (1924). "Zur Axiomatik der Speziellen Relativitätstheorie". Sitzungsberichte de Preussischen Akedemie der Wissenschaften. Physikalish-mathematische Klasse. 13: 12–27.
  • Castagnino, Mario (1971). "The Riemannian structure of space-time as a consequence of a measurement method". Journal of Mathematical Physics. 12 (10): 2203–2211. doi:10.1063/1.1665522.
  • Earman, John (1989), World Enough and Space-Time, Cambridge, Massachusetts: MIT Press
  • Ehlers, J.; Pirani, F.A.E.; Schild, A. (1972), "Ch.4 - The Geometry of Free Fall and Light Propagation", in O'Raifertaigh, L. (ed.), General Relativity. Papers in honour of J.L. Synge., Oxford: Clarendon Press, pp. 63–82
  • Einstein, Albert (1993), "Geometrie und Erfahrung", in Seelig, Carl (ed.), Mein Weltbild, Frankfurt: Ullstein, pp. 119–127
  • Enosh, M. (1971). "Characterization of Gravitational Theories by Cumulative Effects in Free Particle Motion and in the Behaviour of Clocks". Annals of Physics. 69: 279–296. doi:10.1016/0003-4916(72)90010-3. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  • Freudenthal, Hans, ed. (1978), Raumtheorie, Darmstadt: Wissenschaftliche Buchgesellschaft
  • Friedman, Michael (1983), Foundations of Space-Time Theories, Princeton: Princeton University Press
  • Grünbaum, Adolf (1974), Philosophical Problems of Space and Time, Dordrecht: D. Reidel
  • Hehl, F.W. (1989). "Progress in Metric-affine Gauge Theories of Gravity with Local Scale Invariance". Foundations of Physics. 19 (9): 1075–1100. doi:10.1007/BF01883159. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  • Hehl, F.W. (1995). "Metric-affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors and Breaking of Dilation Invariance". Physics Reports. 258 (1–2): 1–171. arXiv:gr-qc/9402012. doi:10.1016/0370-1573(94)00111-F. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  • Kronheimer, E.H. (1967). "On the Structure of Causal Spaces". Proceedings of the Cambridge Philosophical Society. 63 (2): 481–501. doi:10.1017/S030500410004144X. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  • Lebrun, Claude (December 1998). "Book Review – The Nature of Space and Time". Notices of the AMS. American Mathematical Society: 1479–1481.
  • Ludwig, Günther (1974 (1978, 1985)), "Band 1: Raum, Zeit, Mechanik", Einführung in die Grundlagen der Theoretischen Physik, Braunschweig: Friedrich Vieweg & Sohn Verlagsgesellschaft {{citation}}: Check date values in: |year= (help)
  • Malament, David (1986), "Gravity and Spatial Geometry", in Marcus; Barcan (eds.), Logic, Methodology and Philosophy of Science VII, Amsterdam: Elsevier Science, pp. 405–411
  • Lebrun, Claude (December 1998). "Book Review – The Nature of Space and Time". Notices of the AMS. American Mathematical Society: 1479–1481.
  • Novirov, Igor D. (1995), "Black Holes, Wormholes and Time Machines", in Occhionero, Franco (ed.), Birth of the Universe and Fundamental Physics, Heidelberg: Springer, pp. 373–378
  • Pankai, Joshi (1993), Global Aspects in Gravitation and Cosmology, Oxford: Clarendon Press
  • Penrose, Roger (1968), "Structure of Space-Time", in DeWitt, Cecile M.; Wheeler, John A. (eds.), Battelle Rencontres, New York: W.A. Benjamin, pp. 121–235
  • Penrose, Roger; Rindler, Wolfgang (1984), Spinors and space-time Vol.1 – Two-spinor calculus and relativistic fields, Cambridge: Cambridge University Press
  • Pirani, F.A.E. (1973), "Building Space-Time from Light Rays and Free Particles", Symposia Mathematica Vol XII, London, New York: Academic Press, pp. 67–83
  • Reichenbach, Hans (1924), Axiomatik der relativistischen Raum-Zeit-Lehre, Braunschweig: Vieweg
  • Robb, Alfred A. (1936), Geometry of Time and Space, London: Cambridge University Press
  • Robertson, H.P. (1964), "Geometry as a Branch of Physics", in Smart, J.J.C. (ed.), Problems of Space and Time, New York: Macmillan, pp. 231–250
  • Schelb, Udo (1997), Zur physikalischen Begründung der Raum-Zeit-Geometrie, Paderborn: Paderborn University (Habilitation Thesis)
  • Schrödinger, Erwin (1987), "Die Struktur der Raumzeit", in Audretsch, J. (ed.), Darmstadt: Wissenschaftliche Buchgesellschaft {{citation}}: Missing or empty |title= (help)
  • Schutz, John W. (1973), Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time, Berlin: Springer
  • Sklar, Lawrence (1974), Space, Time and spacetime, Berkeley: University of California Press
  • Suppes, Patrick (1959), "Axioms for Relativistic Kinematics with or without Parity", in Henkin, L.; Suppes, P.; Tarski, A. (eds.), Symposium on the Axiomatic Method, with Special Reference to Geometry and Physics, Amsterdam: North Holland, pp. 291–307
  • Suppes, Patrick (1973), "Some open Problems in the Philosophy of Space and Time", in Suppes, P. (ed.), Space, Time and Geometry, Dordrecht: D. Reidel, pp. 383–401
  • Synge, John L. (1964), "Introduction to General Relativity", in DeWitt, C.; DeWitt, B. (eds.), Relativity, Groups and TOpologu, New York: Gordon and Breach, pp. 4–88
  • Synge, John L. (1960), Relativity: The General Theory, Amsterdam: North Holland
  • Weyl, Hermann (1921). "Zur Infinitesimalgeometrie: Einordnung der projektiven und konformen Auffassung". Nachr. Von der K. Ges. Wiss. Göttingen, Mathematisch-physikalische Klasse: 99–112.
  • Weyl, Hermann (1923), "Mathematische Analyse des Raumproblems", Berlin: Julius Springer {{citation}}: Missing or empty |title= (help)
  • Weyl, Hermann (1931). "Geometrie und Physik". Die Naturwissenschaften. 19 (3): 49–58. doi:10.1007/BF01516349.
  • Trautman, Andrzej (1972), "Invariance of Lagrangian Systems", in O'Raifertaigh, L. (ed.), General Relativity. Papers in honour of J.L. Synge., Oxford: Clarendon Press, pp. 85–99
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