User:Marc Goossens/EPS Spacetime

THIS ARTICLE IS A PERSONAL DRAFT (intended for publication in 2008)

EPS spacetime derives its name from the researchers Jürgen Ehlers, Felix Pirani and Alfred Schild, who introduced this general relativistic spacetime model in the early 1970's (Ehlers, Pirani & Schild 1972, pp. 63–84).

The Ehlers-Pirani-Schild spacetime construction (short: 'EPS') provides a partial answer to the 'inverse problem of General Relativity' (Castagnino 1971, p. 2203): it builds a physically motivated mathematical spacetime model that leads to the paradigmatic Lorentzian 4-manifold (short: 'L4') of General relativity. In turn, L4 is the generic underlying spacetime model of General Relativity. This makes EPS into a classic example of physically constructive (spacetime) axiomatics; as such, it has received considerable further attention since its first publication.

The importance of EPS lies in its alternative axiomatic definition, which aims to construct L4 in a physically transparent way, based on world lines of free-falling particles and light rays. In this way it provides stronger physical motivation and understanding not only of L4 as such, but also in comparison with more general geometries (as candidates for modeling physical spacetime mathematically). EPS specifically highlighted the potential role of spacetime models based on Weyl space.

By supplying this new axiomatic characterization of the otherwise mathematically familiar L4 structure, EPS also brings relevant new insight even from a strictly mathematical (geometrical) standpoint.

This article discusses the original 1972 formulation of EPS, situated in the Golden age of general relativity. A 1990 review and update is presented in Meister.

Survey of EPS spacetime

Motivation for EPS in non-technical terms

Einstein's General Theory of Relativity uses advanced mathematical ideas. Things like 4-dimensional curved spacetime (precisely the one denoted here by 'L4') are not easy to grasp. Even if one masters the math behind it, the essential physical meaning and content is not obvious.

EPS is one of a series of attempts to clarify the physics behind the math (see spacetime axiomatics). Unfortunately and unavoidably, getting there requires even more abstract math. At first sight, this seems self-defeating; however, some of these mathematical ideas are chosen so as to be closer to 'operational' physical interpretation, representing more elementary physical observation, measurement and construction. In the upshot, EPS ends up with L4, instead of accepting it as starting point: the idea is to rebuild L4 from scratch, using only bricks with intuitively clear physical meaning – to the extent possible, and at the cost of some extra math.

As is typical in axiomatic reconstructions like EPS, one exploits the benefit of hindsight, as the intended result (in this case: L4 spacetime of General Relativity) is already known. So this in no way detracts from Einstein's original feat, on the contrary.

The scope of EPS is limited to the makeup of spacetime itself; the problem of any possible axiomatic derivation or reconstruction of the Einstein field equations governing matter and gravity within such a spacetime model, is left open.

“ The approach shows how quantitative measures of time, angle and distance, and a procedure of parallel displacement (…) can be obtained constructively from 'geomtery free' assumptions about light-rays and freely falling particles; pseudo-Riemannian (or Weylian) geometry is recognized even more clearly than before as the appropriate language for a generalized kinematics which allows for the unavoidable and ever-present 'distortions" called gravitational fields. ” — Jürgen Ehlers (Ehlers 1973, p. 87)

Outline of the EPS construction

The construction of EPS spacetime proceeds in steps as sketched below, each one enriching the axiomatic content of the underlying set of events. This outline summarizes the account of (Ehlers, Pirani & Schild 1972).

Roughly, the underlying idea is the following. From differential geometry, one knows that the geodesics determine 'their affine connection' (assuming torsion to be zero, for instance) and hence a corresponding metric (Nash & Sen 1983, p. 191). Now, in contrast to the metric itself, these geodesics do possess an immediate physical interpretation (as light ray worldines for null geodesics or particle world lines for timelike ones). So in very general terms, one tries to reconstruct the sought-after metric from known geodesics that fulfill certain qualitative criteria (postulates), which are themselves physically meaningful and plausible.

1. Particles and light rays in event space - EPS adopts a set ${\mathcal {M}}$ of events (to become the spacetime manifold) as its backdrop. On this, a set of particles ${\mathcal {P}}$ and a set of light rays ${\mathcal {L}}$ are assumed given. Each particle and each light ray are identified with their 'world line' of events.
2. Smooth radar coordinates for events - As subsets of the space of events, particle and light ray world lines are taken to be smooth one dimensional manifolds. A permissible local coordinate represents time as measured by a (possibly irregular) local clock. Light ray messages between particles P and Q smoothly relate their private time parameters , the timing of echoes received back by P also relating smoothly to that of the message flashes it sent out to Q to begin with.
Using 'radar soundings' in this way, pairs of 'observer' particles set out to map surrounding events by assigning 2 time values each, or a total of 4 coordinates each. Postulating that this process may cover the entire event set, the events form a smooth 4-dimensional space (manifold).
3. Light propagation ensures local validity of special relativistic causality - At each point of spacetime (event), the propagation of light determines an infinitesimal null cone, amounting to a conformal structure ${\mathcal {C}}$ of Lorentz signature. This assertion is stated operationally, demanding that one may (topologically) distinguish between time-like, space-like and null vectors, directions and curves at an event (see example below). Null curves lying on a null hypersurface are singled out as null geodesics.

4. 

   Free particle (projective) geodesics "fill" light ray (conformal) geodesic light cone at any event e, thus forming a Weyl manifold.

   Free falling particles encode influence of gravity on particle motion - Among the ${\mathcal {C}}$-timelike curves, the free-falling particles form a preferred family. Imposing a generalized law of inertia provides a projective structure, with free-fall world lines as its (${\mathcal {C}}$ time-like) geodesics.
5. Light and particle motion agree - In line with physical experience, one assumes that such particles can be made to chase a photon arbitrarily closely, meaning their paths "fill the light cone" so that each ${\mathcal {C}}$ null geodesic is also a ${\mathcal {P}}$ geodesic.
The manifold of spacetime events equipped with both a conformal and a projective structure, compatible in the above sense, is (by definition) a Weyl space or Weyl manifold. A Weyl space possesses a unique affine structure ${\mathcal {A}}$: ${\mathcal {A}}$ geodesics are P geodesics and ${\mathcal {A}}$ parallel displacement preserves ${\mathcal {C}}$ nullity. In a Weyl space, one may construct a "proper time" arc length (up to linear transformation) along non-null curves by purely geometrical means (i.e. using light rays reflected from particles only, so without any need for atomic clocks). In technical terms, one employs affine parallel displacement, and congruence in the tangent space, as defined by ${\mathcal {C}}$. This 'geodesic' clock is known as the Marzke-Wheeler clock (Marzke & Wheeler 1964).
6. Speed of time does not depend on path - A final physical assumption (expressed mathematically as an axiom) ensures the existence of a Lorentzian metric, compatible with ${\mathcal {C}}$.
"Equally spaced clock ticks" along one particle world line are transported to a nearby particle by Einstein simultaneity. Imposing that this must generate (approximately) equidistant ticks also for the second particle and applying the equation of geodesic deviation for the curvature tensor given by ${\mathcal {A}}$, implies (through the vanishing of the Weyl 'track curvature') the existence of a single Lorentzian metric compatible to both ${\mathcal {C}}$ and ${\mathcal {A}}$. This finally 'reduces' Weyl space to L4. Requiring in this way that 'time runs equally fast along all paths' amounts to denying the existence of a 'second clock effect'. Indeed, in (Lorentzian) General Relativity, only the 'time interval' between 2 events is path dependent (i.e. the 'first clock effect'); not the 'speed' of time.

Some elements of the construction in detail

Some 'feel' for the nature of the EPS arguments and constructions in their original formulation is rendered by the following examples, corresponding to steps ii and iii above ^[1].

• 

  Observers Paul and Ria each establish one pair of spacetime radar coordinates for an event e.

  Mapping the universe using radar - Let observer Paul (seated at a particle) illuminate observer Ria with a smooth sequence of light ray (or 'radar') flashes. EPS supposes that relabelling flash times by the arrival time of their corresponding echo blips or vice versa, amounts to a smooth conversion of the time parameter on Paul's world line. Similarly, the message blips from Paul as registered by Ria, smoothly relate 'flash time' at Paul's to 'blip time' at Ria's ^[1].
By sending a light ray to a chosen target event e, A may write down the time of emission and the time of reception of the echo. If B does the same, targeting the same event e, e is identified by these four numbers. This set of "coordinate" numbers identifying e is rather like latitude and longitude on a map, except that there are 4 instead of 2. (Of course, the numbers depend on the observers, and the - possibly irregular - clocks they're using.) Given enough such pairs of observers at different particles, EPS claims (as a postulate) that the entire set of spacetime events may be covered by a coordinate atlas (topology), formed by a consistent patchwork of overlapping coordinate maps obtained in this way.
• Using light propagation to construct the conformal metric^[1] – The first EPS light propagation postulate ensures that the following makes sense. In a small enough neighborhood U of an event e on a particle P, any other event a marks a well defined and unique flash event and a blip event on P. For a time parameter 'centered' at e, i.e. t(e) = 0, one may define the function
$g:a\mapsto -t(e_{flash})t(e_{blip})$. One finds that the 'matrix of second partial derivatives' of $g$, determines a symmetric tensor $g_{ij}:g_{,ij}(e)$ at e. A second light propagation axiom states that the set of light directions at an event topologically separates non-lightlike (non-null) directions into timelike and spacelike ones, and that 'future' and 'past' light vectors are likewise separated. Expressing this with the help of the tensor $g_{ij}$ just obtained, one finds it must be endowed with the desired hyperbolic 'Lorentz' signature, in order for the topological separation condition to be met. Up to a conformal factor, the metric tensor thus found is shown to be independent of the choice of the particle P or of any parameterization involved. This fixes the conformal structure ${\mathcal {C}}$ of spacetime once and for all.

Comments and criticisms

Near a spacetime event e, another event a determines both a radar 'flash' and a 'blip' event along a particle through e. This allows the auxiliary function g to be defined, the second derivatives of which determine the (conformal) metric tensor at e.

EPS has been reviewed by many authors. Some of these comments and criticisms have also led to EPS spacetime#Further developments completions and improvements] ^[2].

• Self-contained geometric axiomatics - With their axiomatic approach, Ehlers, Pirani and Schild wish to improve upon the example set by Synge (Synge 1962). In particular, they want to upgrade Synge's chronometry based on atomic clocks, which are extraneous to the purely geometrical framework of spacetime (Ehlers, Pirani & Schild 1972, pp. 64–65, 69).
Instead, EPS takes the notions of free-falling particles (leading to the projective structure ${\mathcal {P}}$) and light rays (leading to the conformal structure ${\mathcal {C}}$) together with qualitative relationships (like incidence) as its starting point. Not only are these concepts physical, they and the constructions (such as 'clocks' based upon them are also intrinsic or 'native' to the geometry. This makes the resulting physical theory more self-contained – as it need not resort to rigid rulers or atomic clocks for example, arising from other realms of physics. Whether time as measured by such a geometric-gravitational clock agrees with time as measured by an atomic clock, is then a matter for experiment to settle.
• Non-circularity of inertial class axiom - The introduction of some at that point unknown subset of the particles as 'free-falling particles' in step (iv) has been criticized as circular. Meister has argued that the resulting EPS theory does imply verifiable criteria: one can test whether a presumed free particle is indeed inertial (Meister 2004, pp. 56–57). According to the Ludwig meta-theoretical concept, this element in the EPS construction is therefore legitimate.
• Light and free fall more fundamental than metric field and curvature - The construction concludes with the intended Lorentzian metric. So for EPS, the metric field is seen as physically less fundamental than the other geometrical structures given by ${\mathcal {P}}$ and ${\mathcal {C}}$ (Ehlers 1973). The latter more directly characterize the motion of freely falling particles and light rays than the former. Indeed, it is precisely through the observation of these motions that gravity becomes 'tangible'. While highly convenient for subsequent mathematical analysis and calculation, the metric field and its curvature in the L4 formulation are seen as farther removed from physical interpretation.
• Arguments vary in detail and strength - The original EPS paper does not fully elaborate the mathematics for all steps in the construction, such as the topology or the nitty-gritty of building the smooth manifold structure. (This has been further elaborated by [[]Meister]]). The radar coordinates are not further exploited, as has been done later by Schröter-Schelb. Also, some steps come with a stronger physical rationale, others less so, as has been remarked by Schelb (Schelb 1997, p. 89).
For instance, the physical meaning of the auxiliary function used for defining the conformal metric is not very transparent. Regarding the 6th step (reducing the Weyl manifold to a Lorentz one), the original EPS article does not give a preferred formulation of the "Riemannian axiom". The reader is given the choice between the direct mathematical argument that vanishing Weyl curvature singles out a preferred Lorentzian metric, or the 'chronometric' argument cited above. The authors admit that this argument is "distinctly more complicated" than the others (Ehlers, Pirani & Schild 1972, pp. 69). This argument has been reformulated by Schröter and Meister (Meister 2004, pp. 99–107), as presented in EPS construction.
• Weylian spacetimes - EPS and subsequent investigations (ref.) show that Weyl space is the spacetime most readily constructed by geometrical means, the further reduction to the L4 of standard General Relativity being not so straightforward, or indeed subject to empirical verification. This explains why Weyl space has been studied as a candidate for a (generalized) model of spacetime. A Weylian universe would exhibit a 'second clock effect', with atomic clocks possibly deviating from 'gravitational' clocks, such as the Marzke-Wheeler clock (Marzke & Wheeler 1964). See further developments below and Weyl space.
• Quantum mechanics needed after all? - The 'no-second-clock' Riemannian axiom is not a local requirement. Feeling this to be a weakness, Audretsch & Lämmerzahl (Audretsch & Lämmerzahl 1990) introduce alternative requirements instead, consisting of some elements of QM; this is a legitimate alternative, but to the extent that QM adopts some spacetime model as a requisite backdrop, the argument may be circular (Meister 2004, p. 100). Schr = without
• EPS = a class, not a single model - Like L4, to which it is mathematically equivalent [by construction], EPS actually represents an entire class of 'possible' spacetimes: it allows many 'different' (non-isomorphic) specific model realizations, including the typical spacetime 'solutions' of the Einstein field equation. (It is a non-categorical model.)
• Mathematical spin-off and inspiration - Whereas the prime stimulus for the investigation of EPS resides in theoretical physics, it also yields purely mathematical, in this case mainly geometrical insights, as it explores the interplay between different geometrical structures and less common axiomatizations for some of these. Ehlers, Pirani and Schild also point out (Ehlers, Pirani & Schild 1972, p. 65) that the other way around, the method adopted for the EPS construction draws on techniques employed by Helmholz and Lie for deriving the metrics of spaces of constant curvature.
• Precursors and supporters - Constructive axiomatics may not have been a popular arena for active research; still, it enjoyed a broad interest among General Relativity theorists. It is interesting to note that the originators of EPS acknowledge contributions and support from R. P. Geroch, D. Sciama, R. Penrose, I. Prigogine and K. Bleuler. p 83

EPS in a broader context

Further developments

The original EPS publication has been widely quoted, refined and expanded by various authors ^[2]:

• Ehlers and Pirani have each reviewed their original publication in (Ehlers 1973) and (Pirani 1973).
• The projective structure has received attention in its own right. From the original authors, Ehlers and Schild have endeavored to provide a physical interpretation of projective curvature (Ehlers & Schild 1973). Coleman and Korte have also studied this structure in depth (Coleman & Korte 1980), (Coleman & Korte 1981), (Coleman & Korte 1984).
• Many comments (cast, ..)
• Woodhouse has enhanced the causal and differential-topological side of EPS. He also replace light ray world lines as a primitive concept by looking only at an emission – absorption pair of events. He explicitly constructs a topology on the space of events. (Woodhouse 1973)
• V. Perlick has taken up the investigation of spacetime concepts modeled on a Weyl manifold (Perlick 1983), (Perlick 1987).
• In a 1990 master's thesis at Paderborn university, R. Meister has wrapped up the complete axiomatization of EPS with slight variations and filling some gaps, treating spacetime topology and differentiable manifold structure more explicitly (Meister 1990). This formalization of EPS is also cast in meta-theoretical scheme of Günther Ludwig and has later been published in an English translation (Meister 2004).
• Inspired by EPS, Schröter-Schelb spacetime provides a more elaborate 'reconstruction' of L4, where the Lorentz metric arises more immediately by exploiting the manifold atlas of radar maps (Schröter 1988), (Schröter & Schelb 1992).

EPS in the context of spacetime axiomatizations

Main article: Spacetime Axiomatizations

Since Einstein launched his General Theory of Relativity in 1915, repeated efforts have been made to motivate and 'explain' the choice of a 4-dimensional, differentiable manifold equipped with a Lorentzian (or pseudo-Riemannian) metric as its essential mathematical model.

These include intuitive as well as mathematically inspired motivations by authors like H. Weyl and E. Schrödinger, work by Robb, Caratheodory, Suppes, Noll and Schutz. In 1924-1925, Reichenbach contributed significantly to a rigorous physical spacetime rationale. In the 1960's, J. Synge laid the basis for modern axiomatizations of general relativistic spacetime; this was then taken up by Ehlers and his co-authors to develop EPS. (See Spacetime Axiomatizations for an overview, and Schelb (Schelb 1997, pp. 48–94) for a detailed exposé.

EPS in the context of foundations of physics

"A general lesson to be drawn from the development of the theory of relativity is that it is desirable to analyse in detail the various structures inherent in the mathematical models used to describe physical phenomena. The analysis should concentrate on the relation of these structures to experimentally verifiable statements."

Andrzej Trautman (Trautman 1972, p. 85)

Main article: Foundations of physics

The EPS spacetime theory is a prime example of operational-constructive axiomatics of physical theories and a relevant contribution to the foundations of theoretical physics, and the research field opened up by Hilbert's sixth problem. Also, the purely mathematical part of EPS spacetime has been formulated as a (hierarchy of) Bourbaki species of structure, and its full physical theory expressed according to the Ludwig scheme ('REF).

As constructive axiomatic theories go, EPS is among their most well known instances (see sections below). To the extent that EPS fulfills its intention of gaining a better physical understanding of the mathematical model used by the classic physical theory that is General Relativity, EPS itself has assumed the role of a paradigmatic illustration of axiomatic foundations, and of the potential relevance of this field for the whole of physics.

Gravitational clocks in General Relativity

Main article: Clocks in General Relativity

The notion of 'clock" plays a central role in EPS, as it does in General Relativity: 'time' shows up as a parameter along particle world lines; a clock is a device which is mathematically represented as such a parameter assignment in some way. The physical functioning of such a device may be left implicit, or be represented more explicitly.

As opposed to atomic clocks, which are foreign to the initial setting of (macroscopic) spacetime models, certain clock constructions have been proposed that are 'indigenous' to the light and particle geometry of spacetime. These are referred to as gravitational or geodesic clocks. Examples are the Kundt-Hoffmann clock and the Marzke-Wheeler clock, both mentioned as such in the original EPS article (ref).

Sometimes, a weaker notion of 'clock' as merely a means to chronologically order events on a particle world line suffices. Such a device is know as a pre-clock.

See also

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

Notes

1. ^ The sections 'Comments & criticisms' and 'Further developments' essentially follow the account by Schelb (Schelb 1997, pp. 85–90).

1. ^ These are axioms D1 and D2 in the original EPS formulation (Ehlers, Pirani & Schild 1972, pp. 70–71).

1. ^ At that point, a fully rigorous formalization had not yet been achieved, as the authors also point out (Ehlers, Pirani & Schild 1972, p. 69); but see Meister.

1. ^ These are axioms L1 and L2 in the original EPS formulation (Ehlers, Pirani & Schild 1972, pp. 72–74).

References

• Castagnino, Mario.A. (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.

• Audretsch, J. (1990). "The Establishing space-time geometry by means of light rays and particle geometry". Wiss. Zeitschrift der Universität jena, Naturwissenschaftliche Reihe. 39 (1): 9–12. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Coleman, R.A. (1980). "Jet bundles and path structures". Journal of Mathematical Physics. 21 (6): 1340–1351. doi:10.1063/1.524598. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Coleman, R.A. (1981). "Spacetime G structures and their prolongations". Journal of Mathematical Physics. 22 (12): 2598–2611. doi:10.1063/1.524838. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Coleman, R.A. (1984). "Constraint of the nature of inertial motion arising from the universality of free fall and the conformal causal structure of space-time". Journal of Mathematical Physics. 25 (12): 3513–3526. doi:10.1063/1.526122. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• 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

• Ehlers, Jürgen (1973), "The Nature and Structure of Spacetime", in Mehra, J. (ed.), The Physicist's Conception of Nature., Dordrecht: D. Reidel Publishing Company, pp. 71–91

• Ehlers, J. (1973). "Geometry in a Manifold with Projective Structure". Commun. Math. Phys. 32 (2). Springer Verlag: 119–146. doi:10.1007/BF01645651. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

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• Marzke, Robert F.; Wheeler, John A. (1964), "Gravitation as geometry-I: The geometry of space-time and the geometrodynamical standard meter", in Chiu, M.-Y.; Hoffmann, W.F. (eds.), Gravitation and Relativity, New York, Amsterdam: W.A. Benjamin, pp. 40–64

• Meister, Ralf (1990), Eine Neuformulierung der EPS Axiomatik, Paderborn: Paderborn University (Master's Thesis)

• Meister, Ralf (2004), A structural analysis of the Ehlers-Pirani-Schild space-time theory (PDF), Bielefeld: Zentr. für Interdisziplinäre Forschung, Bielefeld University

• Nash, Charles; Sen, Siddharta (1983), Topology and Geometry for Physicists, San Diego, …: Academic Press

• Perlick, Volker (1983), Weyl-Mannigfaltigkeiten als Raum-Zeit-Modelle, Berlin: Technische Universität Berlin (Master's Thesis)

• Perlick, Volker (1987). "Characterization of Standard Clocks by Means of Light Rays and Freely Falling Particles". General Relativity and Gravitation. 19 (11): 1059–1073. doi:10.1007/BF00759142.

• 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

• Schelb, Udo (November 1996). "Distinguishability of Weyl- from Lorentz-spacetimes by classical physical means". General Relativity and Gravitation. 28 (11). Springer Netherlands: 1321–1334. doi:10.1007/BF02109524.

• Schelb, Udo (1997), Zur physikalischen Begründung der Raum-Zeit-Geometrie, Paderborn: Paderborn University (Habilitation Thesis)

• Schelb, Udo (August 1996). "Establishment of the Riemannian structure of space-time by classical means". International Journal of Theoretical Physics. 35 (8). Springer Netherlands: 1767–1788. doi:10.1007/BF02302270.

• Schröter, Joachim (1988). "An axiomatic basis for space-time theory, Part I: Construction of a causal space with coordinates". Reports on Mathematical Physics. 26: 303–333. doi:10.1016/0034-4877(88)90012-2. {{cite journal}}: Check |authorlink= value (help); External link in |authorlink= (help)

• Schröter, Joachim (1992). "An axiomatic basis for space-time theory, Part II: Construction of a C°-manifold". Reports on Mathematical Physics. 31: 5–27. doi:10.1016/0034-4877(92)90003-J. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Synge, John L. (1962), "Relativity Based on Chronometry", in O'Raifertaigh, L. (ed.), Recent Developments in General Relativity. Infeld Festschrift., Warszawa: Pergamon Press and Polish Scientific Publishers, pp. 441–448

• 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

• von Helmholtz, H. (1868). "Über die Tatsachen welche der Geometrie zur Grunde liegen". Nachr. Ges. Wiss. Göttingen.

• Woodhouse, N.M.J. (1973). "The differentiabela nd causal structures of space-time". Journal of Mathematical Physics. 14 (4): 495–501. doi:10.1063/1.1666344.