User:Marc Goossens/Foundations of spacetime

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

The study of spacetime foundations investigates the rationale of physical theories unifying space and time, that provide the backdrop for practically all other physical theories, especially those based on the concepts of the special or general theory of relativity. This research aims to provide a better and in some appropriate sense 'more fundamental' justification of the familiar but intuitively not so accessible structures such as Minkowski (as used in Special Relativity) or Lorentzian (as used in General Relativity) spacetime. Via such a more robust motivation, a better understanding of both the physics as well as the mathematics involved is also acquired.

In its to date most prominent appearance, spacetime foundations research addresses the 'Lorentzian postulate of General Relativity' - i.e. the claim that any spacetime model is some specific form of a causal Lorentz (or pseudo-Riemannian) space (or 'L4') – which by itself is felt to be too comprehensive to allow a clear understanding. Instead, one hopes to replace it by a more detailed series of statements by way of alternative postulates, intended to be more compelling and insightful than the original one, while still leading to the familiar L4. Often, the alternative proposal will also regard the notions of spacetime metric and curvature as less fundamental than for instance the more readily understandable light ray and particle world lines. EPS spacetime, first suggested by Ehlers, Pirani and Schild in 1972, constitutes a notable and paradigmatic example of this.

As the term 'foundations' indicates, the preferred approach in this field will be broadly 'axiomatic', typically in a Hilbertian sense (see David Hilbert and in particular Hilbert's 6th problem), though the degree of rigor actually applied varies. In order to achieve the objective of improving physical understanding, the strongest results are obtained by formulating axioms that possess a direct or at least constructive physical meaning. It is specifically this body of research which also contributes an important chapter, including first-rate examples, to the general understanding of the foundations of physics.

Closely connected to these investigations, and also lying within the scope of spacetime foundations, is the issue of certain possible generalizations of the mathematical model of spacetime itself, and the way it can be mathematically represented in a physical theory (such as Weyl space, for example). In the same vein, a thorough understanding of the underlying principles of spacetime, may also help to explain why it is that the 'smooth manifold' picture of spacetime may well break down at tiny, e.g. Planck length scale, and suggest which shortcomings would have to be addressed. Beyond this, the further issue of devising a physically transparent axiomatics for the second postulate of General Relativity, the Einstein field equations, remains an open problem.

Survey of spacetime foundations research

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

Conventional wisdom regarding theory 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.

Standard formulation of General Relativity – General relativity rests on two pillars: the first is a choice for a specific choice of a geometric structure to describe how gravity exerts its influence on the space of events; the second is a condition that ties this gravitational effect to the distribution of matter and energy in that same space. This condition is none other than the Einstein field equation. The event space, together with its chosen structure is what is commonly called spacetime.

First postulate of General Relativity

GR1

Riemannian hypothesis

Riemannian or spacetime postulate of General Relativity
Spacetime continuum = Lorentz manifold – The set M of spacetime events is a 4-dimensional smooth manifold that is both orientable and time orientable, with underlying Hausdorff topology that is connected and paracompact.
Gravitation = metric field – On M, a nondegenerate metric tensor with Lorentz signature determines the causal (light cone) structure, and via its metric connection also the null (light) geodesics and the timelike geodesic paths of freely moving 'gravitational monopole' objects.

As a starting point for a physical theory on the 'primary' concept of spacetime, this statement is quite intricate.

Aiming for a better understanding of GR1 – Together with the second postulate, the Einstein field equation, GR1 is the established standard model for representing spacetime-including-gravity in physics. The following section illustrates that it shows some deficits that to some degree preclude a transparent physical understanding. It is this weak spot which spacetime foundations theory wants to address.

GR1

Standard general relativity is not 'ultimate' – Though well established, general relativity today cannot be regarded as 'final'. The search for generalizations is mainly driven by issues like the problem of singularities ^[1], the wish for a broader unification of physics ("can other interactions than gravity also be included in a geometry?") ^[1], and the view that physics needs a quantum theory of gravity ^[1].

Raison d'être for spacetime foundations research

Tasks for spacetime foundations

Challenge 1
Can the ubiquitous 'L4'* be built up constructed in clearly motivated, explicit steps?*

Challenge 2
Can such a construction be achieved on the basis of simpler and physically more accessible primitive concepts, thus corroborating the mathematics?

Challenge 3
Can this construction be made imply the restriction to 'reasonable' spacetimes from the outset? Can it physically clarify the Lorentzian reduction (or indicate plausible generalizations)?

Always with due consideration for the triumphs of classical General Relativity (both theoretically and empirically) (ref to WP), it is nonetheless worthwhile to look at the shortcomings regarding the motivation and understanding of its GR1 starting point a bit more closely. From this, one can then glance the challenges any attempt at improving the foundations of spacetime theory ought to address. These are listed in the table on the right by way of tentative "program' for such an undertaking. In one way or another they all boil down to the need to bring the mathematical arguments more directly in relation with the physics they are presumed to describe.

Unclear motivation of chosen mathematical structure – The traditional rationale behind this choice is largely historical-heuristic and, as Schelb notes, mostly negative (Schelb 1997, p. 14): one indicates those assumptions in prior ST theories which are somehow 'too strong' (as candidates for a weaker, 'more general' formulation). Moreover, the heuristics employ unsatisfactory tools such as grids of rotating disks, supposedly rigid measuring rods, which are unpractical or unrealistic in an astrophysical or cosmological setting, or of atomic clocks, which are foreign to the intended geometric construction. (See table: challenge 1.)

“

No theory is complete if it postulates that atomic clocks measure the interval dτ = (-g_αβdx^α dx^β)^1/2 constructed from a particular metric. Atomic clocks are complex systems whose behaviour must be calculated from the fundamental laws of quantum theory and electromagnetism.

”

— Thorne, Misner, Wheeler (Thorne, Misner & Wheeler 1973, p. 1067)

Mathematical choices in GR1 not unequivocal – This issue is mostly a consequence of the preceding one. Despite the fact that the General Theory of Relativity is so familiar and has gathered impressive successes, its GR1 claims described above are not undisputed: almost without exception, each ingredient of the GR1 postulate has been tweaked or challenged in some publication or other. The alternatives considered range from entirely non-Riemannian geometries over non-Hausdorff topologies to discrete models (see below). This goes to show that its physical underpinning is not fully transparent and open to debate. (See table: challenge 2.)

“

In general relativity one is accustomed to begin with the assumptions that space-time is a fairly smooth 4-dimensional manifold carrying a normal hyperbolic Riemannian structure – the metric tensor g_ab whose geodesics represent the paths of light rays and free particles (…). These (GR1) assumptions have been challenged many times, and in a variety of ways.

”

— Felix Pirani (Pirani 1973, p. 67)

Complex mathematical structure is too slack here, so tight there – The GR1 postulate given above envelops a broad field of advanced mathematics. Expanding on Weyl's observations (Weyl 1921), (Weyl 1923), Ehlers, Pirani and Schild enumerate the "topological, differential, conformal, projective, affine and metric structures", which GR1 comprises (Ehlers, Pirani & Schild 1972, pp. 63–84). These authors raise the question of what the physical content of these structures and their mutual and very special 'alignment' according to GR1 actually is, and set out to try and dig it out, or indeed reconstruct it, starting from more primitive concepts (ref below + EPS). So one finds that the math of GR1 on the one hand exhibits generous freedom (such as regarding the choice of topology, which is not further constrained by the field equations either), while on the other hand possessing remarkable, seemingly arbitrary 'coincidence' between various geometrical structures. This begs the question as to the physical meaning of some proposal exploiting this freedom ^[1], or indeed about the reasons behind the ultimate 'reduction' of the geometry to the Lorentzian case. (See table: challenge 3.)

Classification of spacetime theories' rationale and scope

The following classification of the motivational approaches for the choice of a spacetime structure is also due to Udo Schelb (Schelb 1997, pp. 49–59).

Philosophical motivation – Philosophers have shown more interest in abstract epistemological questions about space and time or its geometry, than any corresponding physical or empirical basis for it. This with the exception of Hans Reichenbach, who published a historic and landmark (if physically imperfect) axiomatization (Reichenbach 1924).
Mathematical motivation – In addition to their contributions on the global structures of (general relativistic)spacetime, mathematical geometers have mainly looked for elegant axiomatizations of the Minkowski geometry of special relativity (with contributions from A.A. Robb, Carathéodory, W. Noll, S. Basri, P. Suppes, J.W. Schutz and others), but the selection of constructive axioms in a physical sense has not been their focus.

Roger Penrose

Penrose 1968

Intuitive motivations -- Some authors all but identify mathematical structures with physical space(time) 'itself'. In his survey, Schelb lists Hermann Weyl (Weyl 1931, p. 49) and Erwin Schrödinger (Schrödinger 1987) as examples. Interestingly, this leads them to different preferences (of a projective Weyl structure for Weyl; of a geometry with torsion in Schrödinger's case).
Didactical "textbook" motivations - Adler Bazin Schiffer textbook S 2: finsler and weyl geometries are mentioned
Physical motivation – Apart from Einstein (Einstein 1993), also H.P. Robertson (Robertson 1964) 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. In Einstein's view, only the resulting "sum" (G) + (P) of geometry (G) and its physical principles (P) is subject to empirical verification.

Explicit motivations S 2.3

3-D space motivation

GRT motivations textbooks

For more details, see e.g. (Schelb 1997, pp. 12–47).

Historical overwiew

(Schelb 1997, pp. 55–94).

Axiomatic / constructive SST's for GR

"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. The presence of any structure that has no verifiable link to physical phenomena may be considered as an indication of a defective model… On the other hand, a good understanding of the fundamental structures underlying a theory may suggest new and fruitful generalizations."

Andrzej Trautman (Trautman 1972, p. 85)

In the light of the cited deficits of general relativity—not of its results, but of the formulation of its principles—spacetime theory (SST) sets itself the task to investigate and answer these challenges in a rigorous and systematic way. Andrzej Trautman has outlined its program in his 1972 paper in honor of John Synge, as quoted in the box on the right.

Mathematically, this amounts largely to Castagnino's 'inverse problem' or search for 'SST' as proposed by Schröter and Schelb Precursors Synge

EPS spacetime

epitomize original

EPS enhancements

inertial-class problem Coleman-Korté Meister S-S criticism

Schröter-Schelb spacetime

Generalised SST

Einstein-Cartan Finsler Weyl complex topology signature

Various GRT models Deriving the Schwarzschild solution Reissner-Nordström metric (charged, non-rotating solution) Kerr metric (uncharged, rotating solution) Kerr-Newman metric (charged, rotating solution) BKL singularity (interior solution) Black hole, a general review Schwarzschild coordinates Kruskal-Szekeres coordinates Eddington-Finkelstein coordinates http://en.wikipedia.org/wiki/Pp-wave_spacetime

Notes

^ The Einstein field equation forming the second postulate.
^ Even combined with the field equations, GR1 allows spacetime models with 'nontrivial topological structure', leading for instance to causality violating closed world lines (Novikov 1995).
^ Singularities. In the period 1965-1972, Hawking, Penrose and Geroch proved that the existence of 'singularities' in GR1 spacetime models is all but inevitable. At these singularities, the parameters of the Lorentz manifold model take on infinite values, so that it becomes physically useless.
^ Unification of physics. The hunt for geometrical theories that incorporate other fundamental forces (see classical unified field theories and unified field theory), such as the electroweak interaction, nowadays focuses on gauge theories based on the mathematical structure of fiber bundles. Such a theory that includes gravity has remained elusive.
^ Quantum gravity. Physicists actively explore various ways of formulating gravity as a quantum theory. It is hoped that this will also shed new light on the issues of singularities as well as further unification of physics. See for instance (Hawking & Penrose 1996) but also (Lebrun 1998) harv error: multiple targets (2×): CITEREFLebrun1998 (help), a review of the Hawing & Penrose lectures.
^ See Schelb p 52-53, where he refers to (Grünbaum 1974), (Earman 1989), (Friedman 1983), (Sklar 1974).
^ For more on the space problem and the views of H. Weyl on general relativity, see also (Hawkins 1998), and G-structures.
^ This is precisely the ambition of SST as put forward by [[]Schröter]]; see MG wikiref.
^ These are themselves related to the solution of Hilbert's fifth problem by Hidehiko Yamabe.

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