User:Marc Goossens/Schröter-Schelb Spacetime

Schröter-Schelb spacetime^{[note 1]} ^{[note 2]} or SS-ST in short, is an axiomatic development of the spacetime model underlying General Relativity, the Lorentzian manifold. It is one of several proposals to bring clearer physical motivation to and understanding of this mathematical structure in an axiomatic way.

SS-ST was developed during the 1980's and 1990's by the German physicists Joachim Schröter and Udo Schelb, at Paderborn University ^[1] ^[2] and published in full by Schelb ^[3] in 1997. In contrast to the older top-down Ehlers-Pirani-Schild proposal, Schröter-Schelb axiomatics works bottom-up, starting from primitive physical concepts.

Motivation

Mathematically, the canvas underlying the general theory of relativity takes the form of a Lorentzian manifold. This structure from differential geometry is a continuum of spacetime points (events), which can locally be consistently described by a set of 4 coordinates. At each event a mathematical device, the Lorentzian metric (tensor) or 'metric field' distinguishes light signals and particles, singling out their allowable worldline trajectories. In doing so, this 'metric field' also encodes the influence of gravitation on both particles and light.

This 4-dimensional manifold and its metric field are specialized mathematical concepts. As such they have no immediate physical interpretation or motivation. The particle and light worldlines to which they give rise do possess a clear physical meaning, but mahematically these are derived concepts. Therefore, one should like obtain a clearer physical insight of the meaning of the prior mathematical entity, the metric, itself, and ascertain to what extent we may take the model of a 4-dimensional continuum for granted.

Schröter and Schelb propose to tackle this by reconstructing the Lorentzian manifold, starting from a handful of primitive concepts, reality points, elementary events and signals. In the course of the construction, fastest signals, clocks and directions complete this set of primitives. For each, the physical meaning is described explicitly.^{[note 3]}

Survey of the Schröter-Schelb spacetime (SS-ST) construction

Pretopological spacetime

Rpoints, events, signals

Fastest signals

Clocks

Directions

Topological spacetime

The continuum of events

Radar charts

The differentiable spacetime manifold

Differentiable RP worldlines

Switching charts

Differentiability in all directions

The metric structure of spacetime

Some characteristics of the construction

In the stepwise development summarized above, the originators develop and explore each subsequent refinement of the theory in its own right. This clarifies the precise role and extent of each of the assumptions (axioms).
Indeed, the 1997 Schelb memoir painstakingly erects each of these stages of completion as a separate physical theory, in the strict meta-theoretical sense of the structuralist methodology of Günther Ludwig^[4].

In the same publication, Schelb points out some further aspects of the SS-ST approach to the foundations of spacetime geometry^[5]:

SS-ST sets out to introduce all necessary concepts needed with a direct, operational physical interpretation.
The construction makes no use of (even rudiments of) other physical theories. In particular, no notion of quantum mechanics or atomic clocks is employed in order to define 'good' (i.e. sufficiently regular etc.) clocks.
Starting from scratch, SS-ST also makes no prior mathematical assumptions beyond set theory and the real numbers. Mathematical choices as to whether the metric or the connection is to be taken as the primary notion, or whether a geometry with torsion should be allowed, are avoided: the resulting Lorentzian manifold arises directly from physically motivated axioms.
The SS-ST construction uses or recovers several concepts familiar from spacetime research such as signals, causal structure, the Alexandrov topology etc. These established concepts had mostly been introduced starting from the full structure of a Lorentzian manifold. In contrast, in SS-ST they are derived early on in the process of constructing mathematical spacetime from "first concepts". Causal structures, for example, are usually defined by classifying tangent vectors to curves using light cones, all of which are concepts derived from the manifold and metric structures. SS-ST gets by using properties of signals only. The derived causal structure is subsequently used to build the Alexandrov topology on the set of events.).

Refs used in notes: ^[6] .

Notes

^ Different axiomatizations may lead to slightly different mathematical models. For this reason, they are usefully labeled by the axiomatic system termed after its originators. In the case of SS-ST, the resulting manifold is causal from the outset.
^ Named spacetime models like Schwarzschild, Robertson-Walker, Einstein, Taub-NUT, Kerr, etc., typically refer to a Lorentzian manifold with a given, specified form of the metric tensor. In contrast, foundational spacetime models such as SS-ST may lead to a generalisation or restriction of the Lorentzian manifold itself, leaving open the form of the metric.
^ In a hierarchy of physical theories, spacetime models are among the most basic, as they provide the backdrop for practically all other physical theories, including mechanics, thermodynamics, etc. In other words, from a methodological point of view, a spacetime theory should preferably start from primitive concepts which are directly accessible to human observation, and which may be directly interpreted (without needing complicated heuristics or even prior physical theories, such as quantum mechanics.

Short references

^ Schröter 1988.
^ Schröter 1992.
^ Schelb 1997.
^ Schmidt 2002, p. section 4.2.
^ Schelb 1997, p. 96-97.
^ Schelb 1997, p. 108.

References

Schröter, Joachim (1988). "An Axiomatic Basis of Space-Time Theory. Part I: Construction of a Causal Space with Coordinates". Reports on Mathematical Physics. 26 (3): 303–333. doi:10.1016/0034-4877(88)90012-2.

Schröter, Joachim (1992). "An Axiomatic Basis of Space-Time Theory. Part II: Construction of a C⁰ Manifold". Reports on Mathematical Physics. 31 (31): 5–27. doi:10.1016/0034-4877(92)90003-J. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

Schelb, Udo (August 1997), Zur Physikalischen Begründung der Raum-Zeit-Geometrie, Paderborn: Paderborn University (habiliation thesis){{citation}}: CS1 maint: date and year (link)

Schmidt, Heinz-Jürgen (November 24, 2002; rev. March 5, 2008). "Structuralism in Physics". Stanford Encyclopaedia of Philosphy. {{cite web}}: Check date values in: |date= (help)

[why-st-1] Different axiomatizations may lead to slightly different mathematical models. For this reason, they are usefully labeled by the axiomatic system termed after its originators. In the case of SS-ST, the resulting manifold is causal from the outset.

[why-st2-2] Named spacetime models like Schwarzschild, Robertson-Walker, Einstein, Taub-NUT, Kerr, etc., typically refer to a Lorentzian manifold with a given, specified form of the metric tensor. In contrast, foundational spacetime models such as SS-ST may lead to a generalisation or restriction of the Lorentzian manifold itself, leaving open the form of the metric.

[ST-primitives-6] In a hierarchy of physical theories, spacetime models are among the most basic, as they provide the backdrop for practically all other physical theories, including mechanics, thermodynamics, etc. In other words, from a methodological point of view, a spacetime theory should preferably start from primitive concepts which are directly accessible to human observation, and which may be directly interpreted (without needing complicated heuristics or even prior physical theories, such as quantum mechanics.

[FOOTNOTESchröter1988-3] Schröter 1988.

[FOOTNOTESchröter1992-4] Schröter 1992.

[FOOTNOTESchelb1997-5] Schelb 1997.

[FOOTNOTESchmidt2002section_4.2-7] Schmidt 2002, p. section 4.2.

[FOOTNOTESchelb199796-97-8] Schelb 1997, p. 96-97.

[FOOTNOTESchelb1997108-9] Schelb 1997, p. 108.

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