User:Marc Goossens/The Ehlers-Pirani-Schild Construction

Meister-EPS construction

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 in a series of publications (REF).

EPS spacetime (short: 'EPS') is 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. It is a classic example of physically constructive (spacetime) axiomatics, and has received considerable further attention since its first publication (e.g. in studies by (Woodhouse), (Perlick), (Meister) and (Schröter)).

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 ray. 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).

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.

Survey of the EPS construction
The original formulation by Ehlers, Pirani and Schild establishes the familiar L4 spacetime of General Relativity in 5 major parts:  Events, particles, light rays - The starting ingredients laid out on the table are few. First an as yet featureless set ('space') of points, the backdrop of spacetime events is given. This is the spacetime set.  On this substrate, EPS represents a particle (history) by a set of such points, which will form its world line, and with which the particle is identified. The same holds for a light ray. Particles and light rays thus distinguish two particular (but at this point further unspecified) 'sets of subsets' of the spacetime events.

Events, particles and light rays form the primitive physical notions of the EPS model. As the original authors indicate (ref), particles represent the motions of objects in a classical (non-quantum) sense. These can be billiard balls, artificial satellites, possibly even planets or stars, provided their extent and structure can be ignored, with respect to the specific situation considered.

'Light rays'' are the traces produced by sufficiently localized and short flashes of electromagnetic radiation, emitted in a focused direction. A light ray is the line traced by a light flash.

The processes involving particles and light rays are assumed to take place 'unhindered', in an otherwise empty region of spacetime.

Radar mapping of events (coordinate space) - Now light rays are sent out and reflected by particles. By registering departure and arrival times of these signals along (two) "observer" particles, each spacetime event can be operationally assigned 4 real numbers, which are its radar coordinates for this pair of observers.  Postulating that each particle (and light rays?) sweeps out a one-dimensional, smooth "series" of events, and that there are enough suitable pairs of "observer" particles around for "mapping" all events in a consistent way, the spacetime set now receives the make-up of a smooth, 4-dimensional 'coordinate space'. Mathematically speaking, EPS event space becomes a smooth manifold.

As pointed out by the original authors (ref), some rudimentary (notion of) clock carried by an observer is needed in this step. Such a clock assigns real numbers to events along a particle (observer), as needed to establish radar coordinates. It is central to the geometrical approach proposed by EPS, that such a clock may itself be realized by employing only particles and light rays—that is purely by means of elements native to spacetime geometry. This aspect is further discussed below.

Light propagation fixes causal relationship of neighboring events (conformal space) - EPS goes on to assume that the collection of all light rays passing through any given event e, permits (an observer at) e to separate nearby events into those lying in its past, in its future, or its present—the latter lying out of e's causal reach. 

The precise mathematical formulation is given for 'infinitesimally small' vicinities, i.e. in tangent space, making this step in EPS axiomatics less operational. (Later, a formulation of this axiom has been given (ref) with actual world lines, allowing to prove the original axiom given here as a theorem.)

As for the light rays themselves, still infinitesimally, e distinguishes incoming light rays from outgoing ones. These will turn out to be the future and past light half cones at e.

Physically, we have now reached a true spacetime, with a local causal structure to distinguish between spacelike and timelike separation between events. This is nothing but the local validity of special relativity. (Ehlers p. 85) From a mathematical point of view, the manifold of events is now endowed with a 'conformal structure'. (Global causality, i.e. the non-occurrence of closed timelike curves is 'not' ensured, and if desired, needs to be imposed by additional assumptions.)

Independently from these qualitative assumptions on light propagation, attention now returns to particles. Already at this stage, one may put forward the claim that particles exist that chase light rays arbitrarily closely (Weak Compatibility Axiom, or "particles fill the interior of the lightcone"), loosely tying the particle structure to the light structure.

Free falling particles encode 'pure gravity' (projective space) - Spacetime is meant to provide a suitable backdrop for kinematics, and in keeping with Einstein's ideas, should also embody "gravitation". With this in mind, and next to the 'causal structuring' by means of light rays introduced in the previous step, EPS seeks some feature to represent an 'inertial' reference.  To achieve this, the initial notion of particles needs to be refined. It is postulated that there exists a distinguishable subset of particles, the motion of which is determined solely by an event along the particle (world line) and a direction ('initial velocity') at this event. Intuitively, these free-falling particles provide a reference of "no acceleration".

Physically speaking, this particular set of free particle world lines encodes the combined inertial-gravitational 'guiding field' (ref). On the mathematical side, one obtains a path structure, or equivalently a family of affine connections, amounting a projective structure.

Light propagation and free particle motion agree (Weyl space) - Physical experience indicates that the causal (conformal / light) and inertial (projective / particle) structures are somehow 'mutually consistent'. This consistency is anchored into EPS by the (full) Compatibility Axiom. This axiom sharpens the already mentioned weak compatibility by demanding that freely falling particles alone through an event e suffice to reach any event inside e's light cone, and also that they occur with all speeds slower than but arbitrarily close to light.  Aligning in this way the conformal and projective structures introduced previously, spacetime takes on the mathematical form of a Weyl space or Weyl manifold. The compatibility requirement is indeed equivalent to the Weyl compatibility requirement linking a preferred (zero torsion) Weyl connection of the projective equivalence class to any metric g of the conformal equivalence class by a corresponding 1-form.

(some detail: x geodesics are y geodesics)

With (local) causal structure and free particle kinematics both 'gravitationally coupled', Weyl space may be used to formulate (generalized) General Relativity. The question of how to express the Einstein field equations has been investigated (Perlick).

In general, Weyl spacetime will exhibit a second clock effect in addition to the first clock effect (path dependence of time difference between two events) known in traditional Lorentzian General Relativity. If this extra effect occurs, clocks transported along different paths will not only obtain a different time interval, but also 'tick at differing rate', i.e. experience differing 'time speeds".

Whether this phenomenon really happens in nature would be an issue for experiment to settle. Postulating that it does not, leads to the next and final step in EPS.

Chronometric reduction to Lorentzian space L4 - The (supposed) physical deficit of Weyl space is the existence of a second clock effect. Mathematically, it still lacks the preferred metric available in L4. 

To finally obtain L4, the Weyl space indeed needs to be 'reduced' to the more specialized (pseudo-)Riemannian one. At the mathematical level, this is the case if the Weyl connection is the Levi-Civita connection of some conformal metric g. Or equivalently, if its Weyl curvature or track curvature ('Streckenkrümmung') vanishes.

It can be shown that this is equivalent to the physical requirement – as adopted in the original EPS article – that there be no second clock effect. (ref.) 

Some elements of the construction in detail
Some 'feel' for the nature of the EPS arguments and constructions is rendered by the following examples.

Using light rays, a particle A can send messages to B (flash), and receive the echo "reflected" at B (blip). A and B each use their own (possibly irregular) clocks to register the time of flashes and blips. By this procedure, events (from which a message is sent) along A may be related to events at B (at which corresponding messages from A are received).

Now, two observers, one sitting at particle A, the other at particle B, set out to map their spacetime surroundings. Probing these with light rays, they make use of recording flash-blip timings as just described, for jointly assigning numerical coordinates to events. One event, say e, is fully identified by 4 numbers: A pair of flash-blip recordings reflected at e by A, and a second such pair recorded by B, targeting the same e.

Proposing that there are enough such particle-pairs around to map all spacetime points (events) like this in a consistent way, establishes EPS as a 4-dimensional spacetime, with the coordinate numbers arrived at (locally) by physical (clock!) measurement. <li> At an event p along its world line, a particle A may send a light flash in the direction of a second particle B, and register the return at event p' (along A) of the signal reflected from B, the (radar) echo of the emitted light ray. Meanwile, B can also register the event q, at which the 'message' from A arrives.

By means of some (possibly even irregular) clock, A may measure the (pseudo-)time between emission of the flash and reception of the echo.

BETTER:

Let a particle P illuminate a particle Q with a smooth sequence of light ray (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 P. (EPS axiom D1) Similarly, the message blips from P as registered by Q, smoothly relate flash time at P to blip time at Q. (EPS axiom D2)

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" number 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 A and B, and their 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, formed by a consistent patchwork of overlapping coordinate maps obtained in this way. </li>

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Comments and criticisms
<ul> <li>Like L4, to which it is mathematically equivalent [note], EPS actually represents an entire class of 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.) </li>

<li> EPS takes the physical notions of free-falling particles and light rays and qualitative relationships (like incidence) as its starting point. Mathematically, these define two geometric structures: a projective structure $$\mathcal {P}$$ (given by the particles) and a conformal structure (given by the light rays) $$\mathcal {C}$$. </li>

<li> Extending the mathematical axioms by the physical assumption that both these structures are compatible leads to a Weylian space (with an affine structure). Adding a further assumption leads to 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 $$\mathcal {C}$$. 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 convenient for subsequent mathematical analysis and calculation, the metric field and its curvature are farther removed from physical interpretation. </li>

<li> 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 (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. </li>

<li>better than chronometry (pp 64-65 + 69) </li>

<li>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 </li>

<li>circularity of notion of free-falling particles </li>

<li>some gaps in physical motivation </li>

<li>Weyl space </li>

<li> EPS and many subsequent investigations (ref.) show that this structure is the one most readily constructed by geometrical means. </li>

<li>"no-second-clock Riemannian axiom" is not local; Audretsch & Lämmerzahl introduce alternative requirements, 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 p. 100). </li> </ul>

EPS in the context of 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.

The purely mathematical part of EPS spacetime can be formulated as Bourbaki species of structure, and its full physical theory according to the Ludwig scheme (see section 'REF).

EPS in the context of spacetime axiomatizations
Reichenbach, Synge, …

further developments
formalization of EPS according to the Ludwig scheme

Schröter-Schelb spacetime