International Society for the Advanced Study of Spacetime

Relativity and the Dimensionality of the World

Vesselin Petkov

In my view the major open question in both spacetime physics and philosophy of spacetime is the question of the nature (the ontological status) of spacetime:

Is spacetime (Minkowski spacetime or any other relativistic spacetime) nothing more than a four-dimensional (4D) mathematical space which represents an evolving in time three-dimensional (3D) world or a mathematical model of a 4D world with time entirely given as the fourth dimension?
Stated another way, the question is:
What is the dimensionality of the world according to relativity - 3D or 4D?

For physicists the importance of this question is twofold: (i) in view of the multi-dimensional spaces of modern physics it appears natural to address the question of the nature of spacetime first, and (ii) if the macroscopic world is indeed 4D, Minkowski's program - physical laws might find their most perfect expression as relations between worldlines - should be pursued more rigorously.

The importance of the issue of the ontological status of spacetime for philosophers of physics, philosophers, and everyone who wants to have their world view in accordance with science is much greater since a number of fundamental issues look completely differently in a 3D and a 4D world:

  • Change, passage or temporal becoming have their ordinary meaning only in a 3D world: only a 3D body (which preserves its identity in time as a 3D object) can undergo objective change in a sense that it is the same 3D object that changes. In Minkowski world there is no change since the whole history in time of a 3D body is entirely given as the body's 4D worldline (or worldtube). The very concept "worldtube" shows that at different moments of its history the body is not the same 3D object: at any moment the body is represented by a different 3D object - a different 3D cross section of the body's worldtube. What makes the body the same body is the fact that its worldtube retains its identity as a 4D object in spacetime. The different regions of the body's worldtube are different but this is the same type of change we find when we look at different regions of extended 3D bodies (e.g. a computer keyboard). The fact that 3D bodies are extended in space, whereas a body's worldtube is extended in time is insignificant because in spacetime spatial and time dimensions are equally existent; the spacetime signature + - - - (or - - - +) tells us that spatial and time dimensions are different, but it cannot be interpreted to mean that the fourth (time) dimension does not exist as the spatial dimensions do (if the time dimension were not entirely given like the spatial dimensions Minkowski world would not be 4D). It should be specifically stressed that the 4D world of events (Minkowski spacetime) does not contain 3D objects (which may undergo change) by definition - as the concept "event" is defined in relativity as an object, a field point, or a space point at a given moment of time, two events corresponding to the same physical object are in fact two different 3D objects - the physical object at two different moments of its proper time (or, in other words, two different 3D cross-sections of the object's worldtube).

  • The concept of time flow has a completely different meaning in a 3D and a 4D world. In a 3D world we have the ordinary objective and universal flow of time: events are objectively divided into past, present (occurring simultaneously at the moment "now"), and future. In Minkowski world all events are equally existent and therefore are not objectively divided into past, present, and future. In such a 4D world the only meaningful concept of time flow appears to be the one described by Hermann Weyl: "The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life line of my body, does a certain section of this world come to life as a fleeting image in space which continuously changes in time". Not everyone agrees with Weyl on this. However, any suggestions (i) that the consciousness (implicitly defined by Weyl as the entity that makes us self-aware only at the present moment) plays no role in Minkowski spacetime and (ii) that our consciousness exists everywhere along our worldline (and does not crawl up) lead to contradictions with the way we perceive the world.

  • For the first time the concept of consciousness was needed for the interpretation of a physical theory - Minkowski's 4D formulation of relativity (quantum mechanics came after that); the original 3D formulation of relativity does not need consciousness for its interpretation. Therefore the question of the dimensionality of the world is directly related to the issue of consciousness.

  • Free will may exist only in a 3D world. In Minkowski 4D world there is no free will since the entire history of every object is realized and given as the body's worldtube.

  • The issue of conventionality of simultaneity also looks differently in a 3D and in a 4D world. If the world is 3D, simultaneity cannot be conventional. As a 3D world coincides with the present (everything that exists simultaneously at the moment "now") conventionality of simultaneity immediately implies conventionality of what exists which is clearly unacceptable. In Minkowski 4D world simultaneity is unavoidably conventional - as all events of spacetime are equally existent it is really a matter of convention which 3D cross-section of spacetime we will regard as a set of simultaneous events (for instance, as our 3D world).

  • The open question of inertia may also look differently in a 3D and in a 4D world. In a 3D world inertia is what has been for centuries - an outstanding puzzle. In Minkowski 4D world, however, the ordinary 3D bodies are in fact 4D objects - the bodies' worldtubes. If a body moves by inertia (with constant velocity) its worldtube is a straight line. When the body accelerates it resists its acceleration - an inertial force opposes the external force that accelerates the body. As in spacetime the worldtube of an accelerating body is deformed (it is not a straight line) it is quite natural to ask whether the inertial force can be regarded as originating from a four-dimensional stress in the body's worldtube which arises when the worldtube is deformed. If a body's worldtube is a real 4D object the same 4D stress should arise in it whenever it is deformed, i.e. deviated from its geodesic shape in curved spacetime.

The above list demonstrates, I believe, that the issue of the ontological status of spacetime precedes the issues of temporal becoming, flow of time, free will, and conventionality of simultaneity - all these issues directly depend on what the dimensionality of the world is. It is this fact that makes the question of the nature of spacetime the major open question in spacetime physics and philosophy of spacetime.

In my view, the existing experimental evidence clearly supports the 4D view of the world. If we regard existence as absolute and the world as 3D, then none of the kinematic consequences of special relativity are possible. This shows that the kinematic consequences of special relativity are, in fact, manifestations of the four-dimensionality of the world. What is of crucial importance is that not only the consequences of special relativity would be impossible, but the experiments that confirm them would be impossible if the world were 3D. If this turns out to be really the case then it will be special relativity alone that can resolve the issue of the dimensionality of the world at the macroscopic level.

The arguments showing that the kinematic consequences of special relativity are impossible if (i) the existence of the physical objects is regarded as absolute and (ii) the world is assumed to be 3D are outlined in the paper
"Relativity and the Dimensionality of the World" - HTML
"Relativity and the Dimensionality of the World" - PDF