International Society for the Advanced Study of Spacetime

Spacetime or space?

Oliver Pooley

My question is related to Vesselin Petkov's Relativity and the Dimensionality of the World. He asks whether the spacetime of special relativity is a way of representing a fundamentally 3-dimensional world evolving in time, or whether our world is a world of fundamentally 4-dimensional spatiotemporally extended objects. This is a question asked within the standard mathematical framework of special relativity. It is a question about the proper interpretation of that framework.

But the question whether space or spacetime is fundamental also arises in a different context: the quantization of general relativity. The starting point of the canonical quantization programme is the casting of general relativity into a 3+1 Hamiltonian form, as a theory about the evolution of the geometry of 3-dimensional space over time (Arnowitt, Deser and Misner 1962). This formalism is not by itself incompatible with the view that spacetime is fundamental. It merely shows that one can completely characterize the 4-dimensional geometry of the world in terms of the variation of the geometry a family of 3-dimensional hypersurfaces through spacetime. The arbitrariness of which family of hypersurfaces is chosen can even be taken to support the view that spacetime, not space, is the ultimate reality.

But Julian Barbour has advocated an interpretation of classical general relativity according to which 3-space, rather than spacetime, is the fundamental entity. His interpretation is partly motivated by two very suggestive theoretical results. First, in work he calls ``relativity without relativity'', he and collaborators have shown that only a very few actions from a large class of Hamiltonian theories that are natural from a 3-space point of view turn out to be consistent. The consistent actions include those which allow the construction of a 4-dimensional spacetime geometry. In other words, mathematical consistency alone (almost) gives you an effective spacetime geometry from 3-space principles (Barbour et al. 2002; Anderson 2003).

The second result involves formulating geometrodynamics on a configuration space of conformal three-geometries, rather than the standard configuration space of Riemannian 3-geometries (Anderson et al. 2003; Anderson et al. 2004). In such theories, one has a uniquely preferred foliation of the resulting spacetimes. The availability of such a foliation would seem to greatly enhance the viability of an interpretation that takes a 3-space geometrodynamical formulation of a theory as a better reflection of fundamental ontology than a spacetime formulation.

Barbour has offered a sketch of an interpretation of quantum general relativity that is in line with these classical results (Barbour 1994). There are other interpretations that are compatible with viewing space as more fundamental than spacetime (Page and Wootters 1983). However, others are pursuing approaches to quantum gravity that take spacetime's causal structure as fundamental (Markopoulou and Smolin 1997; Reisenberger and Rovelli 1997; Sorkin 1997). So the question is which will quantum gravity show to be more fundamental: space or spacetime?


R. Arnowitt, S. Deser, and C. Misner (1962), `The Dynamics of General Relativity' in `Gravitation: an Introduction to Current Research', ed. L. Witten, (Wiley, New York, 1962).

J. Barbour, B. Foster, and N. O'Murchadha (2002), `Relativity without relativity', Classical and Quantum Gravity 19: 3217--48; gr-qc/0012089

E. Anderson (2003), `Variations on the Seventh Route to Relativity', Physical Review D 68: 104001; gr-qc/0302035.

E. Anderson, J. Barbour, B. Foster, N. O'Murchadha (2003), `Scale-Invariant Gravity: Geometrodynamics' , Classical and Quantum Gravity 20: 1571; gr-qc/0211022.

E. Anderson, J. B. Barbour, B. Foster, B. Kelleher, N. O'Murchadha (2004), `A first-principles derivation of York scaling and the Lichnerowicz-York equation'; gr-qc/0404099.

J. B. Barbour (1994), `The Timelessness Of Quantum Gravity. II: The Appearance Of Dynamics In Static Configurations', Classical and Quantum Gravity 11: 2875-97.

D. Page and W. K. Wootters (1983), `Evolution Without Evolution: Dynamics Described by Stationary Observables', Phys. Rev. D 27: 2885.

F. Markopoulou and L. Smolin (1997), `Causal Evolution of Spin Networks', Nuclear Physics B 508: 409--30; gr-qc/9702025.

M. Reisenberger and C. Rovelli (1997), `Sum over Surfaces form of Loop Quantum Gravity', Physical Review D 56: 3490--508; gr-qc/9612035.

R. D. Sorkin (1997), `Forks in the Road, on the Way to Quantum Gravity', International Journal of Theoretical Physics 36: 2759--81; gr-qc/9706002.