Is there anything else than spacetime?

Question

Can we say that the only thing that exists is the spacetime and everything in it is expressed as curvature?

Answer 1

In a certain narrow sense, I think the answer to your question is believed by most physicists to be an emphatic no. That is, if by curvature you mean the fourth rank, valence $\left(\begin{array}{c}1\\3\end{array}\right)$ tensor $\mathbf{Riemann}$ that appears (through its contracted rank 2 covariant version the Ricci tensor) in the Einstein field equations.

The reason is simply this: curvature is a property of a differentiable (indeed $C^2$) manifold (all second derivatives of the metric tensor exist and are continuous). So the kind of geometric description here is a classical description of a continuum. But the quest for quantum gravity could be summarized as the search for what microscopic behavior leads to a spacetime described - over everyday length scales - by the Einstein field equations. The Einstein field equations and the associated notion of curvature are believed to be a macroscopic approximation to quantum structure - so they may well "fail" in the same sense that fluid mechanics described by the Navier Stokes equation fails at short length scales: in both cases, the modelled object is probably not a continuum in the way that is logically necessary for the equations to be meaningful: fluids too are ultimately made of "chunks" and are not differential manifolds. But, like the Navier Stokes equations, the Einstein equations are fantastically accurate when the length scales are big enough that the manifold model is meaningful.

Having said this, it is possible that geometric ideas may well apply to a future full quantum gravity theory and it may be that certain notions will be expressible as curvatures (in the same way that one can give a geometric description of electromagnetism, where the vector four-potential becomes the connexion co-efficients of the appropriate manifold)

Many researchers have thought along these lines. In the 19th century, Bernhard Riemann and William Kingdon Clifford thought that matter might be "curvature" in space. More recently, Geon theory was proposed in 1955 by John Archibald Wheeler as a model for elementary particles. A geon is a solution of the Einstein field equations where either gravitational waves alone or gravitational waves and electromagnetic waves together confine themselves owing to the curvature that they source in the Einstein field equations. This is a subtle notion since there is no accepted way to localize the energy density of a gravitational field[1]. A rather elegant description of some of the principles underlying the beginnings of the geon idea is given in Chapter 19 "Mass and Angular Momentum of a Gravitating System" and Chapter 20 "Flux Integrals for 4-Momentum and Angular Momentum" of Misner, Thorne and Wheeler, "Gravitation". What is shown here is that, in an asymptotically flat spacetime (i.e. one that looks more and more like Minkowski spacetime the further one gets from the "origin"), to an observer far off from the origin well inside the "flat" region of the asymptotically flat spacetime, any region around the origin filled with arbitrarily complex gravitational waves with or without electromagnetic waves looks exactly like a "particle" defined wholly and only by a gravitational mass parameter and angular momentum because the gravitational fields always converge to the same form as those of a particle with mass and angular momentum at the origin. Wheeler is also well known for wrestling with and deeply investigating the notion of electric charge as arising topologically from "handles" in spacetime.

[1]. Misner Thorne and Wheeler explicitly deny the possibility of gravitational energy localization through the argument that one can always choose freefall co-ordinates to annul the Christoffel symbols at any given point. No Christoffel co-efficients means no gravitational energy density ascribable to that point, so the existence of a localizable "gravitational energy density" would seem to tell against the equivalence principle.

Answer 2

Is there anything else than spacetime?

There's space and energy and fields and waves, but there is no actual spacetime. It isn't what space is, instead it's an abstract thing. See Ben Crowell's answer here. Objects don't move through spacetime. Objects move through space. The Earth is surrounded by space, not spacetime. Light waves move through space, not spacetime. We live in a world of space and motion, not in a world where there is no motion.

Can we say that the only thing that exists is the spacetime

No, see above. Note though that in this 1929 article Einstein described a field as "a state of space". And there's not a lot of difference between a standing wave and a field. On top of that a gravitational field is comprised of energy, and I don't know how to distinguish this from the field or the space.

and everything in it is expressed as curvature?

IMHO it is possible that in future everything will be expressed as curvature, but not as spacetime curvature. Have a google on electromagnetic geometry and check out things like this by Percy Hammond: "We conclude that the field describes the curvature that characterizes the electromagnetic interaction". People talk about curvature of fibre bundles, but if a field is a state of space, then if a field describes a curvature, what we're dealing with is curved space. Note that Einstein described a gravitational field as space that was "neither homogeneous nor isotropic". This is modelled as curved spacetime, but it's inhomogeneous space rather than curved space. The electromagnetic field isn't.