# Is Planck time the causality rate

## Geometry of the expanding universe

### Research Report 2009 - Max Planck Institute for Gravitational Physics

Quantum Gravity and Unified Theories (Prof. Dr. Hermann Nicolai)

MPI for Gravitational Physics, Golm

Predicting the future is the core task of any physical theory. In order to achieve this, equations have to be formulated whose mathematical structure makes it possible to calculate the future state of a system from given initial conditions. At a sufficiently abstract level, this also applies to quantum theories, in which, due to fundamental restrictions, only probabilities for the occurrence of certain measurement results can be calculated. Remarkably, the elementary requirement that a theory should lead to unambiguous predictions restricts the mathematical structure of space-time not to the space-time geometry currently on which all fundamental physical theories are based, but to a much larger class of geometries. The research focus of our group is the investigation of this entire class of spacetime geometries, which on the one hand enable a consistent movement of matter and on the other hand are themselves described by an unambiguous gravitational dynamics.

A few years ago, the theoretical observations to be made in this context would only have been of mathematical-structural interest. A generalization of the geometrical framework on which the general relativity theory and quantum theories of matter on curved space-time are based should have been rightly regarded as not necessary to explain the observable phenomena. Today, however, this assessment can be questioned from both a theoretical and an experimental point of view.

On the theoretical side, the massive efforts to formulate a theory of quantum gravity have led to surprising insights. Once formulated, a quantum theory of space-time structure replaces the classical theory both quantitatively and conceptually. Special attention will then be given to the circumstances under which a classical description of space-time can be extracted from quantum theory and what form it then takes. It is fair to say that a complete theory of quantum gravity has not yet been presented. Nevertheless, it is already clear today that a purely metric geometry (i.e. a geometry that contains just enough information to measure lengths and angles) will not necessarily cover the entire space-time structure even in the classic limit case. In order to understand this fact in this weak formulation, it is of course sufficient to consider a serious candidate for a quantum gravity theory which, under certain conditions, predicts such a generalized spacetime geometry. Indeed, this is the case in string theory, where a two-form field and a scalar field appear alongside the metric to describe the emergent effective classical geometry. Such generalized geometries as appear in string theory have consequently received intense attention. But if one takes into account the possibility that string theory might not assert itself as a quantum theory of gravitation despite all its promising properties, one is thrown back on the study of all consistent classical geometries, as characterized above. So this is the theoretical motivation of our investigations under the assumption that physical spacetime is actually governed by a quantum theory on a fundamental level.

But physics must, assuming a solid theoretical concept formation and language, deal with phenomena that are in principle or actually accessible to observation. This is where it differs from the humanities mathematics. Indeed, astrophysical observations made by orders of magnitude by space telescope have brought to light an astonishing conclusion. Interpreted within the framework of general relativity, the data collected over the last 15 years suggest that the universe is not only expanding (as theoretically predicted), but that this expansion is progressing faster and faster. How dramatic this accelerated expansion affects our understanding of the world becomes immediately clear when one considers it against the background of the Standard Model of elementary particle physics, the most successful theory of physics in earthly accelerator experiments. Because to explain the observed accelerated expansion of the universe, one has to assume that a fabulous 96 percent of all energy and matter in the universe are of completely unknown "dark" nature, with exotic properties such as negative pressure, and not measurably interacting with standard model matter. This interpretation is a logical possibility. It represents an extreme insofar as it is based on the assumption that the general theory of relativity is completely correct and particle physics is only four percent, to put it deliberately in a striking manner. A more sober and fairer assessment of the observational data would be the insight that the accelerated expansion of the universe indicates that we do not understand fundamental aspects of gravity, particle physics, or probably both. In fact, the close formal interweaving of our theories of matter and space-time geometry speaks in favor of the last of the three variants: everything we infer about the structure of space-time we know from the observation of matter, and all realistic models of matter currently require a basis for their mathematical formulation lying spacetime. How exactly a spacetime geometry has to look like so that the related theories of gravity and elementary particle theory explain an accelerated expansion of the universe without further ad-hoc assumptions, and whether this is even possible is not obvious. Again we are thrown back to the necessity of examining all classical spacetime geometries beyond all too narrow assumptions that at least meet the condition mentioned at the beginning of being able to be based on a predictive theory.

Remarkably, one can actually formulate such a general theory of classical spacetime structures. On the one hand, these go far beyond the metric geometries on which the general theory of relativity is based. On the other hand, and this is probably the surprise, the criterion of predictivity sufficiently restricts the conceivable structures. Indeed, one can formulate a general theory of classical spacetime geometries, but only because of a remarkable interplay between the theory of hyperbolic polynomials and convex analysis on the one hand and algebraic geometry on the other.

Specifically, these elements necessarily come into play as follows. If we first consider a geometry defined by any tensor, we know from the theory of partial differential equations that a field theory can only have a well-posed initial value problem on such a background if a polynomial defined in a certain way from the factors of the highest derivative terms is hyperbolic. In the familiar case of Maxwell's field equations and a metric geometry, the hyperbolicity of said polynomial is given if and only if the metric has Lorentz's signature. Hyperbolicity is the generalization of Lorentz's signature condition to any geometry. In fact, all concepts relating to the causality properties of spacetime and the coupling of point particles can be conceptually extended to generalized geometries. The hyperbolicity alone then still implies, for example, that the massive impulses form a convex cone. The knowledge that there can be no classical spacetime geometry in which photons are unstable is based on this important property. So while all constructions of Lorentz's differential geometry generalize due to hyperbolicity, this requires the use of much more sophisticated mathematics than in the metric case. The knowledge gained in this way also sheds more detailed light on the conceptual structure of spacetime geometry in the familiar case of metric geometry. In particular, it becomes clear that from a higher point of view, all too direct attempts at generalizing space-time geometry to more complex structures must fail. A key insight is that the geometry only has to assume a polynomial structure in momentum space.

The dynamic theory of generalized spacetime is then obtained by finding a representation of the deformation algebra of hypersurfaces, in close analogy to how it was shown in the metric case by Wheeler's school. The canonical theory of gravity that follows from this has a well-posed initial value problem by construction, so it is itself predictive again. If one then demands that classical electrodynamics should be possible in such a space-time, one obtains an excellent dynamics of a so-called area-metric geometry [1]. This result is the only consistent starting point for the explanation of observable phenomena [2,3] in the context of a classical theory of gravity. To complete this program, the consequences of these insights for the possible theories of matter on such spacetime are currently being investigated. Should the resulting overall theory of space-time and matter then contradict observations, it would, however, be shown that there can be no classical extension of the general theory of relativity that is predictive.

It is testimony to the power of the modern mathematical apparatus available to us that, despite only very little reliable information regarding conclusions to be drawn from quantum gravity or the observation of an accelerated expanding universe, definitive and far-reaching statements about the structure of the conceivable classical spacetime can be made meet are.

**Original publications**

**76**, 101501 (2007).

**26**, 035024 (2009).

**79**, 124025 (2009).

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