De Sitter Invariant Special Relativity - Introduction

Introduction

De Sitter himself suggested that space-time curvature might not be due solely to gravity but he did not give any mathematical details of how this could be accomplished. In 1968 Henri Bacry and Jean-Marc Lévy-Leblond showed that the de Sitter group was the most general group compatible with isotropy, homogeneity and boost invariance. Later, Freeman Dyson advocated this as an approach to making the mathematical structure of General Relativity more self-evident.

Minkowski's unification of space and time within special relativity replaces the Galilean group of Newtonian mechanics with the Lorentz group. This is called a unification of space and time because the Lorentz group is simple, while the Galilean group is a semi-direct product of rotations and Galilean boosts. This means that the Lorentz group mixes up space and time so that they cannot be disentangled, while the Galilean group treats time as a parameter with different units of measurement than space.

An analogous thing can be made to happen with the ordinary rotation group in three dimensions. If you imagine a nearly-flat world, one in which pancake-like creatures wander around on a pancake flat world, their conventional unit of height might be the micrometre, since that's how high typical structures are in their world, while their x and y axis could be the meter, because that's the size of their body. Such creatures would describe the basic symmetry structure of their world as SO(2), rotations in the x-y plane. Later on, they might discover rotations into the z axis— and in their every-day experience such rotations would always be by an infinitesimal angle, so that these z-rotations would commute with each other.

The rotations into the z-axis would tilt objects by an infinitesimal amount. The tilt in the x-z plane would be one parameter, and the tilt in the y-z plane another. The symmetry group of this pancake world is SO(2) semidirect product with R2, meaning that a two-dimensional rotation plus two extra parameters, the x-tilt and the y-tilt. The reason it is a semidirect product is that, when you rotate, the x-tilt and the y-tilt rotate into each other, since they form a vector and not two scalars. In this world, the difference in height between two objects at the same x, y would be a rotationally invariant quantity unrelated to length and width. The z coordinate is completely separate from x and y.

But eventually, experiments at large angles would convince the creatures that the actual symmetry of the world is SO(3). Then they would understand that z is really the same as x and y, since they can be mixed up by rotations. The SO(2) semidirect product R2 limit would be understood as the limit that the free parameter, the ratio of the height-unit to the length-unit, approaches 0. The Lorentz group is analogous— it is a simple group that turns into the Galilean group when the unit of time is made long compared to the unit of space, which is the limit .

But the symmetry group of special relativity is not entirely simple because there are still translations. The Lorentz group are the transformations that keep the origin fixed, but translations are not included. The full Poincaré group is the semi-direct product of translations with the Lorentz group. But if you take the unification idea to its logical conclusion then not only are boosts non-commutative but translations should be non-commutative too.

In the pancake world, this would happen if the creatures were living on an enormous sphere, not a plane. In this case, when they wander around their sphere, they would eventually come to realize that translations are not entirely separate from rotations, because if they move around on the surface of a sphere, when they come back to where they started, they find that they have been rotated by the holonomy of parallel transport on the sphere. If the universe is the same everywhere (homogenous) and there are no preferred directions (isotropic), then there are not many options for the symmetry group: they either live on a flat plane, or on a sphere with everywhere constant positive curvature, or on a Lobachevski plane with constant negative curvature. If they are not living on the plane, they can describe positions using dimensionless angles, the same parameters that describe rotations, so that translations and rotations are nominally unified.

In relativity, if translations mix up nontrivially with rotations, but the universe is still homogeneous and isotropic, the only options are that space-time has a uniform scalar curvature. If the curvature is positive, the analog of the sphere case for the two-dimensional creatures, the space-time is de Sitter and the symmetry group of spacetime is a de Sitter group rather than the Poincaré group.

De Sitter special relativity postulates that the empty space has de Sitter symmetry as a fundamental law of nature. This means that spacetime is slightly curved even in the absence of matter or energy. This residual curvature is caused by a positive cosmological constant to be determined by observation. Due to the small magnitude of the constant, then special relativity with the Poincaré group is more than accurate enough for all practical purposes.

Modern proponents of this idea, such as S. Cacciatori, V. Gorini and A. Kamenshchik, have reinterpreted this theory as physics, not just mathematics. They believe that the acceleration of the expansion of the universe is not all due to vacuum energy, but at least partly due to the kinematics of the de Sitter group, which in their view is the correct symmetry group of space time, replacing the Lorentz group.

A modification of this idea allows to change with time, so that inflation may come from the cosmological constant being larger near the big bang than nowadays. It can also be viewed as a different approach to the problem of quantum gravity.