de Sitter meets Planck

by Adriana V. Araujo, Diego F. López and José G. Pereira

The Quest for Consistency in Spacetime Kinematics

Newton’s inception of the theory for the gravitational interaction in 1687 was a landmark for modern physics. In addition to explaining all known gravitational phenomena of that time, Newton’s gravitational theory was consistent with the kinematic rules of the Galilei group, known as Galilei relativity. Such consistency provided an atmosphere of intellectual comfort, which lasted for more than two centuries.


From left to right, José, Adriana and Diego. Click here to see the authors taking advantage of all dimensions of a space section of the universe.

By the mid nineteenth century, most secrets of the electric and  magnetic fields were already unveiled. Those advancements culminated with the publication by Maxwell of a comprehensive treatise on the unification of electricity and magnetism, which became known as Maxwell’s theory. This theory brought to the scene the first inconsistency of our tale. In fact, it became immediately clear that the electromagnetic theory was inconsistent with the Galilei relativity: electromagnetism was claiming for a new relativity. In response to this claim, and with contributions from Lorentz and Poincaré, Einstein published in 1905 the basics of what is know today as Einstein special relativity. According to this theory, for velocities near the velocity of light, spacetime kinematics would no longer be ruled by Galilei, but by the Poincaré group. Most importantly, electromagnetism was consistent with Einstein special relativity! Mission accomplished? Not quite!

Upon removal of this inconsistency, another one emerged. The problem was that Newton’s gravity was inconsistent with the newly introduced Einstein special relativity. It was then necessary to construct a new gravitational theory to recover the consistency with the spacetime kinematics. Kind of messy, isn’t it? So messy that it took Einstein ten years to come up with a solution to this problem. In 1915 Einstein published the fundamentals of “general relativity”, a relativistic theory for gravitation, that is, a gravitational theory consistent with special relativity. With this theory, peace returned to physics, but not for so long.

In the 1920s, quantum mechanics, whose first steps were done at the beginning of the century with Planck’s illuminating ideas, consolidated as a theory governing the physics of microcosm. And here we go again: Einstein special relativity was found to be inconsistent with quantum mechanics! The problem is that, for very high energies, or more precisely, for energies of the order of the Planck energy, quantum mechanics predicts the existence of an invariant length scale, dubbed Planck length, which is sometimes interpreted as the minimum attainable length in Nature. It so happens that Einstein special relativity does not allow the existence of an invariant length, which makes it inconsistent with quantum mechanics. This time, it was quantum mechanics that was claiming for a new relativity.

The first idea that comes to mind whenever searching for a quantum kinematics is that, in order to allow the existence of an invariant length, Lorentz symmetry should be broken down. However, to accept a possible violation of the Lorentz symmetry is not so easy. Lorentz symmetry is well-known to be deeply related to causality, and any violation of the first implies a violation of the second. The point is that causality is one of the most fundamental principles of Physics, and its violation, even if it is assumed to take place at the Planck scale only, is the kind of thing we cannot be sure Nature is prepared to afford.

Considering that we strongly believe in the above arguments, we decided to look for a special relativity that preserves Lorentz symmetry. One possibility in this direction is to assume that, instead of being governed by the Poincaré group, the spacetime kinematics is governed by the de Sitter group. This amounts to replacing the Poincaré invariant Einstein special relativity by a de Sitter invariant special relativity. This theory may be thought of as a generalization of Einstein special relativity for energies near the Planck energy, and as such it holds at all energy scales. While preserving the Lorentz symmetry, it allows the presence of an invariant length, being for this reason consistent with quantum mechanics! Voilà!

Now, if special relativity changes, general relativity will change to what we have called de Sitter modified general relativity. A peculiar characteristic of this theory is that, since the de Sitter group includes a cosmological term Λ, the de Sitter modified general relativity naturally reintroduces Λ into gravitation. There are some important differences though. First, Λ is no longer required to be constant. Furthermore, considering that the role of the de Sitter group is to govern the underlying local kinematics, Λ shows up, not as part of dynamics, but as a kinematic effect. Most importantly, the de Sitter modified Einstein equation has a solution for a universe with accelerated expansion. These are the issues addressed in our most recent publication. Curious about other implications of the de Sitter modified general relativity? Do take a look at the paper!

About the authors:

Adriana V. Araujo, from Venezuela, has just got her Ph.D at Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo

Diego F. López, from Colombia, has just got his M.Sc at Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo

José G. Pereira, from Brazil, is a theoretical physicist at Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo

Read the full article in Classical and Quantum Gravity:
de Sitter-invariant special relativity and the dark energy problem

A Araujo et al 2017 Class. Quantum Grav. 34 115014

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