Gravity and higher spin theory

The phenomenon we observe in the nature which makes the planets orbit the Sun and apple fall on the ground is called gravity. The currently accepted theory that describes it is Einstein gravity named after Albert Einstein who discovered that our space-time is curved correspondingly to the mass that is located on certain place in that space-time. The more mass is located on a small area of space-time the more curved the space-time is. That is how one obtains the areas that have space-time of such curvature, due to such big mass located in the centre, that nothing, even light, can escape it. Which is one way to describe black hole. Luckily the curvature weakens at specific distance from where the heavy mass is located and this is where one finds a so-called horizon of a black hole. Everything that crosses that horizon cannot escape the strong gravity from the heavy mass in the middle, while we remain perfectly safe if we have not crossed the horizon. As excellent as this theory of gravity is, and as excellent its agreement with experiments it has, it still cannot explain all the phenomena that we observe in the nature. One of such phenomena is the notion that the visible stars that orbit the center of galaxy do not move according to the prediction of Einstein gravity. In order for the Einstein gravity to successfully describe this motion of the stars, one has to add significant amount of matter which is not visible and it has not been detected so far. One of the possible explanations of this however, is not to add the same amount of the invisible or dark matter but to modify the theory which describes the rotation of the visible stars around the center of galaxy, i.e. to modify the currently accepted theory of gravity. One such possible modified theory is a gravity theory which conserves angles and not the distances, conformal gravity. One can imagine conformal gravity as a theory which has the same laws on the balloon before and after it has been inflated. Current understanding of the Universe tells us that everything is made from small indivisible quantities, elementary particles. The interaction between these particles creates everything that we see, and forces that we observe happen due to particle carriers travelling through space-time and interacting with other particles. One of the properties they have, which is unique only to them, is specific angular momentum called spin. The spin helps us to differentiate between the particles. When we want to fit gravity theory in this description we find that the particle that carries the gravitation force should have spin two, however such particle has not yet been successfully measured. Furthermore, when we want to fit this particle, graviton, in the consistent theory with all the other particles, we find that we still cannot do that. If we managed to do that we would be a big step closer to consistent description of the Universe and some Grand Unifying Theory. Some suggest that we already have this theory and that it is String theory. String theory suggests that instead of particles everything is built from small vibrating strings. Depending on the frequency on which the string vibrates, the particle with specific spin appears. In order for the String theory to be consistent in limit that we need for the space-time that we see, it is thought that the strings needs to be able to vibrate with infinite number of frequencies. Translated into particle language this would mean that we need to have tower of particles of infinite number of spins. In the limit when these particles become massless, we obtain the theory with massless particles of infinite number of spins. Einstein gravity would then be a special spin-two case of a massless theory which has infinite number of spins. This theory is called Higher spin theory. Analogous theory which in addition has conformal symmetry is conformal higher spin theory, while its special spin-two case is conformal gravity. The main focus of this research are Conformal gravity and Higher spin theory.

If we consider the world around us on smaller and smaller scales, we will see that it is built from tiny particles. The way to identify these particles one from another is by their mass, charge, and spin. Intuitively, we understand mass and charge, while spin we can imagine as property related to rotation around the particle's own axis. The difference to macroscopic picture, is that now the spin is a quantum property. The "rotation" happens instantaneously. The particles that are responsible for gravity, gravitons, have spin two. That means when they rotate 1/2 times they look again the same like in the beginning. The similar particle of spin-s, will rotate 1/s times and look as it did in the beginning. Defining such a quantum theory with gravitons runs into theoretical issues, but if we have a theory with n such particles and let n go to infinity, a theory has a nice theoretic behaviour. We can also look into the symmetry of these particles. One thing that we can observe is that using standard Einstein gravity theory we will need to add dark matter in order to explain the behaviour of galaxies. However, if we consider the gravity theory which has an extra conformal symmetry, we will no longer need that much of dark matter. The research in this project studied such a theory, with spin-s particles and with conformal symmetry. We noticed that even for the lowest spin-1 of the theory we can find a connection to conformal and Einstein gravity. To create the connection between Einstein gravity and conformal gravity, and conformal gravity with conformal spin-1 particle, we look at these theories as if they belong to the same more general theory with additional property of color. In this theory Einstein gravity has color number N = 1, conformal gravity has a color number N = 2, and conformal gravity with a conformal spin-1 particle has a color number N = 3. The approach that was used in studying these theories is a holographic principle. The picture of the holographic principle tells us that the gravity theory in the certain spacetime in d dimensions is encoded in the information at the boundary of that spacetime in d 1 dimensions. Studying the quantum theory at the boundary we can obtain information about the gravity theory in the bulk and vice versa. If we don't know the theory at the boundary, the sophisticated task is in the imposing a set of boundary conditions to the bulk theory to learn about the theory at the boundary. Such boundary theories can play a role of toy models that can teach us about the properties that can appear in our world.