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Concept# Mass in special relativity

Summary

The word "mass" has two meanings in special relativity: invariant mass (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity of the observer. According to the concept of mass–energy equivalence, invariant mass is equivalent to rest energy, while relativistic mass is equivalent to relativistic energy (also called total energy).
The term "relativistic mass" tends not to be used in particle and nuclear physics and is often avoided by writers on special relativity, in favor of referring to the body's relativistic energy. In contrast, "invariant mass" is usually preferred over rest energy. The measurable inertia and the warping of spacetime by a body in a given frame of reference is determined by its relativistic mass, not merely its invariant mass. For example, photons have zero rest mass but contribute to the inertia (and weight in a gravitational field) of any system contai

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An intriguing feature of the Standard Model is that the representations of the unbroken gauge symmetries are vector-like whereas those of the spontaneously broken gauge symmetries are chiral. Here we provide a toy model which shows that a natural explanation of this property could emerge in higher dimensional field theories and discuss the difficulties that arise in the attempt to construct a realistic theory. An interesting aspect of this type of models is that the 4D low energy effective theory is not generically gauge invariant. However, the non-invariant contributions to the observable quantities are very small, of the order of the square of the ratio between the light particle mass scale and the Kaluza-Klein mass scale. Remarkably, when we take the unbroken limit both the chiral asymmetry and the non-invariant terms disappear. © SISSA 2007.

2007Solomon G Shamsuddin Osman Endlich, Alexander Monin, Francesco Riva

Space-time symmetries are a crucial ingredient of any theoretical model in physics. Unlike internal symmetries, which may or may not be gauged and/or spontaneously broken, space-time symmetries do not admit any ambiguity: they are gauged by gravity, and any conceivable physical system (other than the vacuum) is bound to break at least some of them. Motivated by this observation, we study how to couple gravity with the Goldstone fields that non-linearly realize spontaneously broken space-time symmetries. This can be done in complete generality by weakly gauging the Poincare symmetry group in the context of the coset construction. To illustrate the power of this method, we consider three kinds of physical systems coupled to gravity: superfluids, relativistic membranes embedded in a higher dimensional space, and rotating point-like objects. This last system is of particular importance as it can be used to model spinning astrophysical objects like neutron stars and black holes. Our approach provides a systematic and unambiguous parametrization of the degrees of freedom of these systems.

Symmetries are omnipresent and play a fundamental role in the description of Nature. Thanks to them, we have at our disposal nontrivial selection rules that dictate how a theory should be constructed. This thesis, which is naturally divided into two parts, is devoted to the broad physical implications that spacetime symmetries can have on the systems that posses them. In the first part, we focus on local symmetries. We review in detail the techniques of a self-consistent framework -- the coset construction -- that we employed in order to discuss the dynamics of the theories of interest. The merit of this approach lies in that we can make the (spacetime) symmetry group act internally and thus, be effectively separated from coordinate transformations. We investigate under which conditions it is not needed to introduce extra compensating fields to make relativistic as well as nonrelativistic theories invariant under local spacetime symmetries and more precisely under scale (Weyl) transformations. In addition, we clarify the role that the field strength associated with shifts (torsion) plays in this context. We also highlight the difference between the frequently mixed concepts of Weyl and conformal invariance and we demonstrate that not all conformal theories (in flat or curved spacetime), can be coupled to gravity in a Weyl invariant way. Once this ``minimalistic'' treatment for gauging symmetries is left aside, new possibilities appear. Namely, if we consider the Poincar'e group, the presence of the compensating modes leads to nontrivial particle dynamics. We investigate in detail their behavior and we derive constraints such that the theory is free from pathologies. In the second part of the thesis, we make clear that even when not gauged, the presence of spontaneously broken (global) scale invariance can be quite appealing. First of all, it makes possible for the various dimensionful parameters that appear in a theory to be generated dynamically and be sourced by the vacuum expectation value of the Goldstone boson of the nonlinearly realized symmetry -- the dilaton. If the Standard Model of particle physics is embedded into a scale-invariant framework, a number of interesting implications for cosmology arise. As it turns out, the early inflationary stage of our Universe and its present-day acceleration become linked, a connection that might give us some insight into the dark energy dynamics. Moreover, we show that in the context of gravitational theories which are invariant under restricted coordinate transformations, the dilaton instead of being introduced ad hoc, can emerge from the gravitational part of a theory. Finally, we discuss the consequences of the nontrivial way this field emerges in the action.