General relativity. When Xxxxxxxx introduced the theory of Special Relativity (SR) [18, 19] in 1905, it became clear that Xxxxxx’x theory of gravity would need to be modified. This is due to the fact that Newtonian gravity implies that gravitational interaction is transmitted between bodies at infinite speed, in sheer contrast to SR; which states that no physical interaction can travel faster then the speed of light [20, p.1]. Thus, at the end of 1915, Xxxxxxxx postulated General Relativity (GR) [21]; which is what we now consider to be the modern theory of gravitation. Simply put, GR considers that gravity is not a force, and instead it is a mani- festation of the curvature of spacetime. A massive object will cause a “bending” in spacetime, and subsequently that “bending” then controls the movement of phys- ical objects. This can be summarised more beautifully, by the words of Xxxx X. Xxxxxxx [22] as: “Matter tells spacetime how to curve, and spacetime tells matter how to move.” Although these words are beautiful, a far more informative4 description of GR, is presented in it’s mathematical form via the Einstein Field Equations (EFEs). These are defined as pl Gµν = M−2Tµν , (1.1) where Mpl is the reduced Xxxxxx mass, Tµν is the energy-momentum (EM) tensor, and Gµν is the Einstein tensor defined as Gµν = Rµν − 2 Rgµν , (1.2) where Rµν is the Ricci tensor, R is the Ricci scalar, and gµν is the metric tensor. The components of this equation shall be discussed later in the section, but for now it’s sufficient to say that the Einstein tensor encodes the curvature of spacetime, whereas the EM tensor encodes the matter source. One can see that the EFEs are written using tensors that are invariant under coordinate transform, and hence, are a direct consequence of The Principle of General Xxxxxxxxxx; that states the laws of physics must be invariant under coordinate transform, and hence, the same for all observers. GR is in-fact based on two key principles; The Principle of General Xxxxxxxxxx, as stated before, and the Einstein Equivalence Principle (EEP). The EEP states in a free-falling reference frame gravitational force effectively vanishes, and the laws of physics apply just like they do in SR. In other words, the EEP implies that 4Some may consider the mathematical form of GR far more beautiful then any combination of words used to describe it. there exists local inertial frames. However, it should be noted that in a non-uniform gravitational field, these local inertial frames...
General relativity. First proposed in 1915, the theory of general relativity is what is called a metric the- ory of gravity (Xxxxxxxx, 1916). It describes spacetime through a dynamical object, the metric gµ⌫ , detailing its curvature. The evolution of this quantity is connected to the energy content of the system, specied by the energy-momentum tensor Tµ⌫ , and their relationship is formalized by Xxxxxxxx’x eld equations: Gµ⌫ + ugµ⌫ = nTµ⌫ . (1.1) In this expression, Gµ⌫ is called the Einstein tensor, a quantity derived from the metric itself. In addition to this, notice the presence of two constants: n, needed to match the units of Gµ⌫ and Tµ⌫ , and u. The latter is called the cosmological constant, and it has important implications for cosmology that will be discussed later. Here, we want to highlight two predictions of general relativity that are particu- larly relevant. The rst is the accurate prediction of the bending of light in the presence of a massive object along the line of sight. Thanks to the rst observation of this phe- nomenon by Xxxxxx Xxxxxxxxx in 1919 (Xxxxx et al., 1920), gravitational lensing was quickly established as an experimental fact, and, over the years, it became a robust ob- servable that is still used to this day. In this theoretical framework, this unusual behav- ior has an obvious explanation: because photons are expected to follow the geodesics dened by the metric gµ⌫ , the curvature induced by the presence of matter naturally results in a perturbed light path. The second relevant prediction to be highlighted is the existence of gravitational waves. Because the metric is dynamical, perturbations on top of a background prole can propagate after being generated by accelerating compact masses. The measurement of the decaying orbit of a binary pulsar due to the energy deposited in this fashion (Xxxxxx and Xxxxxxxx, 1982) represented the rst indirect ob- servation of gravitational waves and, similarly to xxx xxxxxxx case, it quickly ushered in the birth of a new eld. After a few decades, the interest in this science eventu- ally resulted in the direct detection of these tiny spacetime ripples by the LIGO-Virgo consortium in 2015 (LIGO Scientic Collaboration and Virgo Collaboration, 2016). When applied to the Universe as a whole, Xxxxxxxx’x eld equations are solved under two simple assumptions: the system should have no preferred observer, and it should evolve over time. The rst statement is known as the Copernican principle, and it is understood ...
