C. The general theory of relativity and galaxies¶
Throughout this book, we largely use Newtonian (classical) mechanics and the Newtonian theory of gravity. But for more than a century now, it has been known that Newtonian theory provides only a limited description of reality. Newtonian theory fails both in the limit of small distances and in the limit of high velocities or strong gravitational fields. The presently-prevailing theory at small distances is that of quantum mechanics and has culminated in the standard model of particle physics; quantum theory is an essential ingredient in many processes relevant to galaxies or their observation, but because the dynamics of galaxies and the Universe is dominated by gravity, we can largely get away with ignoring quantum mechanics in this book. However, we cannot so easily ignore the fact that the Newtonian theory that we use throughout this book has been superseded by the general theory of relativity, which describes gravitation and the mechanics of particles on all but the smallest scales (Einstein 1916). In all cases where the predictions from the general theory of relativity (which we will also refer to as “GR”) differ measurably from those of Newtonian gravity, GR has so far always prevailed (e.g, Dyson et al. 1920; Adams 1925; Shapiro et al. 1971; Williams et al. 2004; Gravity Collaboration et al. 2018; Gravity Collaboration et al. 2020; see Adelberger et al. 2003 and Will 2014 for reviews). In this appendix, we therefore explicitly demonstrate how the Newtonian theory that we use throughout this book can be derived from the general theory of relativity. We will also discuss two applications of GR that we use and that cannot be adequately addressed in the Newtonian limit: the gravitational bending of light that gives rise to gravitational lensing and the dynamics of the Universe as a whole that is the basis for our discussion of how galaxies form in the expanding Universe.
The general theory of relativity is rightly considered one of the most satisfying theories in fundamental physics. Rather than describing the motion of particles as having their acceleration set by a mysterious force of gravity without explaining why it determines the acceleration rather than, e.g., the velocity or the jerk, GR explains the motion of both matter and light in a gravitational field as that along the shortest path in the four-dimensional spacetime curved by the presence of matter and energy. Einstein arrived at this intuitive picture through the equivalence principle that was inspired by and is a stronger version of the weak equivalence principle that we discuss in Chapter 2.1. The weak equivalence principle states that all objects fall the same way in an external gravitational field, that is, that the inertial mass that appears in Newton’s second law of motion and the gravitational mass that appears in Newton’s law of gravity are one and the same mass. Einstein generalized this principle to also require that in a region of spacetime that is small enough that tidal forces (i.e., gradients in the gravitational field) are negligible, one cannot distinguish between a uniform acceleration and an external gravitational field; said another way, one cannot locally detect the existence of a gravitational field. In a local volume of spacetime and in the absence of non-gravitational forces, bodies can therefore always be considered to be falling freely, that is, with zero acceleration. And it is then only a small leap to suggest that bodies that are not subject to any non-gravitational forces are always falling freely as they traverse spacetime and that what we think of as the force of gravity is nothing more than the freely-falling path through a curved spacetime.
Taking these simple principles and the intuitive picture of gravity as free-falling motion in a curved spacetime and turning them into a quantitative, mathematical theory of gravitation requires a large amount of mathematics not typically taught to science students. The reward in learning this mathematics is that one understands how Einstein’s field equations, which describe how matter and energy curve spacetime, are essentially the simplest equations that one can write down relating the curvature of spacetime and a generalized notion of energy and momentum. That these equations can describe gravitation to one part in a billion in the solar system (Adelberger et al. 2003) is nothing short of remarkable. Here we simply aim to show how Newtonian gravity emerges in the limit of weak gravitational fields and velocities small compared to the speed of light, how light is bent by weak gravitational fields, and we want to be able to build simple cosmological models to understand galaxy formation in the expanding Universe. We will therefore take a more pragmatic approach that is focused on these applications without elucidating the full structure of the theory, but we refer readers to the many excellent textbooks on GR to learn more about the underpinnings and other applications of the theory (e.g., Carroll 2004).
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