General Relativity (thesis series)

To start with some bonus material, the featured image of the post is from Einstein’s notebook during the period during which he was developing general relativity.

This post is a continuation series of posts goes through the elements of the concordance cosmology: general relativity, inflation, the Big Bang and the expansion of the universe, cold dark matter (CDM) and a positive cosmological constant. I will start with a review of the historical, theoretical, and observational advances that led to each model’s widespread acceptance. The posts are excerpts from my Ph.D. thesis, lightly modified for blog format. No mathematical equations are written, with the intent of making the introductory material accessible to the non-specialist. A full bibliography of references in posts in this series is available.

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This post gives a short historical context of general relativity and its development. Since its debut, the theory of Newtonian dynamics (Newton, 1687) dominated the understanding of gravitational force until Einstein introduced GR (Einstein, 1914, 1916). By 1917, Einstein had published his work on the cosmological consequences of GR (Einstein, 1917). Since then, markedly little has changed in this simple theory. GR is marked by the connection between the geometry of spacetime and mass-energy and momentum, collectively known as stress-energy. The connection between these two seemingly disparate elements manifests itself through the Einstein field equations (EFE). Through the mathematical language of differential geometry, the EFE links the two, and allows a description of the dynamical properties.

John Wheeler put general relativity in a particularly succinct fashion when he stated that in GR “Spacetime tells matter how to move. Matter tells spacetime how to curve.” (Wheeler, 2000). A few highly technical quantities are shortly mentioned by name so that in the context of later presentation, particularly in the context of work modifying general relativity, it can be understood whether a particular mathematical quantity is thought of as characteristic of the geometry of spacetime (“telling matter how to move”) or rather of the stress-energy ‘(‘telling spacetime how to curve”). Quantities such as the Riemann curvature tensor, the Ricci tensor, the connection, and the metric (and the Gauss-Bonnet invariant and the torsion tensor) are all geometrical quantities which characterize the geometrical structure of spacetime. For example the metric defines the length of the geometrical structure of spacetime and the Riemann curvature tensor characterizes the curvature of this structure. Quantities such as the energy-momentum tensor characterize the stress-energy configuration of a system being analyzed.

Early observations in Le Verrier (1859) showed a perihelion shift of Mercury whereby the orbit of Mercury changes in a small manner unaccounted for by Newtonian physics. This gave hints observationally of the need for a deeper theory. GR was additionally motivated by Einstein’s deep theoretical desire to explain why freely falling uncharged objects follow the same paths, and his drive to formulate the laws of special relativity independent of gravitational force, position or velocity (Clifton et al. , 2012).

From this impetus, Einstein developed and presented GR (Einstein, 1916). It was quickly shown that GR explained the perihelion shift as well as predicted the deflection of light by the gravitational effect of the sun. This latter prediction was one of the first to be confirmed, making Einstein an instant celebrity (Crelinsten, 2006). Since then, improved observations of this effect (Shapiro et al. , 2004), current best constraints of the perihelion precession of Mercury (Pireaux & Rozelot, 2003), test masses (Schlamminger et al. , 2008), the Shapiro time delay (Shapiro, 1964; Bertotti et al. , 2003), the Nordtvedt effect (Williams et al. , 2004), the Lense-Thirring effect (Ciufolini & Pavlis, 2004) and pulsars (Weisberg et al. , 2010; Damour & Taylor, 1992; Kramer, 1998), have shown GR to be valid at a range of scales, both internal and external to our solar system. GR has withstood every observational test it has been confronted with for 100 years and remains well-accepted as the core of the concordance cosmological framework (Will, 1993).

GR’s fundamental Einstein field equations (EFE) have a freedom; namely, in the EFE the constant Λ, known as the cosmological constant, can be incorporated if observational data were to demand it (Einstein, 1917). The equations of GR, without such a cosmological constant, are shorter and simpler, which initially disfavored its use. Light is itself affected by gravity and GR shows that gravitational lensing results from the connection between stress-energy with the geometry of spacetime. Gravitational lensing causes the path of light to deviate from a straight line (Bartelmann, 2010). The result is an intervening large mass that can effectively bend light as it travels from its source to its destination. Walsh et al. (1979); Bergmann et al. (1990) observed some of the first astrophysical examples of this lensing phenomenon. A full textbook style and mathematical review of general relativity is not within the scope of this series of blog posts, but an excellent one is contained within Spacetime and Geometry: An Introduction to General Relativity (Carroll, 2004).

This post reviewed GR, the motivations for its development, the connection betweenstress- energy and the geometry of spacetime via the Einstein field equations, the wealth of observational material supporting the validity of GR, the cosmological constant in connection with GR, and the phenomenon of gravitational lensing.