APS Journals

Soon after Albert Einstein published his equation for general relativity, Karl Schwarzschild found a spherically symmetric solution to it, which today is used to describe static black holes. At the center of a Schwarzschild black hole, the curvature becomes infinite. Einstein and Nathan Rosen tried to explain this singularity as a portal to an extended spacetime, leading later to the hypothetical concept of a wormhole. Martin Kruskal eventually showed that the Schwarzschild solution has a smooth horizon and any singularity remains behind it.

J. Robert Oppenheimer and Hartland Snyder studied stellar collapse using general relativity along with a model of the star’s neutron core. They concluded that continued gravitational collapse could not be prevented for a large enough core. Rather, the star would continue collapsing, curving spacetime ever more drastically around itself, forming what later would be called a black hole. They concluded that “The star thus tends to close itself off from any communication with a distant observer; only its gravitational field persists.’’ Oppenheimer and Snyder thus proposed a plausible scenario for the formation of black holes as an end result of stellar evolution. It would take another two decades for the concept of black holes to gain widespread acceptance.

In 1963, nearly half a century after Schwarzschild’s spherically symmetric solution to Einstein’s equation, Roy Kerr obtained an exact solution describing cylindrically symmetric black holes. Kerr’s unique solution is characterized by two parameters, the mass and angular momentum, of the black hole. Since astrophysical bodies typically spin, Kerr’s solution was a major step forward in connecting exact general relativity solutions to realistic astrophysical objects.

How stable are black holes? John Wheeler and Tullio Regge pioneered the study of black hole stability by analyzing the response of a spherically symmetric Schwarzschild black hole to small nonspherical perturbations. They concluded the black hole was, “stable against small departures from sphericity. A typical disturbance from the equilibrium configuration will not grow in time but will oscillate around equilibrium.”

Werner Israel proved that Schwarzschild black holes have no “hair”—no distinguishing properties other than their mass. This result was soon extended to the cases of spinning and electrically charged black holes and led to the “no hair theorem”: All black hole solutions of general relativity can be completely characterized by just three externally observable classical parameters mass, electric charge, and angular momentum.

For over four decades general relativity remained the only theory of gravity compatible with observational data and the weak equivalence principle. This changed in 1961 when Carl Brans and Robert Dicke published a modification of general relativity. The Brans-Dicke theory replaced the inverse Newton’s constant by a field that fluctuates across space and time and has a dimensionless parameter that can be tuned to fit observations. This work showed that general relativity could be embedded into a broader class of models and opened the door to studies of modified theories of gravity. It also allowed for characterization of deviations from general relativity that could be tested observationally.

Two theoretical breakthroughs paved the way for global analyses of spacetime structures without the need for specific solutions of Einstein’s equation. In 1955 Amal Kumar Raychaudhuri introduced evolution equations for curves describing the flow of nearby point particles without making assumptions about the homogeneity, or isotropy of the background spacetime. A few years later Roger Penrose proposed an extremely powerful diagrammatic technique for capturing the global causal structure of any spacetime. These techniques allow the global properties of spacetimes to be gleaned. For example, Penrose was able to show that, rather than being artifacts of specific solutions, spacetime singularities are a generic outcome of gravitational collapse.

General relativity confronted quantum mechanics in the 1970s. The pioneering work of Stephen Hawking showed that quantum fluctuations cause black holes to radiate. Black holes, it was realized, are thermal objects carrying a notion of temperature and entropy. The thermal nature of black holes exposed a remarkable tension between general relativity and quantum mechanics—two pillars of modern physics. Thermal radiation from black holes suggests a loss of information inconsistent with one of the foundational tenets of quantum mechanics, unitary time evolution. The search for a resolution of this tension continues to drive the quest for a consistent quantum theory of gravity.

Only in rare cases can the highly nonlinear equations of general relativity be solved analytically. Many interesting questions in general relativity call for approximations and computational techniques that yield accurate answers when analytic methods are not available. Two seminal successes in answering that call are the Choptuik scaling laws of gravitational collapse and the three papers that revolutionized numerical relativity.

While the cosmological constant is just a parameter in classical general relativity, in quantum field theories the “constant” becomes dynamical and astronomically large. Steven Weinberg’s now classic review provided the clearest statement of the cosmological constant problem: The small observed bound on the cosmological constant is hard to reconcile with theories of particle physics, which predict it should be many orders of magnitude larger. Today, this problem is central to the puzzle of dark energy.

Cosmological inflation, a brief period of exponential growth of the early Universe, was proposed by Alan Guth, and expanded upon by others, to explain some of the puzzling features of observational cosmology (the homogeneity, isotropy, and flatness of the Universe). These papers collectively represent some of the key initial developments of the theory.

Cosmology on large scales is described by Friedmann-Robertson-Walker spacetime, which describes the evolution of a homogeneous and isotropic Universe. Yet, everything we see, from stars to galaxy clusters, is due to perturbations of this background metric. James Bardeen addressed how these physical perturbations can clearly be disentangled from mathematical artifacts.

Tests of general relativity are notoriously difficult, due to the small size of general relativistic corrections to Newtonian gravity in the weak field limit. Yet, ingenious experiments continue to be devised and performed, which so far confirm the theory.

General relativity predicts that photons going up (down) in a gravitational field are red-shifted (blue-shifted). Robert Pound and Glen Rebka managed to confirm this tiny (1 part in 10¹⁵ ) effect over a drop of only 74 feet. A later test by Robert Vessot et al. verified the shift over a distance of 10,000 km.

General relativity is based on the equivalence principle, which implies that gravitational and inertial masses are the same. Over the years, several groups have tested the equivalence principle to high precision, notably in a recent series of elegant experiments by Eric Adelberger and colleagues.

Some modified theories of gravity, such as Brans-Dicke theory, violate the equivalence principle, at least in its most stringent form, due to the Nordvedt effect. This is where the self-energy of a body contributes to its gravitational mass, but not to its inertial mass. This effect becomes important only for very massive bodies, and thus requires astronomical tests, such as the lunar lasing tests of James Williams et al.

Cosmology has entered a precision era. The standard model of cosmology, based on cosmological-constant dark energy and cold dark matter (the ΛCDM model), has withstood many precision tests. Max Tegmark et al. used the power spectrum of galaxies and the cosmic microwave background to greatly constrain cosmological parameters, including the age of the Universe. Sudeep Das et al. were the first to show effects on the cosmic microwave background of gravitational lensing—an Eddington experiment on the scale of the observable Universe.