Closely related to light deflection is the gravitational time delay or Shapiro delay , the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction. Predicted in [73] [74] by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to electromagnetic waves.

On February 11, , the Advanced LIGO team announced that they had directly detected gravitational waves from a pair of black holes merging. The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion animated image to the right. However, for weak fields, a linear approximation can be made.

Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed. Some exact solutions describe gravitational waves without any approximation, e. General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation precession of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction. In general relativity, the apsides of any orbit the point of the orbiting body's closest approach to the system's center of mass will precess ; the orbit is not an ellipse , but akin to an ellipse that rotates on its focus, resulting in a rose curve -like shape see image.

Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a test particle. For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by Urbain Le Verrier in , was important evidence that he had at last identified the correct form of the gravitational field equations. The effect can also be derived by using either the exact Schwarzschild metric describing spacetime around a spherical mass [84] or the much more general post-Newtonian formalism.

According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the Solar System or for ordinary double stars , the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting neutron stars , one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period.

Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the Nobel Prize in physics. Several relativistic effects are directly related to the relativity of direction. Near a rotating mass, there are gravitomagnetic or frame-dragging effects. A distant observer will determine that objects close to the mass get "dragged around".

This is most extreme for rotating black holes where, for any object entering a zone known as the ergosphere , rotation is inevitable. The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target.

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Such effects are known as gravitational lensing. Gravitational lensing has developed into a tool of observational astronomy. It is used to detect the presence and distribution of dark matter , provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data provide valuable insight into the structural evolution of galaxies. Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves see Orbital decay , above.

Detection of these waves is a major goal of current relativity-related research. Observations of gravitational waves promise to complement observations in the electromagnetic spectrum. Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can escape. In the currently accepted models of stellar evolution , neutron stars of around 1. Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.

Black holes are also sought-after targets in the search for gravitational waves cf.

### Introduction

Gravitational waves , above. Merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger "chirp" could be used as a " standard candle " to deduce the distance to the merger events—and hence serve as a probe of cosmic expansion at large distances.

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Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed [] cf. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then, other—similarly impractical—GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes.

In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event A can reach any other location X before light sent out at A to X. In consequence, an exploration of all light worldlines null geodesics yields key information about the spacetime's causal structure. This structure can be displayed using Penrose—Carter diagrams in which infinitely large regions of space and infinite time intervals are shrunk " compactified " so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.

Aware of the importance of causal structure, Roger Penrose and others developed what is known as global geometry. In global geometry, the object of study is not one particular solution or family of solutions to Einstein's equations. Rather, relations that hold true for all geodesics, such as the Raychaudhuri equation , and additional non-specific assumptions about the nature of matter usually in the form of energy conditions are used to derive general results. Using global geometry, some spacetimes can be shown to contain boundaries called horizons , which demarcate one region from the rest of spacetime.

The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space as specified in the hoop conjecture , the relevant length scale is the Schwarzschild radius [] , no light from inside can escape to the outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black hole's horizon , is not a physical barrier. Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution used to describe a static black hole and the axisymmetric Kerr solution used to describe a rotating, stationary black hole, and introducing interesting features such as the ergosphere.

Using global geometry, later studies have revealed more general properties of black holes. With time they become rather simple objects characterized by eleven parameters specifying: electric charge, mass-energy, linear momentum , angular momentum , and location at a specified time. This is stated by the black hole uniqueness theorem : "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans.

Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results having emitted gravitational waves is very simple. Even more remarkably, there is a general set of laws known as black hole mechanics , which is analogous to the laws of thermodynamics.

## General relativity

For instance, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the entropy of a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole e. As thermodynamical objects with non-zero temperature, black holes should emit thermal radiation. Semi-classical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in Planck's law.

This radiation is known as Hawking radiation cf. There are other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed " particle horizon " , and some regions of the future cannot be influenced event horizon.

Another general feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as spacetime singularities , where the paths of light and falling particles come to an abrupt end, and geometry becomes ill-defined.

In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the Ricci scalar , take on infinite values. Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artifact of idealization. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity. It describes the state of matter and geometry everywhere and at every moment in that particular universe.

Due to its general covariance, Einstein's theory is not sufficient by itself to determine the time evolution of the metric tensor.

It must be combined with a coordinate condition , which is analogous to gauge fixing in other field theories. To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. The best-known example is the ADM formalism. The notion of evolution equations is intimately tied in with another aspect of general relativistic physics.

## Table of Contents: Gravity, a geometrical course.

In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass or energy. The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy. Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" ADM mass [] or suitable symmetries Komar mass.

The hope is to obtain a quantity useful for general statements about isolated systems , such as a more precise formulation of the hoop conjecture.

If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to solid state physics , would be the other. Ordinary quantum field theories , which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.

The demand for consistency between a quantum description of matter and a geometric description of spacetime, [] as well as the appearance of singularities where curvature length scales become microscopic , indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.

Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems.

One attempt to overcome these limitations is string theory , a quantum theory not of point particles , but of minute one-dimensional extended objects. Another approach starts with the canonical quantization procedures of quantum theory. Using the initial-value-formulation of general relativity cf. Space is represented by a web-like structure called a spin network , evolving over time in discrete steps. Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced, [] there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman Path Integral approach and Regge Calculus , [] dynamical triangulations , [] causal sets , [] twistor models [] or the path integral based models of quantum cosmology.