Everything about General Relativity totally explained
General relativity or the
General theory of relativity is the
geometric theory of
gravitation published by
Albert Einstein in 1915/16. It unifies
special relativity and
Newton's law of universal gravitation, resulting in a theory in which
gravity is a property of the
geometry of
space and
time. In particular, the
curvature of spacetime is directly related to the
four-momentum (
mass-energy and linear
momentum) of whatever
matter and
radiation is present through the
Einstein field equations, a system of
partial differential equations.
General relativity predicts a number of novel effects relating to the passage of time, the geometry of space, the motion of bodies in
free fall and the propagation of
light. Examples are
gravitational time dilation, the
gravitational redshift of light, and the
gravitational time delay; in numerous
observations and experiments to date, the theory's predictions for these effects have been confirmed. Although
not the only relativistic theory of gravity, general relativity is the simplest such theory that's consistent with the experimental data. Still, a number of open questions remain: the most fundamental is how general relativity can be reconciled with the laws of
quantum physics to produce a complete and self-consistent theory of
quantum gravity.
The theory has important astrophysical applications. Notably, it predicts the existence of
black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive
stars. There is evidence that, indeed, such black holes as well as more massive varieties are responsible for the intense
radiation emitted by certain types of astronomical objects (such as
active galactic nuclei or
microquasars). The bending of light by gravity can lead to the curious phenomenon of multiple images of the same astronomical object being visible in the sky, an effect called
gravitational lensing which has spawned an active new branch of astronomy. General relativity also predicts the existence of
gravitational waves, which have been measured indirectly; a direct measurement is the aim of projects such as
LIGO. In addition, it's the basis of current
cosmological models of an expanding universe.
History
special relativity in 1905, Einstein began to think about how to incorporate
gravity into his new relativistic framework. His considerations led him from a simple thought experiment involving an observer in free fall to the
equivalence principle—in a nutshell, that the physical laws for an observer in free fall are those of special relativity—and from there to a theory in which gravity is described purely in the language of
geometry: from explorations of some consequences of the equivalence principle such as the influence of gravity and acceleration on the propagation of light published in 1907 to the main work in the years 1911 to 1915 with the realization of the role of differential geometry (with help from
Marcel Grossmann on the intricacies of that field of mathematics) and a long search, including detours and false starts, for the field equations relating geometry and the mass-energy content of spacetime. In November of 1915, these efforts culminated in Einstein's presentation to the
Prussian Academy of Science of the
Einstein field equations, which specify how the geometry of space and time is influenced by whatever matter is present.
Already in 1916,
Schwarzschild found a
solution to the Einstein field equations that's nowadays known under his name, laying the groundwork for the description of gravitational collapse and, eventually, the extreme state of matter known as a
black hole. The same year saw the first steps of generalization to electrically charged objects that would result in the
Reissner-Nordström solution. In 1917, Einstein applied his theory to the
universe as a whole, initiating the field of relativistic
cosmology. However, in line with contemporary thinking, he was set on describing a static universe, and he added a new parameter, the
cosmological constant, to his original field equations for that purpose.
When it became clear in 1929 with the work of
Hubble and others that our universe is indeed expanding (and thus better described by expanding cosmological solutions found by
Friedmann in 1922),
Lemaître formulated the earliest version of the
big bang models.
During all that time, general relativity remained something of a curiosity among physical theories. There was evidence that it was indeed to be preferred to Newton's earlier description of gravity: Einstein himself had shown in 1915 how it explained the
anomalous perihelion advance of the planet
Mercury, and a 1919 expedition led by
Eddington had announced confirmation of general relativity's prediction for the deflection of the light of distant stars by the Sun (instantly catapulting Einstein to world fame). Yet it was only with the developments between approximately 1960 and 1975, now known as the
Golden age of general relativity, that the theory entered the mainstream of
theoretical physics and
astrophysics, as both the theoretical basis of
black holes as well as their astrophysical applications (
quasars) became clear, ever more precise solar system tests confirmed the theory's predictive power, and relativistic cosmology, too, became amenable to direct observational tests.
From classical mechanics to general relativity
The structure of general relativity and the way the theory is formulated are best understood by examining its similarities with, and departures from, classical physics.
Geometry of Newtonian gravity
At the base of
classical mechanics there's the notion that in describing a body's motion, we can differentiate between a special type of motion commonly known as free (or
inertial) motion, and deviations from this free motion. Such deviations are caused by external
forces acting on a body in accordance with Newton's second
law of motion which states that the
force acting on a body is equal to that body's (inertial)
mass times its
acceleration. There is a direct connection between the preferred inertial motions and the geometry of
space and
time: in the standard
reference frames of classical mechanics, objects in free motion move along straight
lines at constant
speed. In modern parlance, their paths are
geodesics, or straight
world lines in
spacetime.
Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as
electromagnetism or
friction), can be used to define the geometry of space, as well as a time
coordinate. However, there's an ambiguity once
gravity comes into play. Following from
Newton's law of gravity, and independently verified by experiments such as that of
Eötvös and its successors, there's a
universality of free fall (also known as the weak
equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a
test body in
free fall depends only on its position and initial speed, but not on any of its material properties. A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it's impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard an accelerated rocket.
Given the universality of free fall, there's no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it's the
geodesic motion associated with a specific
connection which depends on the gradient of the
gravitational potential. Space, in this construction, still has the ordinary
Euclidean geometry. However, as can be shown using simple thought experiments following the free-fall trajectories of different test particles, the Newtonian connection isn't
integrable—spacetime is
curved. The result is a geometric formulation of Newtonian gravity using only
covariant concepts, in other words: this description is valid in any desired coordinate system. In this geometric description,
tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.
Relativistic generalization
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of
(special) relativistic mechanics. In the language of
symmetry: where gravity can be neglected, physics is
Lorentz invariant (the defining symmetry of special relativity), not
Galilei invariant (the defining symmetry of classical mechanics). The differences between the two become significant when we're dealing with speeds approaching the
speed of light and high-energy phenomena.
Lorentz symmetry introduces an additional structure, in mathematical terms: a
conformal structure. This is the set of light cones (see the image on the left): For each event A, there's a set of events that can, in principle, either influence or be influenced by A via signals or interactions that don't need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer-independent.
Special relativity is defined in the absence of gravity, so for practical applications, it's a suitable model whenever gravity can be neglected. As gravity comes into play, assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global
inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight
time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.
A priori, it isn't clear whether the new local frames in free fall are indeed those in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on
electromagnetism, and its preferred frames might not be the same as the local free-falling inertial frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift. The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity. The generalization of this statement, namely that the laws of special relativity hold, to good approximation, in freely falling (and non-rotating) reference frames, is known as the Einstein
equivalence principle, and is one of the guiding principles when it comes to generalizing special relativistic physics to include gravity.
With reference to the same experimental data, it becomes clear that time as shown by clocks in a gravitational field—
proper time, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it isn't measured by the
Minkowski metric. As in the Newtonian case, this is suggestive of a more general geometry: where all reference frames in free fall are equivalent, and approximately Minkowskian, we're dealing with a curved generalization of Minkowski space: instead of Minkowskian, assume the
metric tensor to be, more generally, semi-
Riemannian. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, a
Levi-Civita connection. Assume this to be the connection implied by the universality of free fall.
Einstein's equations
While the preceding section shows the relativistic, geometric version of the effects of gravity, there's still the question of the source of gravity. In Newtonian gravity, the source is mass. In special relativity, mass is
equivalent to energy, and in fact turns out to be part of a more inclusive quantity called
energy-momentum tensor, which includes both
energy and
momentum densities as well as
stress (that is,
pressure and shear). Drawing further upon the analogy with geometric Newtonian gravity, it's natural to assume that the
field equation for gravity relates this and the tensor that embodies tidal effects, the
Ricci tensor. Adding a suitable geometric form for the
conservation of energy-momentum, the simplest set of equations are what are called Einstein's (field) equations, which equate the energy-momentum tensor and a specific combination of the Ricci tensor and the metric known as the
Einstein tensor:
seconds, known as an
inflationary phase. While recent measurements of the cosmic background radiation have resulted in first evidence for this scenario, problems remain. There is a bewildering variety of possible inflationary scenarios not restricted by current observations. Also, the question remains what happened in the earliest universe, close to where the classical models predict the big bang
singularity. An authoritative answer would require a complete theory of
quantum gravity, which doesn't exist at the moment (cf. the section on
quantum gravity, below).
Advanced concepts
Causal structure and global geometry
In general relativity, no material body can catch up with or over take a light pulse; no influence from an event A can reach any other location before light sent out at A does so. Hence, 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 important techniques that are now termed
global geometry. In global geometry, the object of study isn't one particular
solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the
Raychaudhuri equation, are utilized in conjunction with non-specific assumptions about the nature of
matter (usually in the form of so-called
energy conditions) to derive general results.
Cosmic partitions: horizons
One of the most striking conclusions that can be drawn from studies of global geometry is the existence of boundaries called
horizons, which demarcate one
spacetime region from the rest of the 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 being the
Schwarzschild radius), spacetime is partitioned into an inside and an outside world:n no light from the inside can escape to the outside, and since, in general relativity, no object can overtake a light pulse, all inside matter is imprisoned as well. However, matter and radiation may cross the horizon into the black hole—clearly showing that horizons are not physical barriers. The resulting object is known as a
black hole, and the surface in question as the black hole's horizon.
Initial black hole studies relied on simplified models obtained from
explicit solutions of
Einstein's equation, 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). Subsequent studies using global geometry have revealed more general properties of black holes: in the long run, they're rather simple objects characterized by eleven parameters specifying
energy,
linear momentum,
angular momentum, location at a specified time and
electric charge. This is stated by the
black hole uniqueness theorems: "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's a general set of laws known as
black hole mechanics, which is analogous to the
laws of thermodynamics. For example, 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 from a rotating black hole (for example by the
Penrose process). There is strong evidence that the laws of black hole mechanics are, in fact, a special case of the laws of thermodynamics, and that the black hole area does indeed denote its entropy: semi-classical calculations indicate that black holes do emit
thermal radiation, with the surface gravity playing the role of temperature in
Planck's law. This radiation is known as
Hawking radiation (cf. the
quantum theory section, below).
Horizons also play a role for other kinds of solutions. In an expanding universe, some regions of the past can be unobservable ("
particle horizon"), and some regions of the future can't be influenced (event horizon). In both cases, the location of the horizon in spacetime depends on the event in question. Even in flat Minkowski space, when described by an accelerated observer (
Rindler space), there will be horizons (associated with a semi-classical radiation known as
Unruh radiation).
Singularities
Another general—and quite disturbing—feature of general relativity is the appearance of spacetime boundaries known as singularities. Ordinary spacetime can be explored by following up on all possible ways that light and particles in free fall can travel (that is, all timelike and lightlike geodesics). But there are spacetimes which fulfill all the requirements of Einstein's theory, yet have "ragged edges"—regions where the paths of light and falling particles come to an abrupt end and geometry becomes ill-defined. By definition, these are
spacetime singularities. In more interesting cases, the geometrical quantities characterizing spacetime curvature (for example the
Ricci scalar) take on infinite values at such "curvature singularities". Well-known examples of spacetimes with future singularities—where
worldlines end—are the
Schwarzschild solution, which describes a singularity inside an eternal static black hole, or the
Kerr solution with its ring-shaped singularity inside an eternal rotating black hole. The
Friedmann-Lemaître-Robertson-Walker solutions, and other spacetimes describing universes, have past singularities on which worldlines begin, namely
big bang singularities.
From these examples, all highly symmetric and thus simplified, one might think the occurrence of singularities to be a consequence of idealization. The famous
singularity theorems proved using the methods of global geometry say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage and also at the beginning of a wide class of expanding universes. However, these theorems say very little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized for example by the so-called
BKL conjecture). The
cosmic censorship hypothesis states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity.
Evolution equations
Each
solution of Einstein's equation encompasses the whole history of a universe—it isn't just some snapshot of how things are, but a whole
spacetime: a statement encompassing the state of matter and geometry everywhere and at every moment in that particular universe. By this token, Einstein's theory appears to be different from most other physical theories, which specify
evolution equations for physical systems: if the system is in a given state at some given moment, the laws of physics allow you to extrapolate its past or future. For
Einstein's equations, there appear to be subtle differences compared with other
fields, for example, they're self-interacting (that is,
non-linear even in the absence of other fields, and they've no fixed background structure—the stage itself evolves as the cosmic drama is played out).
Nevertheless, in order to understand Einstein's equations as
partial differential equations, it's crucial to re-formulate them in a way that describes the evolution of the universe over time. This is achieved by so-called "3+1" formulations, where spacetime is split into three space dimensions and one time dimension, such as the
ADM formalism. These decompositions show that the spacetime evolution equations of general relativity are indeed well-behaved, meaning that solutions always
exist and are
uniquely defined (once suitable initial conditions are specified). Formulations like this are also the basis of
numerical relativity: attempts to simulate the evolution of relativistic spacetimes (notably merging black holes or gravitational collapse) using computers.
Global and quasi-local quantities
The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. 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 for this is that the gravitational field—like any physical field—must be ascribed a certain energy. However, it's 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). If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the so-called
Bondi mass at null infinity. Just as in
classical physics, it can be shown that these masses are positive. Analogous global definitions exist for
momentum and
angular momentum. In addition, there have been a number of attempts to define
quasi-local quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about
isolated systems, such as a more precise formulation of the
hoop conjecture.
Relationship with quantum theory
Along with general relativity,
quantum theory, the basis of our understanding of matter from
elementary particles to
solid state physics, is considered one of the two pillars of modern physics. However, it's still an open question of how the concepts of quantum theory can be reconciled with those of general relativity.
Quantum field theory in curved spacetime
»
The unification of quantum theory and special relativity has led to the highly successful
quantum field theories which form the basis of modern
elementary particle physics. These theories 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.
Short of constructing a theory of quantum gravity, in which all interactions, including general relativity's description of gravity, are formulated within the framework of quantum theory, there's a way to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself: use
classical 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 corresponding models have led to highly interesting results. Most notably, they indicate that black holes emit a blackbody spectrum of particles known as
Hawking radiation, leading to the possibility that
black holes evaporate over time. As briefly mentioned, this radiation plays an important role for the thermodynamics of black holes.
Quantum gravity
Both consistency between a quantum description of matter and a geometric description of spacetime, and the appearance of
singularities involving minute curvature length scales indicate that a full theory of
quantum gravity is needed for an adequate description of the interior of black holes and time evolution close to the big bang: a theory in which gravity and the associated geometry of spacetime are described in the language of quantum theory. Despite major efforts in this direction, no complete and consistent theory of quantum gravity is currently known. There are, however, a number of promising candidates.
However, starting with the usual quantum field theories used in
elementary particle physics to describe interactions, while leading to an acceptable
effective (quantum) field theory of gravity at low energies, results in models devoid of all predictive power at very high energies.
One attempt to overcome these limitations is to formulate a quantum theory not of
point particles, but of minute one-dimensional extended objects:
string theory. The theory promises to be a
unified description of all particles and interactions, including gravity; the price to pay are unusual features such as six
extra dimensions of space in addition to the usual three. In what is called the
second superstring revolution, it was conjectured that both string theory and a unification of general relativity and
supersymmetry known as
supergravity form part of a hypothesized eleven-dimensional model known as
M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.
Another approach to quantum gravity starts with the
canonical quantization procedures of quantum theory. Starting with the initial-value-formulation of general relativity (cf. the section on evolution equations,
above), the result is an analogue of the
Schrödinger equation: the
Wheeler-deWitt equation which, regrettably, turns out to be ill-defined. A major break-through came with the introduction of what are now known as
Ashtekar variables, resulting in what is known as
Loop quantum gravity. In this theory, space is represented by a network 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 example being dynamical triangulations, causal sets,
twistor models or the
path-integral based models of
quantum cosmology.
Currently, there's still no complete and consistent quantum theory of gravity, and the candidate models still need to overcome major formal and conceptual problems. They also face the common problem that, as yet, there's no way to put quantum gravity predictions to experimental tests, although there's hope for this to change as future data from cosmological observations and particle physics experiments becomes available.
Current status
General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed every unambiguous observational and experimental test to which it has been subjected. Still, there are strong indications the theory is incomplete.
The problem of quantum gravity, and the associated question of the reality of
spacetime singularities, remain open. Observational data like that for dark energy and dark matter could indicate the need for new physics, and while the so-called
Pioneer anomaly might yet admit of a conventional explanation, it, too, could be a harbinger of new physics. Even while staying within the frame of Einstein's theory, general relativity is rich with possibilities for further exploration: mathematical relativists explore the nature of singularities and the fundamental properties Einstein's equations, ever more comprehensive computer simulations of specific spacetimes (such as those describing merging black holes) are run, and the race for the first direct detection of gravitational waves continues apace, with opportunities to test the theory beyond the limited approximations it has been tested so far even in the
binary pulsar measurements. More than ninety years after the theory was first published, general relativity remains a highly active area of research.
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