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171-452: Wormholes are consistent with the general theory of relativity , but whether they actually exist is unknown. Many scientists postulate that wormholes are merely projections of a fourth spatial dimension , analogous to how a two-dimensional (2D) being could experience only part of a three-dimensional (3D) object. A well-known analogy of such constructs is provided by the Klein bottle , displaying

342-425: A Minkowski spacetime contains a compact region Ω, and if the topology of Ω is of the form Ω ~ S × Σ, where Σ is a three-manifold of the nontrivial topology, whose boundary has the topology of the form ∂Σ ~ S, and if, furthermore, the hypersurfaces Σ are all spacelike, then the region Ω contains a quasi-permanent intrauniverse wormhole. Geometrically, wormholes can be described as regions of spacetime that constrain

513-454: A Penrose diagram of a Schwarzschild black hole . In the Penrose diagram, an object traveling faster than light will cross the black hole and will emerge from another end into a different space, time or universe. This will be an inter-universal wormhole. Theories of wormhole metrics describe the spacetime geometry of a wormhole and serve as theoretical models for time travel. An example of

684-514: A charged black hole . While such wormholes, if possible, may be limited to transfers of information, humanly traversable wormholes may exist if reality can broadly be described by the Randall–Sundrum model 2 , a brane -based theory consistent with string theory . Einstein–Rosen bridges, also known as ER bridges (named after Albert Einstein and Nathan Rosen ), are connections between areas of space that can be modeled as vacuum solutions to

855-403: A gravity well . No ordinary (massive or massless) particle can have a worldline lying on the paraboloid, since all distances on it are spacelike (this is a cross-section at one moment of time, so any particle moving on it would have an infinite velocity ). A tachyon could have a spacelike worldline that lies entirely on a single paraboloid. However, even in that case its geodesic path is not

1026-543: 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). Since Einstein's equations are non-linear , arbitrarily strong gravitational waves do not obey linear superposition , making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe

1197-499: A spacetime foam in a general relativistic spacetime manifold depicted by a Lorentzian manifold , and Euclidean wormholes (named after Euclidean manifold , a structure of Riemannian manifold ). The Casimir effect shows that quantum field theory allows the energy density in certain regions of space to be negative relative to the ordinary matter vacuum energy , and it has been shown theoretically that quantum field theory allows states where energy can be arbitrarily negative at

1368-450: A "wormhole". Wormholes have been defined both geometrically and topologically . From a topological point of view, an intra-universe wormhole (a wormhole between two points in the same universe) is a compact region of spacetime whose boundary is topologically trivial, but whose interior is not simply connected . Formalizing this idea leads to definitions such as the following, taken from Matt Visser's Lorentzian Wormholes (1996). If

1539-681: A (traversable) wormhole metric is the following: first presented by Ellis (see Ellis wormhole ) as a special case of the Ellis drainhole . One type of non-traversable wormhole metric is the Schwarzschild solution (see the first diagram): The original Einstein–Rosen bridge was described in an article published in July 1935. For the Schwarzschild spherically symmetric static solution where d s {\displaystyle ds}

1710-515: A billion. The Schwarzschild solution is named in honour of Karl Schwarzschild , who found the exact solution in 1915 and published it in January 1916, a little more than a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial flat space solution . Schwarzschild died shortly after his paper

1881-570: A body in accordance with Newton's second law of motion , which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration . The preferred inertial motions are related to 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 , straight world lines in curved spacetime . Conversely, one might expect that inertial motions, once identified by observing

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2052-560: A computer, or by considering small perturbations of exact solutions. In the field of numerical relativity , powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes. In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities . Approximate solutions may also be found by perturbation theories such as linearized gravity and its generalization,

2223-508: A curiosity among physical theories. It was clearly superior to Newtonian gravity , being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters (" fudge factors "), and in 1919 an expedition led by Eddington confirmed general relativity's prediction for

2394-530: A curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection , and this is, in fact,

2565-539: A curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow. The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler , spacetime tells matter how to move; matter tells spacetime how to curve. While general relativity replaces

2736-501: A distribution of energy that violates various energy conditions , such as the null energy condition along with the weak, strong, and dominant energy conditions. However, it is known that quantum effects can lead to small measurable violations of the null energy condition, and many physicists believe that the required negative energy may actually be possible due to the Casimir effect in quantum physics. Although early calculations suggested

2907-487: A few months after Schwarzschild published his solution, and was rediscovered by Albert Einstein and his colleague Nathan Rosen, who published their result in 1935. However, in 1962, John Archibald Wheeler and Robert W. Fuller published a paper showing that this type of wormhole is unstable if it connects two parts of the same universe, and that it will pinch off too quickly for light (or any particle moving slower than light) that falls in from one exterior region to make it to

3078-453: A force is exerted to try to keep it there; this occurs because spacetime has been curved so much that the direction of cause and effect (the particle's future light cone ) points into the singularity. The surface r = r s demarcates what is called the event horizon of the black hole. It represents the point past which light can no longer escape the gravitational field. Any physical object whose radius R becomes less than or equal to

3249-418: A free-falling particle (following a geodesic in the spacetime). In order to satisfy this requirement, it turns out that in addition to the black hole interior region that particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region that allows us to extrapolate the trajectories of particles that an outside observer sees rising up away from

3420-429: A given point. Many physicists, such as Stephen Hawking , Kip Thorne , and others, argued that such effects might make it possible to stabilize a traversable wormhole. The only known natural process that is theoretically predicted to form a wormhole in the context of general relativity and quantum mechanics was put forth by Juan Maldacena and Leonard Susskind in their ER = EPR conjecture. The quantum foam hypothesis

3591-597: A gravitational field (cf. below ). 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 , a crucial guiding principle for generalizing special-relativistic physics to include gravity. The same experimental data shows that time as measured by clocks in

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3762-471: 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 is not measured by the Minkowski metric . As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with

3933-399: A hole when rendered in three dimensions but not in four or higher dimensions. In 1995, Matt Visser suggested there may be many wormholes in the universe if cosmic strings with negative mass were generated in the early universe . Some physicists, such as Kip Thorne , have suggested how to make wormholes artificially. For a simplified notion of a wormhole, space can be visualized as

4104-471: A hyperplane r = 2 m {\displaystyle r=2m} or u = 0 {\displaystyle u=0} in which g {\displaystyle g} vanishes. We call such a connection between the two sheets a "bridge". For the combined field, gravity and electricity, Einstein and Rosen derived the following Schwarzschild static spherically symmetric solution where ε {\displaystyle \varepsilon }

4275-448: A less extreme version of the more dramatic case in which a particle passes through the event horizon and dwells inside it forever. Intermediate between the case of Mercury and the case of an object falling past the event horizon, there are exotic possibilities such as knife-edge orbits, in which the satellite can be made to execute an arbitrarily large number of nearly circular orbits, after which it flies back outward. The isometry group of

4446-450: A massive central body M is given by A conservative total force can then be obtained as its negative gradient where L is the angular momentum . The first term represents the force of Newtonian gravity , which is described by the inverse-square law. The second term represents the centrifugal force in the circular motion. The third term represents the relativistic effect. There are alternatives to general relativity built upon

4617-431: A metric, a spherically symmetric solution of Einstein's equations, which we now know is coordinate transformation of the Schwarzschild metric, Gullstrand–Painlevé coordinates , in which there was no singularity at r = r s . They, however, did not recognize that their solutions were just coordinate transforms, and in fact used their solution to argue that Einstein's theory was wrong. In 1924 Arthur Eddington produced

4788-409: A model of closed timelike curves can have internal inconsistencies as it will lead to strange phenomena like distinguishing non-orthogonal quantum states and distinguishing proper and improper mixture. Accordingly, the destructive positive feedback loop of virtual particles circulating through a wormhole time machine, a result indicated by semi-classical calculations, is averted. A particle returning from

4959-430: A more realistic black hole that forms at some particular time from a collapsing star would require a different metric. When the infalling stellar matter is added to a diagram of a black hole's geography, it removes the part of the diagram corresponding to the white hole interior region, along with the part of the diagram corresponding to the other universe. The Einstein–Rosen bridge was discovered by Ludwig Flamm in 1916,

5130-717: A number of exact solutions are known, although only a few have direct physical applications. The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution , the Reissner–Nordström solution and the Kerr metric , each corresponding to a certain type of black hole in an otherwise empty universe, and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes , each describing an expanding cosmos. Exact solutions of great theoretical interest include

5301-442: A prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics . These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall , and the propagation of light, and include gravitational time dilation , gravitational lensing ,

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5472-642: A stable circular orbit with r > 3 r s . Circular orbits with r between 1.5 r s and 3 r s are unstable, and no circular orbits exist for r < 1.5 r s . The circular orbit of minimum radius 1.5 r s corresponds to an orbital velocity approaching the speed of light. It is possible for a particle to have a constant value of r between r s and 1.5 r s , but only if some force acts to keep it there. Noncircular orbits, such as Mercury 's, dwell longer at small radii than would be expected in Newtonian gravity . This can be seen as

5643-410: A tool for teaching general relativity. For this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter , is also known as a Morris–Thorne wormhole . Later, other types of traversable wormholes were discovered as allowable solutions to the equations of general relativity, including a variety analyzed in a 1989 paper by Matt Visser, in which a path through

5814-410: A traversable wormhole might hypothetically work in the following way: One end of the wormhole is accelerated to some significant fraction of the speed of light, perhaps with some advanced propulsion system , and then brought back to the point of origin. Alternatively, another way is to take one entrance of the wormhole and move it to within the gravitational field of an object that has higher gravity than

5985-401: A two-dimensional surface. In this case, a wormhole would appear as a hole in that surface, lead into a 3D tube (the inside surface of a cylinder ), then re-emerge at another location on the 2D surface with a hole similar to the entrance. An actual wormhole would be analogous to this, but with the spatial dimensions raised by one. For example, instead of circular holes on a 2Dimensional plane ,

6156-470: A unified description of gravity as a geometric property of space and time , or four-dimensional spacetime . In particular, the curvature of spacetime is directly related to the energy and momentum of whatever present matter and radiation . The relation is specified by the Einstein field equations , a system of second-order partial differential equations . Newton's law of universal gravitation , which describes classical gravity, can be seen as

6327-490: A university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated." General relativity can be understood by examining its similarities with and departures from classical physics. The first step

6498-473: A very large amount of negative energy would be required, later calculations showed that the amount of negative energy can be made arbitrarily small. In 1993, Matt Visser argued that the two mouths of a wormhole with such an induced clock difference could not be brought together without inducing quantum field and gravitational effects that would either make the wormhole collapse or the two mouths repel each other, or otherwise prevent information from passing through

6669-539: A wave train traveling through empty space or Gowdy universes , varieties of an expanding cosmos filled with gravitational waves. But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models. 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

6840-432: A wormhole could connect these two points by folding that plane (⁠ i.e. the paper) so the points are touching. In this way, it would be much easier to traverse the distance since the two points are now touching. In 1928, German mathematician, philosopher and theoretical physicist Hermann Weyl proposed a wormhole hypothesis of matter in connection with mass analysis of electromagnetic field energy; however, he did not use

7011-412: A wormhole to travel back to a time earlier than when the wormhole was first converted into a time "machine". Until this time it could not have been noticed or have been used. To see why exotic matter is required, consider an incoming light front traveling along geodesics, which then crosses the wormhole and re-expands on the other side. The expansion goes from negative to positive. As the wormhole neck

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7182-442: A wormhole whose length is shorter than the distance between them outside the wormhole, the time taken to traverse it could be less than the time it would take a light beam to make the journey if it took a path through the space outside the wormhole. However, a light beam traveling through the same wormhole would beat the traveler. If traversable wormholes exist, they might allow time travel . A proposed time-travel machine using

7353-526: Is Minkowskian , and the laws of physics exhibit local Lorentz invariance . The core concept of general-relativistic model-building is that of a solution of Einstein's equations . Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi- Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular,

7524-524: Is 3 dimensional Euclidean space, and S 2 ⊂ E 3 {\displaystyle S^{2}\subset E^{3}} is the two sphere. The rotation group S O ( 3 ) = S O ( E 3 ) {\displaystyle \mathrm {SO} (3)=\mathrm {SO} (E^{3})} acts on the E 3 − O {\displaystyle E^{3}-O} or S 2 {\displaystyle S^{2}} factor as rotations around

7695-471: Is a form of gravitational soliton . The spatial curvature of the Schwarzschild solution for r > r s can be visualized as the graphic shows. Consider a constant time equatorial slice H through the Schwarzschild solution by fixing θ = ⁠ π / 2 ⁠ , t = constant, and letting the remaining Schwarzschild coordinates ( r , φ ) vary. Imagine now that there is an additional Euclidean dimension w , which has no physical reality (it

7866-452: Is a mathematical surface which is significant in determining the black hole's properties. Any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole. The solution of the Einstein field equations is valid for any mass M , so in principle (within the theory of general relativity) a Schwarzschild black hole of any mass could exist if conditions became sufficiently favorable to allow for its formation. In

8037-423: Is a scalar parameter of motion (e.g. the proper time ), and Γ μ α β {\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }} are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and

8208-480: Is a solution manifold of Einstein's field equations for a vacuum spacetime, modified by inclusion of a scalar field minimally coupled to the Ricci tensor with antiorthodox polarity (negative instead of positive). (Ellis specifically rejected referring to the scalar field as 'exotic' because of the antiorthodox coupling, finding arguments for doing so unpersuasive.) The solution depends on two parameters: m , which fixes

8379-445: Is 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 an enclosed room, it

8550-406: Is an illusion however; it is an instance of what is called a coordinate singularity . As the name implies, the singularity arises from a bad choice of coordinates or coordinate conditions . When changing to a different coordinate system (for example Lemaître coordinates , Eddington–Finkelstein coordinates , Kruskal–Szekeres coordinates , Novikov coordinates, or Gullstrand–Painlevé coordinates )

8721-446: Is an intrinsic curvature singularity. It also seems to have a singularity on the event horizon r = r s . Depending on the point of view, the metric is therefore defined only on the exterior region r > r s {\displaystyle r>r_{\text{s}}} , only on the interior region r < r s {\displaystyle r<r_{\text{s}}} or their disjoint union. However,

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8892-402: Is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred 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, that is, the way in which the frequency of light shifts as the light propagates through

9063-405: Is called a Schwarzschild black hole . It is a perfectly valid solution of the Einstein field equations, although (like other black holes) it has rather bizarre properties. For r < r s the Schwarzschild radial coordinate r becomes timelike and the time coordinate t becomes spacelike . A curve at constant r is no longer a possible worldline of a particle or observer, not even if

9234-432: Is characterized by a surrounding spherical boundary, called the event horizon , which is situated at the Schwarzschild radius ( r s {\displaystyle r_{\text{s}}} ), often called the radius of a black hole. The boundary is not a physical surface, and a person who fell through the event horizon (before being torn apart by tidal forces) would not notice any physical surface at that position; it

9405-499: Is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which 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. As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics,

9576-405: Is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, 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

9747-723: Is extremely small. For example, the Schwarzschild radius r s ( Earth ) {\displaystyle r_{\text{s}}^{({\text{Earth}})}} of the Earth is roughly 8.9 mm , while the Sun, which is 3.3 × 10 times as massive has a Schwarzschild radius r s ( Sun ) {\displaystyle r_{\text{s}}^{({\text{Sun}})}} of approximately 3.0 km. The ratio becomes large only in close proximity to black holes and other ultra-dense objects such as neutron stars . The radial coordinate turns out to have physical significance as

9918-491: Is free from singularities for all finite points in the space of the two sheets General relativity General relativity , also known as the general theory of relativity , and as Einstein's theory of gravity , is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics . General relativity generalizes special relativity and refines Newton's law of universal gravitation , providing

10089-445: Is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration. Given the universality of free fall, there is no observable distinction between inertial motion and motion under

10260-508: Is known as gravitational time dilation. Gravitational redshift has been measured in the laboratory and using astronomical observations. Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks , while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS). Tests in stronger gravitational fields are provided by

10431-404: Is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor , which includes both energy and momentum densities as well as stress : pressure and shear. Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that

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10602-456: Is merely a limiting case of (special) relativistic mechanics. In the language of symmetry : where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group , which includes translations, rotations, boosts and reflections.) The differences between

10773-418: Is more of a path through time rather than it is a device that itself moves through time, and it would not allow the technology itself to be moved backward in time. According to current theories on the nature of wormholes, construction of a traversable wormhole would require the existence of a substance with negative energy, often referred to as " exotic matter ". More technically, the wormhole spacetime requires

10944-453: Is not part of spacetime). Then replace the ( r , φ ) plane with a surface dimpled in the w direction according to the equation ( Flamm's paraboloid ) This surface has the property that distances measured within it match distances in the Schwarzschild metric, because with the definition of w above, Thus, Flamm's paraboloid is useful for visualizing the spatial curvature of the Schwarzschild metric. It should not, however, be confused with

11115-430: Is now associated with electrically charged black holes . In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant —to match that observational presumption. By 1929, however, the work of Hubble and others had shown that

11286-508: Is of finite size, we would not expect caustics to develop, at least within the vicinity of the neck. According to the optical Raychaudhuri's theorem , this requires a violation of the averaged null energy condition . Quantum effects such as the Casimir effect cannot violate the averaged null energy condition in any neighborhood of space with zero curvature, but calculations in semiclassical gravity suggest that quantum effects may be able to violate this condition in curved spacetime. Although it

11457-401: Is possible with R gravity, a form of f ( R ) gravity . The impossibility of faster-than-light relative speed applies only locally. Wormholes might allow effective superluminal ( faster-than-light ) travel by ensuring that the speed of light is not exceeded locally at any time. While traveling through a wormhole, subluminal (slower-than-light) speeds are used. If two points are connected by

11628-404: Is possible. A possible resolution to the paradoxes resulting from wormhole-enabled time travel rests on the many-worlds interpretation of quantum mechanics . In 1991 David Deutsch showed that quantum theory is fully consistent (in the sense that the so-called density matrix can be made free of discontinuities) in spacetimes with closed timelike curves. However, later it was shown that such

11799-615: Is sometimes used to suggest that tiny wormholes might appear and disappear spontaneously at the Planck scale , and stable versions of such wormholes have been suggested as dark matter candidates. It has also been proposed that, if a tiny wormhole held open by a negative mass cosmic string had appeared around the time of the Big Bang , it could have been inflated to macroscopic size by cosmic inflation . Lorentzian traversable wormholes would allow travel in both directions from one part of

11970-495: Is the Shapiro Time 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. In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on

12141-429: Is the electric charge. The field equations without denominators in the case when m = 0 {\displaystyle m=0} can be written In order to eliminate singularities, if one replaces r {\displaystyle r} by u {\displaystyle u} according to the equation: and with m = 0 {\displaystyle m=0} one obtains The solution

12312-417: Is the metric on the two sphere, i.e. ⁠ d Ω 2 = ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) {\displaystyle {d\Omega }^{2}=\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)} ⁠ . Furthermore, The Schwarzschild metric has a singularity for r = 0 , which

12483-444: Is the one that is related to the gravitational fields of stars and planets. The interior Schwarzschild solution with 0 ≤ r < r s , which contains the singularity at r = 0 , is completely separated from the outer patch by the singularity at r = r s . The Schwarzschild coordinates therefore give no physical connection between the two patches, which may be viewed as separate solutions. The singularity at r = r s

12654-409: Is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories. General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication. Assuming that

12825-573: Is the proper time and c = 1 {\displaystyle c=1} . If one replaces r {\displaystyle r} with u {\displaystyle u} according to u 2 = r − 2 m {\displaystyle u^{2}=r-2m} The four-dimensional space is described mathematically by two congruent parts or "sheets", corresponding to u > 0 {\displaystyle u>0} and u < 0 {\displaystyle u<0} , which are joined by

12996-471: Is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity. At the base of classical mechanics is the notion that a body 's motion can be described as a combination of free (or inertial ) motion, and deviations from this free motion. Such deviations are caused by external forces acting on

13167-459: Is used for the metric of a unit radius 2-dimensional sphere. Moreover, in each entry R and T denote alternative choices of radial and time coordinate for the particular coordinates. Note, the R or T may vary from entry to entry. The Kruskal–Szekeres coordinates have the form to which the Belinski–Zakharov transform can be applied. This implies that the Schwarzschild black hole

13338-507: Is valid for r > R {\displaystyle r>R} . To describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at ⁠ r = R {\displaystyle r=R} ⁠ , such as the interior Schwarzschild metric . In Schwarzschild coordinates ( t , r , θ , ϕ ) {\displaystyle (t,r,\theta ,\phi )}

13509-477: Is written Letting the surface be described by the function w = w ( r ) , the Euclidean metric can be written as Comparing this with the Schwarzschild metric in the equatorial plane ( θ = ⁠ π / 2 ⁠ ) at a fixed time ( t = constant, dt = 0 ) yields an integral expression for w ( r ) : whose solution is Flamm's paraboloid. A particle orbiting in the Schwarzschild metric can have

13680-434: The Einstein field equations , and that are now understood to be intrinsic parts of the maximally extended version of the Schwarzschild metric describing an eternal black hole with no charge and no rotation. Here, "maximally extended" refers to the idea that the spacetime should not have any "edges": it should be possible to continue this path arbitrarily far into the particle's future or past for any possible trajectory of

13851-439: The Einstein field equations . In Schwarzschild's original paper, he put what we now call the event horizon at the origin of his coordinate system. In this paper he also introduced what is now known as the Schwarzschild radial coordinate ( r in the equations above), as an auxiliary variable. In his equations, Schwarzschild was using a different radial coordinate that was zero at the Schwarzschild radius. A more complete analysis of

14022-432: The Einstein notation , meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a test particle whose motion is described by the geodesic equation. In general relativity, the effective gravitational potential energy of an object of mass m revolving around

14193-609: The Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub–NUT solution (a model universe that is homogeneous , but anisotropic ), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture ). Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on

14364-541: The Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present. A version of non-Euclidean geometry , called Riemannian geometry , enabled Einstein to develop general relativity by providing

14535-483: The Schwarzschild metric (also known as the Schwarzschild solution ) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets , including Earth and

14706-631: The field equation for gravity relates this tensor and the Ricci tensor , which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy –momentum corresponds to the statement that the energy–momentum tensor is divergence -free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved- manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of

14877-506: The gravitational redshift of light, the Shapiro time delay and singularities / black holes . So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology , thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite

15048-473: The post-Newtonian expansion , both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity. An extension of this expansion

15219-454: The scalar gravitational potential of classical physics by a symmetric rank -two tensor , the latter reduces to the former in certain limiting cases . For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation. As it is constructed using tensors, general relativity exhibits general covariance : its laws—and further laws formulated within

15390-429: The summation convention is used for repeated indices α {\displaystyle \alpha } and β {\displaystyle \beta } . The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to Newton's laws of motion which likewise provide formulae for the acceleration of a particle. This equation of motion employs

15561-420: The torsion tensor , as a dynamic variable. Torsion naturally accounts for the quantum-mechanical, intrinsic angular momentum ( spin ) of matter. The minimal coupling between torsion and Dirac spinors generates a repulsive spin–spin interaction that is significant in fermionic matter at extremely high densities. Such an interaction prevents the formation of a gravitational singularity (e.g. a black hole). Instead,

15732-419: The "proper distance between two events that occur simultaneously relative to the radially moving geodesic clocks, the two events lying on the same radial coordinate line". The Schwarzschild solution is analogous to a classical Newtonian theory of gravity that corresponds to the gravitational field around a point particle. Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in

15903-482: 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 1859, was important evidence that he had at last identified the correct form of the gravitational field equations. Schwarzschild metric In Einstein 's theory of general relativity ,

16074-778: The Poincaré group, it has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time reversed and spatially inverted. The Ricci curvature scalar and the Ricci curvature tensor are both zero. Non-zero components of the Riemann curvature tensor are given by from which one can see that R γ α γ β = 0 {\displaystyle R^{\gamma }{}_{\alpha \gamma \beta }=0} . Six of these formulas are Eq. 5.13 in Carroll and imply

16245-464: The Riemann tensor are not displayed. To understand the physical meaning of these quantities, it is useful to express the curvature tensor in an orthonormal basis. In an orthonormal basis of an observer the non-zero components in geometric units are Again, components which are obtainable by the symmetries of the Riemann tensor are not displayed. These results are invariant to any Lorentz boost, thus

16416-513: The Schwarzchild metric is ⁠ R × O ( 3 ) × { ± 1 } {\displaystyle \mathbb {R} \times \mathrm {O} (3)\times \{\pm 1\}} ⁠ , where O ( 3 ) {\displaystyle \mathrm {O} (3)} is the orthogonal group of rotations and reflections in three dimensions, R {\displaystyle \mathbb {R} } comprises

16587-796: The Schwarzschild metric (or equivalently, the line element for proper time ) has the form d s 2 = c 2 d τ 2 = ( 1 − r s r ) c 2 d t 2 − ( 1 − r s r ) − 1 d r 2 − r 2 d Ω 2 , {\displaystyle {ds}^{2}=c^{2}\,{d\tau }^{2}=\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)c^{2}\,dt^{2}-\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}-r^{2}{d\Omega }^{2},} where d Ω 2 {\displaystyle {d\Omega }^{2}}

16758-421: The Schwarzschild metric as an event horizon , i.e., a hypersurface in spacetime that can be crossed in only one direction. The Schwarzschild solution appears to have singularities at r = 0 and r = r s ; some of the metric components "blow up" (entail division by zero or multiplication by infinity) at these radii. Since the Schwarzschild metric is expected to be valid only for those radii larger than

16929-458: The Schwarzschild metric, again showing that the singularity at r = r s was a coordinate artifact and that it represented two horizons. A similar result was later rediscovered by George Szekeres , and independently Martin Kruskal . The new coordinates nowadays known as Kruskal–Szekeres coordinates were much simpler than Synge's but both provided a single set of coordinates that covered

17100-472: The Schwarzschild radius has undergone gravitational collapse and become a black hole. The Schwarzschild solution can be expressed in a range of different choices of coordinates besides the Schwarzschild coordinates used above. Different choices tend to highlight different features of the solution. The table below shows some popular choices. In table above, some shorthand has been introduced for brevity. The speed of light c has been set to one . The notation

17271-539: The Sun. It was found by Karl Schwarzschild in 1916. According to Birkhoff's theorem , the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has neither electric charge nor angular momentum (non-rotating). A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass. The Schwarzschild black hole

17442-413: 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 is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment ), there

17613-408: The center O {\displaystyle O} , while leaving the first R {\displaystyle \mathbb {R} } factor unchanged. The Schwarzschild metric is a solution of Einstein's field equations in empty space, meaning that it is valid only outside the gravitating body. That is, for a spherical body of radius R {\displaystyle R} the solution

17784-522: The collapsing matter reaches an enormous but finite density and rebounds, forming the other side of the bridge. Although Schwarzschild wormholes are not traversable in both directions, their existence inspired Kip Thorne to imagine traversable wormholes created by holding the "throat" of a Schwarzschild wormhole open with exotic matter (material that has negative mass/energy). Other non-traversable wormholes include Lorentzian wormholes (first proposed by John Archibald Wheeler in 1957), wormholes creating

17955-712: The components do not change for non-static observers. The geodesic deviation equation shows that the tidal acceleration between two observers separated by ξ j ^ {\displaystyle \xi ^{\hat {j}}} is D 2 ξ j ^ / D τ 2 = − R j ^ t ^ k ^ t ^ ξ k ^ {\displaystyle D^{2}\xi ^{\hat {j}}/D\tau ^{2}=-R^{\hat {j}}{}_{{\hat {t}}{\hat {k}}{\hat {t}}}\xi ^{\hat {k}}} , so

18126-406: The connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates , the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish). Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source

18297-511: The deflection of starlight by the Sun during the total solar eclipse of 29 May 1919 , instantly making Einstein famous. Yet the theory remained outside the mainstream of theoretical physics and astrophysics until developments between approximately 1960 and 1975, now known as the golden age of general relativity . Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations. Ever more precise solar system tests confirmed

18468-452: 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

18639-555: The energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations: G μ ν ≡ R μ ν − 1 2 R g μ ν = κ T μ ν {\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,} On

18810-601: The entire spacetime. However, perhaps due to the obscurity of the journals in which the papers of Lemaître and Synge were published their conclusions went unnoticed, with many of the major players in the field including Einstein believing that the singularity at the Schwarzschild radius was physical. Synge's later derivation of the Kruskal–Szekeres metric solution, which was motivated by a desire to avoid "using 'bad' [Schwarzschild] coordinates to obtain 'good' [Kruskal–Szekeres] coordinates", has been generally under-appreciated in

18981-413: The entry and exit points could be visualized as spherical holes in 3D space leading into a four-dimensional "tube" similar to a spherinder . Another way to imagine wormholes is to take a sheet of paper and draw two somewhat distant points on one side of the paper. The sheet of paper represents a plane in the spacetime continuum , and the two points represent a distance to be traveled, but theoretically,

19152-446: The equivalence principle holds, gravity influences the passage of time. Light sent down into a gravity well is blueshifted , whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted ; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect

19323-499: The event horizon. And just as there are two separate interior regions of the maximally extended spacetime, there are also two separate exterior regions, sometimes called two different "universes", with the second universe allowing us to extrapolate some possible particle trajectories in the two interior regions. This means that the interior black hole region can contain a mix of particles that fell in from either universe (and thus an observer who fell in from one universe might be able to see

19494-456: The exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by 10 − 21 {\displaystyle 10^{-21}} or less. 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.g.,

19665-408: The exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion), several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity. Closely related to light deflection

19836-412: The first coordinate transformation ( Eddington–Finkelstein coordinates ) that showed that the singularity at r = r s was a coordinate artifact, although he also seems to have been unaware of the significance of this discovery. Later, in 1932, Georges Lemaître gave a different coordinate transformation ( Lemaître coordinates ) to the same effect and was the first to recognize that this implied that

20007-433: The first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric . This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, eventually resulting in the Reissner–Nordström solution , which

20178-745: The future does not return to its universe of origination but to a parallel universe. This suggests that a wormhole time machine with an exceedingly short time jump is a theoretical bridge between contemporaneous parallel universes. Because a wormhole time-machine introduces a type of nonlinearity into quantum theory, this sort of communication between parallel universes is consistent with Joseph Polchinski 's proposal of an Everett phone (named after Hugh Everett ) in Steven Weinberg 's formulation of nonlinear quantum mechanics. The possibility of communication between parallel universes has been dubbed interuniversal travel . Wormhole can also be depicted in

20349-412: The general relativistic framework—take on the same form in all coordinate systems . Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent . It thus satisfies a more stringent general principle of relativity , namely that the laws of physics are the same for all observers. Locally , as expressed in the equivalence principle, spacetime

20520-484: The geometry of space. Predicted in 1916 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 11 February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from

20691-441: 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. In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure or conformal geometry. Special relativity

20862-464: The incremental deformation of closed surfaces. For example, in Enrico Rodrigo's The Physics of Stargates, a wormhole is defined informally as: a region of spacetime containing a " world tube " (the time evolution of a closed surface) that cannot be continuously deformed (shrunk) to a world line (the time evolution of a point or observer). The first type of wormhole solution discovered

21033-446: 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 is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential . Space, in this construction, still has

21204-512: The introduction of a number of alternative theories , general relativity continues to be the simplest theory consistent with experimental data . Reconciliation of general relativity with the laws of quantum physics remains a problem, however, as there is a lack of a self-consistent theory of quantum gravity . It is not yet known how gravity can be unified with the three non-gravitational forces: strong , weak and electromagnetic . Einstein's theory has astrophysical implications, including

21375-417: The key mathematical framework on which he fit his physical ideas of gravity. This idea was pointed out by mathematician Marcel Grossmann and published by Grossmann and Einstein in 1913. The Einstein field equations are nonlinear and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist Karl Schwarzschild found

21546-410: 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 is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory

21717-457: The left-hand side is the Einstein tensor , G μ ν {\displaystyle G_{\mu \nu }} , which is symmetric and a specific divergence-free combination of the Ricci tensor R μ ν {\displaystyle R_{\mu \nu }} and the metric. In particular, is the curvature scalar. The Ricci tensor itself is related to

21888-477: The light of stars or distant quasars being deflected as it passes the Sun . This and related predictions follow from the fact that light follows what is called a light-like or null geodesic —a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity. As one examines suitable model spacetimes (either

22059-431: The light that fell in from the other one), and likewise particles from the interior white hole region can escape into either universe. All four regions can be seen in a spacetime diagram that uses Kruskal–Szekeres coordinates . In this spacetime, it is possible to come up with coordinate systems such that if a hypersurface of constant time (a set of points that all have the same time coordinate, such that every point on

22230-413: The literature, but was adopted by Chandrasekhar in his black hole monograph. Real progress was made in the 1960s when the mathematically rigorous formulation cast in terms of differential geometry entered the field of general relativity, allowing more exact definitions of what it means for a Lorentzian manifold to be singular. This led to definitive identification of the r = r s singularity in

22401-455: The matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present. Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless,

22572-410: The metric becomes regular at r = r s and can extend the external patch to values of r smaller than r s . Using a different coordinate transformation one can then relate the extended external patch to the inner patch. The case r = 0 is different, however. If one asks that the solution be valid for all r one runs into a true physical singularity, or gravitational singularity , at

22743-450: The metric is actually non-singular across the event horizon, as one sees in suitable coordinates (see below). For ⁠ r ≫ r s {\displaystyle r\gg r_{\text{s}}} ⁠ , the Schwarzschild metric is asymptotic to the standard Lorentz metric on Minkowski space. For almost all astrophysical objects, the ratio r s R {\displaystyle {\frac {r_{\text{s}}}{R}}}

22914-466: The metric), spacetime itself is then no longer well-defined. Furthermore, Sbierski showed the metric cannot be extended even in a continuous manner. For a long time it was thought that such a solution was non-physical. However, a greater understanding of general relativity led to the realization that such singularities were a generic feature of the theory and not just an exotic special case. The Schwarzschild solution, taken to be valid for all r > 0 ,

23085-442: The more general Riemann curvature tensor as On the right-hand side, κ {\displaystyle \kappa } is a constant and T μ ν {\displaystyle T_{\mu \nu }} is the energy–momentum tensor. All tensors are written in abstract index notation . Matching the theory's prediction to observational results for planetary orbits or, equivalently, assuring that

23256-424: The most beautiful of all existing physical theories. Henri Poincaré 's 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that gravitational waves propagate at

23427-432: The observation of binary pulsars . All results are in agreement with general relativity. However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid. General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing

23598-459: The ordinary Euclidean geometry . However, space time as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable . From this, one can deduce that spacetime

23769-536: The origin. To see that this is a true singularity one must look at quantities that are independent of the choice of coordinates. One such important quantity is the Kretschmann invariant , which is given by At r = 0 the curvature becomes infinite, indicating the presence of a singularity. At this point the metric cannot be extended in a smooth manner (the Kretschmann invariant involves second derivatives of

23940-458: The other 6 by R α β γ δ = g α κ g β λ R λ κ δ γ {\displaystyle R^{\alpha }{}_{\beta \gamma \delta }=g^{\alpha \kappa }g_{\beta \lambda }R^{\lambda }{}_{\kappa \delta \gamma }} . Components which are obtainable by other symmetries of

24111-473: The other entrance, and then return it to a position near the other entrance. For both these methods, time dilation causes the end of the wormhole that has been moved to have aged less, or become "younger", than the stationary end as seen by an external observer; however, time connects differently through the wormhole than outside it, so that synchronized clocks at either end of the wormhole will always remain synchronized as seen by an observer passing through

24282-475: The other exterior region. According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular Schwarzschild black hole. In the Einstein–Cartan –Sciama–Kibble theory of gravity, however, it forms a regular Einstein–Rosen bridge. This theory extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part,

24453-421: The prediction of black holes —regions of space in which space and time are distorted in such a way that nothing, not even light , can escape from them. Black holes are the end-state for massive stars . Microquasars and active galactic nuclei are believed to be stellar black holes and supermassive black holes . It also predicts gravitational lensing , where the bending of light results in multiple images of

24624-511: The preface to Relativity: The Special and the General Theory , Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of

24795-430: The principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory. Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency. In

24966-443: The radius R of the gravitating body, there is no problem as long as R > r s . For ordinary stars and planets this is always the case. For example, the radius of the Sun is approximately 700 000  km , while its Schwarzschild radius is only 3 km . The singularity at r = r s divides the Schwarzschild coordinates in two disconnected patches . The exterior Schwarzschild solution with r > r s

25137-419: The same distant astronomical phenomenon. Other predictions include the existence of gravitational waves , which have been observed directly by the physics collaboration LIGO and other observatories. In addition, general relativity has provided the base of cosmological models of an expanding universe . Widely acknowledged as a theory of extraordinary beauty , general relativity has often been described as

25308-446: The same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Whitehead's theory , Brans–Dicke theory , teleparallelism , f ( R ) gravity and Einstein–Cartan theory . The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how

25479-415: The singularity at r = r s was not physical. In 1939 Howard Robertson showed that a free falling observer descending in the Schwarzschild metric would cross the r = r s singularity in a finite amount of proper time even though this would take an infinite amount of time in terms of coordinate time t . In 1950, John Synge produced a paper that showed the maximal analytic extension of

25650-408: The singularity structure was given by David Hilbert in the following year, identifying the singularities both at r = 0 and r = r s . Although there was general consensus that the singularity at r = 0 was a 'genuine' physical singularity, the nature of the singularity at r = r s remained unclear. In 1921, Paul Painlevé and in 1922 Allvar Gullstrand independently produced

25821-472: The speed of light in vacuum. When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations, In general relativity, the world line of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic. The geodesic equation is: where s {\displaystyle s}

25992-399: The speed of light. Soon afterwards, Einstein started thinking about how to incorporate gravity into his relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall (FFO), he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to

26163-469: The strength of its gravitational field, and n , which determines the curvature of its spatial cross sections. When m is set equal to 0, the drainhole's gravitational field vanishes. What is left is the Ellis wormhole , a nongravitating, purely geometric, traversable wormhole. Kip Thorne and his graduate student Mike Morris independently discovered in 1988 the Ellis wormhole and argued for its use as

26334-421: The surface has a space-like separation, giving what is called a 'space-like surface') is picked and an "embedding diagram" drawn depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an "Einstein–Rosen bridge". The Schwarzschild metric describes an idealized black hole that exists eternally from the perspective of external observers;

26505-488: The term "wormhole" (he spoke of "one-dimensional tubes" instead). American theoretical physicist John Archibald Wheeler (inspired by Weyl's work) coined the term "wormhole" in a 1957 paper he wrote with Charles W. Misner : This analysis forces one to consider situations ... where there is a net flux of lines of force, through what topologists would call "a handle " of the multiply-connected space, and what physicists might perhaps be excused for more vividly terming

26676-518: The theory can be used for model-building. General relativity is a metric theory of gravitation. At its core are Einstein's equations , which describe the relation between the geometry of a four-dimensional pseudo-Riemannian manifold representing spacetime, and the energy–momentum contained in that spacetime. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall , orbital motion, and spacecraft trajectories ), correspond to inertial motion within

26847-644: The theory's predictive power, and relativistic cosmology also became amenable to direct observational tests. General relativity has acquired a reputation as a theory of extraordinary beauty. Subrahmanyan Chandrasekhar has noted that at multiple levels, general relativity exhibits what Francis Bacon has termed a "strangeness in the proportion" ( i.e . elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time versus matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were

27018-475: The time translations, and { ± 1 } {\displaystyle \{\pm 1\}} is the group generated by time reversal. This is thus the subgroup of the ten-dimensional Poincaré group which takes the time axis (trajectory of the star) to itself. It omits the spatial translations (three dimensions) and boosts (three dimensions). It retains the time translations (one dimension) and rotations (three dimensions). Thus it has four dimensions. Like

27189-403: The trajectory one gets through a "rubber sheet" analogy of gravitational well: in particular, if the dimple is drawn pointing upward rather than downward, the tachyon's geodesic path still curves toward the central mass, not away. See the gravity well article for more information. Flamm's paraboloid may be derived as follows. The Euclidean metric in the cylindrical coordinates ( r , φ , w )

27360-487: The two become significant when dealing with speeds approaching the speed of light , and with high-energy phenomena. With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event A , there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in

27531-489: The universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which the universe has evolved from an extremely hot and dense earlier state. Einstein later declared the cosmological constant the biggest blunder of his life. During that period, general relativity remained something of

27702-488: The universe to another part of that same universe very quickly or would allow travel from one universe to another. The possibility of traversable wormholes in general relativity was first demonstrated in a 1973 paper by Homer Ellis and independently in a 1973 paper by K. A. Bronnikov. Ellis analyzed the topology and the geodesics of the Ellis drainhole , showing it to be geodesically complete, horizonless, singularity-free, and fully traversable in both directions. The drainhole

27873-716: The vicinity of a Schwarzschild black hole, space curves so much that even light rays are deflected, and very nearby light can be deflected so much that it travels several times around the black hole. The Schwarzschild metric is a spherically symmetric Lorentzian metric (here, with signature convention (+, -, -, -) ), defined on (a subset of) R × ( E 3 − O ) ≅ R × ( 0 , ∞ ) × S 2 {\displaystyle \mathbb {R} \times \left(E^{3}-O\right)\cong \mathbb {R} \times (0,\infty )\times S^{2}} where E 3 {\displaystyle E^{3}}

28044-438: The weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant κ {\displaystyle \kappa } is found to be κ = 8 π G c 4 {\textstyle \kappa ={\frac {8\pi G}{c^{4}}}} , where G {\displaystyle G} is the Newtonian constant of gravitation and c {\displaystyle c}

28215-426: The wormhole can be made where the traversing path does not pass through a region of exotic matter. However, in the pure Gauss–Bonnet gravity (a modification to general relativity involving extra spatial dimensions which is sometimes studied in the context of brane cosmology ) exotic matter is not needed in order for wormholes to exist—they can exist even with no matter. A type held open by negative mass cosmic strings

28386-411: The wormhole, no matter how the two ends move around. This means that an observer entering the "younger" end would exit the "older" end at a time when it was the same age as the "younger" end, effectively going back in time as seen by an observer from the outside. One significant limitation of such a time machine is that it is only possible to go as far back in time as the initial creation of the machine; it

28557-462: The wormhole. Because of this, the two mouths could not be brought close enough for causality violation to take place. However, in a 1997 paper, Visser hypothesized that a complex " Roman ring " (named after Tom Roman) configuration of an N number of wormholes arranged in a symmetric polygon could still act as a time machine, although he concludes that this is more likely a flaw in classical quantum gravity theory rather than proof that causality violation

28728-428: Was hoped recently that quantum effects could not violate an achronal version of the averaged null energy condition, violations have nevertheless been found, so it remains an open possibility that quantum effects might be used to support a wormhole. In some hypotheses where general relativity is modified , it is possible to have a wormhole that does not collapse without having to resort to exotic matter. For example, this

28899-480: Was published, as a result of a disease (thought to be pemphigus ) he developed while serving in the German army during World War I . Johannes Droste in 1916 independently produced the same solution as Schwarzschild, using a simpler, more direct derivation. In the early years of general relativity there was a lot of confusion about the nature of the singularities found in the Schwarzschild and other solutions of

29070-527: Was put forth by Visser in collaboration with Cramer et al. , in which it was proposed that such wormholes could have been naturally created in the early universe. Wormholes connect two points in spacetime, which means that they would in principle allow travel in time , as well as in space. In 1988, Morris, Thorne and Yurtsever worked out how to convert a wormhole traversing space into one traversing time by accelerating one of its two mouths. However, according to general relativity, it would not be possible to use

29241-667: Was the Schwarzschild wormhole, which would be present in the Schwarzschild metric describing an eternal black hole , but it was found that it would collapse too quickly for anything to cross from one end to the other. Wormholes that could be crossed in both directions, known as traversable wormholes , were thought to be possible only if exotic matter with negative energy density could be used to stabilize them. However, physicists later reported that microscopic traversable wormholes may be possible and not require any exotic matter, instead requiring only electrically charged fermionic matter with small enough mass that it cannot collapse into

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