Wormhole
A wormhole is a speculative structure linking disparate points in spacetime, is based on a special solution of the Einstein field equations solved using a Jacobian matrix and determinant. A wormhole can be visualized as a tunnel with two ends, each at separate points in spacetime. More it is a transcendental bijection of the spacetime continuum, an asymptotic projection of the Calabi–Yau manifold manifesting itself in Anti-de Sitter space. Wormholes are consistent with the general theory of relativity, but whether wormholes exist remains to be seen. A wormhole could connect long distances such as a billion light years or more, short distances such as a few meters, different universes, or different points in time. For a simplified notion of a wormhole, space can be visualized as a two-dimensional surface. In this case, a wormhole would appear as a hole in that surface, lead into a 3D tube re-emerge at another location on the 2D surface with a hole similar to the entrance. An actual wormhole would be analogous with the spatial dimensions raised by one.
For example, instead of circular holes on a 2D plane, the entry and exit points could be visualized as spheres in 3D space. 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, the two points represent a distance to be traveled, however theoretically a wormhole could connect these two points by folding that plane so the points are touching. In this way it would be much easier to traverse the distance. In 1928, Hermann Weyl proposed a wormhole hypothesis of matter in connection with mass analysis of electromagnetic field energy. American theoretical physicist John Archibald Wheeler coined the term "wormhole" in a 1957 paper co-authored by Charles 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, what physicists might be excused for more vividly terming a "wormhole".
Wormholes have been defined both topologically. From a topological point of view, an intra-universe wormhole is a compact region of spacetime whose boundary is topologically trivial, but whose interior is not connected. Formalizing this idea leads to definitions such as the following, taken from Matt Visser's Lorentzian Wormholes. If a Minkowski spacetime contains a compact region Ω, if the topology of Ω is of the form Ω ~ R × Σ, where Σ is a three-manifold of the nontrivial topology, whose boundary has topology of the form ∂Σ ~ S2, if, the hypersurfaces Σ are all spacelike the region Ω contains a quasipermanent intrauniverse wormhole. Geometrically, wormholes can be described as regions of spacetime that constrain 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" that cannot be continuously deformed to a world line; the equations of the theory of general relativity have valid solutions.
The first type of wormhole solution discovered 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 for anything to cross from one end to the other. Wormholes that could be crossed in both directions, known as traversable wormholes, would only be possible if exotic matter with negative energy density could be used to stabilize them. Schwarzschild wormholes known as Einstein–Rosen bridges, are connections between areas of space that can be modeled as vacuum solutions to the Einstein field equations, 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 a free-falling particle.
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 the event horizon. And just as there are two separate interior regions of the maximally extended spacetime, there are 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, 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
General relativity
General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present; the relation is specified by the Einstein field equations, a system of partial differential equations. Some predictions of general relativity differ from those of classical physics concerning the passage of time, the geometry of space, the motion of bodies in free fall, the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the gravitational time delay; the predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date.
Although general relativity is not the only relativistic theory of gravity, it is the simplest theory, consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity. Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not light, can escape—as an end-state for massive stars. There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes; the bending of light by gravity can lead to the phenomenon of gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity predicts the existence of gravitational waves, which have since been observed directly by the physics collaboration LIGO.
In addition, general relativity is the basis of current cosmological models of a expanding universe. Acknowledged as a theory of extraordinary beauty, general relativity has been described as the most beautiful of all existing physical theories. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall, 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 the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations; these equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, form the core of Einstein's general theory of relativity. The Einstein field equations are nonlinear and difficult to solve.
Einstein used approximation methods in working out initial predictions of the theory. But as early as 1916, the astrophysicist Karl Schwarzschild found 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, 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, which resulted in the Reissner–Nordström solution, 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, the work of Hubble and others had shown that our universe is expanding; this is 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 our universe has evolved from an hot and dense earlier state. Einstein declared the cosmological constant the biggest blunder of his life. During that period, general relativity remained something of a curiosity among physical theories, it was superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein himself had shown in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters. A 1919 expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of May 29, 1919, making Einstein famous, yet the theory entered the mainstream of theoretical physics and astrophysics only with the developments between 1960 and 1975, now known as the golden age of general relativity. Physicists began to understand the concept of a black hole, to identify quasars as one of these objects' astrophysical manifestations.
More precise solar system tests confirmed the theory's predictive power, relativistic cosmology, became amenable to direct observational tests. Over the years, general relativity has acqui
Time dilation
According to the theory of relativity, time dilation is a difference in the elapsed time measured by two observers, either due to a velocity difference relative to each other, or by being differently situated relative to a gravitational field. As a result of the nature of spacetime, a clock, moving relative to an observer will be measured to tick slower than a clock, at rest in the observer's own frame of reference. A clock, under the influence of a stronger gravitational field than an observer's will be measured to tick slower than the observer's own clock; such time dilation has been demonstrated, for instance by small disparities in a pair of atomic clocks after one of them is sent on a space trip, or by clocks on the Space Shuttle running slower than reference clocks on Earth, or clocks on GPS and Galileo satellites running faster. Time dilation has been the subject of science fiction works, as it technically provides the means for forward time travel. Time dilation by the Lorentz factor was predicted by several authors at the turn of the 20th century.
Joseph Larmor, at least for electrons orbiting a nucleus, wrote "... individual electrons describe corresponding parts of their orbits in times shorter for the system in the ratio: 1 − v 2 c 2 ". Emil Cohn related this formula to the rate of clocks. In the context of special relativity it was shown by Albert Einstein that this effect concerns the nature of time itself, he was the first to point out its reciprocity or symmetry. Subsequently, Hermann Minkowski introduced the concept of proper time which further clarified the meaning of time dilation. Special relativity indicates that, for an observer in an inertial frame of reference, a clock, moving relative to him will be measured to tick slower than a clock, at rest in his frame of reference; this case is sometimes called special relativistic time dilation. The faster the relative velocity, the greater the time dilation between one another, with the rate of time reaching zero as one approaches the speed of light; this causes massless particles that travel at the speed of light to be unaffected by the passage of time.
Theoretically, time dilation would make it possible for passengers in a fast-moving vehicle to advance further into the future in a short period of their own time. For sufficiently high speeds, the effect is dramatic. For example, one year of travel might correspond to ten years on Earth. Indeed, a constant 1 g acceleration would permit humans to travel through the entire known Universe in one human lifetime.. With current technology limiting the velocity of space travel, the differences experienced in practice are minuscule: after 6 months on the International Space Station an astronaut would have aged about 0.005 seconds less than those on Earth. The cosmonauts Sergei Krikalev and Sergei Avdeyev both experienced time dilation of about 20 milliseconds compared to time that passed on Earth. Time dilation can be inferred from the observed constancy of the speed of light in all reference frames dictated by the second postulate of special relativity; this constancy of the speed of light means that, counter to intuition, speeds of material objects and light are not additive.
It is not possible to make the speed of light appear greater by moving towards or away from the light source. Consider a simple clock consisting of two mirrors A and B, between which a light pulse is bouncing; the separation of the mirrors is L and the clock ticks once each time the light pulse hits either of the mirrors. In the frame in which the clock is at rest, the light pulse traces out a path of length 2L and the period of the clock is 2L divided by the speed of light: Δ t = 2 L c. From the frame of reference of a moving observer traveling at the speed v relative to the resting frame of the clock, the light pulse is seen as tracing out a longer, angled path. Keeping the speed of light constant for all inertial observers, requires a lengthening of the period of this clock from the moving observer's perspective; that is to say, in a frame moving relative to the local clock, this clock will appear to be running more slowly. Straightforward application of the Pythagorean theorem leads to the well-known prediction of special relativity: The total time for the light pulse to trace its path is given by Δ t ′ = 2 D c.
The length of the half path can be calculated as a function of known quantities as D = 2 + L 2. Elimination of the variables D and L from these three equations results in Δ t ′ = Δ t 1 − v 2 c 2, which expresses the fact that the moving observer's period of the clock Δ t ′
Tests of general relativity
Tests of general relativity serve to establish observational evidence for the theory of general relativity. The first three tests, proposed by Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury, the bending of light in gravitational fields, the gravitational redshift; the precession of Mercury was known. A program of more accurate tests starting in 1959 tested the various predictions of general relativity with a further degree of accuracy in the weak gravitational field limit limiting possible deviations from the theory. In the 1970s, additional tests began to be made, starting with Irwin Shapiro's measurement of the relativistic time delay in radar signal travel time near the sun. Beginning in 1974, Hulse and others have studied the behaviour of binary pulsars experiencing much stronger gravitational fields than those found in the Solar System. Both in the weak field limit and with the stronger fields present in systems of binary pulsars the predictions of general relativity have been well tested locally.
In February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from a black hole merger. This discovery, along with additional detections announced in June 2016 and June 2017, tested general relativity in the strong field limit, observing to date no deviations from theory. Albert Einstein proposed three tests of general relativity, subsequently called the classical tests of general relativity, in 1916: the perihelion precession of Mercury's orbit the deflection of light by the Sun the gravitational redshift of lightIn the letter to the London Times on November 28, 1919, he described the theory of relativity and thanked his English colleagues for their understanding and testing of his work, he mentioned three classical tests with comments: "The chief attraction of the theory lies in its logical completeness. If a single one of the conclusions drawn from it proves wrong, it must be given up. Under Newtonian physics, a two-body system consisting of a lone object orbiting a spherical mass would trace out an ellipse with the center of mass of the system at a focus.
The point of closest approach, called the periapsis, is fixed. A number of effects in the Solar System cause the perihelia of planets to precess around the Sun; the principal cause is the presence of other planets. Another effect is solar oblateness. Mercury deviates from the precession predicted from these Newtonian effects; this anomalous rate of precession of the perihelion of Mercury's orbit was first recognized in 1859 as a problem in celestial mechanics, by Urbain Le Verrier. His reanalysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848 showed that the actual rate of the precession disagreed from that predicted from Newton's theory by 38″ per tropical century. A number of ad hoc and unsuccessful solutions were proposed, but they tended to introduce more problems. In general relativity, this remaining precession, or change of orientation of the orbital ellipse within its orbital plane, is explained by gravitation being mediated by the curvature of spacetime.
Einstein showed that general relativity agrees with the observed amount of perihelion shift. This was a powerful factor motivating the adoption of general relativity. Although earlier measurements of planetary orbits were made using conventional telescopes, more accurate measurements are now made with radar; the total observed precession of Mercury is 574.10″±0.65 per century relative to the inertial ICRF. This precession can be attributed to the following causes: The correction by 42.98″ is 3/2 multiple of classical prediction with PPN parameters γ = β = 1. Thus the effect can be explained by general relativity. More recent calculations based on more precise measurements have not materially changed the situation. In general relativity the perihelion shift σ, expressed in radians per revolution, is given by: σ = 24 π 3 L 2 T 2 c 2, where L is the semi-major axis, T is the orbital period, c is the speed of light, e is the orbital eccentricity; the other planets experience perihelion shifts as well, since they are farther from the Sun and have longer periods, their shifts are lower, could not be observed until long after Mercury's.
For example, the perihelion shift of Earth's orbit due to general relativity is of 3.84″ per century, Venus's is 8.62″. Both values have now been measured, with results in good agreement with theory; the periapsis shift has now been measured for binary pulsar systems, with PSR 1913+16 amounting to 4.2º per year. These observations are consistent with general relativity, it i
Gravity well
A gravity well or gravitational well is a conceptual model of the gravitational field surrounding a body in space – the more massive the body, the deeper and more extensive the gravity well associated with it. The Sun is massive, relative to other bodies in the Solar System, so the corresponding gravity well that surrounds it appears "deep" and far-reaching; the gravity wells of asteroids and small moons, are depicted as shallow. Anything at the center of mass of a planet or moon is considered to be at the bottom of that celestial body's gravity well, so escaping the effects of gravity from such a planet or moon is sometimes called "climbing out of the gravity well"; the deeper a gravity well is, the more energy any space-bound "climber" must use to escape it. In astrophysics, a gravity well is the gravitational potential field around a massive body. Other types of potential wells include magnetic potential wells. Physical models of gravity wells are sometimes used to illustrate orbital mechanics.
Gravity wells are confused with embedding diagrams used in general relativity theory, but the two concepts are distinctly separate and not directly related. If G is the universal gravitational constant, the external gravitational potential of a spherically symmetric body of mass M is given by the formula: Φ = − G M | x |. A plot of this function in two dimensions is shown in the figure; this plot has been completed with an interior potential proportional to |x|2, corresponding to an object of uniform density, but this interior potential is irrelevant since the orbit of a test particle cannot intersect the body. The potential function has a hyperbolic cross section. A black hole would not have this "closing" dip due to its size being only determined by its event horizon. In a uniform gravitational field, the gravitational potential at a point is proportional to the height, thus if the graph of a gravitational potential Φ is constructed as a physical surface and placed in a uniform gravitational field so that the actual field points in the −Φ direction each point on the surface will have an actual gravitational potential proportional to the value of Φ at that point.
As a result, an object constrained to move on the surface will have the same equation of motion as an object moving in the potential field Φ itself. Gravity wells constructed on this principle can be found in many science museums. There are several sources of inaccuracy in this model: The friction between the object and the surface has no analogue in vacuum; this effect can be reduced by using a rolling ball instead of a sliding block. The object's vertical motion contributes kinetic energy; this effect can be reduced by making the gravity well shallower. A rolling ball's rotational kinetic energy has no analogue; this effect can be reduced by concentrating the ball's mass near its center so that the moment of inertia is small compared to mr². A ball's center of mass is not located on the surface but at a fixed distance r, which changes its potential energy by an amount depending on the slope of the surface at that point. For balls of a fixed size, this effect can be eliminated by constructing the surface so that the center of the ball, rather than the surface itself, lies on the graph of Φ.
Consider an idealized rubber sheet suspended in a uniform gravitational field normal to the sheet. In equilibrium, the elastic tension in each part of the sheet must be equal and opposite to the gravitational pull on that part of the sheet; the mass density may be viewed as intrinsic to the sheet or as belonging to objects resting on top of the sheet. This equilibrium condition is identical in form to the gravitational Poisson equation ∇ 2 Φ = 4 π G ρ where Φ is the gravitational potential and ρ is the mass density. Thus, to a first approximation, a massive object placed on a rubber sheet will deform the sheet into a shaped gravity well, a second test object placed near the first will gravitate toward it in an approximation of the correct force law. Both the rigid gravity well and the rubber-sheet model are misidentified as models of general relativity due to an accidental resemblance to general relativistic embedding diagrams, Einstein's employment of gravitational "curvature" bending the path of light he described as a prediction of general relativity.
The embedding diagram most found in textbooks, an isometric embedding of a constant-time equatorial slice of the Schwarzschild metric in Euclidean 3-dimensional space, superficially resembles a gravity well. However, embedding diagrams fundamentally differ from gravity wells. Embedding takes a shape, but a potential plot has a distinguished "downward" direction. Turning a gravity well "upside down" by negating the potential of the attractive force, turns it into a repulsive force. However, turning a Schwarzschild embedding upside down, by rotating it, h
Special relativity
In physics, special relativity is the accepted and experimentally well-confirmed physical theory regarding the relationship between space and time. In Albert Einstein's original pedagogical treatment, it is based on two postulates: the laws of physics are invariant in all inertial systems. Special relativity was proposed by Albert Einstein in a paper published 26 September 1905 titled "On the Electrodynamics of Moving Bodies"; the inconsistency of Newtonian mechanics with Maxwell's equations of electromagnetism and the lack of experimental confirmation for a hypothesized luminiferous aether led to the development of special relativity, which corrects mechanics to handle situations involving all motions and those at a significant fraction of the speed of light. Today, special relativity is the most accurate model of motion at any speed when gravitational effects are negligible. So, the Newtonian mechanics model is still valid as a simple and high accuracy approximation at low velocities relative to the speed of light.
Special relativity implies a wide range of consequences, which have been experimentally verified, including length contraction, time dilation, relativistic mass, mass–energy equivalence, a universal speed limit, the speed of causality and relativity of simultaneity. It has replaced the conventional notion of an absolute universal time with the notion of a time, dependent on reference frame and spatial position. Rather than an invariant time interval between two events, there is an invariant spacetime interval. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy, as expressed in the mass–energy equivalence formula E = mc2, where c is the speed of light in a vacuum. A defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other. Rather and time are interwoven into a single continuum known as "spacetime".
Events that occur at the same time for one observer can occur at different times for another. Not until Einstein developed general relativity, introducing a curved spacetime to incorporate gravity, was the phrase "special relativity" employed. A translation, used is "restricted relativity"; the theory is "special" in that it only applies in the special case where the spacetime is flat, i.e. the curvature of spacetime, described by the energy-momentum tensor and causing gravity, is negligible. In order to accommodate gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some outdated descriptions, is capable of handling accelerations as well as accelerated frames of reference; as Galilean relativity is now accepted to be an approximation of special relativity, valid for low speeds, special relativity is considered an approximation of general relativity, valid for weak gravitational fields, i.e. at a sufficiently small scale and in conditions of free fall. Whereas general relativity incorporates noneuclidean geometry in order to represent gravitational effects as the geometric curvature of spacetime, special relativity is restricted to the flat spacetime known as Minkowski space.
As long as the universe can be modeled as a pseudo-Riemannian manifold, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this curved spacetime. Galileo Galilei had postulated that there is no absolute and well-defined state of rest, a principle now called Galileo's principle of relativity. Einstein extended this principle so that it accounted for the constant speed of light, a phenomenon, observed in the Michelson–Morley experiment, he postulated that it holds for all the laws of physics, including both the laws of mechanics and of electrodynamics. Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the known laws of either mechanics or electrodynamics; these propositions were the constancy of the speed of light in a vacuum and the independence of physical laws from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as: The Principle of Relativity – the laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.
The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity c, independent of the state of motion of the emitting body". That is, light in vacuum propagates with the speed c in at least one system of inertial coordinates, regardless of the state of motion of the light source; the constancy of the speed of light was motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous ether. There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment. In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acce
Gravitational redshift
In Einstein's general theory of relativity, the gravitational redshift is the phenomenon that clocks in a gravitational field tick slower when observed by a distant observer. More the term refers to the shift of wavelength of a photon to longer wavelength when observed from a point in a lower gravitational field. In the latter case the'clock' is the frequency of the photon and a lower frequency is the same as a longer wavelength; the gravitational redshift is a simple consequence of Einstein's equivalence principle and was found by Einstein eight years before the full theory of relativity. Observing the gravitational redshift in the solar system is one of the classical tests of general relativity. Gravitational redshifts are an important effect in satellite-based navigation systems such as GPS. If the effects of general relativity were not taken into account, such systems would not work at all. Einstein's theory of general relativity incorporates the equivalence principle, which can be stated in various different ways.
One such statement is that gravitational effects are locally undetectable for a free-falling observer. Therefore, in a laboratory experiment at the surface of the earth, all gravitational effects should be equivalent to the effects that would have been observed if the laboratory had been accelerating through outer space at g. One consequence is a gravitational Doppler effect. If a light pulse is emitted at the floor of the laboratory a free-falling observer says that by the time it reaches the ceiling, the ceiling has accelerated away from it, therefore when observed by a detector fixed to the ceiling, it will be observed to have been Doppler shifted toward the red end of the spectrum; this shift, which the free-falling observer considers to be a kinematical Doppler shift, is thought of by the laboratory observer as a gravitational redshift. Such an effect was verified in the 1959 Pound–Rebka experiment. In a case such as this, where the gravitational field is uniform, the change in wavelength is given by Δ λ λ ≈ g Δ y c 2, where Δ y is the change in height.
Since this prediction arises directly from the equivalence principle, it does not require any of the mathematical apparatus of general relativity, its verification does not support general relativity over any other theory that incorporates the equivalence principle. When the field is not uniform, the simplest and most useful case to consider is that of a spherically symmetric field. By Birkhoff's theorem, such a field is described in general relativity by the Schwarzschild metric, d τ 2 = d t 2 + …, where d τ is the clock time of an observer at distance R from the center, d t is the time measured by an observer at infinity, r s is the Schwarzschild radius 2 G M / c 2, "..." represents terms that vanish if the observer is at rest, G is Newton's gravitational constant, M the mass of the gravitating body, c the speed of light. The result is that frequencies and wavelengths are shifted according to the ratio λ ∞ λ e = − 1 / 2, where λ ∞ is the wavelength of the light as measured by the observer at infinity, λ e is the wavelength measured at the source of emission, R e radius at which the photon is emitted.
This can be related to the redshift parameter conventionally defined as z = λ ∞ / λ e − 1. In the case where neither the emitter nor the observer is at infinity, the transitivity of Doppler shifts allows us to generalize the result to λ 1 / λ 2 = 1 / 2; the redshift formula for the frequency ν = c / λ is ν o / ν e = λ e / λ o. When R 1 − R 2 {\displ