"that has no charge or angular momentum. A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass. Introduction to the mathematics of general relativity The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches"
"relativity has modified the notion of time. But from a fixed Lorentz observer's viewpoint time remains a distinguished, absolute, external, global parameter. The Newtonian notion of time essentially carries over to special relativistic systems, hidden in the spacetime structure. Though classically spacetime appears to be an absolute background, general relativity reveals that spacetime is actually dynamical; gravity is a manifestation of spacetime geometry. Matter reacts with spacetime: Also, spacetime can interact with itself (e.g. gravitational waves). The dynamical nature of spacetime has a vast array of consequences. The dynamical nature of spacetime, via the hole argument, implies that the theory"
"difference is expected in a relativistic theory, however, due to the change in space-time between frames, as discussed next. Relativity takes the Lorentz transformation of space-time suggested by invariance of Maxwell's equations and imposes it upon dynamics as well (a revision of Newton's laws of motion). In this example, the Lorentz transformation affects the ""x""-direction only (the relative motion of the two frames is along the ""x""-direction). The relations connecting time and space are ( ""primes"" denote the moving conductor frame ) : These transformations lead to a change in the ""y""-component of a force: That is, within Lorentz invariance,"
"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 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"
"from the star, the spacetime interval for Newtonian gravitation is one for which only the time coefficient is variable: The formula_148 coefficient in front of formula_149 describes the curvature of time in Newtonian gravitation, and this curvature completely accounts for all Newtonian gravitational effects. As expected, this correction factor is directly proportional to formula_150 and formula_151, and because of the formula_152 in the denominator, the correction factor increases as one approaches the gravitating body, meaning that time is curved. But general relativity is a theory of curved space ""and"" curved time, so if there are terms modifying the spatial components"
"Like the force carriers of the other forces (see charged black hole), gravitation plays a role in general relativity, in defining the spacetime in which events take place. In some descriptions energy modifies the ""shape"" of spacetime itself, and gravity is a result of this shape, an idea which at first glance may appear hard to match with the idea of a force acting between particles. Because the diffeomorphism invariance of the theory does not allow any particular space-time background to be singled out as the ""true"" space-time background, general relativity is said to be background-independent. In contrast, the Standard"
"and are incompatible with modern thermodynamics. In general relativity, the effects of gravitation are ascribed to spacetime curvature instead of to a force. The starting point for general relativity is the equivalence principle, which equates free fall with inertial motion. The issue that this creates is that free-falling objects can accelerate with respect to each other. In Newtonian physics, no such acceleration can occur unless at least one of the objects is being operated on by a force (and therefore is not moving inertially). To deal with this difficulty, Einstein proposed that spacetime is curved by matter, and that free-falling"
"of tests of Albert Einstein's general theory of relativity, performed by using radio waves transmitted from the ""Cassini"" space probe. The radio scientists measured a frequency shift in the radio waves to and from the spacecraft, as those passed close to the Sun. According to the general theory of relativity, a massive object like the Sun causes space-time to curve, causing a beam of radiowaves (or light, or any form of electromagnetic radiation) that passes by the Sun to travel farther (known as the Shapiro time delay). Although some measurable deviations from the values calculated using the general theory of"
"the fact that gravity in general relativity bends both time and space, not just space. In normal circumstances, gravity bends time so slightly that the differences between Newtonian gravity and general relativity are detectable only with precise instruments. de Sitter space involves a variation of general relativity in which spacetime is slightly curved in the absence of matter or energy. This is analogous to the relationship between Euclidean geometry and non-Euclidean geometry. An intrinsic curvature of spacetime in the absence of matter or energy is modeled by the cosmological constant in general relativity. This corresponds to the vacuum having an"
"Einstein's equivalence principle implied that a free fall within a uniform gravitational field is equivalent to inertial motion. By extending special relativity's effects into three dimensions, general relativity extended length contraction into space contraction, conceiving of 4D space-time as the gravitational field that alters geometrically and sets all local objects' pathways. Even massless energy exerts gravitational motion on local objects by ""curving"" the geometrical ""surface"" of 4D space-time. Yet unless the energy is vast, its relativistic effects of contracting space and slowing time are negligible when merely predicting motion. Although general relativity is embraced as the more explanatory theory via"
"Einstein's theory of relativity, which involves a profound analysis of time and space, introduced radical changes, not only in our basic concepts, but also in our modes of physical reasoning. The essence of Einstein's theory was to adopt a relational approach to the notions of time and space, which mathematically can be expressed through the Lorentz space-time transformations. Although the mathematical structure of the Lorentz ether theory, which leaves the speed of light ""in vacuo"", ""c"", a universal constant, is equivalent to that of Einstein's, there is nevertheless a drastic difference in the way to conceive it. On the one"
"force being exerted on them, instead of this being due to the force of gravity as is the case in classical mechanics. This is incompatible with classical mechanics and special relativity because in those theories inertially moving objects cannot accelerate with respect to each other, but objects in free fall do so. To resolve this difficulty Einstein first proposed that spacetime is curved. In 1915, he devised the Einstein field equations which relate the curvature of spacetime with the mass, energy, and any momentum within it. Some of the consequences of general relativity are: Technically, general relativity is a theory"
"In other words, if in the theory we change the time ""t"" to ""t""+100 years (or indeed any other time shift) the theoretical predictions do not change. In Einstein's general relativity, coordinates like ""x"", ""y"", ""z"", and ""t"" are not only ""relative"" in the global sense of translations like formula_6, rotations, etc., but become completely arbitrary, so that, for example, one can define an entirely new time-like coordinate according to some arbitrary rule such as formula_7, where formula_8 has units of time, and yet Einstein's equations will have the same form. Invariance of the form of an equation under an"
"Light-time correction Light-time correction is a displacement in the ""apparent position"" of a celestial object from its ""true position"" (or ""geometric position"") caused by the object's motion during the time it takes its light to reach an observer. Light-time correction occurs in principle during the observation of any moving object, because the speed of light is finite. The magnitude and direction of the displacement in position depends upon the distance of the object from the observer and the motion of the object, and is measured at the instant at which the object's light reaches the observer. It is ""independent"" of"
"be shortened in the direction that they are moving with respect to the observer. Subsequently, Einstein worked on a general theory of relativity, which is a theory of how gravity interacts with spacetime. Instead of viewing gravity as a force field acting in spacetime, Einstein suggested that it modifies the geometric structure of spacetime itself. According to the general theory, time goes more slowly at places with lower gravitational potentials and rays of light bend in the presence of a gravitational field. Scientists have studied the behaviour of binary pulsars, confirming the predictions of Einstein's theories, and non-Euclidean geometry is"
"predicted by general relativity, the presence of a mass deforms spacetime in such a way that the paths taken by particles bend towards the mass. At the event horizon of a black hole, this deformation becomes so strong that there are no paths that lead away from the black hole. To a distant observer, clocks near a black hole would appear to tick more slowly than those further away from the black hole. Due to this effect, known as gravitational time dilation, an object falling into a black hole appears to slow as it approaches the event horizon, taking an"
"was the subject of the Gravity Probe A experiment. Under Einstein's theory of general relativity, matter distorts the surrounding spacetime, so that space gets bent similarly to the way a sheet of fabric would bend if a bowling ball were dropped in the middle of the sheet. But the distortion manifests itself in the time direction as well: time would appear for a distant observer to flow more slowly in the vicinity of a massive object. For example, the metric, surrounding a spherically symmetric gravitating body, has a smaller coefficient at formula_1 closer to the body, which means slower rate"
"General relativity is often explained in terms of curved spacetime. We can picture the universe as the curved surface of a membrane like a soap film that changes dynamically in time. The same picture can help us understand how event symmetry would be broken. A soap bubble is made from molecules that interact via forces that depend on the orientations of the molecules and the distance between them. If you wrote down the equations of motion for all the molecules in terms of their positions, velocities and orientations, then those equations would be unchanged in form under any permutation of"
"defined separately from each other. Rather, space 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 that has often been used is ""restricted relativity""; ""special"" really means ""special case"". 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,"
"field, or something similar. This has been achieved by Einstein's theory of general relativity, in which gravitational interaction is mediated by deformation of space-time geometry. Matter warps the geometry of space-time, and these effects are—as with electric and magnetic fields—propagated at the speed of light. Thus, in the presence of matter, space-time becomes non-Euclidean, resolving the apparent conflict between Newton's proof of the conservation of angular momentum and Einstein's theory of special relativity. Mach's question regarding the bulging of rotating bodies is resolved because local space-time geometry is informing a rotating body about the rest of the universe. In Newton's"
"spacetime. As a consequence of this curvature, it is not a given in general relativity that inertial objects moving at a particular rate with respect to each other will continue to do so. This phenomenon of geodesic deviation means that inertial frames of reference do not exist globally as they do in Newtonian mechanics and special relativity. However, the general theory reduces to the special theory over sufficiently small regions of spacetime, where curvature effects become less important and the earlier inertial frame arguments can come back into play. Consequently, modern special relativity is now sometimes described as only a"
"has changed over the history of the universe. (The same question has also been asked of the fine structure constant.) One interesting cause of such change would be change over time in the strength of the strong force. Astronomical searches for time-varying ""μ"" have typically examined the Lyman series and Werner transitions of molecular hydrogen which, given a sufficiently large redshift, occur in the optical region and so can be observed with ground-based spectrographs. If ""μ"" were to change, then the change in the wavelength ""λ"" of each rest frame wavelength can be parameterised as: where Δ""μ""/""μ"" is the proportional"
"the way the permutations of spacetime would make us believe. In the case of general relativity the gravitational force acts as a background field that controls the measurement properties of spacetime. In ordinary circumstances the geometry of space is flat and Euclidean and the diffeomorphism invariance of general relativity is hidden thanks to this background field. Only in the extreme proximity of a violent collision of black holes would the flexibility of spacetime become apparent. In a similar way, event symmetry could be hidden by a background field that determines not just the geometry of spacetime, but also its topology."
"time is sometimes considered an imaginary coordinate) and in general relativity (where different signs are assigned to time and space components of spacetime metric). Furthermore, in Einstein's general theory of relativity, it is postulated that space-time is geometrically distorted- ""curved"" -near to gravitationally significant masses. One consequence of this postulate, which follows from the equations of general relativity, is the prediction of moving ripples of space-time, called gravitational waves. While indirect evidence for these waves has been found (in the motions of the Hulse–Taylor binary system, for example) experiments attempting to directly measure these waves are ongoing at the LIGO"
"observable evidence for time dilation in special relativity and gravitational time dilation in general relativity, for example in the famous and easy-to-replicate observation of atmospheric muon decay. The theory of relativity states that the speed of light is invariant for all observers in any frame of reference; that is, it is always the same. Time dilation is a direct consequence of the invariance of the speed of light. Time dilation may be regarded in a limited sense as ""time travel into the future"": a person may use time dilation so that a small amount of proper time passes for them,"
"in describing the concept of curvature in four-dimensional space-time. Historically, this mathematical split begins with the disparity between Einstein's theories of relativity, which saw physics through the lens of causality, and classical physics, which interpreted the structure of space-time to be random and inherent. The prevailing notion about cosmic change is that the universe is expanding at a constantly accelerating rate; moreover, it is understood that as one traces the universe's history backwards one finds that it was, at one point, far denser. If true, the Rainbow gravity theory prohibits a singularity such as that which is postulated in the"
"that are invariant under the gauge transformations of the theory. Then, solving only the constraint formula_44 and determining the Dirac observables with respect to it leads us back to the ADM phase space with constraints formula_45. The dynamics of general relativity is generated by the constraints, it can be shown that six Einstein equations describing time evolution (really a gauge transformation) can be obtained by calculating the Poisson brackets of the three-metric and its conjugate momentum with a linear combination of the spatial diffeomorphism and Hamiltonian constraint. The vanishing of the constraints, giving the physical phase space, are the four"
"Gravitomagnetic time delay In according to general relativity, a massive spinning body endowed with angular momentum ""S"" will alter the space-time fabric around it in such a way that several effects on moving test particles and propagating electromagnetic waves occur. In particular, the direction of motion with respect to the sense of rotation of the central body is relevant because co-and counter-propagating waves carry a ""gravitomagnetic"" time delay Δ""t"" which could be, in principle, be measured if ""S"" is known. On the contrary, if the validity of general relativity is assumed, it is possible to use Δ""t"" to measure ""S""."
"experiments, for example STEP, are planned for more accurate experiments in space. Formulations of the equivalence principle include: In general relativity, the effects of gravitation are ascribed to spacetime curvature instead of a force. The starting point for general relativity is the equivalence principle, which equates free fall with inertial motion and describes free-falling inertial objects as being accelerated relative to non-inertial observers on the ground. In Newtonian physics, however, no such acceleration can occur unless at least one of the objects is being operated on by a force. Einstein proposed that spacetime is curved by matter, and that free-falling"
"Einstein's special theory of relativity (and, by extension, the general theory) predicts time dilation that could be interpreted as time travel. The theory states that, relative to a stationary observer, time appears to pass more slowly for faster-moving bodies: for example, a moving clock will appear to run slow; as a clock approaches the speed of light its hands will appear to nearly stop moving. The effects of this sort of time dilation are discussed further in the popular ""twin paradox"". These results are experimentally observable and affect the operation of GPS satellites and other high-tech systems used in daily"
"Shapiro time delay The Shapiro time delay effect, or gravitational time delay effect, is one of the four classic solar-system tests of general relativity. Radar signals passing near a massive object take slightly longer to travel to a target and longer to return than they would if the mass of the object were not present. The time delay is caused by spacetime dilation, which increases the path length. In an article entitled ""Fourth Test of General Relativity"", Shapiro wrote: Because, according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along"
"Newton's theory due to general relativity. An extension of this expansion 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 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"
"Jacobson, ""et al."" Einstein aether theory In physics the Einstein æther theory, also called æ-theory, is a generally covariant modification of general relativity which describes a spacetime endowed with both a metric and a unit timelike vector field named the æther. The theory has a preferred reference frame and hence violates Lorentz invariance. Einstein-æther theories were popularized by Maurizio Gasperini in a series of papers, such as ""Singularity Prevention and Broken Lorentz Symmetry"" in the 1980s. In addition to the metric of general relativity these theories also included a scalar field which intuitively corresponded to a universal notion of time."