More precisely, it is caused by a specific property of material objects: their mass. In Einstein's theory and related theories of gravitation , curvature at every point in spacetime is also caused by whatever matter is present. Here, too, mass is a key property in determining the gravitational influence of matter. But in a relativistic theory of gravity, mass cannot be the only source of gravity. Relativity links mass with energy, and energy with momentum.

In relativity, mass and energy are two different ways of describing one physical quantity. If a physical system has energy, it also has the corresponding mass, and vice versa. In particular, all properties of a body that are associated with energy, such as its temperature or the binding energy of systems such as nuclei or molecules , contribute to that body's mass, and hence act as sources of gravity. In special relativity, energy is closely connected to momentum. Just as space and time are, in that theory, different aspects of a more comprehensive entity called spacetime, energy and momentum are merely different aspects of a unified, four-dimensional quantity that physicists call four-momentum.

In consequence, if energy is a source of gravity, momentum must be a source as well. The same is true for quantities that are directly related to energy and momentum, namely internal pressure and tension. Taken together, in general relativity it is mass, energy, momentum, pressure and tension that serve as sources of gravity: they are how matter tells spacetime how to curve. In the theory's mathematical formulation, all these quantities are but aspects of a more general physical quantity called the energy—momentum tensor.

Einstein's equations are the centerpiece of general relativity. They provide a precise formulation of the relationship between spacetime geometry and the properties of matter, using the language of mathematics. More concretely, they are formulated using the concepts of Riemannian geometry , in which the geometric properties of a space or a spacetime are described by a quantity called a metric.

The metric encodes the information needed to compute the fundamental geometric notions of distance and angle in a curved space or spacetime. A spherical surface like that of the Earth provides a simple example. The location of any point on the surface can be described by two coordinates: the geographic latitude and longitude. Coordinates therefore do not provide enough information to describe the geometry of a spherical surface, or indeed the geometry of any more complicated space or spacetime. That information is precisely what is encoded in the metric, which is a function defined at each point of the surface or space, or spacetime and relates coordinate differences to differences in distance.

All other quantities that are of interest in geometry, such as the length of any given curve, or the angle at which two curves meet, can be computed from this metric function. The metric function and its rate of change from point to point can be used to define a geometrical quantity called the Riemann curvature tensor , which describes exactly how the space or spacetime is curved at each point. In general relativity, the metric and the Riemann curvature tensor are quantities defined at each point in spacetime.

As has already been mentioned, the matter content of the spacetime defines another quantity, the energy—momentum tensor T , and the principle that "spacetime tells matter how to move, and matter tells spacetime how to curve" means that these quantities must be related to each other. Einstein formulated this relation by using the Riemann curvature tensor and the metric to define another geometrical quantity G , now called the Einstein tensor , which describes some aspects of the way spacetime is curved. Einstein's equation then states that. Here, G is the gravitational constant of Newtonian gravity, and c is the speed of light from special relativity.

This equation is often referred to in the plural as Einstein's equations , since the quantities G and T are each determined by several functions of the coordinates of spacetime, and the equations equate each of these component functions. The simplest solution is the uncurved Minkowski spacetime , the spacetime described by special relativity. No scientific theory is apodictically true ; each is a model that must be checked by experiment. Newton's law of gravity was accepted because it accounted for the motion of planets and moons in the Solar System with considerable accuracy.

As the precision of experimental measurements gradually improved, some discrepancies with Newton's predictions were observed, and these were accounted for in the general theory of relativity. Similarly, the predictions of general relativity must also be checked with experiment, and Einstein himself devised three tests now known as the classical tests of the theory:. Of these tests, only the perihelion advance of Mercury was known prior to Einstein's final publication of general relativity in The subsequent experimental confirmation of his other predictions, especially the first measurements of the deflection of light by the sun in , catapulted Einstein to international stardom.

Further tests of general relativity include precision measurements of the Shapiro effect or gravitational time delay for light, most recently in by the Cassini space probe. One set of tests focuses on effects predicted by general relativity for the behavior of gyroscopes travelling through space. One of these effects, geodetic precession , has been tested with the Lunar Laser Ranging Experiment high-precision measurements of the orbit of the Moon.

## Space-time | An Introduction to Einstein's Theory of Gravity | Taylor & Francis Group

Another, which is related to rotating masses, is called frame-dragging. The geodetic and frame-dragging effects were both tested by the Gravity Probe B satellite experiment launched in , with results confirming relativity to within 0. By cosmic standards, gravity throughout the solar system is weak. Since the differences between the predictions of Einstein's and Newton's theories are most pronounced when gravity is strong, physicists have long been interested in testing various relativistic effects in a setting with comparatively strong gravitational fields.

This has become possible thanks to precision observations of binary pulsars. In such a star system, two highly compact neutron stars orbit each other. These beams strike the Earth at very regular intervals, similarly to the way that the rotating beam of a lighthouse means that an observer sees the lighthouse blink, and can be observed as a highly regular series of pulses. General relativity predicts specific deviations from the regularity of these radio pulses. For instance, at times when the radio waves pass close to the other neutron star, they should be deflected by the star's gravitational field.

The observed pulse patterns are impressively close to those predicted by general relativity. One particular set of observations is related to eminently useful practical applications, namely to satellite navigation systems such as the Global Positioning System that are used both for precise positioning and timekeeping. Such systems rely on two sets of atomic clocks : clocks aboard satellites orbiting the Earth, and reference clocks stationed on the Earth's surface. General relativity predicts that these two sets of clocks should tick at slightly different rates, due to their different motions an effect already predicted by special relativity and their different positions within the Earth's gravitational field.

In order to ensure the system's accuracy, the satellite clocks are either slowed down by a relativistic factor, or that same factor is made part of the evaluation algorithm. In turn, tests of the system's accuracy especially the very thorough measurements that are part of the definition of universal coordinated time are testament to the validity of the relativistic predictions.

A number of other tests have probed the validity of various versions of the equivalence principle ; strictly speaking, all measurements of gravitational time dilation are tests of the weak version of that principle , not of general relativity itself. So far, general relativity has passed all observational tests. Models based on general relativity play an important role in astrophysics ; the success of these models is further testament to the theory's validity.

Since light is deflected in a gravitational field, it is possible for the light of a distant object to reach an observer along two or more paths. For instance, light of a very distant object such as a quasar can pass along one side of a massive galaxy and be deflected slightly so as to reach an observer on Earth, while light passing along the opposite side of that same galaxy is deflected as well, reaching the same observer from a slightly different direction.

As a result, that particular observer will see one astronomical object in two different places in the night sky. This kind of focussing is well known when it comes to optical lenses , and hence the corresponding gravitational effect is called gravitational lensing. Observational astronomy uses lensing effects as an important tool to infer properties of the lensing object. Even in cases where that object is not directly visible, the shape of a lensed image provides information about the mass distribution responsible for the light deflection.

In particular, gravitational lensing provides one way to measure the distribution of dark matter , which does not give off light and can be observed only by its gravitational effects. One particularly interesting application are large-scale observations, where the lensing masses are spread out over a significant fraction of the observable universe, and can be used to obtain information about the large-scale properties and evolution of our cosmos. Gravitational waves , a direct consequence of Einstein's theory, are distortions of geometry that propagate at the speed of light, and can be thought of as ripples in spacetime.

They should not be confused with the gravity waves of fluid dynamics , which are a different concept. In February , the Advanced LIGO team announced that they had directly observed gravitational waves from a black hole merger. Indirectly, the effect of gravitational waves had been detected in observations of specific binary stars. Such pairs of stars orbit each other and, as they do so, gradually lose energy by emitting gravitational waves. In such a system, one of the orbiting stars is a pulsar. This has two consequences: a pulsar is an extremely dense object known as a neutron star , for which gravitational wave emission is much stronger than for ordinary stars.

Also, a pulsar emits a narrow beam of electromagnetic radiation from its magnetic poles. As the pulsar rotates, its beam sweeps over the Earth, where it is seen as a regular series of radio pulses, just as a ship at sea observes regular flashes of light from the rotating light in a lighthouse. This regular pattern of radio pulses functions as a highly accurate "clock". It can be used to time the double star's orbital period, and it reacts sensitively to distortions of spacetime in its immediate neighborhood. Since then, several other binary pulsars have been found.

The most useful are those in which both stars are pulsars, since they provide accurate tests of general relativity. Currently, a number of land-based gravitational wave detectors are in operation, and a mission to launch a space-based detector, LISA , is currently under development, with a precursor mission LISA Pathfinder which was launched in Gravitational wave observations can be used to obtain information about compact objects such as neutron stars and black holes , and also to probe the state of the early universe fractions of a second after the Big Bang.

Certain types of black holes are thought to be the final state in the evolution of massive stars. On the other hand, supermassive black holes with the mass of millions or billions of Suns are assumed to reside in the cores of most galaxies , and they play a key role in current models of how galaxies have formed over the past billions of years. Matter falling onto a compact object is one of the most efficient mechanisms for releasing energy in the form of radiation , and matter falling onto black holes is thought to be responsible for some of the brightest astronomical phenomena imaginable.

Notable examples of great interest to astronomers are quasars and other types of active galactic nuclei. Under the right conditions, falling matter accumulating around a black hole can lead to the formation of jets , in which focused beams of matter are flung away into space at speeds near that of light. There are several properties that make black holes most promising sources of gravitational waves. One reason is that black holes are the most compact objects that can orbit each other as part of a binary system; as a result, the gravitational waves emitted by such a system are especially strong.

Another reason follows from what are called black-hole uniqueness theorems : over time, black holes retain only a minimal set of distinguishing features these theorems have become known as "no-hair" theorems , regardless of the starting geometric shape.

## Einstein’s Space-Time

For instance, in the long term, the collapse of a hypothetical matter cube will not result in a cube-shaped black hole. Instead, the resulting black hole will be indistinguishable from a black hole formed by the collapse of a spherical mass. In its transition to a spherical shape, the black hole formed by the collapse of a more complicated shape will emit gravitational waves. One of the most important aspects of general relativity is that it can be applied to the universe as a whole.

A key point is that, on large scales, our universe appears to be constructed along very simple lines: all current observations suggest that, on average, the structure of the cosmos should be approximately the same, regardless of an observer's location or direction of observation: the universe is approximately homogeneous and isotropic. Such comparatively simple universes can be described by simple solutions of Einstein's equations. The current cosmological models of the universe are obtained by combining these simple solutions to general relativity with theories describing the properties of the universe's matter content, namely thermodynamics , nuclear- and particle physics.

Einstein's equations can be generalized by adding a term called the cosmological constant. When this term is present, empty space itself acts as a source of attractive or, less commonly, repulsive gravity. Einstein originally introduced this term in his pioneering paper on cosmology, with a very specific motivation: contemporary cosmological thought held the universe to be static, and the additional term was required for constructing static model universes within the framework of general relativity.

When it became apparent that the universe is not static, but expanding, Einstein was quick to discard this additional term. General relativity is very successful in providing a framework for accurate models which describe an impressive array of physical phenomena. On the other hand, there are many interesting open questions, and in particular, the theory as a whole is almost certainly incomplete.

In contrast to all other modern theories of fundamental interactions , general relativity is a classical theory: it does not include the effects of quantum physics. The quest for a quantum version of general relativity addresses one of the most fundamental open questions in physics. While there are promising candidates for such a theory of quantum gravity , notably string theory and loop quantum gravity , there is at present no consistent and complete theory. It has long been hoped that a theory of quantum gravity would also eliminate another problematic feature of general relativity: the presence of spacetime singularities.

These singularities are boundaries "sharp edges" of spacetime at which geometry becomes ill-defined, with the consequence that general relativity itself loses its predictive power. Furthermore, there are so-called singularity theorems which predict that such singularities must exist within the universe if the laws of general relativity were to hold without any quantum modifications. The best-known examples are the singularities associated with the model universes that describe black holes and the beginning of the universe.

Other attempts to modify general relativity have been made in the context of cosmology. In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely dark energy and dark matter. There have been several controversial proposals to remove the need for these enigmatic forms of matter and energy, by modifying the laws governing gravity and the dynamics of cosmic expansion , for example modified Newtonian dynamics.

Beyond the challenges of quantum effects and cosmology, research on general relativity is rich with possibilities for further exploration: mathematical relativists explore the nature of singularities and the fundamental properties of Einstein's equations, [46] and ever more comprehensive computer simulations of specific spacetimes such as those describing merging black holes are run.

Additional resources, including more advanced material, can be found in General relativity resources. From Wikipedia, the free encyclopedia. Theory of gravity by Albert Einstein. This article is a non-technical introduction to the subject. For the main encyclopedia article, see General relativity.

Introduction History. Fundamental concepts. Principle of relativity Theory of relativity Frame of reference Inertial frame of reference Rest frame Center-of-momentum frame Equivalence principle Mass—energy equivalence Special relativity Doubly special relativity de Sitter invariant special relativity World line Riemannian geometry. Equations Formalisms. Main article: Equivalence principle.

Physics portal Astronomy portal. General relativity Introduction to the mathematics of general relativity Introduction to special relativity History of general relativity Tests of general relativity Numerical relativity Derivations of the Lorentz transformations List of books on general relativity. A precis of Newtonian gravity can be found in Schutz , chapters 2—4. It is impossible to say whether the problem of Newtonian gravity crossed Einstein's mind before , but, by his own admission, his first serious attempts to reconcile that theory with special relativity date to that year, cf.

Pais , p. Norton Janssen , p. Einstein himself also explains this in section XX of his non-technical book Einstein Following earlier ideas by Ernst Mach , Einstein also explored centrifugal forces and their gravitational analogue, cf. Stachel He considered an object "suspended" by a rope from the ceiling of a room aboard an accelerating rocket: from inside the room it looks as if gravitation is pulling the object down with a force proportional to its mass, but from outside the rocket it looks as if the rope is simply transferring the acceleration of the rocket to the object, and must therefore exert just the "force" to do so.

For simple derivations of this, see Harrison A spacelike spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer. A spacelike spacetime interval hence provides a measure of proper distance , i.

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In Euclidean space having spatial dimensions only , the set of points equidistant using the Euclidean metric from some point form a circle in two dimensions or a sphere in three dimensions. These equations describe two families of hyperbolae in an x — ct spacetime diagram, which are termed invariant hyperbolae. Each timelike interval generates a hyperboloid of one sheet, while each spacelike interval generates a hyperboloid of two sheets. Note on nomenclature: The magenta hyperbolae, which cross the x axis, are termed timelike in contrast to spacelike hyperbolae because all "distances" to the origin along the hyperbola are timelike intervals.

Different world lines represent clocks moving at different speeds. A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time. For a clock traveling at 0. This illustrates the phenomenon known as time dilation. Clocks that travel faster take longer in the observer frame to tick out the same amount of proper time, and they travel further along the x—axis than they would have without time dilation.

Length contraction , like time dilation, is a manifestation of the relativity of simultaneity. Measurement of length requires measurement of the spacetime interval between two events that are simultaneous in one's frame of reference. But events that are simultaneous in one frame of reference are, in general, not simultaneous in other frames of reference. The edges of the blue band represent the world lines of the rod's two endpoints. But to an observer in frame S, events O and B are not simultaneous.

To measure length, the observer in frame S measures the endpoints of the rod as projected onto the x -axis along their world lines. The projection of the rod's world sheet onto the x axis yields the foreshortened length OC. In the same way that each observer measures the other's clocks as running slow, each observer measures the other's rulers as being contracted.

Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts. How two clocks can run both slower than the other, is an important question that "goes to the heart of understanding special relativity. Basically, this apparent contradiction stems from not correctly taking into account the different settings of the necessary, related measurements. These settings allow for a consistent explanation of the only apparent contradiction. It is not about the abstract ticking of two identical clocks, but about how to measure in one frame the temporal distance of two ticks of a moving clock.

It turns out that in mutually observing the duration between ticks of clocks, each moving in the respective frame, different sets of clocks must be involved. In order to measure in frame S the tick duration of a moving clock W' at rest in S' , one uses two additional, synchronized clocks W 1 and W 2 at rest in two arbitrarily fixed points in S with the spatial distance d. Conversely, for judging in frame S' the temporal distance of two events on a moving clock W at rest in S , one needs two clocks at rest in S'. Obviously, the necessary recordings for the two judgements, with "one moving clock" and "two clocks at rest" in respectively S or S', involves two different sets, each with three clocks.

Since there are different sets of clocks involved in the measurements, there is no inherent necessity that the measurements be reciprocally "consistent" such that, if one observer measures the moving clock to be slow, the other observer measures the one's clock to be fast. This shows that the time interval OD is longer than OA , showing that the "moving" clock runs slower.

In the lower picture the frame S is moving with velocity - v in the frame S' at rest. The word "measure" is important. In classical physics an observer cannot affect an observed object, but the object's state of motion can affect the observer's observations of the object. Many introductions to special relativity illustrate the differences between Galilean relativity and special relativity by posing a series of "paradoxes". These paradoxes are, in fact, ill-posed problems, resulting from our unfamiliarity with velocities comparable to the speed of light.

The remedy is to solve many problems in special relativity and to become familiar with its so-called counter-intuitive predictions. The geometrical approach to studying spacetime is considered one of the best methods for developing a modern intuition. The twin paradox is a thought experiment involving identical twins, one of whom makes a journey into space in a high-speed rocket, returning home to find that the twin who remained on Earth has aged more.

This result appears puzzling because each twin observes the other twin as moving, and so at first glance, it would appear that each should find the other to have aged less. The twin paradox sidesteps the justification for mutual time dilation presented above by avoiding the requirement for a third clock.

The impression that a paradox exists stems from a misunderstanding of what special relativity states. Special relativity does not declare all frames of reference to be equivalent, only inertial frames. The traveling twin's frame is not inertial during periods when she is accelerating. Furthermore, the difference between the twins is observationally detectable: the traveling twin needs to fire her rockets to be able to return home, while the stay-at-home twin does not. Deeper analysis is needed before we can understand why these distinctions should result in a difference in the twins' ages.

Consider the spacetime diagram of Fig. This presents the simple case of a twin going straight out along the x axis and immediately turning back. From the standpoint of the stay-at-home twin, there is nothing puzzling about the twin paradox at all. The proper time measured along the traveling twin's world line from O to C, plus the proper time measured from C to B, is less than the stay-at-home twin's proper time measured from O to A to B.

More complex trajectories require integrating the proper time between the respective events along the curve i. For the rest of this discussion, we adopt Weiss's nomenclature, designating the stay-at-home twin as Terence and the traveling twin as Stella. We had previously noted that Stella is not in an inertial frame. Given this fact, it is sometimes stated that full resolution of the twin paradox requires general relativity. This is not true. Although general relativity is not required to analyze the twin paradox, application of the Equivalence Principle of general relativity does provide some additional insight into the subject.

We had previously noted that Stella is not stationary in an inertial frame. Analyzed in Stella's rest frame, she is motionless for the entire trip. When she is coasting her rest frame is inertial, and Terence's clock will appear to run slow. But when she fires her rockets for the turnaround, her rest frame is an accelerated frame and she experiences a force which is pushing her as if she were in a gravitational field. Terence will appear to be high up in that field and because of gravitational time dilation , his clock will appear to run fast, so much so that the net result will be that Terence has aged more than Stella when they are back together.

Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence, including Newton's theory. This introductory section has focused on the spacetime of special relativity, since it is the easiest to describe. Minkowski spacetime is flat, takes no account of gravity, is uniform throughout, and serves as nothing more than a static background for the events that take place in it.

The presence of gravity greatly complicates the description of spacetime. In general relativity, spacetime is no longer a static background, but actively interacts with the physical systems that it contains. Spacetime curves in the presence of matter, can propagate waves, bends light, and exhibits a host of other phenomena. A basic goal is to be able to compare measurements made by observers in relative motion.

Say we have an observer O in frame S who has measured the time and space coordinates of an event, assigning this event three Cartesian coordinates and the time as measured on his lattice of synchronized clocks x , y , z , t see Fig. Within the train, a passenger shoots a bullet with a speed of 0.

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The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0. This is in accordance with our naive expectations. What is its velocity u with respect to frame S? The composition of velocities is quite different in relativistic spacetime. To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light,. What is the composite velocity u of the bullet relative to the platform, as represented by the blue arrow? Referring to Fig. The relativistic formula for addition of velocities presented above exhibits several important features:.

We had previously discussed, in qualitative terms, time dilation and length contraction. It is straightforward to obtain quantitative expressions for these effects. Next, we note that for any v greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one. The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mids, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary addition of velocities.

To transform the coordinates of an event from one frame to another in special relativity, we use the Lorentz transformations. We are, in general, always concerned with the space and time differences between events. Note on nomenclature: Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames.

Different authors call one or the other set of transformations the "inverse" set. Example: Terence and Stella are at an Earth-to-Mars space race. Terence is an official at the starting line, while Stella is a participant. The distance from Earth to Mars is light-seconds about There have been many dozens of derivations of the Lorentz transformations since Einstein's original work in , each with its particular focus. Although Einstein's derivation was based on the invariance of the speed of light, there are other physical principles that may serve as starting points. Ultimately, these alternative starting points can be considered different expressions of the underlying principle of locality , which states that the influence that one particle exerts on another can not be transmitted instantaneously.

The derivation given here and illustrated in Fig. Or the other way around, of course. The Lorentz transformations have a mathematical property called linearity, since x ' and t ' are obtained as linear combinations of x and t , with no higher powers involved. The linearity of the transformation reflects a fundamental property of spacetime that we tacitly assumed while performing the derivation, namely, that the properties of inertial frames of reference are independent of location and time.

In the absence of gravity, spacetime looks the same everywhere. Another observer's conventions will do just as well. A result of linearity is that if two Lorentz transformations are applied sequentially, the result is also a Lorentz transformation. Example: Terence observes Stella speeding away from him at 0. Stella, in her frame, observes Ursula traveling away from her at 0.

The Doppler effect is the change in frequency or wavelength of a wave for a receiver and source in relative motion. We are ignoring scenarios where they move along intermediate angles. The classical Doppler analysis deals with waves that are propagating in a medium, such as sound waves or water ripples, and which are transmitted between sources and receivers that are moving towards or away from each other.

The analysis of such waves depends on whether the source, the receiver, or both are moving relative to the medium. Light, unlike sound or water ripples, does not propagate through a medium, and there is no distinction between a source moving away from the receiver or a receiver moving away from the source. Hence, the relativistic Doppler effect is given by [34] : 58— Suppose that a source and a receiver, both approaching each other in uniform inertial motion along non-intersecting lines, are at their closest approach to each other.

It would appear that the classical analysis predicts that the receiver detects no Doppler shift. Due to subtleties in the analysis, that expectation is not necessarily true. Nevertheless, when appropriately defined, transverse Doppler shift is a relativistic effect that has no classical analog. The subtleties are these: [38] : — In scenario a , the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time i.

The source observes the receiver as being illuminated by light of frequency f ' , but also observes the receiver as having a time-dilated clock. In scenario b the illustration shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on.

Scenarios c and d can be analyzed by simple time dilation arguments. The only seeming complication is that the orbiting objects are in accelerated motion. However, if an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation. The converse, however, is not true. In classical mechanics, the state of motion of a particle is characterized by its mass and its velocity. It is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum cannot change.

In relativistic mechanics, the momentum vector is extended to four dimensions. In exploring the properties of the spacetime momentum, we start, in Fig. In the rest frame, the spatial component of the momentum is zero, i. It is apparent that the space and time components of the four-momentum go to infinity as the velocity of the moving frame approaches c. We will use this information shortly to obtain an expression for the four-momentum.

Light particles, or photons, travel at the speed of c , the constant that is conventionally known as the speed of light. This statement is not a tautology, since many modern formulations of relativity do not start with constant speed of light as a postulate. Photons therefore propagate along a light-like world line and, in appropriate units, have equal space and time components for every observer. Photons travel at the speed of light, yet have finite momentum and energy. This result can be derived by inspection of Fig. Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to several famous conclusions.

The second term is just an expression for the kinetic energy of the particle. Mass indeed appears to be another form of energy. The concept of relativistic mass that Einstein introduced in , m rel , although amply validated every day in particle accelerators around the globe or indeed in any instrumentation whose use depends on high velocity particles, such as electron microscopes, [39] old-fashioned color television sets, etc.

Relativistic mass, for instance, plays no role in general relativity. For this reason, as well as for pedagogical concerns, most physicists currently prefer a different terminology when referring to the relationship between mass and energy. The term "mass" by itself refers to the rest mass or invariant mass , and is equal to the invariant length of the relativistic momentum vector.

Expressed as a formula,. This formula applies to all particles, massless as well as massive. Using an uppercase P to represent the four-momentum and a lowercase p to denote the spatial momentum, the four-momentum may be written as. In physics, conservation laws state that certain particular measurable properties of an isolated physical system do not change as the system evolves over time.

In , Emmy Noether discovered that underlying each conservation law is a fundamental symmetry of nature. In this section, we examine the Newtonian views of conservation of mass, momentum and energy from a relativistic perspective. To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension. In Newtonian mechanics, two extreme cases of this problem may be distinguished yielding mathematics of minimum complexity:.

For both cases 1 and 2 , momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat. The four-momentum is, as expected, a conserved quantity. However, the invariant mass of the fused particle, given by the point where the invariant hyperbola of the total momentum intersects the energy axis, is not equal to the sum of the invariant masses of the individual particles that collided. Looking at the events of this scenario in reverse sequence, we see that non-conservation of mass is a common occurrence: when an unstable elementary particle spontaneously decays into two lighter particles, total energy is conserved, but the mass is not.

Part of the mass is converted into kinetic energy. The freedom to choose any frame in which to perform an analysis allows us to pick one which may be particularly convenient. For analysis of momentum and energy problems, the most convenient frame is usually the " center-of-momentum frame " also called the zero-momentum frame, or COM frame. This is the frame in which the space component of the system's total momentum is zero.

In the lab frame, the daughter particles are preferentially emitted in a direction oriented along the original particle's trajectory. In the COM frame, however, the two daughter particles are emitted in opposite directions, although their masses and the magnitude of their velocities are generally not the same.

In a Newtonian analysis of interacting particles, transformation between frames is simple because all that is necessary is to apply the Galilean transformation to all velocities. If the total momentum of an interacting system of particles is observed to be conserved in one frame, it will likewise be observed to be conserved in any other frame. In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition.

### Sir Isaac Newton's Concept of Light as a Particle

In the one-dimensional case of a completely elastic collision with no loss of kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities. In the case of a completely inelastic collision with total loss of kinetic energy, the outgoing velocities of the rebounding particles will be zero. Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum.

As we have discussed in the previous section on four-momentum , this second option was what he chose. The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications.

Kinetic energy converted into heat or internal potential energy shows up as an increase in mass. A charged pion is a particle of mass It is unstable, and decays into a muon of mass The difference between the pion mass and the muon mass is Because of its negligible mass, a neutrino travels at very nearly the speed of light. To conserve momentum, the muon has the same value of the space component of the neutrino's momentum, but in the opposite direction.

Algebraic analyses of the energetics of this decay reaction are available online, [42] so Fig. The energy of the neutrino is Most of the energy is carried off by the near-zero-mass neutrino. The topics in this section are of significantly greater technical difficulty than those in the preceding sections and are not essential for understanding Introduction to curved spacetime. Lorentz transformations relate coordinates of events in one reference frame to those of another frame.

Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas. This nonlinearity is an artifact of our choice of parameters. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other.

The natural functions for expressing these relationships are the hyperbolic analogs of the trigonometric functions. The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula; [6] : 47— The Lorentz transformations take a simple form when expressed in terms of rapidity.

Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called boosts. In other words, Lorentz boosts represent hyperbolic rotations in Minkowski spacetime. The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage. Indeed, none of the elementary derivations of special relativity require them. Working exclusively with such objects leads to formulas that are manifestly relativistically invariant, which is a considerable advantage in non-trivial contexts.

For instance, demonstrating relativistic invariance of Maxwell's equations in their usual form is not trivial, while it is merely a routine calculation really no more than an observation using the field strength tensor formulation. The study of tensors is outside the scope of this article, which provides only a basic discussion of spacetime.

As usual, when we write x , t , etc. The last three components of a 4—vector must be a standard vector in three-dimensional space. As expected, the final components of the above 4-vectors are all standard 3-vectors corresponding to spatial 3-momentum , 3-force etc. The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. As noted in the previous discussion of energy and momentum conservation , Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving 4-vectors rather than give up on conservation of momentum.

Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving 4-vectors require the use of tensors with appropriate rank, which themselves can be thought of as being built up from 4-vectors. It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames.

Actually, accelerating objects can generally be analyzed without needing to deal with accelerating frames at all. It is only when gravitation is significant that general relativity is required. Properly handling accelerating frames does require some care, however. The difference between special and general relativity is that 1 In special relativity, all velocities are relative, but acceleration is absolute. To accommodate this difference, general relativity uses curved spacetime. The Dewan—Beran—Bell spaceship paradox Bell's spaceship paradox is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues.

They are connected by a string which is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration. The main article for this section recounts how, when the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.

The problem with the first argument is that there is no "frame of the spaceships. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity. A spacetime diagram Fig. They are comoving and inertial before and after this phase.

The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig. The "paradox", as it were, comes from the way that Bell constructed his example. As shown in Fig. Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as event horizons. In the text accompanying Fig. During periods of positive acceleration, the traveler's velocity just approaches the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases.

At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows:. Stella lifts off at time 0, her spacecraft accelerating at 0. Every twenty hours, Terence radios updates to Stella about the situation at home solid green lines. Stella receives these regular transmissions, but the increasing distance offset in part by time dilation causes her to receive Terence's communications later and later as measured on her clock, and she never receives any communications from Terence after hours on his clock dashed green lines.

After hours according to Terence's clock, Stella enters a dark region. She has traveled outside Terence's timelike future. On the other hand, Terence can continue to receive Stella's messages to him indefinitely. He just has to wait long enough. Spacetime has been divided into distinct regions separated by an apparent event horizon. So long as Stella continues to accelerate, she can never know what takes place behind this horizon. Newton's theories assumed that motion takes place against the backdrop of a rigid Euclidean reference frame that extends throughout all space and all time.

Gravity is mediated by a mysterious force, acting instantaneously across a distance, whose actions are independent of the intervening space. Nor is there any such thing as a force of gravitation, only the structure of spacetime itself. In spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun.

Instead, the satellite moves through space only in response to local conditions. Since spacetime is everywhere locally flat when considered on a sufficiently small scale, the satellite is always following a straight line in its local inertial frame. We say that the satellite always follows along the path of a geodesic.

No evidence of gravitation can be discovered following alongside the motions of a single particle. In any analysis of spacetime, evidence of gravitation requires that one observe the relative accelerations of two bodies or two separated particles. The tidal accelerations that these particles exhibit with respect to each other do not require forces for their explanation.

Rather, Einstein described them in terms of the geometry of spacetime, i. These tidal accelerations are strictly local. It is the cumulative total effect of many local manifestations of curvature that result in the appearance of a gravitational force acting at a long range from Earth. To go from the elementary description above of curved spacetime to a complete description of gravitation requires tensor calculus and differential geometry, topics both requiring considerable study.

Without these mathematical tools, it is possible to write about general relativity, but it is not possible to demonstrate any non-trivial derivations. Rather than this section attempting to offer a yet another relatively non-mathematical presentation about general relativity, the reader is referred to the featured Wikipedia articles Introduction to general relativity and General relativity. Instead, the focus in this section will be to explore a handful of elementary scenarios that serve to give somewhat of the flavor of general relativity.

In the discussion of special relativity, forces played no more than a background role. Special relativity assumes the ability to define inertial frames that fill all of spacetime, all of whose clocks run at the same rate as the clock at the origin. Is this really possible? In a nonuniform gravitational field, experiment dictates that the answer is no. Gravitational fields make it impossible to construct a global inertial frame.

In small enough regions of spacetime, local inertial frames are still possible.

General relativity involves the systematic stitching together of these local frames into a more general picture of spacetime. Shortly after the publication of the general theory in , a number of scientists pointed out that general relativity predicts the existence of gravitational redshift. Einstein himself suggested the following thought experiment : i Assume that a tower of height h Fig. A photon climbing in Earth's gravitational field loses energy and is redshifted.

Light has an associated frequency, and this frequency may be used to drive the workings of a clock. The gravitational redshift leads to an important conclusion about time itself: Gravity makes time run slower. Suppose we build two identical clocks whose rates are controlled by some stable atomic transition. Place one clock on top of the tower, while the other clock remains on the ground.

An experimenter on top of the tower observes that signals from the ground clock are lower in frequency than those of the clock next to her on the tower. Light going up the tower is just a wave, and it is impossible for wave crests to disappear on the way up. Exactly as many oscillations of light arrive at the top of the tower as were emitted at the bottom.

The experimenter concludes that the ground clock is running slow, and can confirm this by bringing the tower clock down to compare side-by-side with the ground clock. Clocks in a gravitational field do not all run at the same rate. Experiments such as the Pound—Rebka experiment have firmly established curvature of the time component of spacetime. The Pound—Rebka experiment says nothing about curvature of the space component of spacetime. But the theoretical arguments predicting gravitational time dilation do not depend on the details of general relativity at all.

Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence. A standard demonstration in general relativity is to show how, in the " Newtonian limit " i. Newtonian gravitation is a theory of curved time. General relativity is a theory of curved time and curved space. Given G as the gravitational constant, M as the mass of a Newtonian star, and orbiting bodies of insignificant mass at distance r from the star, the spacetime interval for Newtonian gravitation is one for which only the time coefficient is variable: [20] : — But general relativity is a theory of curved space and curved time, so if there are terms modifying the spatial components of the spacetime interval presented above, shouldn't their effects be seen on, say, planetary and satellite orbits due to curvature correction factors applied to the spatial terms?

The answer is that they are seen, but the effects are tiny. Despite the minuteness of the spatial terms, the first indications that something was wrong with Newtonian gravitation were discovered over a century-and-a-half ago. In , Urbain Le Verrier , in an analysis of available timed observations of transits of Mercury over the Sun's disk from to , reported that known physics could not explain the orbit of Mercury, unless there possibly existed a planet or asteroid belt within the orbit of Mercury.

The perihelion of Mercury's orbit exhibited an excess rate of precession over that which could be explained by the tugs of the other planets.

As the famous astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing wobbles in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical new planet.

This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed. In , Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the curvature of spacetime. Curvature in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct. The most spectacular of Einstein's predictions was his calculation that the curvature terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body.

Its movement in space is equal to its movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign curvature in its spatial components. The story of the Eddington eclipse expedition and Einstein's rise to fame is well told elsewhere. In Newton's theory of gravitation , the only source of gravitational force is mass.

In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. One important conclusion to be derived from the equations is that, colloquially speaking, gravity itself creates gravity. In general relativity, the energy of the gravitational field feeds back into creation of the gravitational field.

This makes the equations nonlinear and hard to solve in anything other than weak field cases. In special relativity, mass-energy is closely connected to momentum. As we have discussed earlier in the section on Energy and momentum , just as space and time are different aspects of a more comprehensive entity called spacetime, mass-energy and momentum are merely different aspects of a unified, four-dimensional quantity called four-momentum.

In consequence, if mass-energy is a source of gravity, momentum must also be a source. The inclusion of momentum as a source of gravity leads to the prediction that moving or rotating masses can generate fields analogous to the magnetic fields generated by moving charges, a phenomenon known as gravitomagnetism. It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges.

An eloquent demonstration of this was presented by Feynman in volume II, chapter 13—6 of his Lectures on Physics , available online. Because of the symmetry of the setup, the net force on the central particle is zero. Since the physical situation has not changed, only the frame in which things are observed, the test particle should not be attracted towards either stream. But it is not at all clear that the forces exerted on the test particle are equal. All of these effects together would seemingly demand that the test particle be drawn towards the bottom stream.

The test particle is not drawn to the bottom stream because of a velocity-dependent force that serves to repel a particle that is moving in the same direction as the bottom stream. This velocity-dependent gravitational effect is gravitomagnetism. Matter in motion through a gravitomagnetic field is hence subject to so-called frame-dragging effects analogous to electromagnetic induction.

It has been proposed that such gravitomagnetic forces underlie the generation of the relativistic jets Fig. Quantities that are directly related to energy and momentum should be sources of gravity as well, namely internal pressure and stress. Taken together, mass-energy , momentum, pressure and stress all serve as sources of gravity: Collectively, they are what tells spacetime how to curve.

General relativity predicts that pressure acts as a gravitational source with exactly the same strength as mass-energy density. The inclusion of pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation. For example, the pressure term sets a maximum limit to the mass of a neutron star. The more massive a neutron star, the more pressure is required to support its weight against gravity. The increased pressure, however, adds to the gravity acting on the star's mass.

Above a certain mass determined by the Tolman—Oppenheimer—Volkoff limit , the process becomes runaway and the neutron star collapses to a black hole. The stress terms become highly significant when performing calculations such as hydrodynamic simulations of core-collapse supernovae. These predictions for the roles of pressure, momentum and stress as sources of spacetime curvature are elegant and play an important role in theory.

In regards to pressure, the early universe was radiation dominated, [60] and it is highly unlikely that any of the relevant cosmological data e. Likewise, the mathematical consistency of the Einstein field equations would be broken if the stress terms didn't contribute as a source of gravity. All that is well and good, but are there any direct , quantitative experimental or observational measurements that confirm that these terms contribute to gravity with the correct strength?

The classic experiment to measure the strength of a gravitational source i. Two small but dense balls are suspended on a fine wire, making a torsion balance. Bringing two large test masses close to the balls introduces a detectable torque. Given the dimensions of the apparatus and the measurable spring constant of the torsion wire, the gravitational constant G can be determined.

To study pressure effects by compressing the test masses is hopeless, because attainable laboratory pressures are insignificant in comparison with the mass-energy of a metal ball. Kreuzer did a Cavendish experiment using a Teflon mass suspended in a mixture of the liquids trichloroethylene and dibromoethane having the same buoyant density as the Teflon Fig. Although Kreuzer originally considered this experiment merely to be a test of the ratio of active mass to passive mass, Clifford Will reinterpreted the experiment as a fundamental test of the coupling of sources to gravitational fields.

In , Bartlett and Van Buren noted that lunar laser ranging had detected a 2-km offset between the moon's center of figure and its center of mass. This indicates an asymmetry in the distribution of Fe abundant in the Moon's core and Al abundant in its crust and mantle. If pressure did not contribute equally to spacetime curvature as does mass-energy, the moon would not be in the orbit predicted by classical mechanics. The mission aim was to measure spacetime curvature near Earth, with particular emphasis on gravitomagnetism. The much smaller frame-dragging effect which is due to gravitomagnetism, and is also known as Lense—Thirring precession was difficult to measure because of unexpected charge effects causing variable drift in the gyroscopes.

A realist would say that Einstein discovered spacetime to be non-Euclidean. A conventionalist would say that Einstein merely found it more convenient to use non-Euclidean geometry. The conventionalist would maintain that Einstein's analysis said nothing about what the geometry of spacetime really is. In response to the first question, a number of authors including Deser, Grishchuk, Rosen, Weinberg, etc. Those theories are variously called "bi-metric gravitation", the "field-theoretical approach to general relativity", and so forth.

The flat spacetime paradigm posits that matter creates a gravitational field that causes rulers to shrink when they are turned from circumferential orientation to radial, and that causes the ticking rates of clocks to dilate. The flat spacetime paradigm is fully equivalent to the curved spacetime paradigm in that they both represent the same physical phenomena. However, their mathematical formulations are entirely different. Working physicists routinely switch between using curved and flat spacetime techniques depending on the requirements of the problem.

The flat spacetime paradigm turns out to be especially convenient when performing approximate calculations in weak fields. Hence, flat spacetime techniques will be used when solving gravitational wave problems, while curved spacetime techniques will be used in the analysis of black holes. Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds , smooth manifolds with a Riemannian metric , i. This gives, in particular, local notions of angle , length of curves , surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture " Ueber die Hypothesen, welche der Geometrie zu Grunde liegen " "On the Hypotheses on which Geometry is Based". It is a very broad and abstract generalization of the differential geometry of surfaces in R 3.

Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein 's general theory of relativity , made profound impact on group theory and representation theory , as well as analysis , and spurred the development of algebraic and differential topology.

The metric determines the geometry of spacetime , as well as determining the geodesics of particles and light beams. About each point event on this manifold, coordinate charts are used to represent observers in reference frames. Usually, many overlapping coordinate charts are needed to cover a manifold. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection.

The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally. In general, they will disagree about the exact location and timing of this impact, i. Although their kinematic descriptions will differ, dynamical physical laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these and all other physical laws must take the same form in all coordinate systems.