relativity
Relativity
A general theory of physics, primarily conceived by Albert Einstein, which involves a profound analysis of time and space, leading to a generalization of physical laws, with far-reaching implications in important branches of physics and in cosmology. Historically, the theory developed in two stages. Einstein's initial formulation in 1905 (now known as the special, or restricted, theory of relativity) does not treat gravitation; and one of the two principles on which it is based, the principle of relativity (the other being the principle of the constancy of the speed of light), stipulates the form invariance of physical laws only for inertial reference systems. Both restrictions were removed by Einstein in his general theory of relativity developed in 1915, which exploits a deep-seated equivalence between inertial and gravitational effects, and leads to a successful “relativistic” generalization of Isaac Newton's theory of gravitation.
Special theory
The key feature of the theory of special relativity is the elimination of an absolute notion of simultaneity in favor of the notion that all observers always measure light to have the same velocity, in vacuum, c, independently of their own motion. The impetus for the development of the theory arose from the theory of electricity and magnetism developed by J. C. Maxwell. This theory accounted for all observed phenomena involving electric and magnetic fields and also predicted that disturbances in these fields would propagate as waves with a definite speed, c, in vacuum. These electromagnetic waves predicted in Maxwell's theory successfully accounted for the existence of light and other forms of electromagnetic radiation. However, the presence of a definite speed, c, posed a difficulty, since if one inertial observer measures light to have velocity c, it would be expected that another inertial observer, moving toward the light ray with velocity v with respect to the first, would measure the light to have velocity c + v. Hence, it initially was taken for granted that there must be a preferred rest frame (often referred to as the ether) in which Maxwell's equations would be valid, and only in that frame would light be seen to travel with velocity c. However, this viewpoint was greatly shaken by the 1887 experiment of A. A. Michelson and E. W. Morley, which failed to detect any motion of the Earth through the ether. By radically altering some previously held beliefs concerning the structure of space and time, the theory of special relativity allows Maxwell's equations to hold, and light to propagate with velocity c, in all frames of reference, thereby making Maxwell's theory consistent with the null result of Michelson and Morley. See Electromagnetic radiation, Light, Maxwell's equations
Simultaneity in prerelativity physics
The most dramatic aspect of the theory of special relativity is its overthrowing of the notion that there is a well-defined, observer-independent meaning to the notion of simultaneity. The following terminology will be introduced: An event is a point of space at an instant of time. Since it takes four numbers to specify an event—one for the time at which the event occurred and three for its spatial position—it follows that the set of all events constitutes a four-dimensional continuum, which is referred to as space-time.
A space-time diagram (Fig. 1) is a plot of events in space-time, with time, t, represented by the vertical axis and two spatial directions (x, y) represented by the horizontal axes. (The z direction is not shown.) For any event A shown in the diagram, there are many other events in this diagram—say, an event B—having the property that an observer or material body starting at event B can, in principle, be present at event A. The collection of all such events constitutes the past of event A. Similarly, there are many events—say, an event C—having the property that an observer or material body starting at event A can, in principle, be present at event C. These events constitute the future of A. Finally, there remain some events in space-time which lie neither to past nor future of A. In prerelativity physics, these events are assumed to make a three-dimensional set, and they are referred to as the events which are simultaneous with event A.
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In both prerelativity physics and special relativity, an inertial observer is one who is not acted upon by any external forces. In both theories it is assumed that any inertial observer, can build a rigid cartesian grid of meter sticks, all of which intersect each other at right angles. Observer may then label the points on this cartesian grid by the coordinates (x, y, z) representing the distance of the point from along the three orthogonal directions of the grid. A clock may then be placed at each grid point. In prerelativity physics, these clocks may be synchronized by requiring that they start simultaneously with each other. Any event in space-time may then be labeled by the four numbers t, x, y, z, where t is the time of the event as determined by the synchronized clock situated at that grid point. See Frame of reference
It is of interest to compare the coordinate labelings given to events in space-time by two inertial observers, and ′, who are in relative motion. The relationship occurring in prerelativity physics is called a galilean transformation. In the simple case where ′ moves with velocity v in the x direction with respect to, and these observers meet at the event A labeled by (t, x, y, z) = (t′, x′, y′, z′) = (0, 0, 0, 0), with the axes of the grid of meter sticks carried by ′ aligned (that is, not rotated) with respect to those of, the transformation is given by Eqs. (1).
Causal structure in special relativity
In special relativity there is a different causal relationship between an arbitrary event A and other events in space-time (Fig. 2). As in prerelativity physics, there are many events, B, which lie to the past of A. There also are many events which lie to the future of A. However, there is now a much larger class of events which lie neither to the past nor to the future of A. These events are referred to as being spacelike-related to A.
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The most striking feature of this causal structure (Fig. 2) is the absence of any three-dimensional surface of simultaneity. Indeed, the closest analog to the surface of simultaneity in prerelativity physics is the double-cone-shaped surface that marks the boundaries of the past and future of event A. This surface comprises the paths in space-time of all light rays which pass through event A, and for this reason it is referred to as the light cone of A. Thus, the statement that the events lying to the future of A are contained within the light cone of A is equivalent to the statement that a material body present at event A can never overtake a light ray emitted at event A. In special relativity, the light cone of an event A replaces the surface of simultaneity with event A as the absolute, observer-independent structure of space-time related to causality.
As in prerelativity physics, it is assumed in special relativity that an inertial observer, can build a rigid grid of meter sticks, place clocks at the grid points, and label events in space-time by global inertial coordinates (t, x, y, z). The only difference from the procedure used for the construction of the similar coordinates in prerelativity physics is that the synchronization of clocks is now nontrivial, since the causal structure of space-time no longer defines an absolute notion of simultaneity. Nevertheless, any pair of clocks in 's system can be synchronized—and thereby all of 's clocks can be synchronized—by having an assistant stationed half-way between the clocks send a signal to the two clocks in a symmetrical manner. This synchronization of clocks allows to define a notion of simultaneity; that is, may declare events A1 and A2 to be simultaneous if time readings t1 and t2 of the synchronized clocks at events A1 and A2 satisfy t1 = t2. However, events judged by to be simultaneous will, in general, be judged by ′ to be nonsimultaneous.
The key assumptions of special relativity are encapsulated by the following two postulates.
Postulate 1: The laws of physics do not distinguish between inertial observers; in particular, no inertial observer can be said to be at rest in an absolute sense. Thus, if observer writes down equations describing laws of physics obeyed by physically measurable quantities in her global inertial coordinate system (t, x, y, z), then the form of these equations must be identical when written down by observer ′ in his global inertial coordinates (t′, x′, y′, z′).
Postulate 2: All inertial observers (independent of their relative motion) must always obtain the same value, c, when they measure the velocity of light in vacuum. In particular, the path of a light ray in space-time must be independent of the motion of the emitter of the light ray. Furthermore, no material body can have a velocity greater than c.
The precise relationship between the labeling of events in space-time by the coordinate systems of two inertial observers, and ′, in special relativity is given by the Lorentz transformation formulas. In the simple case where ′ moves with velocity v in the x direction with respect to and crosses 's world line at the event labeled by (t, x, y, z) = (t′, x′, y′, z′) = (0, 0, 0, 0) with spatial axes aligned, the Lorentz transformation is given by Eqs. (2a)–(2d). Equation (2a) shows explicitly that and ′ disagree over simultaneity.
Space-time geometry
A key question both in prerelativity physics and in special relativity is what quantities, describing the space-time relationships between events, are observer independent. Such quantities having observer-independent status may be viewed as describing the fundamental, intrinsic structure of space-time.
It has already been seen that in special relativity the time interval, Δt, between two events is no longer observer independent. Furthermore, since different inertial observers disagree over simultaneity, the spatial interval between two simultaneous events is not even a well-defined concept, and cannot be observer independent. Remarkably, however, in special relativity, all inertial observers agree upon the value of the space-time interval, I, between any two events, where I is defined by Eq. (3).
The formulation of special relativity as a theory of the lorentzian geometry of space-time is of great importance for the further development of the theory, since it makes possible the generalization which describes gravitation. The lorentzian geometry defined by Eq. (3) is a flat geometry, wherein initially parallel geodesics remain parallel forever. The theory of general relativity accounts for the effects of gravitation by allowing the lorentzian geometry of space-time to be curved. See Space-time
Consequences
The theory of special relativity makes many important predictions, the most striking of which concern properties of time. One such effect, known as time dilation, is predicted directly by the Lorentz transformation formula (2a). If observer carries a clock, then the event at which 's clock reads time &tgr; would be labeled by her as (&tgr;, 0, 0, 0). According to Eq. (2a), the observer ′ would label the event as (t ′, x ′, 0, 0) where t ′ is given by Eq. (4).
The decay of unstable elementary particles provides an important direct application of the time dilation effect. If a particle is observed to have a decay lifetime T when it is at rest, special relativity predicts that its observed lifetime will increase according to Eq. (6) when it is moving. Exactly such an increase is routinely observed in experiments using particle accelerators, where particle velocities can be made to be extremely close to c. See Particle accelerator
An even more striking prediction of special relativity is the clock paradox: Two identical clocks which start together at an event A, undergo different motions, and then rejoin at event B will, in general, register different total elapsed time in going from A to B. This effect is the lorentzian geometry analog of the mundane fact in euclidean geometry that different paths between two points can have different total lengths. See Clock paradox
General theory
One of the basic tenets of special relativity is that no physical effect can propagate with a velocity greater than the speed of light, c, which represents a universal speed limit. On the other hand, classical gravitational theory describes the gravitational field of a body throughout space as a function of its instantaneous position, which is equivalent to the assumption that gravitational effects propagate with an infinite velocity. Thus, special relativity and classical gravitational theory are inconsistent, and a modified theory of gravity is necessary.
Principle of equivalence
It had long been considered a fundamental question why bodies of different mass fall with the same acceleration in a gravitational field. This situation was explained by Newton with the statement that both the gravitational force on a body and its inertial resistance to acceleration are proportional to its mass.
Newton's explanation is more in the nature of an ad hoc description. A deeper and more natural explanation occurred to Einstein. There are numerous forces other than gravity which are mass-proportional. These generally arise due to the use of accelerated coordinate systems to describe the motion, for example, the centrifugal force encountered in a rotating coordinate system. If an observer in the gravitational field of the Earth and another in an accelerating elevator or rocket in free space both drop a test body, they will both observe it to accelerate relative to the floor. According to classical theory, the Earth-based observer would attribute this to a gravitational force and the elevator-based observer would attribute it to the accelerated floor overtaking the uniformly moving body. In both cases the motion is identical, and in particular the acceleration is independent of the mass of the test body. Einstein elevated this fact to a general principle, the principle of equivalence; the principle states that on a local scale all physical effects of a gravitational field are indistinguishable from the physical effects of an accelerated coordinate system. This profound principle is the physical cornerstone of the theory of general relativity. From the point of view of the principle of equivalence, it is obvious why the motion of a test body in a gravitational field is independent of its mass. But the principle applies not only to mechanics but to all physical phenomena and thereby has profound consequences for electromagnetic and other nonmechanical phenomena. See Centrifugal force
Tensor field equations
The close connection between gravity and accelerating coordinate systems convinced Einstein that gravity is fundamentally a geometric phenomenon. Because of this, it is naturally described by the mathematics of higher-dimensional abstract geometry. This geometry involves systems of equations, called tensor equations, that are manifestly independent of the coordinate system. Tensors are a simple generalization of vectors.
The space-time of relativity contains one covariant second-rank tensor of particularly great importance, called the metric tensor gμ&ngr;, which is a generalization of the Lorentz metric of special relativity, introduced in Eq. (3). Nearby points in space-time, called events, which are separated by coordinate distances dxμ have an invariant physical separation whose square, called the line element, is defined by Eq. (5).
Tensor equations are equations in which one tensor of a given type is set equal to another of the same type. The field equations of general relativity are tensor equations for the metric tensor, which completely describes the geometry of the space. The Riemann tensor (or curvature tensor), Rμβν α , plays a central role in the geometric structure of a space; if it is zero, the space is termed flat and has no gravitational field; if nonzero, the space is termed curved, and a gravitational field is present. In terms of the contracted Riemann tensor, that is, a Riemann tensor summed over α = β, the Einstein field equations for empty space are given by Eq. (6). (6) 
The field equations are a set of 10 second-order partial differential equations since the four-by-four symmetric tensor Rμ&ngr; has 10 independent components; they are to be solved for the metric tensor. A solution in a given coordinate system defines an Einstein space-time. The curvature of this space corresponds to the intrinsic presence of a gravitational field. Thus the concept of a field of mechanical force in classical gravitational theory is replaced by the geometric concept of curved space in relativity theory.
In a nonempty region of space the field equations (6) must be modified to include a tensor representing the matter or energy content of space, the energy-momentum tensor Tμ&ngr;. The modified equations are Eq. (7), where
Cosmological term
The field equations were given in the form of Eq. (7) by Einstein in 1916. However, they can be consistently generalized by the addition of another term on the left side, which he called the cosmological term, &Lgr;gμ&ngr;. The more general equations are (8). The constant &Lgr; is called the cosmological constant.
Einstein introduced the cosmological term in 1917 in order to obtain mathematical models of the universe that were independent of time, since it was then believed that the universe was static. When it was discovered in 1929 that the universe is expanding, as evidenced by the Doppler shifts of distant galaxies, Einstein abandoned the cosmological term. However, interest in the cosmological constant has revived in connection with a serious inconsistency between relativity and quantum theory involving the quantum energy of the vacuum, and with observations since 1998 of type Ia supernovae which suggest that the expansion of the universe is accelerating.
Motion of test bodies
The path of a test body is a generalization of a straight line in euclidean space; it is the shortest “distance” (in terms of intervals ds) between points in space-time, known as a geodesic. General relativity theory possesses an extraordinary property: because the field equations are nonlinear, unlike those of newtonian theory, the motion of a test body in a gravitational field is not arbitrary since the body itself has mass and contributes to the field. Indeed, the field equations are so restrictive that the geodesic equation of motion is a necessary consequence and need not be treated as a separate postulate.
Schwarzschild solution
A very important solution of the field equations was obtained by K. Schwarzschild in 1916, surprisingly soon after the inception of general relativity. This solution represents the field in free space around a spherically symmetric body such as the Sun. It is the basis for a relativistic description of the solar system and most of the experimental tests of general relativity which have been carried out.
Gravitational redshift
Electromagnetic radiation of a given frequency emitted in a gravitational field will appear to an outside observer to have a lower frequency; that is, it will be redshifted. The redshift can be derived from the principle of equivalence. The most accurate test of the redshift to date was performed using a hydrogen maser on a rocket. Comparison of the maser frequency with Earth-based masers gave a measured redshift in agreement with theory to about 1 part in 104.
Perihelion shift
The equations of motion can be solved for a planet considered as a test body in the Schwarzschild field of the Sun. As should be expected, the orbits obtained are very similar to the ellipses of classical theory. However, the ellipse rotates very slowly in the plane of the orbit so that the perihelion, the point of closest approach of the planet to the Sun, is at a slightly different angular position for each orbit. This shift is extremely small. It is greatest in the case of the planet Mercury, whose perihelion advance is predicted to be 43 seconds of arc in a century. This agrees with the discrepancy between classical theory and observation, which was well known for many years before the discovery of general relativity.
Deflection of light
The principle of equivalence suggests an extraordinary phenomenon of gravity. Light or other electromagnetic radiation crossing the Einstein elevator horizontally will appear to be deflected downward in a parabolic arc because of the upward acceleration of the elevator. The same phenomenon must occur for light in the gravitational field of the Sun; it must be deflected toward the Sun. A calculation of this deflection gives 1.75 seconds for the net deflection of starlight grazing the edge of the Sun. Modern measurements, made by tracking quasars as they pass near or behind the Sun, find the deflection to be within 1% of the value predicted by general relativity.
In 1936 Einstein observed that if two stars were exactly lined up with the Earth, the more distant star would appear as a ring of light, distorted from its point appearance by the lens effect of the gravitational field of the nearer star. It was soon pointed out that a very similar phenomenon was much more likely to occur for entire galaxies instead of individual stars. Many candidates for such gravitational lens systems have been found.
Radio time delay
In the curved space around the Sun the distance between points in space, for example between two planets, is not the same as it would be in flat space. In particular, the round-trip travel time of a radar signal sent between the Earth and the planet will be measurably increased by the curvature effect when the Earth, the Sun, and the planet are approximately lined up. Using a transponder on the Viking spacecraft, the time delay was found to agree with the predictions of general relativity to an accuracy of about one-half of 1%.