Stress–energy tensor. Contravariant components of the stress-energy tensor. Definition[edit] The stress–energy tensor involves the use of superscripted variables (not exponents; see tensor index notation and Einstein summation notation). If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: x0 = t, x1 = x, x2 = y, and x3 = z, where t is time in seconds, and x, y, and z are distances in meters. In some alternative theories like Einstein–Cartan theory, the stress–energy tensor may not be perfectly symmetric because of a nonzero spin tensor, which geometrically corresponds to a nonzero torsion tensor. Identifying the components of the tensor[edit] Because the stress–energy tensor is of rank two, its components can be displayed in 4 x 4 matrix form: In the following, i and k range from 1 through 3. and for an electromagnetic field in otherwise empty space this component is where E and B are the electric and magnetic fields, respectively.[3] The components where is:
World line. In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" (such as an orbit in space or a trajectory of a truck on a road map) by the time dimension, and typically encompasses a large area of spacetime wherein perceptually straight paths are recalculated to show their (relatively) more absolute position states — to reveal the nature of special relativity or gravitational interactions.
The idea of world lines originates in physics and was pioneered by Hermann Minkowski. The term is now most often used in relativity theories (i.e., special relativity and general relativity). However, world lines are a general way of representing the course of events. Usage in physics[edit] , upwards and the space coordinate, say horizontally. A curve M in [spacetime] is called a worldline of a particle if its tangent is future timelike at each point.
(where. Twin paradox. In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as traveling, and so, according to an incorrect naive application of time dilation, each should paradoxically find the other to have aged more slowly. However, this scenario can be resolved within the standard framework of special relativity: Acceleration is not relative, unlike position and velocity, and one twin is accelerated more than the other. Therefore the Twin paradox is not a paradox in the sense of a logical contradiction. The twin paradox has been verified experimentally by precise measurements of atomic clocks flown in aircraft and satellites. For example, gravitational time dilation and special relativity together have been used to explain the Hafele–Keating experiment.
Time dilation. Time dilation explains why two working clocks will report different times after different accelerations. For example, ISS astronauts return from missions having aged slightly less than they would have been if they had remained on Earth, and GPS satellites work because they adjust for similar bending of spacetime to coordinate with systems on Earth.[1] An accurate clock at rest with respect to one observer may be measured to tick at a different rate when compared to a second observer's own equally accurate clocks.
This effect arises neither from technical aspects of the clocks nor from the fact that signals need time to propagate, but from the nature of spacetime itself. Overview[edit] In theory, and to make a clearer example, time dilation could affect planned meetings for astronauts with advanced technologies and greater travel speeds. The astronauts would have to set their clocks to count exactly 80 years, whereas mission control – back on Earth – might need to count 81 years. Speed of light. Principle of relativity. In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. For example, in the framework of special relativity the Maxwell equations have the same form in all inertial frames of reference.
In the framework of general relativity the Maxwell equations or the Einstein field equations have the same form in arbitrary frames of reference. Several principles of relativity have been successfully applied throughout science, whether implicitly (as in Newtonian mechanics) or explicitly (as in Albert Einstein's special relativity and general relativity). History of relativity[edit] Basic relativity principles[edit] Certain principles of relativity have been widely assumed in most scientific disciplines. Any principle of relativity prescribes a symmetry in natural law: that is, the laws must look the same to one observer as they do to another. Special principle of relativity[edit] See also[edit]
Special relativity. Relativity of simultaneity. On spaceships, map-clocks may look unsync'ed. Event B is simultaneous with A in the green reference frame, but it occurred before in the blue frame, and will occur later in the red frame. Events A, B, and C occur in different order depending on the motion of the observer. The white line represents a plane of simultaneity being moved from the past to the future. If we imagine one reference frame assigns precisely the same time to two events that are at different points in space, a reference frame that is moving relative to the first will generally assign different times to the two events.
This is illustrated in the ladder paradox, a thought experiment which uses the example of a ladder moving at high speed through a garage. A mathematical form of the relativity of simultaneity ("local time") was introduced by Hendrik Lorentz in 1892, and physically interpreted (to first order in v/c) as the result of a synchronization using light signals by Henri Poincaré in 1900. Spacetime diagrams[edit] Spacetime. In non-relativistic classical mechanics, the use of Euclidean space instead of spacetime is appropriate, as time is treated as universal and constant, being independent of the state of motion of an observer. [disambiguation needed] In relativistic contexts, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer and also on the strength of gravitational fields, which can slow the passage of time for an object as seen by an observer outside the field.
Until the beginning of the 20th century, time was believed to be independent of motion, progressing at a fixed rate in all reference frames; however, later experiments revealed that time slows at higher speeds of the reference frame relative to another reference frame. Such slowing, called time dilation, is explained in special relativity theory. Spacetime in literature[edit] Mathematical concept[edit] is that.
Invariant mass. The invariant mass, rest mass, intrinsic mass, proper mass, or (in the case of bound systems or objects observed in their center of momentum frame) simply mass, is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations. If a center of momentum frame exists for the system, then the invariant mass of a system is simply the total energy divided by the speed of light squared. In other reference frames, the energy of the system increases, but system momentum is subtracted from this, so that the invariant mass remains unchanged.
Systems whose four-momentum is a null vector (for example a single photon or many photons moving in exactly the same direction) have zero invariant mass, and are referred to as massless. A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized tachyon), and these do not appear to exist. . Invariant mass. Frame of reference. Different aspects of "frame of reference"[edit] The need to distinguish between the various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in Cartesian frame of reference. Sometimes the state of motion is emphasized, as in rotating frame of reference. Sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference.[1] In this article, the term observational frame of reference is used when emphasis is upon the state of motion rather than upon the coordinate choice or the character of the observations or observational apparatus.
In this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. And [R, R′, etc.]:[5] . Proper time. The dark blue vertical line represents an inertial observer measuring a coordinate time interval t between events E1 and E2. The red curve represents a clock measuring its proper time τ between the same two events. In terms of four-dimensional spacetime, proper time is analogous to arc length in three-dimensional (Euclidean) space.
By convention, proper time is usually represented by the Greek letter τ (tau) to distinguish it from coordinate time represented by t or T. By contrast, coordinate time is the time between two events as measured by a distant observer using that observer's own method of assigning a time to an event. In the special case of an inertial observer in special relativity, the time is measured using the observer's clock and the observer's definition of simultaneity. The concept of proper time was introduced by Hermann Minkowski in 1908,[1] and is a feature of Minkowski diagrams. Mathematical formalism[edit] In special relativity[edit] In general relativity[edit] where . . . Proper length. The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on the assumption that the locations of all points involved are measured simultaneously.
But in the theory of relativity, the notion of simultaneity is dependent on the observer. Proper distance provide an invariant measure, whose value is the same for all observers. Proper distance is analogous to proper time. Proper distance between two events[edit] In special relativity, the proper distance between two spacelike-separated events is the distance between the two events, as measured in an inertial frame of reference in which the events are simultaneous. Where Δt is the difference in the temporal coordinates of the two events,Δx, Δy, and Δz are differences in the linear, orthogonal, spatial coordinates of the two events, andc is the speed of light.
Proper length or rest length[edit] Proper distance of a path[edit] See also[edit] Gravitomagnetism. Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity.
Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles. The analogy and equations differing only by some small factors were first published in 1893, before general relativity, by Oliver Heaviside as a separate theory expanding Newton's law.[1] Background[edit] ...or equivalently currentI, same field profile, and field generation due to rotation.
Physical analogues of fields[2] Indirect validations of gravitomagnetic effects have been derived from analyses of relativistic jets. Gravitation. Gravitation, or gravity, is a natural phenomenon by which all physical bodies attract each other. It is most commonly recognized and experienced as the agent that gives weight to physical objects, and causes physical objects to fall toward the ground when dropped from a height.
During the grand unification epoch, gravity separated from the electronuclear force. Gravity is the weakest of the four fundamental forces, and appears to have unlimited range (unlike the strong or weak force). The gravitational force is approximately 10-38 times the strength of the strong force (i.e., gravity is 38 orders of magnitude weaker), 10-36 times the strength of the electromagnetic force, and 10-29 times the strength of the weak force. As a consequence, gravity has a negligible influence on the behavior of sub-atomic particles, and plays no role in determining the internal properties of everyday matter.
History of gravitational theory Scientific revolution Newton's theory of gravitation General relativity. Minkowski space. In theoretical physics, Minkowski space is often contrasted with Euclidean space. While a Euclidean space has only spacelike dimensions, a Minkowski space also has one timelike dimension. The isometry group of a Euclidean space is the Euclidean group and for a Minkowski space it is the Poincaré group. History[edit] In 1905 (published 1906) it was noted by Henri Poincaré that, by taking time to be the imaginary part of the fourth spacetime coordinate √−1 ct, a Lorentz transformation can be regarded as a rotation of coordinates in a four-dimensional Euclidean space with three real coordinates representing space, and one imaginary coordinate, representing time, as the fourth dimension.
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. For further historical information see references Galison (1979), Corry (1997), Walter (1999). Structure[edit] The Minkowski inner product[edit] Standard basis[edit] where. Minkowski diagram. Minkowski diagram with resting frame (x,t), moving frame (x′,t′), light cone, and hyperbolas marking out time and space with respect to the origin. The Minkowski diagram, also known as a spacetime diagram, was developed in 1908 by Hermann Minkowski and provides an illustration of the properties of space and time in the special theory of relativity. It allows a quantitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical equations. The term Minkowski diagram is used in both a generic and particular sense. In general, a Minkowski diagram is a graphic depiction of a portion of Minkowski space, often where space has been curtailed to a single dimension.
Basics[edit] A photon moving right at the origin corresponds to the yellow track of events, a straight line with a slope of 45°. For simplification in Minkowski diagrams, usually only events in a universe of one space dimension and one time dimension are considered. History and in. Metric (mathematics) Lorentz transformation. Pseudo-Riemannian manifold. Length contraction. Invariant (physics) Inertial frame of reference. Geodesic (general relativity) Four-vector. Four-momentum. Equivalence principle. Einstein manifold. Lorentz covariance.