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Stress–energy tensor. Contravariant components of the stress-energy tensor.

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. World line. In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime.

World line

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). 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.

Twin paradox

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.

Time dilation

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] 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.

Principle of relativity

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.

General relativity

Special relativity. Relativity of simultaneity. On spaceships, map-clocks may look unsync'ed.

Relativity of simultaneity

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. 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. 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.

Invariant mass

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. Proper time. The dark blue vertical line represents an inertial observer measuring a coordinate time interval t between events E1 and E2.

Proper time

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. Proper length. The measurement of lengths is more complicated in the theory of relativity than in classical mechanics.

Proper length

In classical mechanics, lengths are measured based on the assumption that the locations of all points involved are measured simultaneously. 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] 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. 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] 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. Metric (mathematics) In differential geometry, the word "metric" may refer to a bilinear form that may be defined from the tangent vectors of a differentiable manifold onto a scalar, allowing distances along curves to be determined through integration. It is more properly termed a metric tensor. d : X × X → R.

Lorentz transformation. Pseudo-Riemannian manifold. In differential geometry, a pseudo-Riemannian manifold [1][2] (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite. Instead a weaker condition of nondegeneracy is imposed. Introduction[edit] Manifolds[edit] Length contraction. Invariant (physics) Note: Invariance, does not imply not varying, it pertains to a condition where there is no variation of the system under observation, and the only applicable condition, is the instantaneous condition. Invariance pertains to now(). Inertial frame of reference. All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration.

Geodesic (general relativity) Four-vector. Four-momentum. Equivalence principle. Einstein manifold. Lorentz covariance.