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. 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). 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. Time dilation is caused by differences in either gravity or relative velocity. Relative velocity time dilation[edit] Speed of light.

The speed at which light propagates through transparent materials, such as glass or air, is less than c.

The ratio between c and the speed v at which light travels in a material is called the refractive index n of the material (n = c / v). For example, for visible light the refractive index of glass is typically around 1.5, meaning that light in glass travels at c / 1.5 ≈ 200000 km/s; the refractive index of air for visible light is 1.000293, so the speed of light in air is 299705 km/s or about 88 km/s slower than c. In most practical cases, light and other electromagnetic waves can be thought of as moving "instantaneously", but for long distances and very sensitive measurements their finite speed has noticeable effects. Numerical value, notation, and units. 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).

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.

Spacetime. 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. 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. Proper length. 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] 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. 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. 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(). Now(+1), to a condition where all variations are solely due the internal variables, with no external aspects imparting nor removing energy (Newton´s law of motion: a system in motion continues in motion, unless an external force imparts or removes energy). That condition is met by using the partial derivative function, ∂f(internal)xf(external) and presuming/setting f(external)=constant, leading to ∂f(external)=1 using the chain rule.

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.