Reference frames, coordinates and the Lorentz transformation
Main article: Lorentz transformation In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in uniform motion with respect to the other. In classical physics (Galilean relativity), the only conversion believed necessary was x' = x − vt, describing how the origin of one observer's coordinate system Diagram 1. Changing views of spacetime along the 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" by the time dimension, and typically encompasses a large area of spacetime wherein perceptually straight of a rapidly accelerating observer. In this animation, the vertical direction indicates time and the horizontal direction indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The lower quarter of the diagram shows the events that are visible to the observer, and the upper quarter shows the light cone A Light cone is the path that a flash of light would take through spacetime. As time progresses, the light from the flash spreads out in a circle, and the result is a cone. In reality, there are three space dimensions, so the light actually forms a sphere in space, and the light cone is actually a fourth dimensional shape, but it's much easier to- those that will be able to see the observer. The small dots are arbitrary events in spacetime. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates.Relativity theory depends on "reference frames". The term reference frame as used here is an observational perspective in space at rest, or in uniform motion, from which a position can be measured along 3 spatial axes. In addition, a reference frame has the ability to determine measurements of the time of events using a 'clock' (any reference device with uniform periodicity).
An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: it is a "point" in space-time In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions. According to certain Euclidean space perceptions, the universe has three. Since the speed of light is constant in relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.
For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four space-time coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame S.
In relativity theory we often want to calculate the position of a point from a different reference point.
Suppose we have a second reference frame S', whose spatial axes and clock exactly coincide with that of S at time zero, but it is moving at a constant velocity with respect to S along the -axis.
Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything is always moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore S and S' are not comoving.
Let's define the event In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions. According to certain Euclidean space perceptions, the universe has three to have space-time coordinates in system S and in S'. Then the Lorentz transformation In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in uniform motion with respect to the other. In classical physics (Galilean relativity), the only conversion believed necessary was x' = x − vt, describing how the origin of one observer's coordinate system specifies that these coordinates are related in the following way:
where is called the Lorentz factor The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ. It gets its name from its earlier appearance in Lorentzian electrodynamics. The Lorentz and is the speed of light The speed of light is a fundamental physical constant of spacetime, the speed at which electromagnetic radiation, including visible light, travels in free space. It is an upper bound on the rate of transfer of matter and information between places. The speed of light is usually denoted by the symbol c in a vacuum.
The and coordinates are unaffected, only the and axes transformed. These Lorentz transformations form a one-parameter group from the real line R to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is injective then φ(R), the image, will be a subgroup of G that is isomorphic to R as additive group. That is, we start knowing only that of linear mappings In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is in especially common use, for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides, that parameter being called rapidity In relativity rapidity is an alternative to speed as a method of measuring motion. For low speeds, rapidity and speed are proportional, but for high speeds, rapidity takes a larger value. The rapidity of light is infinite.
A quantity invariant under Lorentz transformations In physics, the Lorentz transformation precisely describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frame of reference. It reflects the surprising fact that observers moving at different velocities report different distances, passage of time, and in is known as a Lorentz scalar In physics a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. A Lorentz scalar is generated from vectors and tensors. While the vectors and tensors are altered by Lorentz transformations, scalars are unchanged.
The Lorentz transformation given above is for the particular case in which the velocity v of S' with respect to S is parallel to the -axis. We now give the Lorentz transformation in the general case. Suppose the velocity of S' with respect to S is . Denote the space-time coordinates of an event in S by (instead of (t,x,y,z)). Then the coordinates of this event in S' are given by:
where denotes the transpose of , , and denotes the projection onto the direction of .
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