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Concept
Lorentz factor
Summary
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativity, and it arises in derivations of the Lorentz transformations. The name originates from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist Hendrik Lorentz.
It is generally denoted γ (the Greek lowercase letter gamma). Sometimes (especially in discussion of superluminal motion) the factor is written as Γ (Greek uppercase-gamma) rather than γ.
The Lorentz factor γ is defined as
where:
v is the relative velocity between inertial reference frames,
c is the speed of light in vacuum,
β is the ratio of v to c,
t is coordinate time,
τ is the proper time for an observer (measuring time intervals in the observer's own frame).
This is the most frequently used form in practice, though not the only one (see below for alternative forms).
To complement the definition, some authors define the reciprocal
see velocity addition formula.
Following is a list of formulae from Special relativity which use γ as a shorthand:
The Lorentz transformation: The simplest case is a boost in the x-direction (more general forms including arbitrary directions and rotations not listed here), which describes how spacetime coordinates change from one inertial frame using coordinates (x, y, z, t) to another (x, y, z, t) with relative velocity v:
Corollaries of the above transformations are the results:
Time dilation: The time (∆t) between two ticks as measured in the frame in which the clock is moving, is longer than the time (∆t) between these ticks as measured in the rest frame of the clock:
Length contraction: The length (∆x) of an object as measured in the frame in which it is moving, is shorter than its length (∆x) in its own rest frame:
Applying conservation of momentum and energy leads to these results:
Relativistic mass: The mass m of an object in motion is dependent on and the rest mass m0:
Relativistic momentum: The relativistic momentum relation takes the same form as for classical momentum, but using the above relativistic mass:
Relativistic kinetic energy: The relativistic kinetic energy relation takes the slightly modified form: As is a function of , the non-relativistic limit gives , as expected from Newtonian considerations.
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Ontological neighbourhood
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Rapidity
In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates. For one-dimensional motion, rapidities are additive whereas velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velocity are proportional but, for higher velocities, rapidity takes a larger value, with the rapidity of light being infinite.
Time dilation
Time dilation is the difference in elapsed time as measured by two clocks, either due to a relative velocity between them (special relativity) or due to a difference in gravitational potential between their locations (general relativity). When unspecified, "time dilation" usually refers to the effect due to velocity. After compensating for varying signal delays due to the changing distance between an observer and a moving clock (i.e. Doppler effect), the observer will measure the moving clock as ticking slower than a clock that is at rest in the observer's own reference frame.
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total energy and momentum 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 equal to its total mass in that "rest frame".