# Lorentz’ Lambda

Lorentz Transformations

In physics, the Lorentz transformations (or transformation) are coordinate transformations between two coordinate frames that move at constant velocity relative to each other. The transformations are named after Hendrik Lorentz, a Dutch physicist in electrodynamics.

The most common form of the transformation is expressed as:

where (t, x, y, z) and (t′, x′, y′, z′) represent an event’s coordinates in two frames with relative velocity v, c is the speed of light, and the Lorentz factor:

Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating, moving in curved paths, rotational motion with constant angular velocity, etc.). The term “Lorentz transformations” only refers to transformations between inertial frames, usually in the context of special relativity.

In each reference frame, an observer can use a local coordinate system (most exclusively Cartesian coordinates in this context) to measure lengths, and a clock to measure time intervals. An observer is a real or imaginary entity that can take measurements, say humans, or any other living organism—or even robots and computers. An event is something that happens at a point in space at an instant of time, or more formally a point in spacetime. The transformations connect the space and time coordinates of an event as measured by an observer in each frame.

They supersede the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). The Galilean transformation is a good approximation only at relative speeds much smaller than the speed of light. Lorentz transformations have a number of counterintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events, but always such that the speed of light is the same in all inertial reference frames. The invariance of light speed is one of the postulates of special relativity.

Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. The Lorentz transformation is in accordance with special relativity, but was derived before special relativity.

The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space, the mathematical model of spacetime in special relativity, the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.

Coordinate transformation

A “stationary” observer in frame F defines events with coordinates t, x, y, z. Another frame F moves with velocity v relative to F, and an observer in this “moving” frame F defines events using the coordinates t′, x′, y′, z.

The coordinate axes in each frame are parallel (the x and x axes are parallel, the y and y axes are parallel, and the z and z axes are parallel), remain mutually perpendicular, and relative motion is along the coincident xx′ axes. At t = t′ = 0, the origins of both coordinate systems are the same, (x, y, z) = (x′, y′, z′) = (0, 0, 0). In other words, the times and positions are coincident at this event. If all these hold, then the coordinate systems are said to be in standard configuration, or synchronized.

If an observer in F records an event t, x, y, z, then an observer in F records the same event with coordinates[10]

 Lorentz boost (x direction)

where v is the relative velocity between frames in the x-direction, c is the speed of light, and

(lowercase gamma) is the Lorentz factor.

Here, v is the parameter of the transformation, for a given boost it is a constant number, but can take a continuous range of values. In the setup used here, positive relative velocity v > 0 is motion along the positive directions of the xx axes, zero relative velocity v = 0 is no relative motion, while negative relative velocity v < 0 is relative motion along the negative directions of the xx axes. The magnitude of relative velocity v cannot equal or exceed c, so only subluminal speeds c < v < c are allowed. The corresponding range of γ is 1 ≤ γ < ∞.

The transformations are not defined if v is outside these limits. At the speed of light (v = c) γ is infinite, and faster than light (v > c) γ is a complex number, each of which make the transformations unphysical. The space and time coordinates are measurable quantities and numerically must be real numbers.

As an active transformation, an observer in F′ notices the coordinates of the event to be “boosted” in the negative directions of the xx axes, because of the v in the transformations. This has the equivalent effect of the coordinate system F′ boosted in the positive directions of the xx axes, while the event does not change and is simply represented in another coordinate system, a passive transformation.

The inverse relations (t, x, y, z in terms of t′, x′, y′, z) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. Here F is the “stationary” frame while F is the “moving” frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations from F to F must take exactly the same form as the transformations from F to F. The only difference is F moves with velocity v relative to F (i.e., the relative velocity has the same magnitude but is oppositely directed).

The Lorentz Factor

The Lorentz factor or Lorentz term is the factor by which time, length, and relativistic mass 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.

Due to its ubiquity, it is generally denoted with the symbol γ (Greek lowercase gamma). Sometimes (especially in discussion of superluminal motion) the factor is written as Γ (Greek uppercase-gamma) rather than γ.

Definition

The Lorentz factor is defined as:[2]

where:

• v is the relative velocity between inertial reference frames,
• β is the ratio of v to the speed of light c.
• τ is the proper time for an observer (measuring time intervals in the observer’s own frame),
• t is coordinate time
• c is the speed of light in a vacuum.

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:[3]

Occurrence

Following is a list of formulae from Special relativity which use γ as a shorthand:[2][4]

• 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:
Numerical values

Lorentz factor γ as a function of velocity. Its initial value is 1 (when v = 0); and as velocity approaches the speed of light (vc) γ increases without bound (γ → ∞).

α (Lorentz factor inverse) as a function of velocity – a circular arc.

In the table below, the left-hand column shows speeds as different fractions of the speed of light (i.e. in units of c). The middle column shows the corresponding Lorentz factor, the final is the reciprocal. Values in bold are exact.

Speed (units of c) Lorentz factor Reciprocal

0.000 1.000 1.000
0.050 1.001 0.999
0.100 1.005 0.995
0.150 1.011 0.989
0.200 1.021 0.980
0.250 1.033 0.968
0.300 1.048 0.954
0.400 1.091 0.917
0.500 1.155 0.866
0.600 1.250 0.800
0.700 1.400 0.714
0.750 1.512 0.661
0.800 1.667 0.600
0.866 2.000 0.500
0.900 2.294 0.436
0.990 7.089 0.141
0.999 22.366 0.045
0.99995 100.00 0.010

There are other ways to write the factor. Above, velocity v was used, but related variables such as momentum and rapidity may also be convenient.

Momentum

Solving the previous relativistic momentum equation for γ leads to:

This form is rarely used, although it does appear in the Maxwell–Jüttner distribution.[5]

Rapidity

Applying the definition of rapidity as the following hyperbolic angle φ:[6]

also leads to γ (by use of hyperbolic identities):

Using the property of Lorentz transformation, it can be shown that rapidity is additive, a useful property that velocity does not have. Thus the rapidity parameter forms a one-parameter group, a foundation for physical models.Series expansion (velocity)

The Lorentz factor has the Maclaurin series:

which is a special case of a binomial series.

The approximation γ ≈ 1 + 1/2 β2 may be used to calculate relativistic effects at low speeds. It holds to within 1% error for v < 0.4 c (v < 120,000 km/s), and to within 0.1% error for v < 0.22 c (v < 66,000 km/s).

The truncated versions of this series also allow physicists to prove that special relativity reduces to Newtonian mechanics at low speeds. For example, in special relativity, the following two equations hold:

For γ ≈ 1 and γ ≈ 1 + 1/2 β2, respectively, these reduce to their Newtonian equivalents:

The Lorentz factor equation can also be inverted to yield:

This has an asymptotic form of:

The first two terms are occasionally used to quickly calculate velocities from large γ values. The approximation β ≈ 1 − 1/2 γ−2 holds to within 1% tolerance for γ > 2, and to within 0.1% tolerance for γ > 3.5.

Applications in astronomy

The standard model of long-duration gamma-ray bursts (GRBs) holds that these explosions are ultra-relativistic (initial γ greater than approximately 100), which is invoked to explain the so-called “compactness” problem: absent this ultra-relativistic expansion, the ejecta would be optically thick to pair production at typical peak spectral energies of a few 100 keV, whereas the prompt emission is observed to be non-thermal.[7]

Subatomic particles called muons, have a relatively high lorentz factor and therefore experience extreme time dilation. As an example, muons generally have a half-life of about 2.2 μs which means muons generated from cosmic ray collisions at about 10 km up in the atmosphere should be non-detectable on the ground due to their decay rate. However, it has been found that ~10% of muons are still detected on the surface, thereby proving that to be detectable they have had their decay rates slow down relative to our inertial frame of reference.

Source: en.wikipedia.org/wiki/Lorentz_transformation  en.wikipedia.org/wiki/Lorentz_factor