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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 (lowercase gamma). It gets its name from its earlier appearance in Lorentzian electrodynamics. The Lorentz factor is named after the Dutch physicist Hendrik Lorentz.[1]

It is defined as:

\gamma \equiv \frac{c}{\sqrt{c^2 - u^2}} = \frac{1}{\sqrt{1 - \beta^2}} = \frac{\mathrm{d}t}{\mathrm{d}\tau}

where:

\beta = \frac{u}{c} is the velocity in terms of the speed of light,
u is the velocity as observed in the reference frame where time t is measured
τ is the proper time, and
c is the speed of light.

Contents

Approximations

The Lorentz factor has a Maclaurin series of:

\gamma ( \beta ) = 1 + \frac{1}{2} \beta^2 + \frac{3}{8} \beta^4 + \frac{5}{16} \beta^6 + \frac{35}{128} \beta^8 + ...

The approximation γ ≈ 1 + 1/2 β2 is occasionally 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:

\vec p = \gamma m \vec v
E = \gamma m c^2 \,

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

\vec p = m \vec v
 E = m c^2 + \frac{1}{2} m v^2

The Lorentz factor equation can also be inverted to yield:

\beta = \sqrt{1 - \frac{1}{\gamma^2}}

This has an asymptotic form of:

\beta = 1 - \frac{1}{2} \gamma^{-2} - \frac{1}{8} \gamma^{-4} - \frac{1}{16} \gamma^{-6} - \frac{5}{128} \gamma^{-8} + ...

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.

Values

Lorentz factor as a function of velocity. It starts at value 1 and for v\to c it goes to infinity.
Speed Lorentz factor Reciprocal
β = v / c γ 1 / γ
0.000 1.000 1.000
0.100 1.005 0.995
0.200 1.021 0.980
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
0.800 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

In the above chart, the lefthand column shows speeds as different fractions of the speed of light (c). The middle column shows the corresponding Lorentz factor.

Rapidity

Note that if tanh r = β, then γ = cosh r. Here the hyperbolic angle r is known as the rapidity[2]. 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. Sometimes (especially in discussion of superluminal motion) γ is written as Γ (uppercase-gamma) rather than γ (lowercase-gamma).

The Lorentz factor applies to time dilation, length contraction and relativistic mass relative to rest mass in Special Relativity. An object moving with respect to an observer will be seen to move in slow motion given by multiplying its actual elapsed time by gamma. Its length is measured shorter as though its local length were divided by γ.

γ may also (less often) refer to \frac{\mathrm{d}\tau}{\mathrm{d}t} = \sqrt{1 - \beta^2}. This may make the symbol γ ambiguous, so many authors prefer to avoid possible confusion by writing out the Lorentz term in full.

In particle physics, rapidity is usually defined as (For example, see [3])

y = \frac{1}{2} \ln \left(\frac{E+p_L}{E-p_L}\right)

Derivation

One of the fundamental postulates of Einstein's special theory of relativity is that all inertial observers will measure the same speed of light in vacuum regardless of their relative motion with respect to each other or the source. Imagine two observers: the first, observer A, traveling at a constant speed v with respect to a second inertial reference frame in which observer B is stationary. A points a laser “upward” (perpendicular to the direction of travel). From B's perspective, the light is traveling at an angle. After a period of time tB, A has traveled (from B's perspective) a distance d = vtB; the light had traveled (also from B perspective) a distance d = ctB at an angle. The upward component of the path dt of the light can be solved by the Pythagorean theorem.

d_t = \sqrt{(c t _B)^2 - (v t_B)^2}

Factoring out ctB gives,

d_t = c t _B\sqrt{1 - {\left(\frac{v}{c}\right)}^2}

The distance that A sees the light travel is dt = ctA and equating this with dt calculated from B reference frame gives,

ct_A = ct_B \sqrt{1 - {\left(\frac{v}{c}\right)}^2}

which simplifies to

t_A = t_B\sqrt{1 - {\left(\frac{v}{c}\right)}^2}

See also

References

  1. ^ One universe, by Neil deGrasse Tyson, Charles Tsun-Chu Liu, and Robert Irion.
  2. ^ Kinematics, by J.D. Jackson, See page 7 for definition of rapidity.
  3. ^ Introduction to High-Energy Heavy-Ion Collisions, by Cheuk-Yin Wong, See page 17 for definition of rapidity.
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Wiktionary

Up to date as of January 15, 2010

Definition from Wiktionary, a free dictionary

English

Noun

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Wikipedia

Lorentz factor

  1. (physics) the factor, used in special relativity, to calculate the degree of time dilation, length contraction and relativistic mass of an object moving relative to an observer

Simple English

The Lorentz Factor is the name of the factor by which time, length, and "relativistic mass" change for an object while that object is moving and is often written \gamma (gamma). This number is determined by the object's speed in the following way:

\gamma = \frac{1}{\sqrt{1 - (\frac{v}{c})^2}}

Where v is the speed of the object and c is the speed of light (expressed in the same units as your speed). This quantity (the fraction of c) is often labeled \beta (beta) and so the above equation can be rewritten:

\gamma = \frac{1}{\sqrt{1 - \beta^2}}

Contents

Classical Relativity

Classical Relativity is the idea that if you throw a ball at 50 mph while running at 5 mph, the ball will now travel 55 mph. Of course, the ball still moves away from you at 50 mph, so if one were to ask you, you saw the ball traveling 50 mph. Meanwhile, your friend Bob saw that you happened to be running at 5 mph. He would say that the ball was traveling 55 mph. Both of you are right, you just happened to be moving with the ball. The speed of light, c, is 670,616,629 mph. So if you're in a car traveling half the speed of light (0.5c) and you turn on your headlights, the light moves away from you at 1 c... or is it 1.5 c? It ends up that c is c no matter what. The next section explains why it's not c - 0.5c.

Time Dilation

When a clock is in motion, it ticks slower by a (usually small) factor of \gamma. The famous twin paradox says that if there were two twins and twin A stayed on earth while twin B traveled near c for a few years, when twin B got back to earth, he would be many years younger than twin A (because he experienced less time). For example, if twin B left when he was 20 and traveled at .9c for 10 years, then we he got back to earth, twin B would be 30 (20 years + 10 years) and twin A would be almost 43:

20 + (10*\frac{1}{\sqrt{1-.9^2}}) = 42.9416

Twin B wouldn't notice that time had slowed at all. To him, if he looked out a window, it would seem like the universe was moving past him, and therefore slower (remember, to him, he's at rest). So time is relative.

Length Contraction

Another way that the universe changes to make the speed of light not change is that it makes things shorter when they travel. During twin B's journey (see: Time Dilation), he would notice something strange about the universe. He would notice that it got shorter (contracted in the direction of his motion). And the factor by which things get shorter is \gamma.

Relativistic Mass

Something called relativistic mass also increases. It doesn't actually make things heavier, it just makes them harder to push. So by the time you reach 0.9999c, you need a very big force to make you go faster. This makes it impossible for anything to reach the speed of light.

Real Consequences

The most significant consequence of these "relativistic effects" for humans is time dilation. If we were to build a space ship capable of traveling very close to the speed of light, the passengers of the ship could travel millions of light years in years or even months, measured with a clock on board the space ship. Because time is relative though, a clock on the observer would see the trip take millions of years.


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