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. 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*.

# γ

d
t

d
τ

=

1

c
2

c
2

−

v

2

c
2

=

1

1
−

β

2

=

1

v

2

v

4

v

6

v

8

.
.
.

{\displaystyle
\gamma =
\frac{dt}{d\tau} =
\sqrt{\frac{1}{\frac{c^2}{c^2} - \frac{\mathbf{v}^2}{c^2}}} =
\sqrt{\frac{1}{1 - \mathbf{\beta}^2}} =
\sqrt{1 + \mathbf{v}^2 + \mathbf{v}^4 + \mathbf{v}^6 + \mathbf{v}^8 ...}
}

where:

* ***v*** is the relative velocity between inertial reference frames,
* *c* is the *speed of light in a vacuum*,
* ***β*** is the **DIMENSIONLESS** 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).

The derivatives of gamma are:

γ
˙

=

d
γ

d

β

=

v

c

γ
3

dimensionless vector

γ
˙

=

d
γ

d
t

=

d
γ

d

β

d

β

d
t

=

v

c

γ
3

⋅

a

c

vector with units of

1
t

γ
˙

=

d
γ

d
τ

=

d
γ

d

β

d

β

d
t

d
t

d
τ

=

v

c

γ
3

⋅

a

c

⋅
γ

vector with units of

1
t

{\displaystyle
\begin{array}{lllll}
\dot\gamma &= \frac{d \gamma}{d\mathbf{\beta}} & &= \frac{\mathbf{v}}{c} \gamma^3 & \text{dimensionless vector} \\

\dot\gamma &= \frac{d \gamma}{d t} &= \frac{d \gamma}{d\mathbf{\beta}} \frac{d\mathbf{\beta}}{dt} &= \frac{\mathbf{v}}{c} \gamma^3 \cdot \frac{\mathbf{a}}{c} & \text{vector with units of} \frac{1}{t} \\

\dot\gamma &= \frac{d \gamma}{d \tau} &= \frac{d \gamma}{d\mathbf{\beta}} \frac{d\mathbf{\beta}}{dt} \frac{dt}{d \tau} &= \frac{\mathbf{v}}{c} \gamma^3 \cdot \frac{\mathbf{a}}{c} \cdot \gamma & \text{vector with units of} \frac{1}{t}
\end{array}
}

Since

γ
−
1
≈

β

2

2

{\displaystyle \gamma - 1 \approx \frac{\mathbf{\beta}^2}{2}}

 for small values of v it is not surprising that:

E
kinetic

=
m
⋅
(
γ
−
1
)

c
2

≈

m

β

2

c
2

2

≈

m

v

2

2

{\displaystyle E\_\text{kinetic} = m \cdot (\gamma - 1 ) c^2 \approx \frac{m \mathbf{\beta}^2 c^2}{2} \approx \frac{m \mathbf{v}^2}{2}}

## Inverse of gamma

See also: A Geometric View of Complex Trigonometric Functions

Gamma is defined as:

# γ

d
t

d
τ

=

1

c
2

c
2

−

v

2

c
2

{\displaystyle
\gamma =
\frac{dt}{d\tau} =
\sqrt{\frac{1}{\frac{c^2}{c^2} - \frac{\mathbf{v}^2}{c^2}}}
}

To complement the definition, some authors define the reciprocal

# α

1
γ

=

c
2

c
2

−

v

2

c
2

=

1
−

β

2

;

{\displaystyle
\alpha = \frac{1}{\gamma} =
\sqrt{\frac{c^2}{c^2} - \frac{\mathbf{v}^2}{c^2}} \ =
\sqrt{1- \mathbf{\beta}^2} ;
}

Hence

α
2

=

c
2

c
2

−

v

2

c
2

{\displaystyle \alpha^2 = \frac{c^2}{c^2} - \frac{\mathbf{v}^2}{c^2}}

I would have preferred it to have been defined as

α
^

2

c
2

v

2

c
2

=

c
2

c
2

{\displaystyle \frac{\hat\alpha^2}{c^2} + \frac{\mathbf{v}^2}{c^2} = \frac{c^2}{c^2}}

This looks a great deal like the Pythagorean theorem for a unit circle. If c is the hypotenuse and v is the sine then

α
^

{\displaystyle \hat\alpha}

 is the cosine and

γ

{\displaystyle \gamma}

 is the secant.

Therefore

α
^

{\displaystyle \hat\alpha}

 is a velocity that is c when the particle is not moving but approaches zero when the particle approaches the speed of light. It is the "speed" at which the particle is moving through time. (

τ

/

t

{\displaystyle \tau / t}

)

α

{\displaystyle \alpha}
 sin(θ)
  arcsin(v)
 tan(θ)

α

{\displaystyle \alpha}

 = v

γ

{\displaystyle \gamma}

 = proper velocity

multiplying by m we get:

 m\*sin(θ)
 sec(θ)-1

γ

{\displaystyle \gamma}

-1 = kinetic energy (topmost image to the right)

and

 ex+iθ | = ex (cos(θ) + isin(θ))

In analogy to four-position, four-velocity, and four-acceleration we can create the following vectors based on alpha and the derivative with respect to ordinary coordinate time (t). The Pythagorean theorem is

c
2

=

t
2

x
2

y
2

z
2

{\displaystyle c^2 = t^2 + x^2 + y^2 + z^2}

.

[

c
τ
(
t
)

x
(
t
)

y
(
t
)

z
(
t
)

]

,

[

α
c

v
x

(
t
)

v
y

(
t
)

v
z

(
t
)

]

,

[

a
v

/

α
c

a
x

(
t
)

a
y

(
t
)

a
z

(
t
)

]

[

m

c
2

−
m

v
2

m

v
x
2

(
t
)

m

v
y
2

(
t
)

m

v
z
2

(
t
)

]

,

{\displaystyle
\begin{bmatrix}
c \tau(t) \\ x(t) \\ y(t) \\ z(t)
\end{bmatrix} ,
\begin{bmatrix}
\alpha c \\ v\_x(t) \\ v\_y(t) \\ v\_z(t)
\end{bmatrix} ,
\begin{bmatrix}
a v / \alpha c \\ a\_x(t) \\ a\_y(t) \\ a\_z(t)
\end{bmatrix} \quad \quad
\begin{bmatrix}
m c^2 - m v^2 \\ m v\_x^2(t) \\ m v\_y^2(t) \\ m v\_z^2(t)
\end{bmatrix} ,
}

Four-position, four-velocity, and four-acceleration work in a similar way but based on gamma and the derivative with respect to proper time. Also they exist on the Unit hyperbola instead of the unit circle. The Pythagorean theorem becomes

−

c
2

=
−

t
2

x
2

y
2

z
2

{\displaystyle -c^2 = -t^2 + x^2 + y^2 + z^2}

 and the resulting metric signature becomes (− + + +). See Minkowski\_space#Metric\_signature. (Time can be thought of as imaginary distance)

[

c
t
(
τ
)

x
(
t
(
τ
)
)

y
(
t
(
τ
)
)

z
(
t
(
τ
)
)

]

[

γ
c

γ

v
x

(
t
)

γ

v
y

(
t
)

γ

v
z

(
t
)

]

[

γ
4

a
v

c
2

⋅
c

γ
4

a
v

c
2

⋅

v
x

(
t
)
γ
⋅
γ

a
x

(
t
)

γ
4

a
v

c
2

⋅

v
y

(
t
)
γ
⋅
γ

a
y

(
t
)

γ
4

a
v

c
2

⋅

v
z

(
t
)
γ
⋅
γ

a
z

(
t
)

]

{\displaystyle
\begin{bmatrix}
ct(\tau) \\ x(t(\tau)) \\ y(t(\tau)) \\ z(t(\tau))
\end{bmatrix}
\begin{bmatrix}
\gamma c \\ \gamma v\_x(t) \\ \gamma v\_y(t) \\ \gamma v\_z(t)
\end{bmatrix}
\left[ \begin{array}{ll}
\gamma^4\tfrac{\mathbf{av}}{c^2} \cdot c \\
\gamma^4\tfrac{\mathbf{av}}{c^2} \cdot v\_x(t) + \gamma \cdot \gamma a\_x(t) \\
\gamma^4\tfrac{\mathbf{av}}{c^2} \cdot v\_y(t) + \gamma \cdot \gamma a\_y(t) \\
\gamma^4\tfrac{\mathbf{av}}{c^2} \cdot v\_z(t) + \gamma \cdot \gamma a\_z(t) \\
\end{array} \right]
}

## Occurrence

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*:

t
′

=
γ

(

t
−

v
x

c
2

)

,

{\displaystyle t' = \gamma \left( t - \frac{vx}{c^2} \right ), }

x
′

=
γ

(

x
−
v
t

)

.

{\displaystyle x' = \gamma \left( x - vt \right ). }

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:

Δ

t
′

=
γ
Δ
t
.

{\displaystyle \Delta t' = \gamma \Delta t.}
* **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:

Δ

x
′

=
Δ
x

/

γ
.

{\displaystyle \Delta x' = \Delta x/\gamma.}

Applying conservation of momentum and energy leads to these results:

* **Relativistic mass:** The mass *m* of an object in motion is dependent on

γ

{\displaystyle \gamma}

 and the rest mass *m*0:

# m
γ

m
0

.

{\displaystyle m = \gamma m\_0.}
* **Relativistic momentum:** The relativistic momentum relation takes the same form as for classical momentum, but using the above relativistic mass:

p
→

=
m

v
→

=
γ

m
0

v
→

.

{\displaystyle \vec p = m \vec v = \gamma m\_0 \vec v.}
* **Relativistic kinetic energy:** The relativistic kinetic energy relation takes the slightly modified form:

E
k

=
E
−

E
0

=
(
γ
−
1
)

m
0

c
2

{\displaystyle E\_k = E - E\_0 = (\gamma - 1) m\_0 c^2}

As

γ

{\displaystyle \gamma}

 is a function of

v
c

{\displaystyle \tfrac{v}{c}}

, the non-relativistic limit gives

lim

c
→
∞

E
k

=

1
2

m
0

v
2

{\displaystyle \lim\_{c\to\infty}E\_k=\tfrac{1}{2}m\_0v^2}

, as expected from Newtonian considerations.
## Numerical values

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.

## Alternative representations

Main articles: Momentum, Rapidity
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

# γ

1

(

p

m
0

c

)

2

{\displaystyle \gamma = \sqrt{1+\left ( \frac{p}{m\_0 c} \right )^2 } }

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

### Rapidity

Applying the definition of rapidity as the hyperbolic angle

φ

{\displaystyle \varphi}

:

tanh
⁡
# φ
β

{\displaystyle \tanh \varphi = \beta}

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

# γ
cosh
⁡
# φ

1

1
−

tanh
2

⁡
φ

=

1

1
−

β
2

.

{\displaystyle \gamma = \cosh \varphi = \frac{1}{\sqrt{1 - \tanh^2 \varphi}} = \frac{1}{\sqrt{1 - \beta^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.

### Series expansion (velocity)

The Lorentz factor has the Maclaurin series:

γ

=

1

1
−

β
2

=

∑

# n
0

∞

β

2
n

∏

# k
1

n

(

2
k
−
1

2
k

)

=
1

1
2

β
2

3
8

β
4

5
16

β
6

35
128

β
8

63
256

β

10

⋯
,

{\displaystyle \begin{align}
\gamma & = \dfrac{1}{\sqrt{1 - \beta^2}} \\
& = \sum\_{n=0}^{\infty} \beta^{2n}\prod\_{k=1}^n \left(\dfrac{2k - 1}{2k}\right) \\
& = 1 + \tfrac12 \beta^2 + \tfrac38 \beta^4 + \tfrac{5}{16} \beta^6 + \tfrac{35}{128} \beta^8 + \tfrac{63}{256} \beta^{10} + \cdots ,\\
\end{align}}

which is a special case of a binomial series.

The approximation *γ* ≈ 1 + 12 *β*2 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:

p
→

=
γ
m

v
→

,

E

=
γ
m

c
2

.

{\displaystyle \begin{align}
\vec p & = \gamma m \vec v, \\
E & = \gamma m c^2.
\end{align}}

For *γ* ≈ 1 and *γ* ≈ 1 + 12 *β*2, respectively, these reduce to their Newtonian equivalents:

p
→

=
m

v
→

,

E

=
m

c
2

1
2

m

v
2

.

{\displaystyle \begin{align}
\vec p & = m \vec v, \\
E & = m c^2 + \tfrac12 m v^2.
\end{align}}

The Lorentz factor equation can also be inverted to yield

# β

1
−

1

γ
2

.

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

This has an asymptotic form

# β
1
−

1
2

γ

−
2

−

1
8

γ

−
4

−

1
16

γ

−
6

−

5
128

γ

−
8

⋯

{\displaystyle \beta = 1 - \tfrac12 \gamma^{-2} - \tfrac18 \gamma^{-4} - \tfrac{1}{16} \gamma^{-6} - \tfrac{5}{128} \gamma^{-8} + \cdots}

.
The first two terms are occasionally used to quickly calculate velocities from large *γ* values. The approximation *β* ≈ 1 − 12 *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

{\displaystyle \gamma}

 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.

Subatomic particles called muons, have a relatively high Lorentz factor and therefore experience extreme time dilation. As an example, muons generally have a mean lifetime 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.

## See also

* Inertial frame of reference
* Pseudorapidity
* Proper velocity