Alternative representations Lorentz factor

1 alternative representations

1.1 momentum
1.2 rapidity
1.3 series expansion (velocity)

alternative representations

there other ways write factor. above, velocity v used, related variables such momentum , rapidity may convenient.

momentum

solving previous relativistic momentum equation γ leads to:

γ
=


1
+


(


p


m

0


c



)


2






{\displaystyle \gamma ={\sqrt {1+\left({\frac {p}{m_{0}c}}\right)^{2}}}}

this form used, appear in maxwell–jüttner distribution.

rapidity

applying definition of rapidity following hyperbolic angle φ:

tanh
⁡
φ
=
β




{\displaystyle \tanh \varphi =\beta \,\!}

also leads 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 property of lorentz transformation, can shown rapidity additive, useful property velocity not have. rapidity parameter forms one-parameter group, foundation physical models.

series expansion (velocity)

the lorentz factor has 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


+
⋯






{\displaystyle {\begin{aligned}\gamma &={\dfrac {1}{\sqrt {1-\beta ^{2}}}}\\&=\sum _{n=0}^{\infty }\beta ^{2n}\prod _{k=1}^{n}\left({\dfrac {2k-1}{2k}}\right)\\&=1+{\tfrac {1}{2}}\beta ^{2}+{\tfrac {3}{8}}\beta ^{4}+{\tfrac {5}{16}}\beta ^{6}+{\tfrac {35}{128}}\beta ^{8}+\cdots \\\end{aligned}}}

which special case of binomial series.

the approximation γ ≈ 1 + /2 β may used calculate relativistic effects @ low speeds. holds within 1% error v < 0.4 c (v < 120,000 km/s), , within 0.1% error v < 0.22 c (v < 66,000 km/s).

the truncated versions of series allow physicists prove special relativity reduces newtonian mechanics @ low speeds. example, in special relativity, following 2 equations hold:

p
→



=
γ
m



v
→





{\displaystyle {\vec {p}}=\gamma m{\vec {v}}}






e
=
γ
m

c

2





{\displaystyle e=\gamma mc^{2}\,}

for γ ≈ 1 , γ ≈ 1 + /2 β, respectively, these reduce newtonian equivalents:

p
→



=
m



v
→





{\displaystyle {\vec {p}}=m{\vec {v}}}






e
=
m

c

2


+



1
2



m

v

2




{\displaystyle e=mc^{2}+{\tfrac {1}{2}}mv^{2}}

the lorentz factor equation can inverted yield:

β
=


1
−


1

γ

2








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

this has asymptotic form of:

β
=
1
−



1
2




γ

−
2


−



1
8




γ

−
4


−



1
16




γ

−
6


−



5
128




γ

−
8


+
⋯


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

the first 2 terms used calculate velocities large γ values. approximation β ≈ 1 - /2 γ holds within 1% tolerance γ > 2, , within 0.1% tolerance γ > 3.5.