Relativistic addition theorem for speeds

The relativistic addition theorem for speeds says how the speed of an object in a certain frame of reference is to be determined if the object moves at a speed with respect to a second frame of reference, which itself moves at a speed with respect to the first . The theorem can be derived from the Lorentz transformation for inertial systems moving against each other . ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {u}}'}$ ${\displaystyle {\vec {v}}}$

In classical mechanics , velocities are added vectorially ( ) and therefore have no upper limit. But since, according to the special theory of relativity, the speed of an object can not exceed the speed of light , the classical equations can only be an approximation. Differences become noticeable when one or both of the velocities to be added are no longer negligibly small compared to the speed of light. ${\displaystyle {\vec {u}}={\vec {u}}'+{\vec {v}}}$ ${\displaystyle c}$

The relativistic addition theorem for speeds has been confirmed by measurements.

definition

Diagram for the relativistic addition of the rectified speeds and in each case expressed in fractions of the speed of light (explanations see article text). The contour lines also show the resulting speed normalized (gradation changed for ). The greater the two initial speeds , the more the result deviates from the arithmetic addition: the resulting speed will not exceed the speed of light either. ${\displaystyle u_{x}'}$ ${\displaystyle v,}$
${\displaystyle c}$
${\displaystyle u_{x},}$ ${\displaystyle c}$
${\displaystyle {\tfrac {u_{x}}{c}}>0{,}9}$

An observer moves towards the observer with the speed in the direction of the -axis. For the observer a body moves with the speed u ' Then this body has the speed u with the components for the observer ${\displaystyle {\mathcal {B}}^{\prime }}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle v}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {B}}^{\prime }}$ ${\displaystyle =(u_{x}^{\prime },u_{y}^{\prime },u_{z}^{\prime })\,.}$ ${\displaystyle {\mathcal {B}}}$

${\displaystyle u_{x}={\dfrac {u_{x}'+v}{1+{\dfrac {u_{x}'\,v}{c^{2}}}}}\qquad \qquad \Leftrightarrow {\dfrac {u_{x}}{c}}={\dfrac {{\dfrac {u_{x}'}{c}}+{\dfrac {v}{c}}}{1+{\dfrac {u_{x}'}{c}}\cdot {\dfrac {v}{c}}}}}$

${\displaystyle u_{y}={\dfrac {u_{y}'{\sqrt {1-\left({\dfrac {v}{c}}\right)^{2}}}}{1+{\dfrac {u_{x}'\,v}{c^{2}}}}}=u_{y}'\,{\dfrac {1}{\gamma \left(1+{\dfrac {u_{x}'\,v}{c^{2}}}\right)}}}$

${\displaystyle u_{z}={\dfrac {u_{z}'{\sqrt {1-\left({\dfrac {v}{c}}\right)^{2}}}}{1+{\dfrac {u_{x}'\,v}{c^{2}}}}}=u_{z}'\,{\dfrac {1}{\gamma \left(1+{\dfrac {u_{x}'\,v}{c^{2}}}\right)}}}$

With

• the speed of light and ${\displaystyle c}$
• the Lorentz factor (which is always greater than or equal to 1)

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

Expressed without coordinates: The resulting speed results from the simple addition of the speeds ( ) with the following modifications: ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {u}}'+{\vec {v}}}$

• The speed is smaller by a factor . ${\displaystyle {\vec {u}}}$ ${\displaystyle 1+{\tfrac {{\vec {u}}'\cdot {\vec {v}}}{c^{2}}}}$
• The components of the velocity perpendicular to are also smaller by a factor . ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle 1/\gamma }$

interpretation

Are the speeds involved very small compared to the speed of light

${\displaystyle v\ll c\Leftrightarrow {\frac {v}{c}}\ll 1\,,}$

the denominator (and also the term under the root in the numerator) hardly differs from 1

${\displaystyle \Rightarrow 1+{\frac {u_{x}'\,v}{c^{2}}}\approx 1\,,\qquad {\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}={\frac {1}{\gamma }}\approx 1\,,}$

and the usual non-relativistic speed addition results as a good approximation:

${\displaystyle {\begin{aligned}\Rightarrow u_{x}&\approx u_{x}'+v\\u_{y}&\approx u_{y}'\\u_{z}&\approx u_{z}'\,.\end{aligned}}}$

Example: in a train that is moving , a person is walking in the direction of travel relative to the train. The speed of the person measured by an observer standing at the embankment is just 0.17  nm / h slower than that obtained with simple addition . For comparison: the diameter of an atom is in the order of magnitude of 0.1 nm. This means that the “train runner” gets almost two atomic diameters less far an hour than one would expect with non-relativistic calculations - which is a distance covered by 205 km is certainly negligible - quite apart from the law of valid digits, which is often overlooked by laypeople . ${\displaystyle v=200\ \mathrm {km/h} }$ ${\displaystyle {\mathcal {B}}^{\prime }}$ ${\displaystyle u_{x}^{\prime }=5\ \mathrm {km/h} }$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle u_{x}}$ ${\displaystyle u_{x}^{\prime }+v=205\ \mathrm {km/h} }$

For speeds close to the speed of light, however, there are significant deviations from the non-relativistic addition rule, cf. the following examples.

Inferences

As a consequence of the addition theorem, the speed of light cannot be exceeded even if two speeds are superimposed.

1st example

Be there

${\displaystyle v=0{,}75c\quad }$ and ${\displaystyle \quad u_{x}'=0{,}75c\,.}$

Then

${\displaystyle u_{x}={\frac {0{,}75c+0{,}75c}{1+0{,}75\cdot 0{,}75}}={\frac {1{,}5c}{1{,}5625}}=0{,}96c<c}$

and not about 1.5 c .

2nd example

If the speed u 'is equal to the speed of light for the observer , then it is also the same for the observer ${\displaystyle {\mathcal {B}}^{\prime }}$ ${\displaystyle {\mathcal {B}}.}$

Are for example

${\displaystyle u_{x}'=0\,,\quad u_{y}'=c\,,\quad u_{z}'=0\,.}$

Then surrender

${\displaystyle u_{x}=v\,,\quad u_{y}=c/\gamma \,,\quad u_{z}=0\,.}$

So follows

${\displaystyle {\sqrt {u_{x}^{2}+u_{y}^{2}+u_{z}^{2}}}={\sqrt {v^{2}+c^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)}}={\sqrt {c^{2}}}=c\,.}$

Derivation

To keep the formula simple, all speeds are given as multiples of the speed of light in natural units . Then time and length have the same unit of measurement and the dimensionless speed of light is ${\displaystyle c=1.}$

From the inverse Lorentz transformation (replacement of by - ) ${\displaystyle v}$ ${\displaystyle v}$

${\displaystyle t={\frac {t'+v\,x'}{\sqrt {1-v^{2}}}}\ ,\quad x={\frac {x'+v\,t'}{\sqrt {1-v^{2}}}}\ ,\quad y=y'\ ,\quad z=z'}$

it follows for the differentials, since the transformation is linear,

${\displaystyle \mathrm {d} t={\frac {\mathrm {d} t'+v\,\mathrm {d} x'}{\sqrt {1-v^{2}}}}\ ,\quad \mathrm {d} x={\frac {\mathrm {d} x'+v\,\mathrm {d} t'}{\sqrt {1-v^{2}}}}\ ,\quad \mathrm {d} y=\mathrm {d} y'\ ,\quad \mathrm {d} z=\mathrm {d} z'\,.}$

Therefore it follows for the velocities that the observer determines ${\displaystyle {\mathcal {B}}}$

${\displaystyle u_{x}={\frac {\mathrm {d} x}{\mathrm {d} t}}={\frac {\mathrm {d} x'+v\,\mathrm {d} t'}{\mathrm {d} t'+v\,\mathrm {d} x'}}={\frac {{\frac {\mathrm {d} x'}{\mathrm {d} t'}}+v}{1+v\,{\frac {\mathrm {d} x'}{\mathrm {d} t'}}}}={\frac {u_{x}'+v}{1+v\,u_{x}'}}\ ,}$

${\displaystyle u_{y}={\frac {\mathrm {d} y}{\mathrm {d} t}}={\frac {\mathrm {d} y'{\sqrt {1-v^{2}}}}{\mathrm {d} t'+v\,\mathrm {d} x'}}={\frac {{\frac {\mathrm {d} y'}{\mathrm {d} t'}}{\sqrt {1-v^{2}}}}{1+v\,{\frac {\mathrm {d} x'}{\mathrm {d} t'}}}}={\frac {u_{y}'{\sqrt {1-v^{2}}}}{1+v\,u_{x}'}}\ ,}$

${\displaystyle u_{z}={\frac {\mathrm {d} z}{\mathrm {d} t}}={\frac {\mathrm {d} z'{\sqrt {1-v^{2}}}}{\mathrm {d} t'+v\,\mathrm {d} x'}}={\frac {{\frac {\mathrm {d} z'}{\mathrm {d} t'}}{\sqrt {1-v^{2}}}}{1+v\,{\frac {\mathrm {d} x'}{\mathrm {d} t'}}}}={\frac {u_{z}'{\sqrt {1-v^{2}}}}{1+v\,u_{x}'}}\ .}$

Broken down according to the deleted variables, the following relationships result:

${\displaystyle u_{x}'={\frac {u_{x}-v}{1-v\,u_{x}}}\ ,\quad u_{y}'={\frac {u_{y}{\sqrt {1-v^{2}}}}{1-v\,u_{x}}}\ ,\quad u_{z}'={\frac {u_{z}{\sqrt {1-v^{2}}}}{1-v\,u_{x}}}\ .}$

Web links

Wikibooks: Special Theory of Relativity  - Learning and Teaching Materials