Acceleration (special theory of relativity)

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Accelerations in the special theory of relativity (SRT) are, as in Newtonian mechanics , derivatives of the velocity with respect to time . Sincethe concept of time in the SRT is more complexdue to the Lorentz transformation and time dilation , more complex definitions of acceleration follow from this. The SRT as the theory of the “flat” Minkowski spacetime remains valid even in the presence of accelerations. The general theory of relativity is onlyrequiredwhen a space-time curvature occurs due to the energy-momentum tensor (which is determined by the mass ). For practical applications such as experiments in particle accelerators , the space-time curvature due to the earth's mass is negligible, so the SRT remains valid to a sufficient approximation.

For accelerations in three dimensions, transformation formulas for the coordinates of an accelerated observer can be derived from the point of view of an external inertial system. This acceleration is called triple acceleration or coordinate acceleration . In addition, the self- acceleration measured by a moving acceleration sensor can be calculated. Another formalism is the four acceleration , the components of which are connected in different inertial systems by Lorentz transformations. It can be used to formulate equations of motion that combine acceleration and force . Equations for different types of acceleration and their curved world lines follow from these formulas through integration . Well-known cases are the hyperbolic movement for constant longitudinal self-acceleration and uniform circular movement for constant transverse self-acceleration. In addition, it is possible to describe these movements in accelerated reference systems within the framework of the SRT, in which effects occur that are analogous to homogeneous gravitational fields (which are formally similar to the real, inhomogeneous gravitational fields of the curved space-time of the GTR). These are, for example, the Rindler coordinates for the hyperbolic movement and the Born or Langevink coordinates for the uniform circular movement.

The relativistic equations for describing accelerations were established in the early days of the SRT and presented in textbooks by Max von Laue (1911, 1921) and Wolfgang Pauli (1921). Individual equations of motion and acceleration transformations can be found in the works of Hendrik Antoon Lorentz (1899, 1904), Henri Poincaré (1905), Albert Einstein (1905) and Max Planck (1906). Acceleration of four, self-acceleration, hyperbolic movement, circular movement, accelerated reference systems and Born's rigidity were analyzed by Einstein (1907), Hermann Minkowski (1907, 1908), Max Born (1909), Gustav Herglotz (1909), Arnold Sommerfeld (1910), von Laue ( 1911) and Friedrich Kottler (1912, 1914), see section on history .

Triple acceleration

In Newtonian mechanics as well as in SRT, the usual triple acceleration or coordinate acceleration is the first derivative of the speed according to the coordinate time (the time indicated by clocks that are permanently at rest in an inertial system and synchronized with one another) or the second derivative of the location according to the coordinate time: ${\displaystyle \mathbf {a} =\left(a_{x},\ a_{y},\ a_{z}\right)}$ ${\displaystyle \mathbf {u} =\left(u_{x},\ u_{y},\ u_{z}\right)}$ ${\displaystyle t}$ ${\displaystyle \mathbf {r} =\left(x,\ y,\ z\right)}$

${\displaystyle \mathbf {a} ={\frac {d\mathbf {u} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}}$.

However, the theories differ sharply in their predictions regarding the transformation of the triple acceleration of an object between different inertial systems. In Newtonian mechanics, time is absolutely in agreement with the Galileo transformation , which is why the three-fold acceleration derived from it is the same in all inertial systems: ${\displaystyle t'=t}$

${\displaystyle \mathbf {a} =\mathbf {a} '}$.

In contrast to this, both and in the SRT depend on the Lorentz transformation, which is why the triple acceleration and its components turn out differently in different inertial systems. If the relative speed of the inertial systems points in the x-direction with and if the Lorentz factor is, then the Lorentz transformation is known to have the form ${\displaystyle \mathbf {r} }$ ${\displaystyle t}$ ${\displaystyle \mathbf {a} }$ ${\displaystyle v=v_{x}}$ ${\displaystyle \gamma _{v}=1/{\sqrt {1-v^{2}/c^{2}}}}$

${\displaystyle {\begin{array}{c|c}{\begin{aligned}x'&=\gamma _{v}(x-vt)\\y'&=y\\z'&=z\\t^{\prime }&=\gamma _{v}\left(t-{\frac {v}{c^{2}}}x\right)\end{aligned}}&{\begin{aligned}x&=\gamma _{v}(x'+vt')\\y&=y'\\z&=z'\\t&=\gamma _{v}\left(t'+{\frac {v}{c^{2}}}x'\right)\end{aligned}}\end{array}}}$		( 1a )

or for any speed with the norm : ${\displaystyle \mathbf {v} =\left(v_{x},\ v_{y},\ v_{z}\right)}$ ${\displaystyle |\mathbf {v} |=v}$

${\displaystyle {\begin{array}{c|c}{\begin{aligned}\mathbf {r} '&=\mathbf {r} +\mathbf {v} \left[{\frac {\left(\mathbf {r\cdot v} \right)}{v^{2}}}\left(\gamma _{v}-1\right)-t\gamma _{v}\right]\\t^{\prime }&=\gamma _{v}\left(t-{\frac {\mathbf {r\cdot v} }{c^{2}}}\right)\end{aligned}}&{\begin{aligned}\mathbf {r} &=\mathbf {r} '+\mathbf {v} \left[{\frac {\left(\mathbf {r'\cdot v} \right)}{v^{2}}}\left(\gamma _{v}-1\right)+t'\gamma _{v}\right]\\t&=\gamma _{v}\left(t'+{\frac {\mathbf {r'\cdot v} }{c^{2}}}\right)\end{aligned}}\end{array}}}$		( 1b )

In order to find the transformation of the three-way acceleration, the spatial coordinates and the Lorentz transformation are derived from and , from which the transformation of the three-way speed (also known as relativistic speed addition ) follows between and , and finally by a further derivation after and follows the transformation of the three-way acceleration between and . Starting with ( ${\displaystyle \mathbf {r} }$ ${\displaystyle \mathbf {r} '}$ ${\displaystyle t}$ ${\displaystyle t'}$ ${\displaystyle \mathbf {u} }$ ${\displaystyle \mathbf {u} '}$ ${\displaystyle t}$ ${\displaystyle t'}$ ${\displaystyle \mathbf {a} }$ ${\displaystyle \mathbf {a} '}$^1a ) , one obtains the transformation for the case that the accelerations are either parallel (x-direction) or perpendicular (y-, z-direction) to the velocity:

${\displaystyle {\begin{array}{c|c}{\begin{aligned}a_{x}^{\prime }&={\frac {a_{x}}{\gamma _{v}^{3}\left(1-{\frac {u_{x}v}{c^{2}}}\right)^{3}}}\\a_{y}^{\prime }&={\frac {a_{y}}{\gamma _{v}^{2}\left(1-{\frac {u_{x}v}{c^{2}}}\right)^{2}}}+{\frac {a_{x}{\frac {u_{y}v}{c^{2}}}}{\gamma _{v}^{2}\left(1-{\frac {u_{x}v}{c^{2}}}\right)^{3}}}\\a_{z}^{\prime }&={\frac {a_{z}}{\gamma _{v}^{2}\left(1-{\frac {u_{x}v}{c^{2}}}\right)^{2}}}+{\frac {a_{x}{\frac {u_{z}v}{c^{2}}}}{\gamma _{v}^{2}\left(1-{\frac {u_{x}v}{c^{2}}}\right)^{3}}}\end{aligned}}&{\begin{aligned}a_{x}&={\frac {a_{x}^{\prime }}{\gamma _{v}^{3}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)^{3}}}\\a_{y}&={\frac {a_{y}^{\prime }}{\gamma _{v}^{2}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)^{2}}}-{\frac {a_{x}^{\prime }{\frac {u_{y}^{\prime }v}{c^{2}}}}{\gamma _{v}^{2}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)^{3}}}\\a_{z}&={\frac {a_{z}^{\prime }}{\gamma _{v}^{2}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)^{2}}}-{\frac {a_{x}^{\prime }{\frac {u_{z}^{\prime }v}{c^{2}}}}{\gamma _{v}^{2}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)^{3}}}\end{aligned}}\end{array}}}$		( 1c )

or starting with (^1b ) this process gives the result for the general case of arbitrary directions of velocities and accelerations:

${\displaystyle {\begin{aligned}\mathbf {a} '&={\frac {\mathbf {a} }{\gamma _{v}^{2}\left(1-{\frac {\mathbf {v\cdot u} }{c^{2}}}\right)^{2}}}-{\frac {\mathbf {(a\cdot v)v} \left(\gamma _{v}-1\right)}{v^{2}\gamma _{v}^{3}\left(1-{\frac {\mathbf {v\cdot u} }{c^{2}}}\right)^{3}}}+{\frac {\mathbf {(a\cdot v)u} }{c^{2}\gamma _{v}^{2}\left(1-{\frac {\mathbf {v\cdot u} }{c^{2}}}\right)^{3}}}\\\mathbf {a} &={\frac {\mathbf {a} '}{\gamma _{v}^{2}\left(1+{\frac {\mathbf {v\cdot u} '}{c^{2}}}\right)^{2}}}-{\frac {\mathbf {(a'\cdot v)v} \left(\gamma _{v}-1\right)}{v^{2}\gamma _{v}^{3}\left(1+{\frac {\mathbf {v\cdot u} '}{c^{2}}}\right)^{3}}}-{\frac {\mathbf {(a'\cdot v)u} '}{c^{2}\gamma _{v}^{2}\left(1+{\frac {\mathbf {v\cdot u} '}{c^{2}}}\right)^{3}}}\end{aligned}}}$		( 1d )

This means that if two inertial systems and with the relative speed are given, then the acceleration of an object is measured with the current speed , whereas the same object has the acceleration and the current speed . Just like the speed addition, these acceleration transformations also ensure that the resulting speed of an accelerated object can never reach or exceed the speed of light from the point of view of any inertial system . ${\displaystyle S}$ ${\displaystyle S'}$ ${\displaystyle \mathbf {v} }$ ${\displaystyle S}$ ${\displaystyle \mathbf {a} }$ ${\displaystyle \mathbf {u} }$ ${\displaystyle S'}$ ${\displaystyle \mathbf {a} '}$ ${\displaystyle \mathbf {u} '}$

Four acceleration

In the theory of relativity it is often advantageous to use four-vectors instead of three- vectors , whereby here the derivation does not take place according to the coordinate time , but according to the proper time (i.e. the time measured by a clock that moves with the object). Starting from the four-man position , one derives the four-man speed , and another derivative gives the four-man acceleration : ${\displaystyle t}$ ${\displaystyle \mathbf {\tau } }$ ${\displaystyle \mathbf {R} }$ ${\displaystyle \mathbf {U} }$ ${\displaystyle \mathbf {A} =\left(A_{t},\ A_{x},\ A_{y},\ A_{z}\right)=\left(A_{t},\ \mathbf {A} _{r}\right)}$

${\displaystyle {\begin{aligned}\mathbf {A} &={\frac {d\mathbf {U} }{d\tau }}={\frac {d^{2}\mathbf {R} }{d\tau ^{2}}}=\left(c{\frac {d^{2}t}{d\tau ^{2}}},\ {\frac {d^{2}\mathbf {r} }{d\tau ^{2}}}\right)\\&=\left(\gamma ^{4}{\frac {\mathbf {u} \cdot \mathbf {a} }{c}},\ \gamma ^{4}{\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}+\gamma ^{2}\mathbf {a} \right)\end{aligned}}}$		( 2 )

where the triple acceleration of the object and its instantaneous speed is with the norm and the corresponding Lorentz factor. If only the spatial part is considered, if the speed points in the x-direction, and the accelerations are either parallel (x-direction) or perpendicular (y-, z-direction) to the speed, this expression simplifies to: ${\displaystyle \mathbf {a} }$ ${\displaystyle \mathbf {u} }$ ${\displaystyle |\mathbf {u} |=u}$ ${\displaystyle \gamma =1/{\sqrt {1-u^{2}/c^{2}}}}$

${\displaystyle \mathbf {A} _{r}=\mathbf {a} \left(\gamma ^{4},\ \gamma ^{2},\ \gamma ^{2}\right)}$

In contrast to the three-way acceleration discussed above, it is not necessary to introduce a new transformation of the four-way acceleration, because as with all four-vectors, the components of are also connected to one another by ordinary Lorentz transformations. By replacing with in ( ${\displaystyle \mathbf {A} }$ ${\displaystyle x,\ y,\ z,\ ct}$ ${\displaystyle A_{x},\ A_{y},\ A_{z},\ A_{t}}$^1a ) follows:

${\displaystyle {\begin{array}{c|c}{\begin{aligned}A_{x}^{\prime }&=\gamma _{v}\left(A_{x}-{\frac {v}{c}}A_{t}\right)\\A_{y}^{\prime }&=A_{y}\\A_{z}^{\prime }&=A_{z}\\A_{t}^{\prime }&=\gamma _{v}\left(A_{t}-{\frac {v}{c}}A_{x}\right)\end{aligned}}&{\begin{aligned}A_{x}&=\gamma _{v}\left(A_{x}^{\prime }+{\frac {v}{c}}A_{t}^{\prime }\right)\\A_{y}&=A_{y}^{\prime }\\A_{z}&=A_{z}^{\prime }\\A_{t}&=\gamma _{v}\left(A_{t}^{\prime }+{\frac {v}{c}}A_{x}^{\prime }\right)\end{aligned}}\end{array}}}$

or by replacing with in ( ${\displaystyle \mathbf {r} ,\ ct}$ ${\displaystyle \mathbf {A} _{r},\ A_{t}}$^1b ) for any direction from : ${\displaystyle \mathbf {v} }$

${\displaystyle {\begin{array}{c|c}{\begin{aligned}\mathbf {A} {}_{r}^{\prime }&=\mathbf {A} _{r}+{\frac {\mathbf {v} }{c}}\left[{\frac {\left(\mathbf {A} _{r}\cdot \mathbf {v} \right)c}{v^{2}}}\left(\gamma _{v}-1\right)-A_{t}\gamma _{v}\right]\\A_{t}^{\prime }&=\gamma _{v}\left(A_{t}-{\frac {\mathbf {A} _{r}\cdot \mathbf {v} }{c}}\right)\end{aligned}}&{\begin{aligned}\mathbf {A} _{r}&=\mathbf {A} {}_{r}^{\prime }+{\frac {\mathbf {v} }{c}}\left[{\frac {\left(\mathbf {A} {}_{r}^{\prime }\cdot \mathbf {v} \right)c}{v^{2}}}\left(\gamma _{v}-1\right)+A_{t}^{\prime }\gamma _{v}\right]\\A_{t}&=\gamma _{v}\left(A_{t}^{\prime }+{\frac {\mathbf {A} {}_{r}^{\prime }\cdot \mathbf {v} }{c}}\right)\end{aligned}}\end{array}}}$,

On the other hand, the square of the amount with the signature and its norm is invariant, so: ${\displaystyle \mathbf {A} ^{2}=-A_{t}^{2}+\mathbf {A} _{r}^{2}}$ ${\displaystyle -,+,+,+}$ ${\displaystyle |\mathbf {A} |={\sqrt {\mathbf {A} ^{2}}}}$

${\displaystyle |\mathbf {A} '|=|\mathbf {A} |={\sqrt {\gamma ^{4}\left[\mathbf {a} ^{2}+\gamma ^{2}\left({\frac {\mathbf {u} \cdot \mathbf {a} }{c}}\right)^{2}\right]}}}$.		( 3 )

Self-acceleration

In infinitesimal time intervals there is always an inertial system which has the same speed as the accelerated body and in which the Lorentz transformation is valid. The triple acceleration occurring in such momentary inertial systems can be read off directly from an acceleration sensor that moves with it, and is referred to as self-acceleration or acceleration at rest. The relationship between in a momentary inertial frame and in an external inertial frame follows from ( ${\displaystyle \mathbf {a} ^{0}=\left(a_{x}^{0},\ a_{y}^{0},\ a_{z}^{0}\right)}$ ${\displaystyle \mathbf {a} ^{0}}$ ${\displaystyle S'}$ ${\displaystyle \mathbf {a} }$ ${\displaystyle S}$^1c , ^1d ) By setting , , and . If the speed points in the x-direction and the accelerations are either parallel (x-direction) or perpendicular (y-, z-direction) to the speed, it follows from ( ${\displaystyle \mathbf {a} '=\mathbf {a} ^{0}}$ ${\displaystyle \mathbf {u} '=0}$ ${\displaystyle \mathbf {u} =\mathbf {v} }$ ${\displaystyle \gamma =\gamma _{v}}$ ${\displaystyle u=u_{x}=v=v_{x}}$^1c ) :

${\displaystyle {\begin{array}{c|c|cc}{\begin{aligned}a_{x}^{0}&={\frac {a_{x}}{\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}}}\\a_{y}^{0}&={\frac {a_{y}}{1-{\frac {u^{2}}{c^{2}}}}}\\a_{z}^{0}&={\frac {a_{z}}{1-{\frac {u^{2}}{c^{2}}}}}\end{aligned}}&{\begin{aligned}a_{x}&=a_{x}^{0}\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}\\a_{y}&=a_{y}^{0}\left(1-{\frac {u^{2}}{c^{2}}}\right)\\a_{z}&=a_{z}^{0}\left(1-{\frac {u^{2}}{c^{2}}}\right)\end{aligned}}&{\text{oder}}&{\begin{aligned}\mathbf {a} ^{0}&=\mathbf {a} \left(\gamma ^{3},\ \gamma ^{2},\ \gamma ^{2}\right)\\\mathbf {a} &=\mathbf {\mathbf {a} } ^{0}\left({\frac {1}{\gamma ^{3}}},\ {\frac {1}{\gamma ^{2}}},\ {\frac {1}{\gamma ^{2}}}\right)\end{aligned}}\end{array}}}$		( 4a )

Generalized in the sense of (^1d ) for any speed with the norm : ${\displaystyle \mathbf {u} }$ ${\displaystyle |\mathbf {u} |=u}$

${\displaystyle {\begin{aligned}\mathbf {a} ^{0}&=\gamma ^{2}\left[\mathbf {a} +{\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{u^{2}}}\left(\gamma -1\right)\right]\\\mathbf {a} &={\frac {1}{\gamma ^{2}}}\left[\mathbf {a} ^{0}-{\frac {(\mathbf {a} ^{0}\cdot \mathbf {u} )\mathbf {u} }{u^{2}}}\left(1-{\frac {1}{\gamma }}\right)\right]\end{aligned}}}$

There is also a close relationship to the norm of the four-point acceleration: Since it is invariant, it can also be determined in a momentarily moving inertial system, in which applies , and with it follows : ${\displaystyle S'}$ ${\displaystyle \mathbf {A} _{r}^{\prime }=\mathbf {a} ^{0}}$ ${\displaystyle dt'/d\tau =1}$ ${\displaystyle d^{2}t'/d\tau ^{2}=A_{t}^{\prime }=0}$

${\displaystyle |\mathbf {A} '|={\sqrt {0+\left.\mathbf {a} ^{0}\right.^{2}}}=|\mathbf {a} ^{0}|}$.		( 4b )

The norm of the four-speed acceleration corresponds to the norm of the self-acceleration. It can therefore establish a connection with (³ ) Are produced, thus an alternative method for the determination of the relationship between in and in given is ${\displaystyle \mathbf {a} ^{0}}$ ${\displaystyle S'}$ ${\displaystyle \mathbf {a} }$ ${\displaystyle S}$

${\displaystyle |\mathbf {a} ^{0}|=|\mathbf {A} |={\sqrt {\gamma ^{4}\left[\mathbf {a} ^{2}+\gamma ^{2}\left({\frac {\mathbf {u} \cdot \mathbf {a} }{c}}\right)^{2}\right]}}}$

from what again (^4a ) follows if the speed points in the x-direction and the accelerations are either parallel (x-direction) or perpendicular (y-, z-direction) to the speed. ${\displaystyle u=u_{x}}$

Acceleration and power

The force of four as a function of the force of three is given. This four force, the four acceleration according to ( ${\displaystyle \mathbf {F} }$ ${\displaystyle \mathbf {f} }$ ${\displaystyle \mathbf {F} =\gamma \left((\mathbf {f} \cdot \mathbf {u} )/c,\ \mathbf {f} \right)}$² ) , and the invariant mass are also connected according to Newton's formula , i.e. ${\displaystyle m}$ ${\displaystyle \mathbf {F} =m\mathbf {A} }$

${\displaystyle \mathbf {F} =\left(\gamma {\frac {\mathbf {f} \cdot \mathbf {u} }{c}},\ \gamma \mathbf {f} \right)=m\mathbf {A} =m\left(\gamma ^{4}\left({\frac {\mathbf {u} \cdot \mathbf {a} }{c}}\right),\ \gamma ^{4}\left({\frac {\mathbf {u} \cdot \mathbf {a} }{c^{2}}}\right)\mathbf {u} +\gamma ^{2}\mathbf {a} \right)}$.

From this follows the relationship between the triple force and triple acceleration for any direction of velocity with:

${\displaystyle {\begin{aligned}\mathbf {f} &=m\gamma ^{3}\left({\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)+m\gamma \mathbf {a} \\\mathbf {a} &={\frac {1}{m\gamma }}\left(\mathbf {f} -{\frac {(\mathbf {f} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)\end{aligned}}}$		( 5a )

If the speed points in the x-direction with , and the accelerations are either parallel (x-direction) or perpendicular (y-, z-direction) to the speed, then it follows: ${\displaystyle u=u_{x}}$

${\displaystyle {\begin{array}{c|c|cc}{\begin{aligned}f_{x}&={\frac {ma_{x}}{\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}}}\\f_{y}&={\frac {ma_{y}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\\f_{z}&={\frac {ma_{z}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\end{aligned}}&{\begin{aligned}a_{x}&={\frac {f_{x}}{m}}\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}\\a_{y}&={\frac {f_{y}}{m}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}\\a_{z}&={\frac {f_{z}}{m}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}\end{aligned}}&{\text{oder}}&{\begin{aligned}\mathbf {f} &=m\mathbf {a} \left(\gamma ^{3},\ \gamma ,\ \gamma \right)\\\mathbf {a} &={\frac {\mathbf {f} }{m}}\left({\frac {1}{\gamma ^{3}}},\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)\end{aligned}}\end{array}}}$		( 5b )

This is why Newton's definition of mass as the ratio of the three force to the three acceleration is disadvantageous in the SRT, because this mass would depend on both the speed and the direction. Therefore, the following mass definitions can only be found in older textbooks:

${\displaystyle m_{\Vert }={\frac {f_{x}}{a_{x}}}=m\gamma ^{3}}$ as "longitudinal mass",

${\displaystyle m_{\perp }={\frac {f_{y}}{a_{y}}}={\frac {f_{z}}{a_{z}}}=m\gamma }$ as "transversal mass".

Equation (^5a ) between triple acceleration and triple force can also be obtained from the well-known relativistic equation of motion:

${\displaystyle \mathbf {f} ={\frac {d\mathbf {p} }{dt}}={\frac {d(m\gamma \mathbf {u} )}{dt}}={\frac {d(m\gamma )}{dt}}\mathbf {u} +m\gamma {\frac {d\mathbf {u} }{dt}}=m\gamma ^{3}\left({\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)+m\gamma \mathbf {a} }$		( 5c )

where the three is impulse . The corresponding transformation of the force of three between in and in (if the speed points in the x-direction with and the accelerations are either parallel (x-direction) or perpendicular (y-, z-direction) to the speed) follows by substituting the transformation formulas for , , , , or from the transformed components of the Lorentz force of four, with the result: ${\displaystyle \mathbf {p} }$ ${\displaystyle \mathbf {f} }$ ${\displaystyle S}$ ${\displaystyle \mathbf {f} '}$ ${\displaystyle S'}$ ${\displaystyle v=v_{x}}$ ${\displaystyle \mathbf {u} }$ ${\displaystyle \mathbf {a} }$ ${\displaystyle m\gamma }$ ${\displaystyle d(m\gamma )/dt}$

${\displaystyle {\begin{array}{c|c}{\begin{aligned}f_{x}^{\prime }&={\frac {f_{x}-{\frac {v}{c^{2}}}(\mathbf {f} \cdot \mathbf {u} )}{1-{\frac {u_{x}v}{c^{2}}}}}\\f_{y}^{\prime }&={\frac {f_{y}}{\gamma _{v}\left(1-{\frac {u_{x}v}{c^{2}}}\right)}}\\f_{z}^{\prime }&={\frac {f_{z}}{\gamma _{v}\left(1-{\frac {u_{x}v}{c^{2}}}\right)}}\end{aligned}}&{\begin{aligned}f_{x}&={\frac {f_{x}^{\prime }+{\frac {v}{c^{2}}}(\mathbf {f} ^{\prime }\cdot \mathbf {u} ^{\prime })}{1+{\frac {u_{x}^{\prime }v}{c^{2}}}}}\\f_{y}&={\frac {f_{y}^{\prime }}{\gamma _{v}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)}}\\f_{z}&={\frac {f_{z}^{\prime }}{\gamma _{v}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)}}\end{aligned}}\end{array}}}$		( 6a )

Or generalized for any directions from and with the norm : ${\displaystyle \mathbf {u} }$ ${\displaystyle \mathbf {v} }$ ${\displaystyle |\mathbf {v} |=v}$

${\displaystyle {\begin{aligned}\mathbf {f} '&={\frac {{\frac {\mathbf {f} }{\gamma _{v}}}-\left\{(\mathbf {f\cdot u} ){\frac {v^{2}}{c^{2}}}-(\mathbf {f\cdot v} )\left(1-{\frac {1}{\gamma _{v}}}\right)\right\}{\frac {\mathbf {v} }{v^{2}}}}{1-{\frac {\mathbf {v\cdot u} }{c^{2}}}}}\\\mathbf {f} &={\frac {{\frac {\mathbf {f} '}{\gamma _{v}}}+\left\{(\mathbf {f'\cdot u} '){\frac {v^{2}}{c^{2}}}+(\mathbf {f'\cdot v} )\left(1-{\frac {1}{\gamma _{v}}}\right)\right\}{\frac {\mathbf {v} }{v^{2}}}}{1+{\frac {\mathbf {v\cdot u'} }{c^{2}}}}}\end{aligned}}}$		( 6b )

Self-acceleration and self-strength

The force measured with a moving spring balance in the momentary inertial system can be referred to as intrinsic force . It follows from ( ${\displaystyle \mathbf {f} ^{0}}$^6a , ^6b ) by setting and as well as and . So according to ( ${\displaystyle \mathbf {f} '=\mathbf {f} ^{0}}$ ${\displaystyle \mathbf {u} '=0}$ ${\displaystyle \mathbf {u} =\mathbf {v} }$ ${\displaystyle \gamma =\gamma _{v}}$^6a ) if the speed points in the x-direction and the accelerations are either parallel (x-direction) or perpendicular (y-, z-direction) to the speed: ${\displaystyle u=u_{x}=v=v_{x}}$

${\displaystyle {\begin{array}{c|c|cc}{\begin{aligned}f_{x}^{0}&=f_{x}\\f_{y}^{0}&={\frac {f_{y}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\\f_{z}^{0}&={\frac {f_{z}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\end{aligned}}&{\begin{aligned}f_{x}&=f_{x}^{0}\\f_{y}&=f_{y}^{0}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}\\f_{z}&=f_{z}^{0}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}\end{aligned}}&{\text{oder}}&{\begin{aligned}\mathbf {f} ^{0}&=\mathbf {f} \left(1,\ \gamma ,\ \gamma \right)\\\mathbf {f} &=\mathbf {f} ^{0}\left(1,\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)\end{aligned}}\end{array}}}$		( 7a )

Generalized according to (^6b ) for any directions from with the norm : ${\displaystyle \mathbf {u} }$ ${\displaystyle |\mathbf {u} |=u}$

${\displaystyle {\begin{aligned}\mathbf {f} ^{0}&=\mathbf {f} \gamma -{\frac {(\mathbf {f} \cdot \mathbf {u} )\mathbf {u} }{u^{2}}}(\gamma -1)\\\mathbf {f} &={\frac {\mathbf {f} ^{0}}{\gamma }}+{\frac {(\mathbf {f} ^{0}\cdot \mathbf {u} )\mathbf {u} }{u^{2}}}\left(1-{\frac {1}{\gamma }}\right)\end{aligned}}}$

Since it is valid in the inertial system that is moving at the moment , the Newtonian relation can be used (this also follows from the above relation , since in the momentary rest system and ), which is why ( ${\displaystyle \gamma =1}$ ${\displaystyle \mathbf {f} ^{0}=m\mathbf {a} ^{0}}$ ${\displaystyle \mathbf {F} =m\mathbf {A} }$ ${\displaystyle \mathbf {F} =\mathbf {f} ^{0}}$ ${\displaystyle \mathbf {A} =\mathbf {a} ^{0}}$^4a , ^5b , ^7a ) can be summarized:

${\displaystyle {\begin{aligned}\mathbf {f} ^{0}&=\mathbf {f} \left(1,\ \gamma ,\ \gamma \right)=m\mathbf {a} ^{0}=m\mathbf {a} \left(\gamma ^{3},\ \gamma ^{2},\ \gamma ^{2}\right)\\\mathbf {f} &=\mathbf {f} ^{0}\left(1,\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)=m\mathbf {a} ^{0}\left(1,\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)=m\mathbf {a} \left(\gamma ^{3},\ \gamma ,\ \gamma \right)\end{aligned}}}$		( 7b )

This also resolves the apparent contradiction in the historical definitions of the transversal mass . Einstein (1905) described the relationship between triple acceleration and self-strength ${\displaystyle m_{\perp }}$

${\displaystyle m_{\perp \ \mathrm {Einstein} }={\frac {f_{y}^{0}}{a_{y}}}={\frac {f_{z}^{0}}{a_{z}}}=m\gamma ^{2}}$,

while Lorentz (1899, 1904) and Planck (1906) described the relationship between three-fold acceleration and three-fold force

${\displaystyle m_{\perp \ \mathrm {Lorentz} }={\frac {f_{y}}{a_{y}}}={\frac {f_{z}}{a_{z}}}=m\gamma }$.

Curved world lines

By integrating the above equations of motion, one obtains the curved world lines of accelerated bodies (the term curvature here refers to the shape of the world lines in Minkowski diagrams, which has nothing to do with the curved spacetime of the GTR). This is related to the so-called clock hypothesis: the proper time of a moving clock is independent of the acceleration, so the time dilation of these clocks from the point of view of other inertial systems only depends on the current relative speed to these systems (see experimental confirmations of the clock hypothesis ). Two simple cases of curved world lines follow by integrating equation (^4a ) for self-acceleration:

a) Hyperbolic movement : The constant, longitudinal self-acceleration according to ( ${\displaystyle \alpha =a_{x}^{0}=a_{x}\gamma ^{3}}$^4a ) leads to the world line

${\displaystyle {\begin{aligned}&t(\tau )={\frac {c}{\alpha }}\sinh {\frac {\alpha \tau }{c}},\quad x(\tau )={\frac {c^{2}}{\alpha }}\left(\cosh {\frac {\alpha \tau }{c}}-1\right),\quad y=0,\quad z=0,\\&\tau (t)={\frac {c}{\alpha }}\ln \left({\sqrt {1+\left({\frac {\alpha t}{c}}\right)^{2}}}+{\frac {\alpha t}{c}}\right),\quad x(t)={\frac {c^{2}}{\alpha }}\left({\sqrt {1+\left({\frac {\alpha t}{c}}\right)^{2}}}-1\right)\end{aligned}}}$		( 8 )

This world line corresponds to the hyperbola equation . These equations are often used to calculate various scenarios such as the twin paradox , Bell's spaceship paradox , or space travel with constant acceleration . ${\displaystyle c^{4}/\alpha ^{2}=\left(x+c^{2}/\alpha \right)^{2}-c^{2}t^{2}}$

b) The constant, transverse self-acceleration according to ( ${\displaystyle a_{y}^{0}=a_{y}\gamma ^{2}}$^4a ) can be understood as centripetal acceleration , which leads to the world line of a body in uniform circular motion:

${\displaystyle {\begin{aligned}x&=r\cos \Omega _{0}t=r\cos \Omega \tau \\y&=r\sin \Omega _{0}t=r\sin \Omega \tau \\z&=z\\t&=\gamma \tau ={\frac {\tau }{\sqrt {1-\left({\frac {r\Omega _{0}}{c}}\right)^{2}}}}=\tau {\sqrt {1+\left({\frac {r\Omega }{c}}\right)^{2}}}\end{aligned}}}$

where is the tangential velocity , the orbital radius , the angular velocity as a function of the coordinate time and as a function of the proper time. ${\displaystyle v=r\Omega _{0}}$ ${\displaystyle r}$ ${\displaystyle \Omega _{0}}$ ${\displaystyle \Omega =\gamma \Omega _{0}}$

A classification of curved world lines follows from the differential geometry of curves in the sense of the Frenet-Serret formulas for Minkowski space-time. It turns out that the hyperbolic movement and the uniform circular movement are special cases of movements with constant curvature and torsion . These bodies also satisfy the condition of Born rigidity , in which the space-time distance between the world lines of their infinitesimally separated components remains constant during acceleration.

Accelerated reference systems in the SRT

Instead of the inertial coordinates, these accelerated movements and curved world lines can also be described by accelerated or curvilinear coordinates . This allows self-reference systems (sometimes referred to as Fermi coordinates or Eigen coordinates) to be defined in which the proper time of the accelerated observer is used as the coordinate time of the entire system. In the rest frame of an observer in hyperbolic motion , hyperbolic coordinates (sometimes called Rindler coordinates ) can be used, or for uniform circular motion, rotating cylindrical coordinates (sometimes called Born or Langevin coordinates ) can be used. In terms of the equivalence principle , the effects occurring in these accelerated reference systems can be interpreted in analogy to the effects in a homogeneous, fictitious gravitational field. This shows that the use of accelerated reference systems already provides important mathematical relationships in the SRT, which later become of fundamental importance in the description of real, inhomogeneous gravitational fields in the sense of the curved spacetime of the GTR.

history

For more information, see von Laue, Pauli, Miller, Zahar, Gourgoulhon, and the historical sources in History of Special Relativity .

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1899: Hendrik Lorentz directs the correct (except for an undefined factor ) relations for the accelerations, forces and masses between a static electrostatic particle system (in a resting ether ) and a system that emerges from the other through a translation, where the Lorentz factor is : ${\displaystyle \epsilon }$ ${\displaystyle S_{0}}$ ${\displaystyle S}$ ${\displaystyle k}$

${\displaystyle {\frac {1}{\epsilon ^{2}}}}$, , For according to ( ${\displaystyle {\frac {1}{k\epsilon ^{2}}}}$ ${\displaystyle {\frac {1}{k\epsilon ^{2}}}}$ ${\displaystyle \mathbf {f} /\mathbf {f} ^{0}}$^7a ) ;

${\displaystyle {\frac {1}{k^{3}\epsilon }}}$, , For according to ( ${\displaystyle {\frac {1}{k^{2}\epsilon }}}$ ${\displaystyle {\frac {1}{k^{2}\epsilon }}}$ ${\displaystyle \mathbf {a} /\mathbf {a} ^{0}}$^4a ) ;

${\displaystyle {\frac {k^{3}}{\epsilon }}}$, , For , that longitudinal and transverse composition of ( ${\displaystyle {\frac {k}{\epsilon }}}$ ${\displaystyle {\frac {k}{\epsilon }}}$ ${\displaystyle \mathbf {f} /(m\mathbf {a} )}$^5b ) ;

Lorentz stated that he had no means of determining the value of . If he had equated it, his expressions would take the exact relativistic form. ${\displaystyle \epsilon }$ ${\displaystyle \epsilon =1}$

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1904: Lorentz derived the previous relations in somewhat more detail, namely with regard to the properties of particles that rest in a system and a relatively moving system : ${\displaystyle \Sigma '}$ ${\displaystyle \Sigma }$

${\displaystyle {\mathfrak {F}}(\Sigma )=\left(l^{2},\ {\frac {l^{2}}{k}},\ {\frac {l^{2}}{k}}\right){\mathfrak {F}}(\Sigma ')}$for as a function of according to ( ${\displaystyle \mathbf {f} }$ ${\displaystyle \mathbf {f} ^{0}}$^7a ) ;

${\displaystyle m{\mathfrak {j}}(\Sigma )=\left(l^{2},\ {\frac {l^{2}}{k}},\ {\frac {l^{2}}{k}}\right)m{\mathfrak {j}}(\Sigma ')}$for as a function of according to ( ${\displaystyle m\mathbf {a} }$ ${\displaystyle m\mathbf {a} ^{0}}$^7b ) ;

${\displaystyle {\mathfrak {j}}(\Sigma )=\left({\frac {l}{k^{3}}},\ {\frac {l}{k^{2}}},\ {\frac {l}{k^{2}}}\right){\mathfrak {j}}(\Sigma ')}$for as a function of according to ( ${\displaystyle \mathbf {a} }$ ${\displaystyle \mathbf {a} ^{0}}$^4a ) ;

${\displaystyle m(\Sigma )=\left(k^{3}l,\ kl,\ kl\right)m(\Sigma ')}$longitudinal and transversal mass as a function of rest mass (^5b , ^7b ) .

This time Lorentz was able to show that what gave his formulas their exact relativistic form. He also formulated the equation of motion ${\displaystyle l=1}$

${\displaystyle {\displaystyle {\mathfrak {F}}={\frac {d{\mathfrak {G}}}{dt}}}}$ with ${\displaystyle {\displaystyle {\mathfrak {G}}={\frac {e^{2}}{6\pi c^{2}R}}kl{\mathfrak {w}}}}$

what equation (^5c ) Corresponds with , , , , , , and as electromagnetic rest mass . He also stated that these equations should not only apply to forces and masses of electrically charged particles, but also to other processes, so that the movement of the earth through the ether remains undetectable. ${\displaystyle \mathbf {f} ={\frac {d\mathbf {p} }{dt}}={\frac {d(m\gamma \mathbf {u} )}{dt}}}$ ${\displaystyle l=1}$ ${\displaystyle {\mathfrak {F}}=\mathbf {f} }$ ${\displaystyle {\mathfrak {G}}=\mathbf {p} }$ ${\displaystyle {\mathfrak {w}}=\mathbf {u} }$ ${\displaystyle k=\gamma }$ ${\displaystyle e^{2}/(6\pi c^{2}R)=m}$

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1905: Henri Poincaré found the transformation of the force of three (^6a ) :

${\displaystyle X_{1}^{\prime }={\frac {k}{l^{3}}}{\frac {\rho }{\rho ^{\prime }}}\left(X_{1}+\epsilon \Sigma X_{1}\xi \right),\quad Y_{1}^{\prime }={\frac {\rho }{\rho ^{\prime }}}{\frac {Y_{1}}{l^{3}}},\quad Z_{1}^{\prime }={\frac {\rho }{\rho ^{\prime }}}{\frac {Z_{1}}{l^{3}}}}$

with , and as Lorentz factor, the charge density. Or in modern notation: , , , and . Like Lorentz, he too bet . ${\displaystyle {\frac {\rho }{\rho ^{\prime }}}={\frac {k}{l^{3}}}(1+\epsilon \xi )}$ ${\displaystyle k}$ ${\displaystyle \rho }$ ${\displaystyle \epsilon =v}$ ${\displaystyle \xi =u_{x}}$ ${\displaystyle \left(X_{1},\ Y_{1},\ Z_{1}\right)=\mathbf {f} }$ ${\displaystyle \Sigma X_{1}\xi =\mathbf {f} \cdot \mathbf {u} }$ ${\displaystyle l=1}$

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1905: Albert Einstein derived the equations of motion based on his SRT, which represent the relationships between equal inertial systems without having to assume a mechanical ether. Einstein first assumed that in a momentary inertial system the equations of motion retain their Newtonian form ${\displaystyle k}$

${\displaystyle \mu {\frac {d^{2}\xi }{d\tau ^{2}}}=\epsilon X',\quad \mu {\frac {d^{2}\eta }{d\tau ^{2}}}=\epsilon Y',\quad \mu {\frac {d^{2}\zeta }{d\tau ^{2}}}=\epsilon Z'}$.

That corresponds , because of and and . By transforming it into a relatively moving system , he obtained the equations for the electrical and magnetic components in this system ${\displaystyle \mathbf {f} ^{0}=m\mathbf {a} ^{0}}$ ${\displaystyle \mu =m}$ ${\displaystyle \left({\frac {d^{2}\xi }{d\tau ^{2}}},\ {\frac {d^{2}\eta }{d\tau ^{2}}},\ {\frac {d^{2}\zeta }{d\tau ^{2}}}\right)=\mathbf {a} ^{0}}$ ${\displaystyle \left(\epsilon X',\ \epsilon Y',\ \epsilon Z'\right)=\mathbf {f} ^{0}}$ ${\displaystyle K}$

${\displaystyle {\frac {d^{2}x}{dt^{2}}}={\frac {\epsilon }{\mu }}{\frac {1}{\beta ^{3}}}X,\quad {\frac {d^{2}y}{dt^{2}}}={\frac {\epsilon }{\mu }}{\frac {1}{\beta }}\left(Y-{\frac {v}{V}}N\right),\quad {\frac {d^{2}z}{dt^{2}}}={\frac {\epsilon }{\mu }}{\frac {1}{\beta }}\left(Z+{\frac {v}{V}}M\right)}$.

This corresponds to (^5b ) with , because and and and . Thereupon Einstein derived the longitudinal and transverse mass, whereby he defined the latter as the ratio between the inherent force in the momentary rest system, which is measured by a spring balance moving along with it, and the triple force in the system : ${\displaystyle \mathbf {a} ={\frac {\mathbf {f} }{m}}\left({\frac {1}{\gamma ^{3}}},\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)}$ ${\displaystyle \mu =m}$ ${\displaystyle \left({\frac {d^{2}x}{dt^{2}}},\ {\frac {d^{2}y}{dt^{2}}},\ {\frac {d^{2}z}{dt^{2}}}\right)=\mathbf {a} }$ ${\displaystyle \left[\epsilon X,\ \epsilon \left(Y-{\frac {v}{V}}N\right),\ \epsilon \left(Z+{\frac {v}{V}}M\right)\right]=\mathbf {f} }$ ${\displaystyle \beta =\gamma }$ ${\displaystyle \left(\epsilon X',\ \epsilon Y',\ \epsilon Z'\right)=\mathbf {f} ^{0}}$ ${\displaystyle K}$

${\displaystyle {\begin{array}{c|c}{\begin{aligned}\mu \beta ^{3}{\frac {d^{2}x}{dt^{2}}}&=\epsilon X=\epsilon X'\\\mu \beta ^{2}{\frac {d^{2}y}{dt^{2}}}&=\epsilon \beta \left(Y-{\frac {v}{V}}N\right)=\epsilon Y'\\\mu \beta ^{2}{\frac {d^{2}z}{dt^{2}}}&=\epsilon \beta \left(Z+{\frac {v}{V}}M\right)=\epsilon Z'\end{aligned}}&{\begin{aligned}{\frac {\mu }{\left({\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}\right)^{3}}}&\ {\text{longitudinale Masse}}\\\\{\frac {\mu }{1-\left({\frac {v}{V}}\right)^{2}}}&\ {\text{transversale Masse}}\end{aligned}}\end{array}}}$

This corresponds to (^7b ) with . ${\displaystyle m\mathbf {a} \left(\gamma ^{3},\ \gamma ^{2},\ \gamma ^{2}\right)=\mathbf {f} \left(1,\ \gamma ,\ \gamma \right)=\mathbf {f} ^{0}}$

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1905: Poincaré leads the transformation of triple acceleration (^1c ) a:

${\displaystyle {\frac {d\xi ^{\prime }}{dt^{\prime }}}={\frac {d\xi }{dt}}{\frac {1}{k^{3}\mu ^{3}}},\quad {\frac {d\eta ^{\prime }}{dt^{\prime }}}={\frac {d\eta }{dt}}{\frac {1}{k^{2}\mu ^{2}}}-{\frac {d\xi }{dt}}{\frac {\eta \epsilon }{k^{2}\mu ^{3}}},\quad {\frac {d\zeta ^{\prime }}{dt^{\prime }}}={\frac {d\zeta }{dt}}{\frac {1}{k^{2}\mu ^{2}}}-{\frac {d\xi }{dt}}{\frac {\zeta \epsilon }{k^{2}\mu ^{3}}}}$

where and and and . ${\displaystyle \left(\xi ,\ \eta ,\ \zeta \right)=\mathbf {u} }$ ${\displaystyle k=\gamma }$ ${\displaystyle \epsilon =v}$ ${\displaystyle \mu =1+\xi \epsilon =1+u_{x}v}$

In addition, he introduced the force of four in the form:

${\displaystyle k_{0}X_{1},\quad k_{0}Y_{1},\quad k_{0}Z_{1},\quad k_{0}T_{1}}$

where and and . ${\displaystyle k_{0}=\gamma _{0}}$ ${\displaystyle \left(X_{1},\ Y_{1},\ Z_{1}\right)=\mathbf {f} }$ ${\displaystyle T_{1}=\Sigma X_{1}\xi =\mathbf {f} \cdot \mathbf {u} }$

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1906: Max Planck derived the equations of motion

${\displaystyle {\frac {m{\ddot {x}}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}=e{\mathfrak {E}}_{x}-{\frac {e{\dot {x}}}{c^{2}}}\left({\dot {x}}{\mathfrak {E}}_{x}+{\dot {y}}{\mathfrak {E}}_{y}+{\dot {z}}{\mathfrak {E}}_{z}\right)+{\frac {e}{c}}\left({\dot {y}}{\mathfrak {H}}_{z}-{\dot {z}}{\mathfrak {H}}_{y}\right)\ {\text{etc.}}}$

With

${\displaystyle e\left({\dot {x}}{\mathfrak {E}}_{x}+{\dot {y}}{\mathfrak {E}}_{y}+{\dot {z}}{\mathfrak {E}}_{z}\right)={\frac {m\left({\dot {x}}{\ddot {x}}+{\dot {y}}{\ddot {y}}+{\dot {z}}{\ddot {z}}\right)}{\left(1-{\frac {q^{2}}{c^{2}}}\right)^{3/2}}}}$ other ${\displaystyle e{\mathfrak {E}}_{x}+{\frac {e}{c}}\left({\dot {y}}{\mathfrak {H}}_{z}-{\dot {z}}{\mathfrak {H}}_{y}\right)=X\ {\text{etc.}}}$

and

${\displaystyle {\frac {d}{dt}}\left\{{\frac {m{\dot {x}}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}\right\}=X\ {\text{etc.}}}$

These equations correspond to (^5c ) with , and and and , in agreement with those of Lorentz (1904). ${\displaystyle \mathbf {f} ={\frac {d\mathbf {p} }{dt}}={\frac {d(m\gamma \mathbf {u} )}{dt}}=m\gamma ^{3}\left({\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)+m\gamma \mathbf {a} }$ ${\displaystyle X=f_{x}}$ ${\displaystyle q=v}$ ${\displaystyle {\dot {x}}{\ddot {x}}+{\dot {y}}{\ddot {y}}+{\dot {z}}{\ddot {z}}=\mathbf {u} \cdot \mathbf {a} }$

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1907: Einstein analyzed a uniformly accelerated reference system and obtained the formulas for the coordinate-dependent speed of light and time dilation, analogous to those of the Kottler-Møller-Rindler coordinates .

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1907: Hermann Minkowski defined the relationship between the quadruple force (which he called the moving force) and the quadruple acceleration.

${\displaystyle m{\frac {d}{d\tau }}{\frac {dx}{d\tau }}=R_{x},\quad m{\frac {d}{d\tau }}{\frac {dy}{d\tau }}=R_{y},\quad m{\frac {d}{d\tau }}{\frac {dz}{d\tau }}=R_{z},\quad m{\frac {d}{d\tau }}{\frac {dt}{d\tau }}=R_{t}}$

accordingly . ${\displaystyle m\mathbf {A} =\mathbf {F} }$

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1908: Minkowski describes the second derivative of the proper time as the acceleration vector (four-point acceleration). He showed that its norm at any point on the world line has the value where the norm of a vector is directed from the center of the corresponding hyperbola of curvature . ${\displaystyle x,y,z,t}$ ${\displaystyle P}$ ${\displaystyle c^{2}/\varrho }$ ${\displaystyle \varrho }$ ${\displaystyle P}$

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1909: Max Born describes the movement with a constant norm of the four-point acceleration as hyperbolic movement, in connection with his study of Born's rigidity. He continued (today referred to as proper velocity or celerity) and as Lorentz factor and as proper time, and the transformation formulas ${\displaystyle p=dx/d\tau }$ ${\displaystyle q=-dt/d\tau ={\sqrt {1+p^{2}/c^{2}}}}$ ${\displaystyle \tau }$

${\displaystyle x=-q\xi ,\quad y=\eta ,\quad z=\zeta ,\quad t={\frac {p}{c^{2}}}\xi }$.

which (^8th ) correspond with and . By eliminating , Born obtained the hyperbola equation and helped to define the norm of this acceleration . He noticed that this can also be understood as a transformation into a hyperbolically accelerated frame of reference. ${\displaystyle \xi =c^{2}/\alpha }$ ${\displaystyle p=c\sinh(\alpha \tau /c)}$ ${\displaystyle p}$ ${\displaystyle x^{2}-c^{2}t^{2}=\xi ^{2}}$ ${\displaystyle b=c^{2}/\xi }$

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1909: Gustav Herglotz extended Born's investigation to all possible cases of rigidly accelerated movements, including uniform circular movements.

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1910: Arnold Sommerfeld brought Born's formulas for the hyperbolic movement into a clearer form as an imaginary time coordinate and as an imaginary angle: ${\displaystyle l=ict}$ ${\displaystyle \varphi }$

${\displaystyle x=r\cos \varphi ,\quad y=y',\quad z=z',\quad l=r\sin \varphi }$

He noticed that when is variable and constant, then they describe the world line of a charged body in hyperbolic motion. But if is constant and variable, then they describe the transformation into the rest system. ${\displaystyle r,y,z}$ ${\displaystyle \varphi }$ ${\displaystyle r,y,z}$ ${\displaystyle \varphi }$

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1911: Sommerfeld explicitly used the expression "self-acceleration" for the first time for the size in what ( ${\displaystyle {\dot {v}}_{0}}$ ${\displaystyle {\dot {v}}={\dot {v}}_{0}\left(1-\beta ^{2}\right)^{3/2}}$^4a ) than the acceleration in the momentary inertial system.

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1911: Herglotz explicitly used the expression "calm acceleration" instead of self-acceleration for the first time. He wrote them in the form and what ( ${\displaystyle \gamma _{l}^{0}=\beta ^{3}\gamma _{l}}$ ${\displaystyle \gamma _{t}^{0}=\beta ^{2}\gamma _{t}}$^4a ) corresponds to where the Lorentz factor is and or are longitudinal and transverse masses. ${\displaystyle \beta }$ ${\displaystyle \gamma _{l}^{0}}$ ${\displaystyle \gamma _{t}^{0}}$

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1911: Max von Laue derived the transformation of triple acceleration in the first edition of "The Relativity Principle"

${\displaystyle {\begin{aligned}{\mathfrak {\dot {q}}}_{x}&=\left({\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}+v{\mathfrak {q}}_{x}^{\prime }}}\right)^{3}{\mathfrak {\dot {q}}}_{x}^{\prime },&{\mathfrak {\dot {q}}}_{y}&=\left({\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}+v{\mathfrak {q}}_{x}^{\prime }}}\right)^{2}\left({\mathfrak {\dot {q}}}_{x}^{\prime }-{\frac {v{\mathfrak {q}}_{y}^{\prime }{\mathfrak {\dot {q}}}_{x}^{\prime }}{c^{2}+v{\mathfrak {q}}_{x}^{\prime }}}\right),\end{aligned}}}$

in accordance with (^1c ) and Poincaré (1905/6). From this she derived the transformation for the acceleration at rest (equivalent to^4a ) and finally got the formulas for the hyperbolic movement according to (^8th ) :

${\displaystyle \pm {\mathfrak {q}}_{x}=\pm {\frac {dx}{dt}}={\frac {cbt}{\sqrt {c^{2}+b^{2}t^{2}}}},\quad \pm \left(x-x_{0}\right)={\frac {c}{b}}{\sqrt {c^{2}+b^{2}t^{2}}},}$

so

${\displaystyle x^{2}-c^{2}t^{2}=x^{2}-u^{2}=c^{4}/b^{2},\quad y=\eta ,\quad z=\zeta }$,

and the transformation into a hyperbolic reference system with imaginary time and angle:

${\displaystyle {\begin{array}{c|c}{\begin{aligned}X&=R\cos \varphi \\L&=R\sin \varphi \end{aligned}}&{\begin{aligned}R^{2}&=X^{2}+L^{2}\\\tan \varphi &={\frac {L}{X}}\end{aligned}}\end{array}}}$.

He also writes the transformation of the power of three as

${\displaystyle {\begin{aligned}{\mathfrak {K}}_{x}&={\frac {{\mathfrak {K}}_{x}^{\prime }+{\frac {v}{c^{2}}}({\mathfrak {q'K'}})}{1+{\frac {v{\mathfrak {q}}_{x}^{\prime }}{c^{2}}}}},&{\mathfrak {K}}_{y}&={\mathfrak {K}}_{y}^{\prime }{\frac {\sqrt {1-\beta ^{2}}}{1+{\frac {v{\mathfrak {q}}_{x}^{\prime }}{c^{2}}}}},&{\mathfrak {K}}_{z}&={\mathfrak {K}}_{z}^{\prime }{\frac {\sqrt {1-\beta ^{2}}}{1+{\frac {v{\mathfrak {q}}_{x}^{\prime }}{c^{2}}}}},\end{aligned}}}$

in accordance with (^6a ) and Poincaré (1905).

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1912-1914: Friedrich Kottler obtained the general covariance of Maxwell's equations , and used four-dimensional Frenet-Serret formulas for the analysis of the rigid Born movements according to Herglotz (1909). This gave him the proper reference systems for the hyperbolic movement and uniform circular movement.

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1913: In the second edition of his book, von Laue replaced the transformation of the three-fold acceleration with Minkowski's acceleration vector, for which he first used the expression "four-fold acceleration". It is given with where the speed of four is. He showed that the norm of the four-speed acceleration corresponds to the acceleration at rest ${\displaystyle {\dot {Y}}={\frac {dY}{d\tau }}}$ ${\displaystyle Y}$ ${\displaystyle {\dot {\mathfrak {q}}}^{0}}$

${\displaystyle |{\dot {Y|}}={\frac {1}{c}}|{\dot {\mathfrak {q}}}^{0}|}$,

in accordance with (^4b ) . Then he derived the same formulas for the transformation of the acceleration at rest, hyperbolic movement, and the hyperbolically moved reference system as in 1911.

literature

• V. Petkov A. Ashtekar: Springer Handbook of Spacetime . Springer, 2014, ISBN 3-642-41992-5 .
• D. Bini, L. Lusanna, B. Mashhoon: Limitations of radar coordinates . In: International Journal of Modern Physics D . 14, No. 8, 2005, pp. 1413-1429. arxiv : gr-qc / 0409052 . doi : 10.1142 / S0218271805006961 .
• R. Ferraro: Einstein's Space-Time: An Introduction to Special and General Relativity . Spectrum, 2007, ISBN 0-387-69946-5 .
• P. Fraundorf: A traveler-centered intro to kinematics . In: Arxiv . 2012, p. IV-B. arxiv : 1206.2877 .
• AP French: Special Relativity . CRC Press, 1968, ISBN 1-4200-7481-4 .
• J. Freund: Special Relativity for Beginners: A Textbook for Undergraduates . World Scientific, 2008, ISBN 981-277-159-X .
• E. Gourgoulhon: Special Relativity in General Frames: From Particles to Astrophysics . Springer, 2013, ISBN 3-642-37276-7 .
• M. von Laue: Die Relativitätstheorie, Volume 1 , fourth edition of "Das Relativitätsprinzip". Edition, Vieweg, 1921 .; First edition 1911, second expanded edition 1913, third expanded edition 1919.
• D. Koks: Explorations in Mathematical Physics . Springer, 2006, ISBN 0-387-30943-8 .
• Kopeikin, S .; Efroimsky, M .; Kaplan, G .: Relativistic Celestial Mechanics of the Solar System . John Wiley & Sons, 2011, ISBN 3-527-40856-8 .
• Arthur I. Miller: Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905-1911) . Addison-Wesley, Reading 1981, ISBN 0-201-04679-2 .
• Misner, CW; Thorne, KS; Wheeler, JA: Gravitation . Freeman, 1973, ISBN 0-7167-0344-0 .
• C. Møller: The theory of relativity . Oxford Clarendon Press, 1955/1952.
• H. Nikolić: Relativistic contraction and related effects in noninertial frames . In: Physical Review A . 61, No. 3, 2000, p. 032109. arxiv : gr-qc / 9904078 . doi : 10.1103 / PhysRevA.61.032109 .
• Wolfgang Pauli : The Relativity Theory . In: Encyclopedia of Mathematical Sciences . 5, No. 2, 1921, pp. 539-776.
• M. Vallisneri M. Pauri: Märzke-Wheeler coordinates for accelerated observers in special relativity . In: Foundations of Physics Letters . 13, No. 5, 2000, pp. 401-425. arxiv : gr-qc / 0006095 . doi : 10.1023 / A: 1007861914639 .
• S. Nir J. Pfeffer: Modern Physics: An Introductory Text . World Scientific, 2012, ISBN 1-908979-57-7 .
• A. Shadowitz: Special relativity , Reprint of 1968. Edition, Courier Dover Publications, 1988, ISBN 0-486-65743-4 .
• F. Rahaman: The Special Theory of Relativity: A Mathematical Approach . Springer, 2014, ISBN 81-322-2080-3 .
• E. Rebhan: Theoretical Physics I . Spectrum, Heidelberg Berlin 1999, ISBN 3-8274-0246-8 .
• W. Rindler: Essential Relativity . Springer, 1977, ISBN 3-540-07970-X .
• JL Synge: Timelike helices in flat space-time . In: Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences . 1966, pp. 27-42. JSTOR 20488646
• RC Tolman: The theory of the relativity of motion . University of California Press , 1917, OCLC 13129939 .
• E. Zahar: Einstein's Revolution: A Study in Heuristic . Open Court Publishing Company, 1989, ISBN 0-8126-9067-2 .

Web links

• Transverse Mass in Einstein's Electrodynamics , Accelerated Travels , Born Rigidity, Acceleration, and Inertia , Does A Uniformly Accelerating Charge Radiate? Mathpages
• Acceleration in Special Relativity , The Relativistic Rocket Physics FAQ

Individual evidence

1. ↑ Misner & Thorne & Wheeler (1973), p. 163: "Accelerated motion and accelerated observers can be analyzed using special relativity."
2. ↑ ^{a b} von Laue (1921)
3. ↑ ^{a b} Pauli (1921)
4. ↑ Sexl & Schmidt (1979), p. 116
5. ^ Møller (1955), p. 41
6. ↑ Tolman (1917), p. 48
7. ^ French (1968), p. 148
8. ^ Zahar (1989), p. 232
9. ↑ Freund (2008), p. 96
10. ^ Kopeikin & Efroimsky & Kaplan (2011), p. 141
11. ↑ Rahaman (2014), p. 77
12. ↑ ^{a b c d} Pauli (1921), p. 627
13. ↑ ^{a b c d} Freund (2008), pp. 267-268
14. ↑ Ashtekar & Petkov (2014), p. 53
15. ↑ Sexl & Schmidt (1979), p. 198, Solution to example 16.1
16. ↑ ^{a b} Ferraro (2007), p. 178
17. ↑ Sexl & Schmidt (1979), p. 121
18. ↑ ^{a b c} Kopeikin & Efroimsky & Kaplan (2011), p. 137
19. ↑ ^{a b c} Rindler (1977), pp. 49-50
20. ↑ ^{a b c d} von Laue (1921), pp. 88-89
21. ^ Rebhan (1999), p. 775
22. ^ Nikolić (2000), eq. 10
23. ↑ Rindler (1977), p. 67
24. ↑ ^{a b c} Sexl & Schmidt (1979), solution of example 16.2, p. 198
25. ↑ ^{a b} Freund (2008), p. 276
26. ↑ ^{a b c} Møller (1955), pp. 74-75
27. ↑ ^{a b} Rindler (1977), pp. 89-90
28. ↑ ^{a b} von Laue (1921), p. 210
29. ^ Pauli (1921), p. 635
30. ↑ ^{a b} Tolman (1917), pp. 73-74
31. ↑ von Laue (1921), p. 113
32. ^ Møller (1955), p. 73
33. ^ Kopeikin & Efroimsky & Kaplan (2011), p. 173
34. ↑ ^{a b} Shadowitz (1968), p. 101
35. ↑ ^{a b} Pfeffer & Nir (2012), p. 115, “In the special case in which the particle is momentarily at rest relative to the observer S, the force he measures will be the proper force ”.
36. ↑ ^{a b} Møller (1955), p. 74
37. ^ Rebhan (1999), p. 818
38. ↑ see Lorentz's 1904 equations and Einstein's 1905 equations in the history section
39. ↑ ^{a b} Mathpages (see links), "Transverse Mass in Einstein's Electrodynamics", eq. 2.3
40. ↑ Rindler (1977), p. 43
41. ^ Koks (2006), section 7.1
42. ^ Fraundorf (2012), section IV-B
43. ↑ PhysicsFAQ (2016), see web links.
44. ^ Pauri & Vallisneri (2000), eq. 13
45. ↑ Bini & Lusanna & Mashhoon (2005), eq. 28.29
46. ↑ Synge (1966)
47. ↑ Pauri & Vallisneri (2000), Appendix A
48. ^ Misner & Thorne & Wheeler (1973), Section 6
49. ↑ ^{a b} Gourgoulhon (2013), entire book
50. ↑ Miller (1981)
51. ↑ Zahar (1989)

Historical work

1. ^ ^{A b c} Hendrik Antoon Lorentz: Simplified Theory of Electrical and Optical Phenomena in Moving Systems . In: Proceedings of the Royal Netherlands Academy of Arts and Sciences . 1, 1899, pp. 427-442.
2. ↑ ^{a b c d e f g} Hendrik Antoon Lorentz: Electromagnetic phenomena in a system moving with any velocity smaller than that of light . In: Proceedings of the Royal Netherlands Academy of Arts and Sciences . 6, 1904, pp. 809-831.
3. ^ ^{A b c} Henri Poincaré: Sur la dynamique de l'électron . In: Comptes rendus hebdomadaires des séances de l'Académie des sciences . 140, 1905, pp. 1504-1508.
4. ^ ^{A b c} Henri Poincaré: Sur la dynamique de l'électron . In: Rendiconti del Circolo matematico di Palermo . 21, pp. 129-176.
5. ↑ ^{a b c} Albert Einstein: On the electrodynamics of moving bodies . In: Annals of Physics . 322, No. 10, 1905, pp. 891-921. ; See also: English translation .
6. ↑ ^{a b c d} Max Planck: The principle of relativity and the basic equations of mechanics . In: Negotiations German Physical Society . 8, 1906, pp. 136-141.
7. ↑ ^{a b} Albert Einstein: About the principle of relativity and the conclusions drawn from it . In: Yearbook of radioactivity and electronics . 4, pp. 411-462. bibcode : 1908JRE ..... 4..411E . ; English translation On the relativity principle and the conclusions drawn from it at Einstein paper project.
8. ^ ^{A b} Hermann Minkowski: Space and Time . Lecture given at the 80th meeting of natural scientists in Cologne on September 21, 1908 . In: Annual report of the German Mathematicians Association . , Leipzig.
9. ↑ ^{a b} Hermann Minkowski: The basic equations for the electromagnetic processes in moving bodies . In: News from the Society of Science in Göttingen, Mathematical-Physical Class . , Pp. 53-111.
10. ↑ ^{a b c} Max Born: The theory of the rigid electron in the kinematics of the principle of relativity . In: Annals of Physics . 335, No. 11, 1909, pp. 1-56. doi : 10.1002 / andp.19093351102 .
11. ↑ ^{a b c} G. Herglotz: About the body that can be described as rigid from the standpoint of the principle of relativity . In: Annals of Physics . 336, No. 2, pp. 393-415. doi : 10.1002 / andp.19103360208 .
12. ↑ ^{a b c d} G. Herglotz: On the mechanics of the deformable body from the standpoint of the theory of relativity . In: Annals of Physics . 341, No. 13, 1911, pp. 493-533. doi : 10.1002 / andp.19113411303 .
13. ^ ^{A b} Arnold Sommerfeld: On the theory of relativity II: Four-dimensional vector analysis . In: Annals of Physics . 338, No. 14, 1910, pp. 649-689. doi : 10.1002 / andp.19103381402 .
14. ↑ ^{a b c d} Arnold Sommerfeld: About the structure of the gamma rays . In: Meeting reports of the mathematical-physical class of the KB Academy of Sciences in Munich . No. 1, 1911, pp. 1-60.
15. ↑ ^{a b c d e} Max von Laue: The principle of relativity . Vieweg, Braunschweig 1911.
16. ↑ ^{a b c} Max von Laue: The principle of relativity , 2nd edition. Edition, Vieweg, Braunschweig 1913.
17. ↑ ^{a b c} Friedrich Kottler: About the space-time lines of the Minkowski world . In: Wiener session reports 2a, Volume 121, 1912, pp. 1659-1759, hdl: 2027 / mdp.39015051107277 Friedrich Kottler: Principle of Relativity and Accelerated Movement . In: Annals of Physics . 349, No. 13, 1914, pp. 701-748. doi : 10.1002 / andp.19143491303 . Friedrich Kottler: Falling reference systems from the standpoint of the principle of relativity . In: Annals of Physics . 350, No. 20, 1914, pp. 481-516. doi : 10.1002 / andp.19143502003 .