Non-inertial reference system

Non-inertial reference system - a reference system that moves with acceleration or rotates relatively inertial. Newton’s second law also does not hold in non-inertial reference frames. In order for the equation of motion of a material point in a non-inertial reference frame in form to coincide with the equation of Newton’s second law, inertial forces are introduced in addition to the “ordinary” forces acting in inertial systems.

Newton's laws are satisfied only in inertial reference systems. Nevertheless, the motion of bodies in non-inertial reference frames can be described by the same equations of motion as in inertial, if along with the forces due to the action of the bodies on each other, the inertia forces are taken into account ^[1] ^[2]

Content

1 In classical mechanics
2 In the general theory of relativity
3 In quantum theory
4 notes
5 Literature

In classical mechanics

Classical mechanics postulates the following two principles:

time is absolute, that is, the time intervals between any two events are the same in all arbitrarily moving reference systems;
space is absolute, that is, the distance between any two material points is the same in all arbitrarily moving reference frames.

These two principles make it possible to write down the equation of motion of a material point with respect to any non-inertial reference frame in which Newton’s first law is not satisfied.

The equation of motion of a material point in a non-inertial reference frame can be represented as ^[3] :

m{\vec {a}}_{r}={\vec {F}}-m{\vec {a}}_{e}-m{\vec {a}}_{k}

{\ displaystyle m {\ vec {a}} _ {r} = {\ vec {F}} - m {\ vec {a}} _ {e} -m {\ vec {a}} _ {k}}

,

or in expanded form:

m{\vec {a}}_{r}={\vec {F}}+2m\left[{\vec {v}}_{r}\times {\vec {\omega }}\right]-m{\frac {d{\vec {v_{0}}}}{dt}}+m\omega ^{2}{\vec {r}}_{\perp }-m\left[{\frac {d{\vec {\omega }}}{dt}}\times {\vec {r}}\right]

{\ displaystyle m {\ vec {a}} _ {r} = {\ vec {F}} + 2m \ left [{\ vec {v}} _ {r} \ times {\ vec {\ omega}} \ right] -m {\ frac {d {\ vec {v_ {0}}}} {dt}} + m \ omega ^ {2} {\ vec {r}} _ {\ perp} -m \ left [{ \ frac {d {\ vec {\ omega}}} {dt}} \ times {\ vec {r}} \ right]}

,

Where $\ m$ ${\ displaystyle \ m}$ - body weight $\ {\vec {a}}_{r}$ ${\ displaystyle \ {\ vec {a}} _ {r}}$ , ${\vec {v}}_{r}$ ${\ displaystyle {\ vec {v}} _ {r}}$ - acceleration and speed of the body relative to the non-inertial reference frame, $\ {\vec {F}}$ ${\ displaystyle \ {\ vec {F}}}$ - the sum of all external forces acting on the body, $\ {\vec {a}}_{e}$ ${\ displaystyle \ {\ vec {a}} _ {e}}$ - portable acceleration of the body, $\ {\vec {a}}_{k}$ ${\ displaystyle \ {\ vec {a}} _ {k}}$ - Coriolis acceleration of the body, $\omega$ ${\ displaystyle \ omega}$ - the angular velocity of the rotational motion of the non-inertial reference system around the instantaneous axis passing through the origin, ${\vec {v_{0}}}$ ${\ displaystyle {\ vec {v_ {0}}}}$ - the speed of movement of the origin of the non-inertial reference frame relative to any inertial reference frame.

This equation can be written in the usual form of Newton’s second law , if we introduce the forces of inertia :

${\vec {F}}_{e}=-m{\vec {a}}_{e}$ ${\ displaystyle {\ vec {F}} _ {e} = - m {\ vec {a}} _ {e}}$ - portable inertia force
${\vec {F}}_{k}=-m{\vec {a}}_{k}$ ${\ displaystyle {\ vec {F}} _ {k} = - m {\ vec {a}} _ {k}}$ - Coriolis force

In non-inertial reference systems, inertia forces arise. The appearance of these forces is a sign of the inertia of the reference frame. ^[four]

In General Theory of Relativity

According to the principle of equivalence of gravitational and inertial forces, it is locally impossible to distinguish which force acts on a given body - gravitational force or inertial force . At the same time, due to the curvature of space-time in its final region, it is impossible to eliminate the tidal forces of gravity by switching to any reference system (see deviation of geodesics ). In this sense, global and even finite inertial reference frames in the general theory of relativity are generally absent, that is, all reference frames are non-inertial.

In quantum theory

In 1976, William Unruh , using the methods of quantum field theory, showed that in non-inertial reference systems there is thermal radiation with a temperature equal to

T_{u}={\frac {\hbar a}{2\pi kc}}

{\ displaystyle T_ {u} = {\ frac {\ hbar a} {2 \ pi kc}}}

,

Where $a$ ${\ displaystyle a}$ - acceleration of the reference system ^[5] . The Unruh effect is absent in inertial reference systems ( $a=0$ ${\ displaystyle a = 0}$ ) The Unruh effect also leads to the fact that in non-inertial reference frames protons acquire a finite lifetime - the possibility of its reverse beta decay into a neutron, positron and neutrino opens up. ^[6] ^[7] ^[8] At the same time, this Unruh radiation has properties that do not completely coincide with the usual thermal radiation, for example, an accelerated quantum-mechanical detector system does not necessarily behave the same as in a thermal bath. ^[9]

Notes

↑ Savelyev I.V. General physics course. T. 1. Mechanics. Molecular physics. - M .: Nauka, 1987 .-- pp. 118-119.
↑ Landsberg G.S. Elementary textbook of physics. Volume 1. Mechanics. Heat. Molecular physics. - M .: Nauka, 1975 .-- C. 292
↑ Sivukhin D.V. §64. Inertia forces with arbitrary accelerated motion of the reference system // General course of physics. - M .: Nauka , 1979. - T. I. Mechanics. - S. 337—347. - 520 s.
↑ Loytsyansky L.G., Lurie A.I. Course in Theoretical Mechanics. Volume 2 Dynamics (Science 1983) Page 433: “non-inertial systems additionally generate special forces, the so-called inertia forces; the appearance of these forces is a sign of the inertia of the reference frame. ”
↑ LCB Crispino, A. Higuchi, GEA Matsas "The Unruh effect and its applications" Reviews of Modern Physics. 2008. Vol. 80. No.3. P.787-838. ( arxiv = 0710.5373
↑ R. Mueller, Decay of accelerated particles , Phys. Rev. D 56 , 953-960 (1997) preprint .
↑ DAT Vanzella and GEA Matsas, Decay of accelerated protons and the existence of the Fulling-Davies-Unruh effect , Phys. Rev. Lett. 87 , 151301 (2001) preprint .
↑ H. Suzuki and K. Yamada, Analytic Evaluation of the Decay Rate for Accelerated Proton , Phys. Rev. D 67,065002 (2003) preprint .
↑ Belinsky V.A., Karnakov B.M., Moore V.D., Narozhniy N. B. // Letters in JETP, 1997.V. 65.P. 861.

Literature

Yavorsky B.M., Detlaf A.A. Handbook of Physics. 2nd ed., Revised. M .: Nauka, 1985.512 s.

[1] Savelyev I.V. General physics course. T. 1. Mechanics. Molecular physics. - M .: Nauka, 1987 .-- pp. 118-119.

[2] Landsberg G.S. Elementary textbook of physics. Volume 1. Mechanics. Heat. Molecular physics. - M .: Nauka, 1975 .-- C. 292

[3] Sivukhin D.V. §64. Inertia forces with arbitrary accelerated motion of the reference system // General course of physics. - M .: Nauka , 1979. - T. I. Mechanics. - S. 337—347. - 520 s.

[4] Loytsyansky L.G., Lurie A.I. Course in Theoretical Mechanics. Volume 2 Dynamics (Science 1983) Page 433: “non-inertial systems additionally generate special forces, the so-called inertia forces; the appearance of these forces is a sign of the inertia of the reference frame. ”

[5] LCB Crispino, A. Higuchi, GEA Matsas "The Unruh effect and its applications" Reviews of Modern Physics. 2008. Vol. 80. No.3. P.787-838. ( arxiv = 0710.5373

[muel-6] R. Mueller, Decay of accelerated particles , Phys. Rev. D 56 , 953-960 (1997) preprint .

[van-7] DAT Vanzella and GEA Matsas, Decay of accelerated protons and the existence of the Fulling-Davies-Unruh effect , Phys. Rev. Lett. 87 , 151301 (2001) preprint .

[suz-8] H. Suzuki and K. Yamada, Analytic Evaluation of the Decay Rate for Accelerated Proton , Phys. Rev. D 67,065002 (2003) preprint .

[9] Belinsky V.A., Karnakov B.M., Moore V.D., Narozhniy N. B. // Letters in JETP, 1997.V. 65.P. 861.