“Hyperbolic motion (relativity)”的意思、由来-开放百科全书

Hermann Minkowski (1908) showed the relation between a point on a worldline and the magnitude of four-acceleration and a "curvature hyperbola" ({{lang-de|Krümmungshyperbel}}).^[2] In the context of Born rigidity, Max Born (1909) subsequently coined the term "hyperbolic motion" ({{lang-de|Hyperbelbewegung}}) for the case of constant magnitude of four-acceleration, then provided a detailed description for charged particles in hyperbolic motion, and introduced the corresponding "hyperbolically accelerated reference system" ({{lang-de|hyperbolisch beschleunigtes Bezugsystem}}).^[3] Born's formulas were simplified and extended by Arnold Sommerfeld (1910).^[4] For early reviews see the textbooks by Max von Laue (1911, 1921)^[5] or Wolfgang Pauli (1921).^[6] See also Galeriu (2015)^[7] or Gourgoulhon (2013),^[4] and Acceleration (special relativity)#History.

Worldline

The proper acceleration

of a particle is defined as the acceleration that a particle "feels" as it accelerates from one inertial reference frame to another. If the proper acceleration is directed parallel to the line of motion, it is related to the ordinary three-acceleration in special relativity

where

is the instantaneous speed of the particle,

the Lorentz factor,

is the speed of light, and

is the coordinate time. Solving for the equation of motion gives the desired formulas, which can be expressed in terms of coordinate time

as well as proper time

. For simplification, all initial values for time, location, and velocity can be set to 0, thus:^[5]^[6]^[7]^[8]^[9]

This gives

, which is a hyperbola in time and the spatial location variable

. In this case, the accelerated object is located at

at time

. If instead there are initial values different from zero, the formulas for hyperbolic motion assume the form:^[10]^[11]^[12]

Rapidity

The worldline for hyperbolic motion (which from now on will be written as a function of proper time) can be simplified in several ways. For instance, the expression

by which the observer is at position

at time

. Furthermore, by setting

and introducing the rapidity

,^[12] the equations for hyperbolic motion reduce to^[14]^[15]

Charged particles in hyperbolic motion

Born (1909),^[3] Sommerfeld (1910),^[14] von Laue (1911),^[5] Pauli (1921)^[6] also formulated the equations for the electromagnetic field of charged particles in hyperbolic motion.^[16] This was extended by Hermann Bondi & Thomas Gold (1955)^[17] and Fulton & Rohrlich (1960)^[18]^[19]

This is related to the controversially^[20]^[21] discussed question, whether charges in perpetual hyperbolic motion do radiate or not, and whether this is consistent with the equivalence principle – even though it's about an ideal situation, because perpetual hyperbolic motion is not possible. While early authors such as Born (1909) or Pauli (1921) argued that no radiation arises, later authors such as Bondi & Gold^[17] and Fulton & Rohrlich^[18]^[19] showed that radiation does indeed arise.

Proper reference frame

In equation ({{equationNote|2}}) for hyperbolic motion, the expression

was constant, whereas the rapidity

was variable. However, as pointed out by Sommerfeld,^[15] one can define

as a variable, while making

constant. This means, that the equations become transformations indicating the simultaneous rest shape of an accelerated body with hyperbolic coordinates

as seen by a comoving observer

By means of this transformation, the proper time becomes the time of the hyperbolically accelerated frame. These coordinates, which are commonly called Rindler coordinates (similar variants are called Kottler-Møller coordinates or Lass coordinates), can be seen as a special case of Fermi coordinates or Proper coordinates, and are often used in connection with the Unruh effect. Using these coordinates, it turns out that observers in hyperbolic motion possesses an apparent event horizon, beyond which no signal can reach them.

Special conformal transformation

A lesser known method for defining a reference frame in hyperbolic motion is the employment of the special conformal transformation, consisting of an inversion, a translation, and another inversion. It is commonly interpreted as a gauge transformation in Minkowski space, though some authors alternatively use it as an acceleration transformation (see Kastrup for a critical historical survey).^[35] It has the form

Using only one spatial dimension by

, and further simplifying by setting

, and using the acceleration

, it follows^[22]

with the hyperbola

. It turns out that at

the time becomes singular, to which Fulton & Rohrlich & Witten^[22] remark that one has to stay away from this limit, while Kastrup^[23] (who is very critical of the acceleration interpretation) remarks that this is one of the strange results of this interpretation.

Notes

1. ^{{harvnb|Misner|Thorne|Wheeler|1973|loc=Chapter 6.}}
2. ^{{Cite journal|author=Minkowski, Hermann|year=1909|title=Raum und Zeit. Vortrag, gehalten auf der 80. Naturforscher-Versammlung zu Köln am 21. September 1908.|trans-title=Wikisource translation: Space and Time|journal=Jahresbericht der Deutschen Mathematiker-Vereinigung|location=Leipzig}}
3. ^¹{{Cite journal|author=Born, Max|year=1909|title=Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips|trans-title=Wikisource translation: The Theory of the Rigid Electron in the Kinematics of the Principle of Relativity|journal=Annalen der Physik|volume=335|issue=11|pages=1–56 |doi=10.1002/andp.19093351102|bibcode=1909AnP...335....1B}}
4. ^{{cite book |author=Gourgoulhon, E. |year=2013 |title=Special Relativity in General Frames: From Particles to Astrophysics |pages=396|publisher=Springer|isbn=3642372767}}
5. ^¹²{{Cite book|author=von Laue, M.|year=1921|title=Die Relativitätstheorie, Band 1|edition=fourth edition of "Das Relativitätsprinzip”|pages=89–90, 155–166|publisher=Vieweg|url=https://archive.org/details/dierelativitts01laueuoft}}; First edition 1911, second expanded edition 1913, third expanded edition 1919.
6. ^¹²{{Citation|author=Pauli, Wolfgang|authorlink=Wolfgang Pauli|year=1921|journal=Encyclopädie der mathematischen Wissenschaften|title= Die Relativitätstheorie|pages=539–776|volume=5|issue=2|url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672%7CLOG_0265}}
In English: {{cite book|author=Pauli, W.|title=Theory of Relativity|journal=Fundamental Theories of Physics|volume=165|publisher=Dover Publications|year=1981|orig-year=1921|isbn=0-486-64152-X}}
7. ^{{Cite book|author=Møller, C.|title=The theory of relativity|year=1955|pages=74–75|publisher=Oxford Clarendon Press|url=https://archive.org/details/theoryofrelativi029229mbp}}
8. ^{{Cite book|author=Rindler, W.|title=Essential Relativity|year=1977|pages=49–50|publisher=Springer|isbn=354007970X}}
9. ^PhysicsFAQ (2016), "Relativistic rocket", see external links
10. ^{{Cite book|author=Gallant, J.|title=Doing Physics with Scientific Notebook: A Problem Solving Approach|year=2012|pages=437–441|publisher=John Wiley & Sons|isbn=0470665971}}
11. ^{{Cite journal|author=Müller, T., King, A., & Adis, D.|year=2006|title=A trip to the end of the universe and the twin "paradox"|journal=American Journal of Physics|volume=76|issue=4|pages=360–373|arxiv=physics/0612126|doi=10.1119/1.2830528|bibcode=2008AmJPh..76..360M}}
12. ^¹{{Cite journal|author=Fraundorf, P. |year=2012|title=A traveler-centered intro to kinematics|journal=|volume=|issue=|pages=IV–B|arxiv=1206.2877|doi=|bibcode=2012arXiv1206.2877F}}
13. ^Pauli (1921), p. 628, used the notation

where

14. ^¹²{{Cite journal |author=Sommerfeld, Arnold |year=1910 |title=Zur Relativitätstheorie II: Vierdimensionale Vektoranalysis|trans-title=Wikisource translation: On the Theory of Relativity II: Four-dimensional Vector Analysis |journal=Annalen der Physik |volume=338 |issue=14 |pages=649–689|doi=10.1002/andp.19103381402|bibcode=1910AnP...338..649S}}
15. ^¹Sommerfeld (1910), pp. 670-671 used the form

and

with the imaginary angle

and imaginary time

.
16. ^¹{{Cite journal|author=Galeriu, C.|year=2017|orig-year=2015|title=Electric charge in hyperbolic motion: the early history|journal=Archive for History of Exact Sciences|volume=71|issue=4|pages=1–16|doi=10.1007/s00407-017-0191-x|arxiv=1509.02504}}
17. ^¹{{Cite journal|author=Bondi, H., & Gold, T.|year=1955|title=The field of a uniformly accelerated charge, with special reference to the problem of gravitational acceleration|journal=Proceedings of the Royal Society of London|volume=229|issue=1178|pages=416–424|doi=10.1098/rspa.1955.0098|bibcode=1955RSPSA.229..416B}}
18. ^¹{{Cite journal|author=Fulton, Thomas; Rohrlich, Fritz|year=1960|title=Classical radiation from a uniformly accelerated charge|journal=Annals of Physics|volume=9|issue=4|pages=499–517|doi=10.1016/0003-4916(60)90105-6|bibcode=1960AnPhy...9..499F}}
19. ^¹{{Cite journal|author=Rohrlich, Fritz|year=1963|title=The principle of equivalence|journal=Annals of Physics|volume=22|issue=2|pages=169–191|doi=10.1016/0003-4916(63)90051-4|bibcode=1963AnPhy..22..169R}}
20. ^{{Cite book|author=Stephen Lyle|title=Uniformly Accelerating Charged Particles: A Threat to the Equivalence Principle|year=2008|publisher=Springer|isbn=3540684778}}
21. ^{{Cite journal|author=Øyvind Grøn|year=2012|title=Review Article: Electrodynamics of Radiating Charges|journal=Advances in Mathematical Physics|volume=2012|pages=528631|doi=10.1155/2012/528631}}
22. ^¹{{Cite journal|author=Fulton, T., Rohrlich, F., & Witten, L.|year=1962|title=Physical consequences of a co-ordinate transformation to a uniformly accelerating frame|journal=Il Nuovo Cimento|volume=26|issue=4|pages=652–671|doi=10.1007/BF02781794|bibcode=1962NCim...26..652F}}
23. ^¹{{Cite journal|author=Kastrup, H. A.|title=On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics|journal=Annalen der Physik|volume=520|issue=9–10|year=2008|pages=631–690|arxiv=0808.2730|doi=10.1002/andp.200810324|bibcode = 2008AnP...520..631K }}

History