Abstract
An extended object composed of discrete particles, or a portion of a continuum, will describe a worldtube enclosing the worldlines of its constituent particles.
I know not with what weapons World War III will be fought, but World War IV will be fought with sticks and stones.
—Albert Einstein.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Strictly speaking \(\lambda =\pm c\tau \) but the worldline should be future-oriented, hence we choose the positive sign. In any case, the worldline (6.13) is symmetric with respect to time reflection \(t \rightarrow -t\).
- 2.
To avoid confusion, pay attention to the rather unfortunate, but common, notation. The 3-velocity \(d\mathbf x /dt\) is denoted with the familiar symbol \(\mathbf v \) and \(u^{\mu }=\left( u^0, \mathbf u \right) \) with \(\mathbf u =\gamma \, \mathbf v \), but the other familiar symbol \(\mathbf p \) denotes \(\gamma \, m \mathbf v \), not \(m\mathbf v \), with \(p^{\mu }=\left( p^0, \mathbf p \right) \).
- 3.
1 ton (of TNT) is equivalent to \(4.2\times 10^9\) J.
- 4.
Some neutral particles coincide with their own antiparticle.
- 5.
The known leptons are divided into three families: the electron e\(^{-}\) and the electron neutrino\(\nu _e\); the muon\(\mu ^{-}\) and the muon neutrino\(\nu _{\mu }\); and the \(\tau ^{-}\) muon and its neutrino\(\nu _{\tau }\), plus their respective antiparticles e\(^{+}, \bar{\nu }_e, \mu ^{+}, \bar{\nu }_{\mu }, \tau ^{+}\), and \(\bar{\nu }_{\tau }\). A lepton number\(+1\) is assigned to the electron and the electron neutrino, a lepton number \(-1\) is carried by their respective antiparticles e\(^{+}\) and \(\bar{\nu }_e\), and all other particles have lepton number zero.
- 6.
In other words, the proper 3-acceleration of a particle is not invariant under Lorentz transformations, while it is invariant under Galilei transformations.
- 7.
High energy cosmic rays are much more energetic than the most energetic particle beams in accelerators. However, one cannot predict their location and trajectory, while particle beams in an accelerator can be aimed precisely at a target and their energies can be controlled.
- 8.
It is not entirely trivial that \(P^{\mu }\) is a 4-vector because it is obtained by adding 4-vectors at a single time, but different observers do not agree that these different 4-vectors at different locations are simultaneous. Nevertheless, it can be shown that \(P^{\mu }\) is indeed a 4-vector [5, 13]. The same can be said of the angular momentum \(L^{\mu \nu }\).
- 9.
This classic exercise recurring in many thermodynamics and Special Relativity textbooks seems to be due originally to Pauli.
References
P. Schneider, J. Ehlers, E.E. Falco, Gravitational Lenses. (Springer, New York, 1999)
R.M. Wald, General Relativity. (Chicago University Press, Chicago, 1984)
S.M. Carroll, Spacetime and Geometry: An Introduction to General Relativity. (Addison-Wesley, San Francisco, 2004)
J.B. Hartle, Gravity: An Introduction to Einstein’s General Relativity. (Addison-Wesley, San Francisco, 2003)
L.D. Landau, E. Lifschitz, The Classical Theory of Fields. (Pergamon Press, Oxford, 1989)
C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation. (Freeman, New York, 1973)
D.L. Bailey, D.W. Townsend, P.E. Valk, M.N. Maisey, Positron Emission Tomography: Basic Sciences. (Springer, New York, 2005)
S.A. Fulling, Phys. Rev. D 7, 2850 (1973)
P.C.W. Davies, J. Phys. A 8, 609 (1975)
W.G. Unruh, Phys. Rev. D 14, 870 (1976)
Large Hadron Collider website http://lhc.web.cern.ch/lhc/
International Linear Collider homepage http://www.linearcollider.org/
W. Rindler, Introduction to Special Relativity, 2nd edn. (Clarendon Press, Oxford, 1991)
D. Prialnik, An Introduction to the Theory of Stellar Structure and Evolution. (Cambridge University Press, Cambridge, 2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Faraoni, V. (2013). Relativistic Mechanics. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01107-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-01107-3_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01106-6
Online ISBN: 978-3-319-01107-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)