Errors in “Time Dilation and the Velocity of Unstable Particles”

Steve Brown

doi:10.4006/1.3025490

Time Dilation and the Velocity of Unstable Particles

Physics Essays ◽

10.4006/1.3028796 ◽

1999 ◽

Vol 12 (4) ◽

pp. 669-670

Author(s):

Robert J. Hannon

Keyword(s):

Time Dilation ◽

Unstable Particles

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Time Dilation in Relativistic Quantum Decay Laws of Moving Unstable Particles

Advances in High Energy Physics ◽

10.1155/2018/7308935 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Filippo Giraldi

Keyword(s):

Decay Rate ◽

Effective Mass ◽

Survival Probability ◽

Relativistic Quantum ◽

Scaling Law ◽

Power Laws ◽

Time Dilation ◽

Linear Momentum ◽

Unstable Particle ◽

Unstable Particles

The relativistic quantum decay laws of moving unstable particles are analyzed for a general class of mass distribution densities which behave as power laws near the (nonvanishing) lower bound μ0 of the mass spectrum. The survival probability Pp(t), the instantaneous mass Mp(t), and the instantaneous decay rate Γp(t) of the moving unstable particle are evaluated over short and long times for an arbitrary value p of the (constant) linear momentum. The ultrarelativistic and nonrelativistic limits are studied. Over long times, the survival probability Pp(t) is approximately related to the survival probability at rest P0(t) by a scaling law. The scaling law can be interpreted as the effect of the relativistic time dilation if the asymptotic value Mp∞ of the instantaneous mass is considered as the effective mass of the unstable particle over long times. The effective mass has magnitude μ0 at rest and moves with linear momentum p or, equivalently, with constant velocity 1/1+μ02/p2. The instantaneous decay rate Γp(t) is approximately independent of the linear momentum p, over long times, and, consequently, is approximately invariant by changing reference frame.

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The Equivalence Principle

10.1093/oso/9780199658633.003.0013 ◽

2018 ◽

Author(s):

David M. Wittman

Keyword(s):

General Relativity ◽

Special Relativity ◽

Equivalence Principle ◽

Time Dilation ◽

Gravitational Redshift ◽

Coordinate Systems ◽

Spacetime Metric

The equivalence principle is an important thinking tool to bootstrap our thinking from the inertial coordinate systems of special relativity to the more complex coordinate systems that must be used in the presence of gravity (general relativity). The equivalence principle posits that at a given event gravity accelerates everything equally, so gravity is equivalent to an accelerating coordinate system.This conjecture is well supported by precise experiments, so we explore the consequences in depth: gravity curves the trajectory of light as it does other projectiles; the effects of gravity disappear in a freely falling laboratory; and gravitymakes time runmore slowly in the basement than in the attic—a gravitational form of time dilation. We show how this is observable via gravitational redshift. Subsequent chapters will build on this to show how the spacetime metric varies with location.

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Energy and Momentum

10.1093/oso/9780199658633.003.0012 ◽

2018 ◽

Author(s):

David M. Wittman

Keyword(s):

Time Dilation ◽

Speed Of Light ◽

Massless Particles ◽

The Everyday ◽

Low Speeds ◽

Energy And Momentum ◽

Deep Relationship ◽

Insight Into ◽

Momentum Relation

Tis chapter explains the famous equation E = mc2 as part of a wider relationship between energy, mass, and momentum. We start by defning energy and momentum in the everyday sense. We then build on the stretching‐triangle picture of spacetime vectors developed in Chapter 11 to see how energy, mass, and momentum have a deep relationship that is not obvious at everyday low speeds. When momentum is zero (a mass is at rest) this energy‐momentum relation simplifes to E = mc2, which implies that mass at rest quietly stores tremendous amounts of energy. Te energymomentum relation also implies that traveling near the speed of light (e.g., to take advantage of time dilation for interstellar journeys) will require tremendous amounts of energy. Finally, we look at the simplifed form of the energy‐momentum relation when the mass is zero. Tis gives us insight into the behavior of massless particles such as the photon.

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