Home | Sign Up | Log In | RSSThursday, 2018-12-13, 7:06 AM


Site menu
Section categories
Education [150]
Videos [1]
Music [0]
Chat Box
learn c language learn c language branching and loop learn c language functions learn c language Pointer Basics how web page works Oracle Solaris 10 Oracle Solaris 10 How Web Pages Work Images How Web Pages Work Adding images&gr How Web Pages Work Introduction to How to Install WAMP How to Host Your Own Website for Fr How to Install the Apache Web Serve Atomic Structure The text provides 000 tons of conversations Introduction to How Radio Works Rad often over millions of miles citizens band radio Introduction to How the Radio Spect Can information travel faster than How F-15s Work by Tom Harris Browse How F/A-18s Work by Robert Valdes B How F/A-18s Work by Robert Valdes B Flying Video Game: In the Cockpit I How F/A-22 Raptors Work by Gary Wol 10 Unidentified Sounds That Scienti 10 Things You Didn't Know About Ein Is glass really a liquid? by Laurie How Radio Works by Marshall Brain B How Radio Works by Marshall Brain B 10 Most Terrifying Vehicle Manufact How a Top Fuel Dragster Works by Ch but can Siri meet our nee How Siri Works by Bernadette Johnso How the Tesla Turbine Works by Will Lecture 1: Inflationary Cosmology: Lecture 2: Inflationary Cosmology: Lecture 4: The Kinematics of the Ho Lecture 5: Cosmological Redshift an Lecture 6: The Dynamics of Homogene Lecture 7: The Dynamics of Homogene Lecture 8: The Dynamics of Homogene Lecture 9: The Dynamics of Homogene Lecture 10: Introduction to Non-Euc IBPS Clerical Cadre Exam Pattern De Institute of Banking Personnel Sele Educational Qualifications: A Degre INTERVIEW Candidates who have been official trailer for Mission Imposs How Relativity Connects Electric an in 1738 Kinetic Theory of Gases: A Brief Re Frames of Reference and Newton’s La attempts to measure the UVa Physics 12/1/07 “Moving Clocks Run Slow” pl More Relativity: The Train and the 12/1/07 The Formula If I walk from Adding Velocities: A Walk on the Tr 3/1/2008 The Story So Far: A Brief Mass and Energy Michael Fowler Energy and Momentum in Lorentz Tran How Relativity Connects Electric an Analyzing Waves on a String Michael by f even earlier than the bra Fermat's Principle of Least Time 9/ Hamilton's Principle and Noether's meaning they have Mechanical Similarity and the Viria Hamilton's Equations 9/10/15 A Dyna A New Way to Write the Action Integ Maupertuis came up with a kind of p Maupertuis' Principle: Minimum Acti Canonical Transformations Point Tra Introduction to Liouville's Theorem Adiabatic Invariants and Action-Ang Hyperbolas Michael Fowler Prelimina Mathematics for Orbits: Ellipses Keplerian Orbits Michael Fowler Pre Newton's equations for particle mot Dynamics of Motion in a Central Pot A Vectorial Approach: Hamilton's Eq Elastic Scattering Michael Fowler B Driven Oscillator Michael Fowler (c Dynamics of a One-Dimensional Cryst I usefor the spring constant (is a a mass on a spring Motion in a Rapidly Oscillating Fie Anharmonic Oscillators Michael Fowl in which the distance betw Motion of a Rigid Body: the Inertia Moments of Inertia: Examples Michae Euler's Angles Michael Fowler Intro or more precisely one our analysis of rotational motion h Euler's Equations Michael Fowler In Motion in a Non-inertial Frame of R Ball Rolling on Tilted Turntable Mi live cricket score roll the ball ba

Total online: 1
Guests: 1
Users: 0
Home » 2015 » August » 20 » Energy and Momentum in Lorentz Transformations Michael Fowler, University of Virginia
7:48 PM
Energy and Momentum in Lorentz Transformations Michael Fowler, University of Virginia

Energy and Momentum in Lorentz Transformations

Michael Fowler, University of Virginia

How Does the Total Energy of a Particle Depend on Speed?

We have a formula for the total energy E = K.E. + rest energy,

so we can see how total energy varies with speed.

The momentum varies with speed as


How Does the Total Energy of a Particle Depend on Momentum?

It turns out to be useful to have a formula for E in terms of p.



hence using p = mv we find


If p is very small, this gives


the usual classical formula.

If p is very large, so c2p2  >> m02c4, the approximate formula is E = cp.

The High Kinetic Energy Limit: Rest Mass Becomes Unimportant!

Notice that this high energy limit is just the energy-momentum relationship Maxwell found to be true for light, for all p.  This could only be true for all  p if m02c4 = 0, that is, m0 = 0. 

Light is in fact composed of “photons”—particles having zero “rest mass”, as we shall discuss later.  The “rest mass” of a photon is meaningless, since they’re never at rest—the energy of a photon

is of the form 0/0, since m0 = 0 and v = c, so “m” can still be nonzero. That is to say, the mass of a photon is really all K.E. mass.

For very fast electrons, such as those produced in high energy accelerators, the additional K.E. mass can be thousands of times the rest mass.  For these particles, we can neglect the rest mass and take E = cp.

Transforming Energy and Momentum to a New Frame

We have shown

Notice we can write this last equation in the form


That is to say, depends only on the rest mass of the particle and the speed of light.  It does not depend on the velocity of the particle, so it must be the same—for a particular particle—in all inertial frames. 

This is reminiscent of the invariance of , the interval squared between two events, under the Lorentz transformations.  One might guess from this that the laws governing the transformation from Ep in one Lorentz frame toE′, p′ in another are similar to those for x,t.  We can actually derive the laws for Ep to check this out. 

As usual, we consider all velocities to be parallel to the x-axis.

We take the frame S'  to be moving in the x-direction at speed v relative to S

Consider a particle of mass m0 (rest mass) moving at u′ in the x′ direction in frame S', and hence at u along x in S, where


The energy and momentum in S' are

and in S:



from which it is easy to show that

Similarly, we can show that

These are the Lorentz transformations for energy and momentum of a particle—it is easy to check that


Photon Energies in Different Frames

For a zero rest mass particle, such as a photon,  E = cp,  E2 – c2p2 = 0 in all frames.



Since E = cpE′ = cp′ we also have


Notice that the ratios of photon energies in the two frames coincides with the ratio of photon frequencies found in the Doppler shift.  As we shall see when we cover quantum mechanics, the photon energy is proportional to the frequency, so these two must of course transform in identical fashion.  But it’s interesting to see it come about this way.  Needless to say, relativity gives us no clue on what the constant of proportionality (Planck’s constant) is: it must be measured experimentally.  But the same constant plays a role in all quantum phenomena, not just those concerned with photons.

Category: Education | Views: 654 | Added by: farrel | Tags: University of Virginia How Does the, Energy and Momentum in Lorentz Tran | Rating: 0.0/0
Total comments: 0
Name *:
Email *:
Code *:
Log In
Entries archive

Copyright eduCampus.tk © 2018
Powered by uCoz