Momentum and Energy

1. Momentum and Energy • Conservation of momentum is one of the most fundamental and most useful concepts of elementary physis • Does it apply in special relativity? • Consider the following collision process. All balls have mass m and speed v before and after collision • Surely this will conserve momentum Before m After m m m Momentum is conserved But what about other frames?

2. Momentum in another frame y’ m m m v m x’ In this frame, momentum is not conserved!

3. The problem with momentum • How can we explain this failure of conservation of momentum? Three choices: • This process is physically impossible; it cannot occur • Conservation of momentum is not a conservation law in S.R. • Our formula for momentum is wrong in S.R. • Why is t given special privilege over other coordinates? • We want something that is like time, but not coordinate dependant in the denominator, like proper time 

4. Does this fix the problem? (1)

5. Does this fix the problem? (2) Momentum is conserved

6. What’s missing here . . . • Hint – we’ve got all three dimensions . . . • But we live in four-dimensional spacetime • Is there a conserved fourth component of momentum? • These quantities are related by Lorentz boost, just as pxand py are related by rotations • If px, pyand pzare conserved, then so must be pt • Is it pt that is conserved, or cpt? • Actually it doesn’t matter; if ptis conserved, so is any multiple of it • We will work with a different multiple • The ERIC quantity • Eric is conserved

7. What is Eric, really? m1 m1 Consider low velocity limit: m2 m2 Not careful enough . . . E is really energy

8. Potential and Kinetic Energy • E is the total energy – it includes both potential and kinetic energy • Total energy is always conserved • Potential energy is the energy something has when it’s not moving • Also called rest energy • More about this later • Kinetic energy is how much extra energy there is due to motion

9. A sample problem . . . In 2004, the total world consumption of energy was 15 TW. How much fuel would have to be consumed each hour to power the world if we could extract 100% of its potential energy?

10. Good vs. Bad Notation • Eric’s notation: • m, called the mass, is a constant, and independent of velocity • It does not increase with velocity • It does not approach infinity as u c • The famous formula E = mc2 is not true except when u = 0. • Einstein never wrote this formula • The quantity m is then sometimes called the relativistic mass - but I won’t use this term. • Satan’s notation: • mm0 is renamed the rest mass • We then define m = m0 as the mass • In this notation, mass m goes to infinity as u c • In this notation, E = mc2 is always true • Do not use this notation/nomenclature in this class

11. Momentum and Energy • For small velocities,   1, and the momentum is just the usual formula • As u c, the momentum and energy both go to infinity • This tells us we can’t reach u = c. E/mc2 p/mc u/c

12. Particle physics units • Forces are usually induced by using electric fields • Charges are usually the same as the electron charge (-e) or simple multiples of this (+e, -2e, etc.) • Electric potentials are measured in volts (V) • Easy to work with units of electron charge times volts, or eV’s (called electron volts) • Metric multiples are also common • Momentum has units of energy/velocity, so it is often expressed in units of eV/c • Mass has units of energy/velocity2, so it is often expressed in units of eV/c2

13. A sample problem . . . A Z-particle at rest (mass mZ = 91.19 GeV/c2) decays to a B-meson and an anti-B meson (both mass mB = 5.279 GeV/c2). What is the energy and velocity of the resulting B-mesons? Z B B • Since the initial Z is at rest, pZ = 0 and EZ= mZc2 = 91.19 GeV • The B-meson and anti-B-meson must have equal & opposite momenta • Since they have the same mass, this means they have equal speeds • They therefore have equal energies Note: Mass is not conserved!

14. Two Useful Formulas Divide them: Written this way, both sides are dimensionless Square and subtract them:

15. Different types of particles • Must the combination E2 – c2p2 always be positive? Maybe m2 needn’t always be positive • If m2 > 0 we call the particle a massive particle • Electrons, protons, rockets, people • E > cp, so u < c: Always slower than light • If m2 = 0 we call the particle a massless particle • Photons (particles of light) • E = cp, so u = c: Always at the speed of light • If m2 < 0 we call the particle a tachyon • No known particles • E < cp, so u > c: Always faster than light

16. Massless and nearly massless particles • First two equations make no sense for massless particles – don’t use them • If mass is very small (mc2 much smaller than E or cp), then often easier to treat as massless • Massless versions of these equations: The Large Electron-Positron collider (LEP) accelerates electrons (me = 0.511MeV/c2) to an energy of E = 100 GeV. How fast are they moving?

17. Sample problem A Z-particle at rest (mZ = 91.2 GeV/c2) decays to an electron (me = 0.511 MeV/c2) with energy 25.0 GeV, a positron (me) with energy 20.0 GeV moving at 71° angle compared to the electron, and an unknown X particle. What is the momentum, energy, mass, and velocity of X? e+ • Electrons are nearly massless 71 Z e- X Conservation of energy: Continued . . .

18. Sample problem continued . . . . . . What is the momentum, energy, mass, and velocity of X? Conservation of momentum: Velocity formula: Mass formula:

19. Lorentz boost of four-momentum • Suppose I measure the momentum and energy of a particle or collection of particles. What does a moving observer measure? • Far easier than the book makes it out • Difference in coordinates (x, y, z, t) satisfy same equations as (x, y, z, t) These formulas work for either a single particle or for the sum of all particles