Understanding Relativistic Kinetic Energy and the Mass-Energy Equivalence

1. TheStern- Gerlach experiment showed that we need a new quantum number: Spin of electron. For an exact discussion we need the Dirac equation(DE). We will not use it yet and instead employ results from perturbation theory obtained form the Hamilton operator of the DE, when we develop it in terms of v/c up to the quadratic term.

2. Refresher for special relativity If I’m in a car, traveling at the speed of light… • If I turn on my headlights, do they work? • Answer: of course—to you, all is normal • you are in an un-accelerated (inertial) frame of reference • all things operate normally in your frame • To the “stationary” outsider, your lights look weird • but then again, so do you (because you’re going so fast) • in fact, at the speed of light, all forward signals you send arrive at the same time you do • And the outside, “stationary” world looks weird to you • But I must inquire: how did you manage to get all the way up to the speed of light?!

3. If I’m in a car, traveling at the speed of light… • If I turn on my headlights, do they workAnswer: of course—to you, all is normal • you are in an un-accelerated (inertial) frame of reference • all things operate normally in your frame • To the “stationary” outsider, your lights look weird • but then again, so do you (because you’re going so fast) • in fact, at the speed of light, all forward signals you send arrive at the same time you do • And the outside, “stationary” world looks weird to you • But I must inquire: how did you manage to get all the way up to the speed of light?! 3

4. What would I experience at light speed? • It is impossible to get a massive thing to travel truly at the speed of light • energy required is mc2, where  as v • so requires infinite energy to get all the way to c • But if you are a massless photon… • to the outside, your clock is stopped • so you arrive at your destination in the same instant you leave your source (by your clock) • across the universe in a perceived instant • makes sense, if to you the outside world’s clock has stopped: you see no “ticks” happen before you hit

5. E = mc2 as a consequence of relativity • Express 4-vector as (ct, x, y, z) • describes an “event”: time and place • time coordinate plus three spatial coordinates • factor of c in time dimension puts time on same footing as space (same units) • We’re always traveling through time • our 4-velocity is (c, 0, 0, 0), when sitting still • moving at speed of light through time dimension • stationary 4-momentum is p = mv(mc, 0, 0, 0) • for a moving particle, p = (mc, px, py, pz) • where px, etc. are the standard momenta in the x, y, and z directions • the time-component times another factor of c is interpreted as energy • conservation of 4-momentum gets energy and momentum conservation in one shot

6. E = mc2, continued •  can be approximated as •  = 1 + ½v2/c2 + …(small stuff at low velocities) • so that the time component of the 4-momentum  c is: • mc2 = mc2 + ½mv2 + … • the second part of which is the familiar kinetic energy • Interpretation is that total energy, E = mc2 • mc2 part is ever-present, and is called “rest mass energy” • kinetic part adds to total energy if in motion • since  sticks to m in 4-momentum, can interpret this to mean mass is effectively increased by motion: m  m • gets harder and harder to accelerate as speed approaches c

7. Experimental Confirmation • in accelerators, particles live longer at high speed • their clocks are running slowly as seen by us • seen daily in particle accelerators worldwide • cosmic rays make muons in the upper atmosphere • these muons only live for about 2 microseconds • if not experiencing time dilation, they would decay before reaching the ground, but they do reach the ground in abundance • We see length contraction of the lunar orbit • squished a bit in the direction of the earth’s travel around the sun • E = mc2 extensively confirmed • nuclear power/bombs • sun’s energy conversion mechanism • bread-and-butter of particle accelerators

8. Relativistic Kinetic Energy

10. Relativistic Kinetic Energy Equation (2.58) does not seem to resemble the classical result for kinetic energy, K = ½mu2. However, if it is correct, we expect it to reduce to the classical result for low speeds. Let’s see if it does. For speeds u << c, we expand in a binomial series as follows: where we have neglected all terms of power (u/c)4 and greater, because u << c. This gives the following equation for the relativistic kinetic energy at low speeds: which is the expected classical result.

11. Total Energy and Rest Energy, Mass-energy Equivalence We rewrite the energy equation in the form The term mc2 is called the rest energy and is denoted by E0. This leaves the sum of the kinetic energy and rest energy to be interpreted as the total energy of the particle. The total energy is denoted by Eand is given by (2.63) (2.64) (2.65)

12. Kinetic Energy-Velocity (Relativistic and Classical )

13. The Equivalence of Mass and Energy • By virtue of the relation for the rest mass of a particle: • we see that there is an equivalence of mass and energy in the sense that “mass and energy are interchangeable” • Thus the terms mass-energy and energy are sometimes used interchangeably.

14. 2.12#70 Challenge problem What is the speed of the electron accelerated to 50 GeV or 8x10^-9 J in the Stanford linear accelerator?

15. Problem 2.12 #75

16. Clicker - Questions • Which of the following is a basic premise of Einstein’s Relativity Theory? • A Your relatives are just like you. • B The speed of light is infinite. • C The speed of light is a constant. • D The speed of your inertial frame is changing. • E The speed of light is 3x108 m/s.

17. Relationship of Energy and Momentum We square this result, multiply by c2, and rearrange the result. We use the equation for  to express β2 and find Expressing  through 

18. Energy and Momentum The first term on the right-hand side is just E2, and the second term is E02. The last equation becomes We rearrange this last equation to find the result we are seeking, a relation between energy and momentum. or Equation (2.70) is a useful result to relate the total energy of a particle with its momentum. The quantities (E2 – p2c2) and m are invariant quantities. Note that when a particle’s velocity is zero and it has no momentum, Equation (2.70) correctly gives E0 as the particle’s total energy. (2.70) (2.71)

19. Useful formulas and from

21. #2.95

22. Massless particles have a speed equal to the speed of light c • Recall that a photon has “zero” rest mass and that equation 2.70, from the last slide, reduces to: E = pc and we may conclude that: • Thus the velocity, u, of a massless particle must be c since, as 0, and it follows that: u = c.