Relativistic Momentum Question Would you expect Newtons second

Relativistic Momentum ØQuestion: Would you expect Newton’s second law F=ma to hold at high velocity? ØAnswer: No, a constant force can accelerate a particle to v>c ØAnswer: No, certainly something is different since in a Galilean transformation the force and acceleration are the same. This is not true for a Lorentz transformation. 1

Relativistic Momentum Ø Galilean transformation n We showed a=a’ and F=F’ Ø Lorentz transformation Ø We might try starting with F=dp/dt 2

Newtonian Relativity Ø Note Newton’s laws are valid in both frames n n The force and acceleration are the same in both frames There is no way to detect which frame is moving and which is at rest 3

Relativistic Momentum ØFrank (F) is in K with a ball of mass m ØMary (M) is in K’ with a ball of mass m ØFrank throws his ball along y with velocity u 0 ØMary throws her ball along –y’ with velocity u 0 Ø The balls collide elastically 4

Relativistic Momentum 5

Relativistic Momentum Ø Frank sees the momentum change of his ball Ø Frank sees for Mary’s ball Ø Frank sees the momentum change of Mary’s ball 6

Addition of Velocities Ø Recall the addition of velocities in special relativity 7

Relativistic Momentum Ø Momentum is not conserved in the y direction! Ø Because we strongly believe in the conservation of momentum, let’s modify the definition of momentum 8

Relativistic Momentum Ø Frank sees the momentum change of his ball Ø Frank sees the momentum change of Mary’s ball 9

Relativistic Momentum ØThus Frank sees that momentum is conserved in the x and y directions using ØLikewise Mary would see that momentum conservation holds in her frame as well 10

Relativistic Momentum ØNotes n n Unfortunately γ is used for both Usually we write them out when they both come into play In these equations m=m 0 is the rest mass Sometime we interpret γm as the relativistic mass but that is not standard 11

Relativistic Energy Ø Let’s assume that the relativistically correct form of Newton’s law is given by Ø The validity of this assumption can be determined by examining its consequences Ø Aside, are Newton’s first and third laws relativistically correct? 12

Relativistic Energy Ø In classical mechanics the work done by a force in moving a particle from one position to another equals the change in kinetic energy 13

Relativistic Energy Ø We can evaluate the integral by writing 14

Relativistic Energy ØThen ØThus T = γmc 2 – mc 2 15

Relativistic Energy ØNow at low speeds we can use the binomial expansion ØTo find as expected 16

Relativistic Energy ØWe define the total energy E n E = kinetic energy plus rest energy E = T + mc 2 n E = γmc 2 n 17

Relativistic Energy Ø We also have for E Ø Thus E 2=p 2 c 2 + m 2 c 4 n E = mc 2 for p=0 n 18