Special relativity energy momentum and mass Physics 123

Special relativity: energy, momentum and mass Physics 123 10/24/2020 Lecture IX 1

Outline • Lorentz transformations • 4 -dimentional energy-momentum • Mass is energy • Doppler shift 10/24/2020 Lecture IX 2

Lorentz transformations System (x’, y’z’, t’) is moving with respect to system (x, y, z, t) with velocity v • • • Galileo x=x’+vt’ y=y’ z=z’ t=t’ • • • 10/24/2020 Lecture IX Lorentz x=g(x’+vt’) y=y’ z=z’ t=g(t’+vx’/c 2) 3

Time dilation • Clocks moving relative to an observer are measured by the observer to run more slowly ( as compared to clocks at rest) • Dt – measured in v=0 frame, Dt 0 - measured in moving frame Hendrik Antoon Lorentz Derived time and space transformations before Einstein 10/24/2020 Lecture IX 4

Twin paradox • Two twins: Joe and Jane. Joe stays on Earth and Jane goes to Pluto at v<~c • Joe observes that Jane's on-board clocks (including her biological one), which run at Jane's proper time, run slowly on both outbound and return leg. He therefore concludes that she will be younger than he will be when she returns. • On the outward leg Jane observes Joe's clock to run slowly, and she observes that it ticks slowly on the return run. So will Jane conclude that Joe will have aged less? And if she does, who is correct? 10/24/2020 Lecture IX 5

Length contraction • No change in directions perpendicular to velocity • The length of an object is measured to be shorter when it is moving relative to the observer than when it is at rest 10/24/2020 Lecture IX 6

4 -dimensional space – time • Add time to space metric: x 1=x, x 2=y, x 3=z, x 4=ict • 4 - dimensional “length”=interval - Lorentz invariant • AB – real – space-like interval, there exists a frame of reference where the two events happen at the same time (t 1=t 2 ), but at different places (r 12≠ 0) • AB – imaginary – time-like interval, there exists a frame of reference where the two events happen at the same place (r 12=0), but at different times (t 1≠t 2) x A x y 10/24/2020 B y Lecture IX 7

Energy, mass and momentum m 0 – mass at rest • Relativistic energy: • Energy at rest E=m 0 c 2 • Kinetic energy: • Relativistic momentum: • 4 -dimensional Energy – momentum – vector: • (pxc, pyc, pzc, i. E) • Lorentz invariant interval: 10/24/2020 Lecture IX 8

Conservation laws • Both energy and momentum are conserved in the relativistic case: • Mass must be considered as an integral component of energy E=gmc 2 10/24/2020 Lecture IX 9

Conservation laws • Energy could be used to create mass • To conserve momentum electron and positron must collide head on. Then Z-boson is produced at rest. 10/24/2020 Lecture IX 10

Conservation laws • Mass could be destroyed and converted into energy • To conserve momentum (zero initially) the photons must be flying in the opposite direction with the same absolute values of momenta 10/24/2020 Lecture IX 11

Mass and energy • Mass and energy are interchangeable • Energy can be used to create mass (matter) • Mass can be destroyed and energy released 10/24/2020 Lecture IX 12

Doppler shift • Light emitted at f 0, l 0 • In the source’s r. f. – the distance between crests is l 0 – The time between crests is t 0=1/f 0= l 0/c • Where are crests in the r. f. moving with speed v wrt source’s r. f. (chasing the wave) – l=c. Dt-v. Dt=(c-v)Dt – Dt=g. Dt 0=g l 0/c – l=(c-v) g l 0/c 10/24/2020 Lecture IX 13

Doppler shift • When the source and the observer move towards each other the wavelength decrease (red violet) • When the source and the observer move away from each other the wavelength increase (violet red) – Redshift – used to measure galaxies velocities universe expansion (Hubble) 10/24/2020 Lecture IX 14