Special relativity: energy, momentum and mass

1. Special relativity:energy, momentum and mass Physics 123 Lecture IX

2. Outline • Lorentz transformations • 4-dimentional energy-momentum • Mass is energy • Doppler shift Lecture IX

3. Lorentz transformations System (x’,y’z’,t’) is moving with respect to system (x,y,z,t) with velocity v • Lorentz • x=g(x’+vt’) • y=y’ • z=z’ • t=g(t’+vx’/c2) • Galileo • x=x’+vt’ • y=y’ • z=z’ • t=t’ Lecture IX

4. 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, Dt0- measured in moving frame Hendrik Antoon Lorentz Derived time and space transformations before Einstein Lecture IX

5. 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? Lecture IX

6. 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 Lecture IX

7. Add time to space metric: x1=x, x2=y, x3=z, x4=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 (t1=t2), but at different places (r12≠0) AB – imaginary – time-like interval, there exists a frame of reference where the two events happen at the same place (r12=0), but at different times (t1≠t2) 4-dimensional space – time A x x B y y Lecture IX

8. m0 – mass at rest Relativistic energy: Energy, mass and momentum • Energy at rest • E=m0c2 • Kinetic energy: • Relativistic momentum: • 4-dimensional Energy –momentum – vector: • (pxc, pyc, pzc, iE) • Lorentz invariant interval: Lecture IX

9. Conservation laws • Both energy and momentum are conserved in the relativistic case: • Mass must be considered as an integral component of energy E=gmc2 Lecture IX

10. 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. Lecture IX

11. 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 Lecture IX

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

13. Doppler shift • Light emitted at f0,l0 • In the source’s r.f. • the distance between crests is l0 • The time between crests is t0=1/f0=l0/c • Where are crests in the r.f. moving with speed v wrt source’s r.f. (chasing the wave) • l=cDt-vDt=(c-v)Dt • Dt=gDt0=gl0/c • l=(c-v) gl0/c Lecture IX

14. 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) Lecture IX