PH 301

1. PH 301 Dr. Cecilia Vogel Lecture 6

2. Review • Lorenz transformation • velocity transformation • relativistic momentum & energy • mass energy Outline

3. Relativistic Momentum • m (sometimes mo) is rest mass • measured when object is at rest • v is object’s velocity • p is the object’s momentum. • This quantity is conserved in all collisions, reactions, etc

4. F=ma? • What is mass? • Mass tells how hard object is to accelerate. • F = ma is a classical equation • Generally F = dp/dt • is equivalent to F=ma, • if p=mv, with m constant, and a = dv/dt • Classically dp/dt=m dv/dt – requires the same force, for the same rate of change of v • Not at high speed!

5. F ≠ ma • As speed changes, numerator and denominator both change • dp/dt ≠ m dv/dt • As v gets close to c, object gets harder and harder to accelerate

6. What is mass? • Rest mass (m) is mass measured when at rest • property of the object • what you look up in text book • Relativistic mass (mrel) shows how hard it is to accelerate the object • increases with speed • p = mrelv • if use relativistic mass

7. Energy • Momentum increases with speed, so does energy. • E = mrelc2 • Kinetic energy • is zero when v=0 • K= gmc2 - mc2 • When v= 0 • Energy is not zero • rest energy = mc2

8. Momentum and Energy Units • Energy SI units: • J = kgm2/s2 = CV • Momentum SI units: • kgm/s • Mass SI unit • kg • Energy can also be given in eV • Momentum can also be given in eV/c • Mass can also be given in eV/c2

9. Momentum Change To increase the speed of electron • from 0.8c to 0.9 c: • from 0.1c to 0.2 c: Same change in speed, but seven times as much force needed.

10. Energy Change To increase the speed of electron • from 0.8c to 0.9 c: • from 0.1c to 0.2 c: Same change in speed, but forty times as much energy needed!

11. to ∞ to ∞ Graphically • It would require infinite force and infinite energy to accelerate anything with mass to the speed of light!

12. Inertial Reference Frames • recall • special relativity (ch 2) • only true for • inertial ref frame is one in which • Newton’s Law of Inertia holds • not accel • no gravity (or “weak” like Earth’s) • To describe grav or accel • use General Relativity

13. Principle • Recall the first Postulates of Special Rel: • All laws of physics the same in all inertial frames • const vel frame indistinguishable from rest frame • The basic postulate of gen rel • The Principle of Equivalence • uniform gravity frame indistinguishable from constant accel frame • equiv accel and g are equal in size, but opposite dir

14. Equivalence • Artificial Gravity • rotating spaceship, with centrip accel = g • feels like home, earth’s grav • Virtual Reality • tilted chair has grav down and back • feels like grav down, and accelforward • Car accelerating forward • you pushed down and back in chair • cup falls down and back • fuzzy dice hang down and back • He balloon floats up and forward • all as if grav had a backward component

15. Effect on Light • Effect of grav on light identical to effect of accel on light. • Light traveling perpendicular to grav field: • consider accel frame • you move a small dist rel to light at first, then larger and larger distances • in your frame, light moves small distance, then larger and larger • light follows curved path • see handout

16. Effect on Light • Effect of grav on light identical to effect of accel on light. • Light traveling parallel to grav field: • consider accel frame • the frame is moving slowly when light emitted • moving faster when received • as if sender and receiver are moving relative to each other • Doppler shift • see handout

17. Gravity • Light’s path is bent by gravity, • light from a star behind the sun travels a curved path, so appears at a different place • gravitational lensing • Light retardation, • pulse sent to Venus • is slowed as it passes the Sun • all of time is slowed by grav not just the freq of light

18. Black Hole • very large stars collapse • their gravity is stronger than the Pauli repulsion of neutrons • the star continues to collapse • no known force can stop it • collapses to zero volume (?) • infinite density (?)

19. PAL – Classical and Relativistic Momentum • In classical mechanics, momentum is proportional to velocity, so if you double the velocity, you double the momentum. Let’s see how this goes at high speed: • 1 a) By what factor does the momentum change when an object’s speed is doubled from 0.05 c to 0.1 c? • 1 b) By what factor does the momentum change when an object’s speed is doubled from 0.3 c to 0.6 c? • 1 c) The faster you go, the (worse or better) classical physics fits reality.

21. PAL – Classical and Relativistic Kinetic Energy • In classical mechanics, kinetic energy is ½mv2. Let’s see how this goes at high speed: • 2 a) What is the percent error you get by using the classical equation instead of the relativistic equation for an object moving at 0.05 c? • 2 b) What is the percent error you get by using the classical equation instead of the relativistic equation for an object moving at 0.6 c? • 2 c) The faster you go, the (worse or better) classical physics fits reality.

22. Spacebarn Paradox The spaceship is 40 m long in its own frame. The spacebarn is traveling at about 0.866c relative to the ship, so it is only 10.01 m long in the ship’s frame. In the ship’s frame, the ship does NOT fit in the spacebarn. In the ship’s frame, will the ship get hit by the doors?