Physics 2 - Lecture 7: Einstein’s Special Relativity - Pham Tan Thi

Nội dung text: Physics 2 - Lecture 7: Einstein’s Special Relativity - Pham Tan Thi

1. Einstein’s Special Relativity Pham Tan Thi, Ph.D. Department of Biomedical Engineering Faculty of Applied Sciences Ho Chi Minh University of Technology
2. Classic Picture for Relative Motion
3. Consider a Situation Reference #1 Speed of light 0.5c
4. Classical and Modern Physics Classical Physics Modern Physics Large, Slow moving Object Small, Fast moving Object • Newtonian Mechanics • Relativistic Mechanics • Electromagnetism and Waves • Quantum Mechanics • Thermodynamics 10% of c
5. History Albert Einstein surprised the world in 1905 when • He theorized that time and distance cannot be measured absolutely • They only have meaning when they are measured relative to something Einstein published his theory in two steps: • Special theory of relativity (1905) ➔ How space and time are interwoven • General theory of relativity (1915) ➔ Effects of gravity on space & time What is “relative” in relativity? • Motion all motions is relative • Measurements of motion (and space & time) make no sense unless we are told what they are being measured relative to What is “absolute” in relativity? • The laws of nature are the same for everyone • The sped of light, c, is the same for everyone
6. Origin of Special Theory of Relativity Albert Einstin (1879 - 1955) • In 1905, Albert Einstein changed our perception of the world forever. • He published the paper on the electrodynamics of a moving body • In this, he presented what is now called the Special Theory of Relativity Albert Einstin, Ann. Phys. 17, 891 (1905).
7. The Special Theory of Relativity • The laws of Physics are known to be unchanged (“invariant”) under rotations. • A rotation mixes the space coordinates but does not change the length of any object. • So there should be a linear transformation.
8. The Special Theory of Relativity We will now examine the physical meaning of this statement, as well as how it came to be proposed by Einstein.
9. Electrodynamics These equations depend on the speed of light, c. • In what frame is this speed to be measured? • It was thought that light propagates via a medium called “ether”, much as sound waves propagate via air or water. • In that case, the speed of light should change when we move with respect to the ether - just as for sound in air. • So c would be the speed of light as measured while one is at rest relative to the ether.
10. Reference Frames A reference frame in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of object in it. Inertial frames Non-Inertial frames • in which no accelerations are observed • that is accelerating with respect to in the absence of external forces an inertial reference frame • that is not accelerating • bodies have acceleration in the • Newton’s laws hold in all inertial absence of applied forces reference frames
11. Inertial Frame K and K’ • K is at rest and K’ is moving with velocity ~v • Axes are parallel • K and K’ are said to be inertial coordinate systems
12. Conditions of the Galilean Transformation • Parallel axes • K’ has a constant relative velocity in the x-direction with respect to K x0 = x ~vt y0 = y z0 = z t0 = t • Time (t) for all observers is a Fundamental invariant, i.e. the same for all inertial observers. • Inverse relation: x = x0 + ~vt y0 = y z0 = z t0 = t
13. The Transition to Modern Relativity • Although Newton’s laws of motion had the same form under the Galilean transformation, Maxwell’s did not. • In 1905, Albert Einstein proposed a fundamental connection between space and time; and that Newton’s law are only an approximation
14. Maxwell’s Equations In Maxwell’s theory, the speed of light, in terms of permeability and permittivity of free space, was given by 1 v = c = pµo"o Thus the velocity of light between moving systems must be a constant.
15. The Michelson-Morley Experiment • Albert Michelson built an extremely precise device called an interferometer to measure the minute phase difference between two light waves traveling in mutually orthogonal directions. • Experiments were performed to compare the speed of light when moving along or against the ether. Albert Abraham Michelson (1852 - 1931)
16. Michelson Interferometer
17. Michelson Interferometer 1. AC is parallel to the motion of the Earth inducing an “ether wind” 2. Light from source S is split by mirror C and D in mutually perpendicular directions 3. After reflection the beams recombine at A with slightly out of phase due to the “ether wind” as viewed by telescope E.
18. The Analysis Assuming the Galilean Transformation Time t1 from A to C and back: l l 2cl 2l 1 t = 1 + 1 = 1 = 1 1 c + v c v c2 v2 c 1 v2/c2 ✓ ◆ Time t2 from A to D and back: 2 2l2 l2 2l22 l2l11 l1 t2 = = = t = t2 t1 =t =2 t2 2t1 = 2 2 pc c 1v v2/ccc2 11 vv2/c2/c22! 1 v2/c2 ! So that the changep in time is: pp p 2 l2 l1 t = t2 t1 = c 1 v2/c2 1 v2/c2 ! p p
19. Result Using the Earth’s orbital speed as: v = 3 x 104 m/s together with l1 = l2 = 1.2 m so that the time difference becomes ∆t’ - ∆t ≈ v2 (l1 + l2)/c3 = 8 x 10-17 s Although a very small number, it was within the experimental range of measurement for light waves.
20. Possible Explanations Many explanations were proposed but the most popular was the ether drag hypothesis • This hypothesis suggested that the Earth somehow “dragged” the ether along as it rotates on its axis and revolves about the sun. • This was contradicted by stellar aberration wherein telescopes had to be titled to observe starlight due to the Earth’s motion. If ether was dragged along, this tilting would not exist.
21. Einstein’s Two Postulates Albert Einstein was only two years old when Michelson reported his first null measurement for the existence of the ether. At the age of 16th, Einstein began thinking about the form of Maxwell’s equations in moving inertial systems. In 1905, at the age of 26th, he published his startling proposal about the principle of relativity, in which he believed to be fundamental.
22. Re-evaluation of Time In Newtonian physics, we previous assumed at t = t’ • Thus with “synchronized” clocks, events in K and K’ can be considered simultaneously BUT Einstein realized that each system must have its own observers with their own clocks and meter sticks • Thus events considered simultaneous in K may not be in K’
23. The Problem of Simultaneity Mary, moving to the right with a speed v, observes events A and B in different order: Mary “sees” event B, then A
24. Lorentz Transformation Equations A more symmetric form: x0 = (x ct) v = y0 = y c 1 = z0 = z 1 v2/c2 t0 = (t x/c) p PROPERTIES OF γ: Recall β = v/c < 1 for all observers 1) γ equals 1 only when v = 0. 2) Graph of γ (note v ≠ c)
25. Derivation • Use the fixed system K and the moving system K’ • At t = 0, the origins and axes of both systems are coincident with system K’ moving to the right along the x axis. • A flashbulb goes off at the origins when t = 0. • According to postulate 2, the speed of light will be c in both systems and the wavefronts observed in both systems must be spherical.
26. Derivation 1. Let x’ = γ(x - vt) so that x = γ’(x’ - vt’) 2. By Einstein’s first postulate: γ = γ’ 3. The wavefront along the x, x’-axes must satisfy: x = ct and x’ = ct’ 4. Thus ct’ = γ(ct - vt) and ct = γ’(ct’ + vt’) 5. Solving the first one above for t’ and substituting into the second gives the following result: 2 v v t0 = t0 1 1+ c c ⇣ ⌘⇣ ⌘ from which we derive: 1 2 = 1 v2/c2
27. The Complete Lorentz Transformation For the moving frame: For the stationary frame: x vt x0 + vt0 x0 = x = 1 2 1 2 y0 = yp y = yp0 z0 = z z = z0 2 2 t (vx/c ) t0 +(vx0/c ) t0 = t = 1 2 1 2 p p
28. Consequence of Lorentz Transformation: Time Dilation Time Dilation: • Clocks in K’ run slow with respect to stationary clocks in K.
29. Time Dilation To understand time dilation, the idea of proper time must be understood: • The term proper time, To, is the time difference between two events occurring at the same position in a system as measured by a clock at that position.
30. Consequence of Lorentz Transformation: Length Contraction Length Contraction: • If something is moving relative to you, its length in the direction that it is moving will seem to shorter that it would if it was not moving. • Lengths in K’ are contracted with respect to the same lengths stationary in K.
31. Consequence of Lorentz Transformation: Length Contraction Length Contraction: • If something is moving relative to you, its length in the direction that it is moving will seem to shorter that it would if it was not moving. • Lengths in K’ are contracted with respect to the same lengths stationary in K.
32. Consequence of Lorentz Transformation: Length Contraction Length Contraction: • If something is moving relative to you, its length in the direction that it is moving will seem to shorter that it would if it was not moving. • Lengths in K’ are contracted with respect to the same lengths stationary in K.
33. Relativistic Energy Due to the new idea of relativistic mass, we must now re-define the concepts of work and energy. We modify Newton’s second law to include our new definition of linear momentum, and force becomes: dp~ d d m~v F~ = = (m~v)= dt dt dt 1 v2/c2 ! For simplicity, let the particle start from rest punder the influence of the force and calculate the kinetic energy K after the work is done. d d W = K = (m~v) ~vdt K = m dt (~v) ~v dt · dt · Z v Z K = m vd(v) 0 Relativistic kinetic energy: Z 1 K = mc2 mc2 = mc2 1 1 v2/c2 ! p
34. Relativistic Energy Even when a particle has no velocity and therefore no kinetic energy, it still has energy by virtue of its mass. The laws of conservation of energy and conservation of mass must be combined into one law: Law of conservation of mass-energy Relativistic energy and momentum: Massless particles: 2 2 2 2 • For a particle having no mass: E = p c + Eo E2 = p2c2 + m2c4 E = pc • For a particle having no mass: v = c
35. Twins Paradox However, this scenario can be resolved within the standard framework of special relativity. The clear implication is that the travelling twin would indeed be younger, but the scenario is complicated by the fact that the travelling twin must be accelerated up to travelling speed, turned around, and decelerated again upon return to Earth. Accelerations are outside the realm of special relativity and require general relativity. The Resolution Frank’s clock is in an inertial system during the entire trip; however, Mary’s clock is not. As long as Mary is traveling at constant speed away from Frank, both of them can argue that the other twin is aging less rapidly When Mary slows down to turn around, she leaves her original inertial system and eventually returns in a completely different inertial system. Mary’s claim is no longer valid, because she does not remain in the same inertial system. There is also no doubt as to who is in the inertial system. Frank feels no acceleration during Mary’s entire trip, but Mary does.