Special Relativity
Galileo Galilei (1564-1642) Title page of Galileo's Dialogue concerning the two chief world systems, published in Florence in February 1632. 2
Galilean Transformation z z!!! r ' = r vt t ' = t x ' = x vt y ' = y z ' = z t ' = t 2 2 d r! ' d r! = 2 2 dt dt 3
Galilean Transformation Time and Space!!! r ' = r vt t ' = t m 2 d r dt! 2 ' =! F m 2 d r dt! 2 =! F 力学规律在各惯性系中相同, 即所有惯性系等价, 不可通过力学现象区分 绝对运动不可测 质量与参照系无关, 物体惯性不受运动影响 4
Inertial Frame and Non-Inertial Frame All Inertial Frame are equivalent.! ma! = F In non-inertial Frame, one must add noninertial force:! ma! = F + @#$% ^ &*! 5
Newton: 存在绝对空间, 惯性系是相对于绝对空间静止或匀速运动的参照系 Ernst Mach(1836-1916): 惯性归因于宇宙平均质量的相互作用 6
How fast does the ball appear to roll? 5 km/s, east. v = 5 km/s 15 km/s, east.* V = 10 km/s * That s just v+v = 15 km/s 7
Consequence
Speed of light The speed of light is a very large number It was first measure by Galileo, before 1638. The first quantitative estimate of the speed of light was made in 1676 by Romer. From the observation that the periods of Jupiter s innermost moon lo appeared to be shorter when the Earth was approaching Jupiter than when receding from it, he concluded that light travels at a finite speed, and estimated that it takes light 22 minutes to cross the diameter of Earth s orbit to obtain an estimate of speed of light of 2200000km/s, 26% lower than the actual value.
James Clerk Maxwell 1831-1879 Maxwell 方程 1864!!! B!! E E =, B = µ 0J + µ 0ε 0 t t! ρ! E =, B = 0 ε 0 10
Maxwell Equation 光速为常数, 与光源的速度无关 c = 1/ µ ε 0 0 Maxwell 方程在伽利略变换下不是不变的 相互运动的两个参照系有不同的电磁学规律 可以利用电学实验验证一个参照系是否运动 实验结果 : 否定 11
electromagnetism In Electromagnetism, Maxwell unified all theories in 1860 s, wrote done a set of fundamental equations: Maxwell equation. The Maxwell Eq. has a remarkable prediction: electromagnetic wave and its speed just equal to the speed of light So it was concluded that the light is an electromagnetic wave!
Speed of light wave According to the wave theory, a wave needs a medium to propagate. Furthermore, it is in the frame that the media is at rest, the wave speed can be calculated from the fundamental theory. What is the medium that supports the propagation of electromagnetic wave? Assume there is an ether In the ether, the light travels with THE speed of light.
Michelson and Morley 若光是在以太中传播的, 则以太速度会影响光的传播速度 C 为光在静止以太中的传播速度 14
Michelson-Morley experiment 1887 ΔC<5km/s The speed of light is the same for all observers Monochromatic light source mirror Half silvered mirror mirror Here there be fringes!!! The Nobel Prize in Physics 1907 15
Along the flow v ' u = c + v flow v v ' u = c v t ' a l l = + c + v c v 2lc = 2 2 c v l 2 1+ c v c 2 2 16
Cross the flow v ' = c v 2 2 t ' = 2 l / c v c 2 2 l v 2 1+ c 2 c 2 2 17
l v Δ t = = c c v c 2 2 2 2 t if v c v = 10, = 10 c 2 4 8 then 2 cδt Δ ϕ = 2π = 2π λ l v λ c 2 2 λ π ϕ 7 7 = 2 10 m, l = 1m Δ = 10 v c 2 2 18
The speed of light is the same for different observers Albert A. Michelson Edward W. Morley The Nobel Prize in Physics 1907 19
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Einstein s hypothesis Michelson-Morley experiment did not find any noticeable motion of the Earth relative to ether! This negative result prompted Einstein to make the following assumption: The speed of light is the largest speed of all motion, and is the same in all inertial frame. Furthermore, he made another an additional assumption replacing Galileo s relativity All laws of physics is the same in all inertial frames. This the foundation of special relativity.
Speed of Light 299792 km/ s, east. Laser 299792 km/s, east.** V = 10 km/s ** Not 299802 km/s! 22
A huge problem to Galilian transformation!
A huge problem to Galilian transformation! In one frame, we have x 2 + y 2 + z 2 = c 2 t 2 In another frame: x 02 + y 02 + z 02 = c 2 t 02 x ' = x vt y ' = y z ' = z t ' = t x 2 2xV t + V 2 t 2 + y 2 + z 2 = c 2 t 2
Lorentz transformation
Event interval 26
Again on space-time rotation 27
Imaginary rotation angle 28
Imaginary rotation angle 29
Light-Cine 30
How will physics be changed?
Special Relativity (1905) 32
Special Relativity (1905) m = m 0 1 v / c 2 2 33
Relativity by examples 34
Length contraction 35
Time dilation 36
Muon s frame 长度收缩 Earth s frame 时间变慢 37
Twin paradox Peter and Paul are twin brothers. When Peter drives a space-ship with high velocity away, its clock slows down and Peter lives longer. When Peter comes back, he will see that his twin brother Paul is much older than him (Space travel) One might argue that Paul also see Peter goes away with high speed, so the effect shall be the other way around. However, Peter will see acceleration and de-acceleration etc, and as such Peter is indeed younger (must use general relativity to explain) 38
Speed transformation Suppose in K frame, the velocity is! " #, the velocity see in the K frame will be! " = (! " # +')/(1 + '! " # +, ) According to this, if an object moves at -, +, in a spaceship with velocity -, +, the object moves at. + relative to the ground. / The light speed in the spaceship is the same as on the ground. Transformation laws for velocity in other directions! 0 =! 0 # 1 2, 1 + 4! " # +,! 6 =! 6 # 1 2, 1 + 4! " # +, 39
Relativistic Momentum & Energy 40
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Relativistic momentum What is the momentum in relativity? It can be shown that it is! = $% & ' This can be done by studying momentum conservation in different reference frames. Thus the effective mass of the particle is % = % & $ 45
Mass and momentum and velocity 46
Newton s equation modified 47
Mass and energy 48
E = mc 2 49
The world's first nuclear-powered task force Task Force 1 USS Enterprise, USS Long Beach & USS Bainbridge in formation in the Mediterranean, 18 June 1964 Enterprise has Einstein's mass energy equivalence formula E=mc² spelled out on its flight deck. Note the distinctive phased array radars in the superstructures of Enterprise and Long Beach. 50
Energy expression again 51
Mass and energy conversion 52
Nuclear Energy: from mass 53
Creating mass 54
Energy-momentum transformation 55
Energy-momentum four-vector 56