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Lecture 2. Relativistic Kinematics, part II

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1 Lecture 2. Relativistic Kinematics, part II
Outline: Length Contraction Relativistic Velocity Addition Relativistic Doppler Effect “Red shift” in the Universe

2 Relativistic effects: length contraction
Question : how long does the signal take to complete the round trip? K0 mirror K An observer in the car’s rest RF : - the proper time interval An observer on the ground : These intervals are related by the time dilation formula: “Moving objects are shortened in the direction of motion”

3 Length Contraction (cont’d)
Of course, the same result follows directly from L.Tr.: Proper length L0 : the length of an object measured in its rest RF ( ). K0 An observer in the RF K moving with respect to the RF K0 with the velocity V directed parallel to the meter stick, measures its length. In order to do that, he/she finds two points x1 and x2 in his/her RF that would simultaneously coincide with the ends of the moving stick (t1 =t2). Comment It’s easier to write L.Tr. for the “proper” length interval in the right-hand side: K observer - the end positions are measured simultaneously in K - moving objects are contracted in the direction of their motion Compare:

4 Length contraction (cont’d)
- moving objects are contracted in the direction of their motion 10 To observe this effect, the relative speed of the reference frames should be large. For the fastest spacecraft, the speed is ~10-4c, and the effect is of an order of 10-8. K 20 Contraction occurs only in the direction of relative motion of RFs! K’ disc at rest the same disc as seen by observer K’

5 Recapitulation: decay of cosmic-ray muons
Muon – an electrically charged unstable elementary particle with a rest energy ~ 207 times greater than the rest energy of an electron. The muon has an average half-life of 2.2 10-6 s. Muons are created at high altitudes due to collisions of fast cosmic-ray particles (mostly protons) with atoms in the Earth atmosphere. (Most cosmic rays are generated in our galaxy, primarily in supernova explosions) N0– the number of muons generated at high altitude In the muon’s rest frame By ignoring relativistic effects (wrong!), we get the decay length: ~20 km altitude In fact, the decay length is much greater, the muons can be detected even at the sea level! Because of the time dilation, in the RF of the lab observer the muon’s lifetime is: N – the number of muons measured in the sea-level lab

6 Decay of cosmic-ray muons in the muon’s RF
Let’s reconsider the same situation, but now our observer moves with the muon (the muon’s rest IRF) N0– the number of muons generated at high altitude We can re-interpret this situation in terms of the length contraction: The life-time in the rest frame: In the muon’s rest frame, the distance to the Earth (~20 km in the Earth’s RF) is significantly shortened: ~20 km altitude The travel time N – the number of muons measured in the sea-level lab becomes comparable with the muon life-time. Thus, again, there is a considerable number of muons (the same as we’ve calculated in the lab RF) that can be detected at the sea level.

7 Problems 1. The nearest star to the Earth is Proxima Centauri, 4.3 light-years away. - at what constant speed must a spacecraft travel from the Earth if it is to reach the star in 2.5 years, as measured by travelers on the spacecraft? - how long does this trip take according to earth observers? Consider two IRFs, K (the Earth) and K’ (the rest RF of the spacecraft). By astronaut's reckoning (K’), the distance to the star is contracted: K K’ and the time of travel is 4.3 years According to earth observers: 2. Consider a disc at rest. We know that the “circumference/diameter” ratio is . Now the disc rotates around its center. If one applies the Lotentz length contraction to the disc, the result would be puzzling: the circumference “shrinks” while the diameter (which is normal to the velocity) remains intact, so “circumference/diameter”   ! What’s going on ???

8 Problem Imagine an alien spaceship traveling so fast that it crosses our galaxy (whose rest diameter is 100,000 light-years) in only 100 years of spaceship time. Observers at rest in the galaxy would say that this is possible because the ship’s speed  is so close to 1 that the proper time it measures between its entry into and departure from the galaxy is much shorter than the galaxy-frame coordinate time (~100,000 ly) between those events. Find the exact value of the speed  that the aliens must have to cross the galaxy in 100 years. and so on… How does it look to the aliens? To them, their clocks are running normally, but the galaxy, which moves backward relative to them at speed  1, is Lorentz contracted. What is the galaxy’s size by aliens’ reckoning?

9 Relativistic Velocity Addition
K observer IRF K: a particle moves a distance dx in a time dt K’ IRF K’: a particle moves a distance dx’ in a time dt’ “+” – anti-parallel “-” - parallel - Galilean velocity addition Speed of light is the largest speed in nature, no body nor any signal can travel with the speed greater than c.

10 Problems K K’ K Yes: 0.83c > 0.75c K’ No: 0.71c < 0.75c
1. A person on a rocket traveling at 0.6c (with respect to the Earth RF) observes a meteor passing him at a speed he measures as 0.6c. How fast is the meteor moving with respect to the Earth? IRF K (rocket): IRF K’ (Earth) moves with respect to K with K Galilean velocity addition: K’ Relativistic velocity addition: 2. As the outlaws escape in their getaway car, which goes 3/4c, the police officer fires a bullet from the pursuit car, which only goes 1/2c. The muzzle velocity of the bullet (relative to the gun) is 1/3c. Does the bullet reach its target (a) according to Galileo, (b) according to Einstein? K Yes: 0.83c > 0.75c IRF K (gun) K’ IRF K’ (Earth) Solve the same problem using IRF K’’ (getaway car). No: 0.71c < 0.75c

11 Doppler Effect for Sound
air (the medium where the waves propagate) observer source of sound v – the speed of an observer with respect to air V – the speed of the source of sound with respect to air f0 – the frequency of sound in the rest frame of the source f – the frequency of sound heard by an observer “+” observer moves toward the source “-” observer moves away from the source “-” source moves toward the observer “+” source moves away from the observer

12 Transverse Doppler Effect for Light
Doppler effect for light - a change in the observed light frequency due to a relative motion of the light source and an observer (no special RF associated with the medium where light propagates!): light wave fronts K 1. Transverse Doppler effect observer - the period of oscillations of the e.-m. field in the rest RF of the source K (the “proper” time interval) K’ - the period of oscillations in the RF of the moving observer f is always smaller than f0 – “red shift” (shift to lower frequencies) The origin of the transverse Doppler effect is time dilation, this is a pure relativistic effect, no counterpart in classical mechanics.

13 Longitudinal Doppler Effect for Light
10 The light source and the observer move away from each other. K light observer V is the velocity of the relative motion of an observer with respect to the light source. K’ an extra time needed for the next wave front to reach an observer - the same time dilation as in the case of the transverse Doppler Effect - “red shift” 20 The light source and the observer approach each other. - “blue shift” (shift toward higher frequencies) The most frequent encounter with Doppler effect in light (microwave): police radar speed detectors (relativistic effects are negligible)

14 Problem A spaceship approaches an asteroid and sends out a radio signal with proper frequency 6.5x109 Hz. The signal bounces off the asteroid’s surface and returns shifted by 5x104 Hz. What is the relative speed of the spaceship and the asteroid? In this situation, there Doppler shift occurs twice. Firstly, the original frequency is received by an asteroid as Secondly, the spaceship receives the reflected signal with the frequency (the asteroid is the “secondary” source of light)

15 the horizon of visibility = infinite red shift
Hubble’s Law (1929) The Universe expands: the larger the distance to an object, the larger the (relative) speed. By measuring the red shift of (identifiable) spectral lines, one can calculate the recessional speed of the light source with respect to the Earth’s observer. According to Hubble's Law, there is a direct proportionality (at least at not too large distances) between the velocity and the distance to the source: V - the observed velocity of the galaxy away from us H0 - Hubble's "constant" (units: s-1) d - the distance to the galaxy (1 Megaparsec=3106 light-yrs) Most recent measurements of H0 ~ 71 ± 2 (km/s)/Mpc. Hubble’s constant gives us the age of the Universe 0: R c0 the horizon of visibility = infinite red shift now t

16 Extreme “red shifts”: quasars and CMBR
Quasars, very bright objects (like ,000 our Galaxies) of a very small size (10-4 of our Galaxy size), believed to be supermassive black holes in the nuclei of distant galaxies. Distance: (2-10)109 light-years [~ (0.8-3)103 Mpc]. Doppler shift: f/f ~ (!) Cosmic Microwave Background Radiation (CMBR) In the standard Big Bang model, the radiation is decoupled from the matter in the Universe about 300,000 years after the Big Bang, when the temperature dropped to the point where neutral atoms form (T~3000K). At this moment, the Universe became transparent for the “primordial” photons. This radiation is coming from all directions and its spectrum is quite distinct from the radiation from stars and galaxies). Currently, the energy of the CMB photons is “red shifted” to ~ 3K (f = f0/1000 !). The sub-mm/THz range contains ~ half of the total luminosity of the Universe and 98% of all the photons emitted since the Big Bang. R. Wilson A. Penzias Nobel 1978 Mather, Smoot, Nobel 2006

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