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Principle of Relativity The Principle of Relativity Relativity and Astrophysics Lecture 04 Terry Herter Outline Continuation of worked problems from textbook The Postulates of Relativity Simultaneity of events Lorentz Contraction Invariance of Spacetime Interval Reading A2290-04 A2290-04 Spacetime Physics: Chapter 3 Principle of Relativity 2 1 Principle of Relativity Test Particles Problem 2-10a 10 kg A test particle is a tool to probe spacetime with out affecting it Its mass should be small enough so that its presence does not affect the motion of other nearby particles Example: 10 kg object may not be a good test particle (depending upon desired accuracy) Acceleration due to gravity only depends on mass! Particles of the same mass but different material, shapes, and sizes in the same location all fall towards Earth with the same acceleration! (Galileo) Allow us to define inertial reference frame – otherwise aluminum would behave differently than wood. Important property of nature. A2290-04 10 cm Force of gravity is: F ma GMm r2 The acceleration is then: 6.673 10 11 m 3 /kg/s 2 10 kg 0.12 m 2 8 2 6.673 10 m/sec a Assuming movement is small so that the force doesn’t change appreciably, the distance moved in 3 min (180 sec) is 1 2 at 2 1 m 180 sec2 6.673 10 8 2 sec 2 0.00108 m s 1.08 mm Principle of Relativity 3 Problem: SP 2-13 Deflection of light by the Sun A falling elevator is an inertial frame therefore a beam of light shot across it will travel in a straight line. However, with respect to the Earth the flash of light is falling because the elevator is falling. Therefore light is deflected by a gravitational field. This effect is very small on the Earth but can be measure for the Sun because gravity is much stronger than on Earth and the travel time is larger. Light from a star Earth Sun A2290-04 A2290-04 Angular deflection, , greatly exaggerated Principle of Relativity 4 2 Principle of Relativity Problem: SP 2-13 (cont’d) A) The “effective time of fall” for a photon is te Dsun 1.4 109 m 4.67 sec c 3 108 m / sec Time for photon to cross Sun’s diameter The velocity of the fall is the maximum acceleration acting for this effective time. vtran asunte 275 m / sec 2 4.67 sec Transverse velocity of the photon 1284 m / sec B) Deflection angle is the ratio of the transverse velocity to the forward speed. vtrans c 4.3 10 6 rad Deflection of photon by the Sun – Off from GR result by factor of 2 0.88 arcseconds A2290-04 Principle of Relativity 5 Problem: SP2-13 Calculus Proof b The photon is accelerated by gravity db v dvT aT dt aT where aT r GM cos r2 ro dvT Sun Deflection is very small so we can assume that light is very close to original path, so vT aT thus vT db GM cos db v v r 2 GM v with r ro sec b ro tan & db ro sec 2 d /2 cos GM 2 ro sec 2 d 2 2 r sec v ro /2 o now But on the previous slide we had v 2GM 1 T v ro c 2 vT asunte GM 2ro 1 2GM 1 2 v c ro c c ro c 2 Checks! A2290-04 A2290-04 Principle of Relativity 6 3 Principle of Relativity Relativity There is no absolute motion. Everything is relative. Suppose two people are alone in space and traveling towards one another: Which one is moving? They can’t tell! Example – A train is moving at 65 mph relative to the tracks. If the people inside the train cannot see out and the track is very smooth, they can not tell they are moving! Example 2 – The earth moves around the sun at 30 km/sec. Can you tell? A2290-04 Principle of Relativity 7 Aside on Albert Einstein Perhaps the greatest physicist of all time. 1905: Published papers on Special Relativity Photoelectric effect (photons carry energy proportional to their frequency) Brownian motion (random movement of particles in a fluid) General Relativity – theory of gravity Critical opalescence Atomic transition probabilities Statistical physics A2290-04 A2290-04 See Wikipedia entry 1879 - 1955 Including relativistic cosmology Explains why the sky is blue Includes “stimulated” emission which led to lasers Bose-Einstein statistics - bosons Principle of Relativity 8 4 Principle of Relativity Special Theory of Relativity Postulates: Albert Einstein (1905) 1879 - 1955 1. The speed of light is the same to all observers, irrespective of their motion. 2. The laws of physics are the same everywhere no matter what the speed of the observer. A2290-04 Principle of Relativity 9 The speed of light is constant. Person in boat moving towards shore shines a beam of light at a person on the shore. Flashlight c vboat A2290-04 A2290-04 The velocity of light as seen by the person in the boat and on the shore is the same, c. Principle of Relativity 10 5 Principle of Relativity Same/Different in Different Frames Not necessarily the same – Same – Space separations, time separations, velocities, accelerations, forces, and field (such as electric and magnetic fields) Physical laws and physical constants in these laws Examples: Speed of light Speed of a particle Charge on an electron Kinetic energy of proton Time between two events Newton’s first law same not necessarily the same same not necessarily the same not necessarily the same same Law of Inertia: A remains at rest or at constant velocity unless acted on by an external, unbalanced force. A2290-04 Principle of Relativity 11 Simultaneity The simultaneity of events is in the eye of the beholder. v A2290-04 A2290-04 A light bulb in the center of a high speed train flashes. An observer on the train sees the light reach the front and back simultaneously Principle of Relativity 12 6 Principle of Relativity From outside the train Simultaneity is in the eye of the beholder. Simultaneity is relative! A2290-04 A2290-04 To the outside observer the light reaches the back first. The back moves towards the light and the front away. Not simultaneous! Flash goes off v Flash reaches back wall first Principle of Relativity v 13 7