Quantum Mechanics A Special Topic in AP Physics at Bow High ...
RELATIVITY A Special Topic in Physics at Bow High School Sources: http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html http://www.phys.unsw.edu.au/einsteinlight/ http://www.thebigview.com/spacetime/ http://www.csupomona.edu/~ajm/materials/twinparadox.html http://abyss.uoregon.edu/~js/ast122/lectures/lec20.html Galileos Principle of Relativity Special and General Relativity Special Relativity provides a framework for translating physical events and laws into forms appropriate for any inertial reference frame. The outcome is that
space and time are not absolute but depend upon the inertial reference frame of the observer! General Relativity extends these ideas into forms appropriate for accelerated reference frames and gravity. http://abyss.uoregon.edu/~js/ast122/lectures/lec20.html The outcome is that the presence of matter causes warps and curves in space-time
General Relativity: Ideas and Confirming Experiments Principle of Equivalence Experiments performed in a uniformly accelerating reference frame with acceleration a are indistinguishable from the same experiments performed in a nonaccelerating reference frame which is situated in a gravitational field where the acceleration of gravity = g = -a = intensity of gravity field. One way of stating this fundamental principle of general relativity is to say that gravitational mass is identical to inertial mass. One of the implications of the principle of equivalence is that since photons have momentum and therefore must be attributed an inertial mass, they must also have a gravitational mass. Thus photons should be deflected by gravity. They should also be impeded in their escape from a gravity field, leading to the gravitational red shift and the inability of light to escape the gravitational field around a black hole.
An observer in the *elevator* cannot distinguish between experiments completed when the the elevator is at rest in a gravitational field and when the elevator is accelerating at a = -g Gravity Bends Light Thought Experiment For an elevator accelerating through space at a = -g: If a light source shines through a pinhole in the side of the elevator, it will appear to the observer that the path of the light bends as it travels through the elevator. The pulse of light travels
straight across the interior of the elevator, but from the perspective of the person in the accelerating elevator the path is bent toward the floor. Hence, by the Principle of Equivalence an observer in an elevator at rest will observe identical results! Thus, we conclude that gravity bends light! The Principle of Equivalence indicates that identical results must be perceived by an observer in an elevator at rest on earth! Confirming Experiments: Gravitational Deflection of Light: Einstein's calculations in his newly developed general relativity indicated that the light from a star
which just grazed the sun should be deflected by 1.75 seconds of arc. It was tested during the total eclipse of 1919 and during most of those which have occurred since. This supported the principle of equivalence and the idea that gravity causes warps and curves in the fabric of space. Gravitational Red Shift: According to the principle of equivalence from general relativity, any frequency shift which can be shown to arise from acceleration of a radiating source could also be produced by the appropriate gravitational field. Confirming Experiment: Gravitational Time Dilation A clock in a gravitational field runs more slowly according to the gravitational time dilation relationship from general relativity . t t o 2GM
1 Rc 2 where t is the time interval measured by a clock far away from the mass and to is the time interval measured at a distance R from the center of the mass. Scout Rocket Experiment In 1976 the Smithsonian Astrophysical Observatory sent aloft a Scout rocket to a height of 10,000 km. At this height, a clock should run 4.5 parts in 1010 faster than one on the Earth. During two hours of free fall from its maximum height, the rocket transmitted timing pulses from a maser oscillator which acted as a clock and which was compared with a similar clock on the ground. This result confirmed the gravitational time dilation relationship to within 0.01%. Special Relativity: Ideas and Confirming Experiments Two Postulates of Special Relativity The Relativity Postulate: Postulate The laws of physics are the same for all observers moving at constant velocity, regardless of what their actual velocity is. An observer moving at constant velocity is said to be in
an inertial reference frame. The Speed of Light Postulate: Postulate The speed of light in a vacuum is the same value (c=3.00x108 m/sec) in all directions and in all inertial reference frames. Another View of Postulate #2: Nothing travels faster than light The speed of light c is said to be the speed limit of the universe because nothing can be accelerated to the speed of light with respect to you. A common way of describing this situation is to say that as an object approaches the speed of light, its mass increases and more force must be exerted to produce a given acceleration. accurate for v c : F ma F m dv dt Applying a force causes the object to speeds up, so eventually its speed will surpass
the speed of light! d (mv) valid for all v : F dt Applying a force causes the object to speed up and/or gain mass! It is generally preferable to say that the relativistic momentum and relativistic energy approach infinity at the speed of light. Since the net applied force is equal to the rate of change of momentum and the work done is equal to the change in energy, it would take an infinite time and an infinite amount of work to accelerate an object to the speed of light. Special Relativity Thought Expt#1: The Relativistic Train The General Idea: Idea If two observers in different inertial reference frames (traveling at constant but different velocities) *see* the same experiment that involves the motion of light and they agree that the speed of light is the same, but that the distance traveled by the light is different, then they will not agree on how much time has elapsed during the experiment.
A Specific Example: The Relativistic Train: Train Consider an experiment where an observer ( ) standing on a very wide train car shines a light across the train car to a mirror ( ) on the other side and measures how long it takes the light to return. Lets consider what is going on from the perspective of the observer that shines the laser and from an observer who watches the experiment from a *stationary* location beside the train tracks ( ): Direction of motion of the Train as seen by the *stationary* observer What does the observer on the train car see? The person on the train car sees the light travel across the train car, bounce off the mirror, and return. The length the light travels will be equal to 2D from their perspective, and the time that passes during the experiment is to. D
c speed of light 2 D dis tan ce light travels to time that elapses c 2D to What does the observer standing beside the tracks see? The person standing beside the tracks sees the motion of the train car during the experiment. The length the light travels will be equal to 2L from their perspective, and the time that passes during the experiment is t. c speed of light 2 L dis tan ce light travels t time that elapses D L v
v L v c 2L t 2 vt 1 where : L vt D 2 2 The two observers agree that the speed of light is the same, but their measurements of the distance that the light has traveled differ (2D .vs. 2L). This requires that the time that has passed during the experiment will be different for
the two observers, with the observer in the *fixed* reference frame measuring a longer interval of time! We can show this algebraically by eliminating the lengths from the equation for L and expressing on of the time intervals in terms of the other: 2 2 2D 2L ct vt cto where : c to t 2 2 2
1 L vt D 2 2 2 v2 to c t v t c t t 1 2 to2 t v2 c 1 2 c 2 2 2 2
2 2 o 2 The elapsed time measured by the stationary observer (standing on the side of the track) will always be greater than the elapsed time measured by the moving observer (who is moving at the same speed as the clock). If their watches are synchronized at the beginning of the experiment, they will disagree at the end of the experiment! Special Relativity Thought Expt#2: The Twin Paradox One of a pair of twins leaves on a high speed space journey during which she travels at a large fraction of the speed of light while the other remains on the Earth. Because of time dilation, time is running more slowly in the spacecraft as seen by the earthbound twin and the traveling twin will find that the earthbound
twin will be older upon return from the journey. Experiments on a commercial airline confirm the existence of a time difference between ground observers and a reference frame moving with respect to them. Helpful Videos: Time Rotation https://www.youtube.com/watch?v=ajhFNcUTJI0 Twin Paradox Primer: https://www.youtube.com/watch?v=Bg9MVRQYmBQ Twin Paradox Final Solution: https://www.youtube.com/watch?v=0iJZ_QGMLD0 Comparing Observations made in Different Reference Frames Suppose we want to compare how observers moving at different, but constant, velocities *view* the results of the same experiment. If an observer moving at a constant velocity (v) in the x-direction sees an object at the space-time location (x,y,z,t), than a person who is not moving will see the object at (x,y,z,t): The Galilean
Transformation is accurate (and what we are used to) if speeds are slow relative to the speed of light The Lorentz transformation incorporates relativistic considerations. Note that it simplifies to the Galilean Transformation if v is much less than c. x x vt; y y; z z ; t t vx x vt c2 x ; y y; z z ; t v2 v2 1 2 1 2 c
c t Proper Time and Time Dilation Proper Time: If two successive events occur at the same place in an inertial reference frame than the time interval, measured on a single clock at that location, is the proper time between the two events. Time Dilation: An observer in any other inertial reference frame will measure a time that is longer than the proper time. If this second observer is moving with a relative speed v, than the measured time interval will be given by: t to v2 1 2 c to Proper Length and Length Contraction Proper Length: The length of an object measured by an observer in the inertial
reference frame in which the object is at rest is called its proper length. Length Contraction: An observer in any other inertial reference frame will measure a length that is shorter than the proper length. If this second observer is moving with a relative speed v, than the measured length will be given by: v 2 Lo L Lo 1 2 c Expt Supporting Special Relativity: Muon Decay The muon is a lepton which decays to form an electron or positron. The lifetime of the muon is 2.20 microseconds. The muon is produced in the upper atmosphere by the decay of pions produced by cosmic rays. The measurement of the flux of muons at the Earth's surface produced an early dilemma because
many more are detected than would be expected, based on their short half-life of 1.56 microseconds. This is a good example of the application of relativistic time dilation to explain the increased particle range for high-speed particles. Observations Made in Different Inertial Reference Frames: Earth .vs. Muon Comparing Observations Made in Different Inertial Reference Frames General & Special Relativity Confirming Experiment: Around-the-World Atomic Clocks In October 1971, Hafele and Keating flew cesium beam atomic clocks around the world twice on regularly scheduled commercial airline flights, once to the East and once to the West. In this experiment, both gravitational
time dilation and kinematic time dilation are significant and are in fact of comparable magnitude. Their predicted and measured time dilation effects were as follows: Predicted: Time difference in ns Eastward Westward Gravitational 144 +/- 14 179 +/- 18 Kinematic -184 +/- 18 96 +/- 10 Net effect
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