Physics 132: Lecture 23 Elements of Physics II Agenda for Today Special Theory of relativity Inertial vs. non-inertial reference frames Postulates of SR Consequences of SR Time dilation Length contraction Physics 201: Lecture 1, Pg 1
Postulates of Special Relativity 1) The laws of Physics have the same form in all inertial reference frames. (Moving with constant velocity) Student: Why does all this weird stuff happen? 1B) The speed of light (in a vacuum) is always c, independent of the observer. Student: how is velocity of light constant irrespective of the position or relative velocity of the observer? Physics 201: Lecture 1, Pg 2
Example Event A occurs at spacetime coordinates x = 300 m, t = 2 s. Event B occurs at spacetime coordinates x = 1200 m, t =6 s s. Could A possibly be the cause of B? x tt 1200m -300m 8 2.25 10 m/s 6μs 2μs Event C occurs at spacetime coordinates x = 2400 m, t = 8 s. Could A possibly be the cause of fc? x x t 2400m -300m 8 3.5 10 m/s 8μs 2μs Physics 201: Lecture 1, Pg 3
Clicker Question 1: At t = 0 an alien spaceship passes by the earth: let this be event A. At t = 13 min (according to synchronized clocks on Earth and Mars) the spaceship passes by Mars, let this be event B. Who measures the proper length between event A and event B? (a) The alien spaceship Student: If the proper time is measured in the frame in which (b) People on Earth the two events happen in the (c) Nobody same place and the proper (d) People on Mars length is measured in the frame (e) Both (b) and (d) in which h the object is at rest, therefore it means that these quantities are measured always in different frames? Physics 201: Lecture 1, Pg 4
Alpha Centauri An astronaut is in a spaceship travelling towards the star Alpha Centauri at a speed of.95 c. Alpha Centauri is 4.3 light years away. t D v 4.3lightyears.95c v =.95 c 4.5years E AC On earth we measure a time of 4.5 years to pass Physics 201: Lecture 1, Pg 5
Length Contraction Let s imagine you are in a spaceship heading to Alpha Centauri, moving at a speed of 0.95c. An observer on the earth measures a time of 4.5 years to make the trip. v =.95 c E AC x =.95c(4.5 years) = 4.3 light years Physics 201: Lecture 1, Pg 6
Length Contraction Now let s look at the frame of the spaceship. In the spaceship you seem to be at rest and the earth and Alpha Centauri moves relative to you with speed 0.95c. You at rest in the ship measure the time of your trip to be 1.4 years. t v 1- c The ships measures the distance between E and AC to be less than observer at rest!!! V =.95c V =.95c 2 2 E v = 0 AC Physics 201: Lecture 1, Pg 7
Length of Position Vector ŷ r xˆ y ˆ x ˆ r Physics 201: Lecture 1, Pg 8
The Spacetime Interval Consider two events that are separated in time by an interval t, and are separated in space by an interval x. Let us define the spacetime interval between the two events to be: The spacetime interval has the same value in all inertial reference frames. 9 Physics 201: Lecture 1, Pg 9
Momentum Other things will change as well. As you can imagine if length and time change so will other quantities. p m 0 v 1 v c 2 2 p is momentum m 0 is rest mass v is relative velocity of objects c is speed of light 10 Physics 201: Lecture 1, Pg 10
Clicker Question 2: A particle is moving at a speed of 0.40 c has a momentum p 0. If the particle s velocity doubles what will happen to it s momentum? (a) It s less than 2 p 0 (b) It s greater than 2 p 0 (c) It s 2 p 0 Physics 201: Lecture 1, Pg 11
Cosmic Speed Limit No causal influence, particle or wave, can travel faster than the speed of light Tachyon t t 0 1- p m0v 0 2 v 2 1 v 2 c 2 c 12 Physics 201: Lecture 1, Pg 12
Mass is equivalent to energy!!! Energy and mass E E = m 2 0 c Energy when at rest (rest energy) tot m 0 c 1 2 v c 2 2 Student: Is there a practical way to get rid of all our trash to power up the city with E=mc^2 equation? 13 Physics 201: Lecture 1, Pg 13
Nuclear Fission Probably the most well-known application of the conservation of total energy is nuclear fission. i A 235 U nucleus absorbs a neutron and then quickly fragments into two smaller nuclei and several extra neutrons. Mass has been lost and converted to an equivalent amount of kinetic energy in the fission products. This generates heat. 14 Physics 201: Lecture 1, Pg 14
Pre-Class Quiz: A quarter-pound hamburger with all the fixings has a mass m = 0.220 kg. The food energy of the hamburger is E = 1.50 MJ. (a) What is the energy equivalent of the mass of the hamburger? (b) By what factor does the energy equivalent exceed the food energy? (a) 20 10 2.0 10 16 J (b) 1.3 10 10 15 Physics 201: Lecture 1, Pg 15
Clicker Question 3: You are in a spaceship with no windows, radios, or other means to check outside. How would you determine if the spaceship is at rest or moving at constant velocity? a) By determining the apparent velocity of light in the spaceship. b) By checking your precision watch. If it s running slow, then the ship is moving. c) By measuring the lengths of objects. in the spaceship. If they are shorter, then the ship is moving. d) You should give up because you ve taken on a impossible task. Physics 201: Lecture 1, Pg 16
Einstein s theories of relativity Theory of special relativity 1905 Deals with reference frames that are not accelerating (IRF) Predicts that strange things happen when objects move very fast Theory of general relativity 1915 More complete theory Deals with reference frames that are accelerating Explains gravity (Newton gave magnitude of gravity) Predicts that strange things happen when masses get very large One of the first times theory proceeded observations Physics 201: Lecture 1, Pg 17
Principle of Equivalence Suppose you are in an elevator with no windows Near surface of earth, not moving a = 9.81 m/s 2 Accelerating upwards a=981m/s 9.81 2
Principle of Equivalence You can do no experiment to figure which is happening The two cases are equivalent Starting from here Einstein formulated general relativity
Principle of Equivalence Pretend we are in the accelerating elevator and light comes by In the elevator you should see
Principle of Equivalence If the two cases are equivalent, gravity should bend light Pretty bold prediction, especially since he just thought it up!
Distant Star Apparent position of Star Light is bent more the closer to the sun you get, must be during eclipse Deflection is greatly exaggerated
General Relativity Einstein s s theory of gravity Gravity is actually the result of curved spacetime Spacetime: Space and time are no longer separate but can be considered part of the same thing g( (4-dimensional) Time and space are no longer absolute Mass curves spacetime Objects simply move in a straight line, however since spacetime is curved the objects might move in a curve Demo Physics 201: Lecture 1, Pg 23
Clicker Question 4: Which of the following statements related to Einstein's general theory of relativity is FALSE? A. It applies to accelerated frames of reference. B. A ray of light can bend in a gravitational field. C. Its predictions agree with Newton's theory of gravity in regions of very strong gravitational fields. Physics 201: Lecture 1, Pg 24
The Relativistic Universe It may seem as though relativity has nothing to do with our daily lives. However, medicine makes use of radioactive materials for imaging and treatment; satellites must take relativistic effects into account in order to function properly; and space exploration would be a disaster if relativistic effects were not handled properly. Physics 201: Lecture 1, Pg 25
General Relativity If the gravitational field is strong enough, light may be bent so much that it cannot escape. An object that is this dense is called a black hole. Calculations show that the radius of a black hole of a given mass will be: Plugging in the numbers shows us that the Earth would have to have a radius of about 0.9 cm in order to be a black hole. Physics 201: Lecture 1, Pg 26