Ch. 26: The Special Theory of Relativity
Simultaneity: Thought Experiment
-If two events are observed to occur at the same time, but one actually took place farther from the observer than the other, then the more distant one must have occurred earlier, and the events were not simultaneous THOUGHT EXPERIMENT: Lightning strikes at two separate places. One observer believes the events are simultaneous--the light has taken the same time to reach her--but another, moving with respect to to the first does not. Here, it is clear that if one observer sees the events as simultaneous, the other cannot, given that the speed of the light is the same for each. *Draw out diagram
Postulates of the Special Theory of Relativity
1. The laws of physics have the same form in all inertial reference frames -there is no experiment you can do in an inertial reference frame to determine if you are at rest or moving uniformly at a constant velocity 2. Light propagates through empty space with speed c independent of the speed of source or observer *The speed of light is in fact the same in all inertial reference frames
The ultimate speed
A basic result of special relativity is that nothing can equal or exceed the speed of light. This would require infinite momentum--not possible for anything with nonzero mass.
Relativist Momentum
Analysis of collisions shows that momentum is conserved if relativistic momentum is written as: (EQUATION) *This is sometimes interpreted as an increase in mass
Length Contraction
Because the spacecraft observers measure the same speed but less time between these two events, they also measure the distance as less. Know the two equations The length of an object moving relative to an observer is measured to be shorter along its direction of motion than when it is at rest. l0= proper length: the length of the object as determined by observers at rest with respect to the object l: the length that will be measured by observers when the object travels past them with speed v. *length contraction only occurs along the direction of motion
What does the γ stand for?
Know equations
Simultaneity
One of the implications of relativity theory is that time is NOT absolute. The time interval between two events, and even whether or not two events are simultaneous, depends on the observer's reference frame *event: something that happens at a particular place and at a particular time
Four-Dimensional Space-Time
Space and time are even more intricately connected. Space has three dimensions, and time is a fourth. When viewed from different reference frames, the space and time coordinates can mix. In a sense the two effects, time dilation and length contraction, balance each other. When viewed from the Earth, what an object seems to lose in size it gains in length of time it lasts. Space, or length, is exchanged for time.
Space Travel--Time Dilation
To reach a start 100 light-years away would not be possible for ordinary mortals (1 yr=9.5x10^15m) In a spaceship traveling at v=0.999c, the time for such a trip would be only about 4.5 yr (plug into equation). Thus time dilation allows such a trip, but the enormous practical problems of achieving such speeds may not be possible to overcome. *All processes, including aging and other life processes, run more slowly for the astronaut as measured by an Earth observer. But to the astronaut, time would pass in a normal way.
Time Dilation Thought Experiment
Using a clock consisting of a light beam and mirrors, shows that moving observers must disagree about the passage of time. *Draw diagram
E=mc^2; Mass; Energy
Einstein found that mass is a form of energy. He showed that the kinetic energy of a particle of mass m traveling at speed v is given by... The second term, mc^2, is constant, it is called the REST ENERGY of the particle -Represents a form of energy that particle has even when at rest.
What is time dilation?
Clocks moving relative to an observer are measured to run more slowly, as compared to clocks at rest. Time is actually measured to pass more slowly in any moving reference frame as compared to your own.
Relativity principle
The basic laws of physics are the same in all inertial reference frames. *Laws are the same, but paths may be different in different reference frames Assumptions: 1. The lengths of objects are the same in one reference frame as in another 2. Time passes as the same rate in different reference frames
Twin paradox: If any inertial frame is just as good as any other, why doesn't the astronaut age faster than the Earth traveling away from him?
The solution to the paradox is that the astronaut's reference frame has not been continuously inertial--he turns around at some point and comes back. It is impossible to do this without accelerating.
What is Δt?
The time interval in a frame moving with respect to the first one.
What is t0?
Where an observer at rest sees the two events occur at the same point in space -proper time The time interval in the frame where two events occur in the same place
Inertial reference frames
Which are reference frames in which Newton's first law is valid: if an object experiences no net force, the object either remains at rest or continuous in motion with constant speed in a straight line. *Earth is rotating and therefore not an inertial reference frame, but can treat it as one for many purposes. A reference frame that moves with constant velocity with respect to an inertial frame is itself also an inertial frame. -When we say that we observe or make measurements from a certain reference frame, it means that we are at rest in that reference frame.
