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| Ch.26.3-.5 | Ch.26.7,-.10 | 27.1-.2 |
| Time dilation. | Length contraction, momentum, energy and kinetic energy. | Introduction to quantum effects. |
Lecture by Dan Dessau on the final points of special relativity.
In an effort to reconcile observed fact that the speed of light in vacuum is the same for all inertial observers, we have found that two inertial observers will measure different amounts of time between the same two events! Time is no longer considered to be an absolute quantity that all observers measure to be the same. Rather, the amount of time that different observers will measure between two events depends upon their uniform motion.
The MAJOR advantage to realizing this fact (and it IS a fact. Many experiments have confirmed it over the last 80 years or so) is that we are on the road to fixing Newton's laws so that they work equally well at speeds either slow, or near the speed of light.
You may remember that the SI system of units is often referred to as the MKS system, or Meters, Kilogram, and Second system. We have just shown through the time dilation example that Seconds is an idea that depends upon relative motion. It turns out that Meters AND Kilograms also depend upon the observer.
Length Contraction
Just as for the case of time dilation, to see length contraction, you need to consider two inertial observers and think carefully about the particular measurements that they do. Imagine that one of the observers has a long stick at rest in their inertial reference frame. A second inertial observer is in relative motion with respect to the first. Here is the picture:
Because Obs.1 is at rest relative to the stick, he has all the time he likes to measure the length via any process he desires. He could pull out a ruler and measure the stick. He could fire light or other objects with known speed and measure how long they take to travel from one end to the other, and so on. As a definition, we call the length as measured by an inertial observer at rest with respect to the rod the proper length.
As an example of an experiment that Obs.1 can do to measure the proper length of the rod, notice that Obs.2 wears a stylish little hat. Obs.1 can set a clock at one end of the stick and a second clock at the other end. The clocks are designed to stop when the stylish hat passes them. Therefore, they record the time when Obs.2 passes each end of the rod:
After the experiment, Obs.1 collects the two clocks, subtracts the 1st time from the second to see how long it took, and determines the length of the rod from the known speed of Obs.2 and the measured time:
Notice that I have listed the time as NOT PROPER, because Obs.1 has used two different clocks, located at two different positions to measure the passing of Obs.2. Proper time is the time measured between two events that happen at the same place so that the time can be measured with a single clock. CLEARLY: Obs.1 did not do a simple Proper time measurement.
Now, what does Obs.2 think about this sequence of events? Well, Obs.2 actually CAN measure a proper time. She holds a clock and measures how long it take for (from her perspective) for the stick to fly past her to the left at -v.
Because she measures the passing of the stick with a single clock and the stick ends pass the same place in her frame, she has measured a proper time. She also knows the speed of the stick in her reference frame, so she calculates a stick length of:
The length she determines is, by definition, NOT the proper length, because the stick is moving relative to her. Using our time dilation result to relate the proper and not proper time allows us to see that Obs.2 and Obs.1 measure different lengths:
Remember, gamma is always greater than or equal to 1. Therefore, if the stick is moving relative to the observer, they measure it to be SHORTER!
OK, so we need to rethink our old ideas about absolute space and time: Both space and time turn out to be different for different inertial observers. And, it turns out the mass too depends upon your inertial frame. Einstein very carefully studied the issues of conservation of momentum and conservation of energy and came to several important conclusions. Here, we simply quote the results:
Relativistic momentum and mass
To preserve conservation of momentum, Einstein showed that our definition of momentum must be adjusted to:
Here, m0 is the mass that you measure if the object is at rest in your frame. It is referred to as the rest mass. Often, we write the momentum in terms of the 'apparent mass',
where this mass is given by:
This apparent mass increases very quickly as the object approaches the speed of light. At the speed of light, the mass would be infinite. Therefore, no force is sufficient to accelerate an object with rest mass to the speed of light. The speed of light is a limiting speed for all massive objects (like spaceships or like us).
Relativistic Energy and Kinetic Energy
To preserve conservation of energy, Einstein showed that the energy of a moving particle is given by:
One of the most interesting things about this result is that it predicts that a particle at rest, still has an energy of:
Therefore, by virtue of having mass, you also have energy! In fact, we now know from MANY experiments that conservation of energy and conservation of mass are directly related. They are basically the same thing. It is now known with exquisite accuracy that when an object loses energy, say during a chemical reaction, that it also loses some mass, and visa versa.
If a massive particle is moving, it also has a kinetic energy. In special relativity, this kinetic energy is just the energy above the rest energy:
Important points
. We think of
this effect as an increase in the mass of the object as it moves
at higher speed. All objects with mass have their mass go towards
infinity as they approach the speed of light. Therefore, no massive
object can actually quite get to the speed of light.
