Special Theory of Relativity Chapter 26 The Newtonian Electron Newtonian Theory (everything we have done so far in class) can be tested at high speeds by accelerating electrons or other charged particles through a potential difference. However, experiments have shown, that no matter the size of the accelerating voltage, the speed of the electron (or any other particle with mass) will always be less then the speed of light. Newton vs. Einstein This means nothing with mass can go faster then 3.0 x 10 8 m/s Universal Speed Limit 3.0x10 8 m/s Since Newtonian theory no longer worked at high speed, another theory was needed. This is where Einstein stepped in. In 1915 Einstein published his general theory of relativity. Even though this theory is what Einstein is mostly known for, it is not what won him a Nobel Prize His Nobel Prize was for his explanation of the Photoelectric Effect in 1921 So if Newtonian Physics is wrong. As long as an object s speed is much less then the speed of light, Newtonian Physics works wonderfully. However, if an object s speed starts to approach the speed of lighome interesting things occur. It is all Relative According to the Special Theory of Relativity, two observers moving relative to each other, will measure different outcomes for the same event This makes it necessary to choose a frame of reference 1
Inertial Frames of Reference are: Any reference frame in which Newtonian Physics is valid. Any reference frame in which objects that experience no forces, move in straight lines at a constanpeed (or not at all). We will be working with these types of reference frames. Example: Two students are playing baseball on a train moving at 100 mi/hr. The pitcher throws the ball at 50 mi/hr. According to a stationary observer, how fast is the ball going? 50 mi/hr Baseball speed = 150 mi/hr 100 mi/hr The batter hits it back at 50 mi/hr. Baseball speed = 50 mi/hr 50 mi/hr 100 mi/hr What if instead of a baseball, it was a light pulse? But remember, nothing can go faster then 3.0x10 8 m/s (including light itself). Therefore, there must be a problem with the classical addition law for velocities. And that is where Einstein s Special Theory of Relativity comes in. Let s kick it up a notch! Relativities Two Postulates 1. The principle of Relativity: All the laws of physics are the same in all inertial reference frames. 2. The constancy of the speed of light: The speed of light in a vacuum has the same value (c = 2.997 924 58x10 8 m/s, rounded to 3.0 in this class) in all inertial reference frames, regardless of the velocity of the observer or the velocity of the source emitting the light. 2
Effects of Relativity Length contraction - Moving rulers are short Time dilation - Moving clocks run slow Length Contraction When viewed by an outside observer, moving objects appear to contract along the direction of motion. For everyday speeds, the amount of contraction is too small to be measured. For relativistic speeds, the contraction is noticeable. A meter stick whizzing past you on a spaceship moving at 87% the speed of light (0.87c) would appear to be only 0.5 m long. What would a baseball thrown at relativistic speeds look like to a fan sitting in the stands? It would contract along the direction of motion. At rest 0.50c.95c Calculating the length contraction: L = L s 1 v 2 L = moving length L s = stationary length (length at rest) v = velocity of the object c = speed of light Example: A meter stick flies past you at 99.5% the speed of light. What is it s apparent length? L = L s 1 v 2 L =1 1.9952 = 1 0.990 L = 0.10 m or 10 cm How do people on spaceships view their meter sticks? 1. They are smaller then usual 2. They are the same 3. They are larger than life But outside viewers sure look funny! 3
You are packing for a trip to another star, and on your journey you will be traveling at a speed of 0.99c. Can you sleep in a smaller cabin then usual, because you will be shorter when you lie down? Explain. Time Dilation Pretend you are in a spaceship at rest in Ms. Stevens class. The clock on the wall reads 12-noon. To say it reads 12 noon is to say that light reflects from the clock and carries the information 12 noon to you in the direction of sight. If you suddenly move your head to the side, the light would miss your eye and continue out into space where another observer mighee it. The observer in space would then later say Oh it is 12 noon on Earth right now But from your point of view, it isn t. Now suppose your spaceship is moving as fast as the speed of light (just pretend!). You would be keeping up with the signal saying 12 noon To you on the spaceship, time at home would appear frozen! Consider a light clock This is in essence, time dilation: Clock that are moving, run slow. Light comes out of the source, hits a mirror and bounces back into a detector. Based on how far it traveled and the speed of light, we could calculate the time it took. 4
Now, put the time clock on a spaceship Since the speed of light is the same for everyone, time must be running slow for the astronaut. Calculating time dilation: 1 v 2 time (observer) = time if standing still (object) v = velocity c = speed of light Stationary object on a moving spaceship. Example: The period of a pendulum is measured to be 3.00 s in the inertial frame of the pendulum. What is the period measured by an observer moving at a speed of 0.95c? 1 v2 3.00 1.95 2 9.6 seconds If you were moving in a spaceship at a high speed relative to Earth, would you notice a difference in your pulse rate? No, there is no relative speed difference between you and your pulse. Would you notice a difference in the pulse rate of the people left on Earth? Yes, it would be slower then usual. The relativistic effect always happens to the other guy. Does time dilation mean that time really does pass more slowly in moving systems or that it only seems to pass more slowly? The slowing of time in moving systems is not merely an illusion resulting from motion. Time really does pass more slowly in a moving system compared with one at relative rest. But this leads to some interesting paradoxes. The Twin Paradox There are two twins, Speedo and Goslo. When they are 20 years old, Speedo, the more adventurous of the two sets off on an epic journey to Planet X, located 20 lightyears away from Earth. Further, his spaceship is capable of reaching a speed of 0.95c relative to the inertial frame of his twin brother back home. After reaching Planet X, Speedo becomes homesick and immediately returns to Earth at the same speed of 0.95c. Upon his return, Speedo is shocked to discover that Goslo is now an old man while Speedo is not. 5
The Paradox From Speedo s point of view, his brother Goslo was the one racing away and then back at 0.95c. Therefore, Goslo should now be younger then Speedo. But that isn t the case. The Resolution Consider a third observer traveling in a spaceship at a constanpeed of 0.50c relative to Goslo. To the third observer, Goslo never changes inertial reference frames (his speed relative to the observer is always the same). The third observer notes however, that Speedo accelerated during his journey, changing his reference frame in the process. To the third observer, the motion of Goslo and Speedo are not the same. Therefore roles played by Goslo and Speedo are noymmetric. So ihould not be surprising that time flows differently for each. Example: How old are Goslo and Speedo when they finally reunite? (remember, Planet X is 20 lightyears away. The are each 20 years old in the beginning, and Speedo travels at 0.95c) For Goslo: Speedo: 2dist. = 2(20ly) v 0.95ly/ y 42 years Age = 42 yrs + 20 yrs Goslo is 62 years old 1 v 2 42yrs = 1 0.95 2 13 years Age = 13yrs + 20 yrs Speedo is 33 years old 6