Special Theory of Relativity I had a better

Special Theory of Relativity “I had a better grasp of things when physics dealt mostly with falling bodies. ”

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.

• This means nothing with mass can go faster then 3. 0 x 108 m/s Universal Speed Limit 3. 0 x 108 m/s

Newton vs. Einstein • 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 light some 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

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 constant speed (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 100 mi/hr Baseball speed = 150 mi/hr

The batter hits it back at 50 mi/hr 100 mi/hr Baseball speed = 50 mi/hr

What if instead of a baseball, it was a light pulse? But remember, nothing can go faster then 3. 0 x 108 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.

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 58 x 108 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.

Relativistic Addition of Velocity • The problem: A motorcycle moving at 0. 80 c with respect to a stationary observer. Then the rider throws a ball forward at 0. 70 c relative to himself. • According to Newton (and classical mechanics), how fast is the ball going?

The solution: Velocity of object relative to a stationary person Velocity of object relative to a moving person Velocity of mover relative to a stationary person Object – what is being “launched” Mover – what does the “launching” Stander – Usually the Earth

Example: A motorcycle moving at 0. 80 c with respect to a stationary observer. Then the rider throws a ball forward at 0. 70 c relative to himself. How fast is the ball going relative to a person standing still? • Vos = ? • Vms = 0. 80 c • Vom = 0. 70 c

Example 2: A spaceship leaves Earth traveling at 0. 71 c relative to Earth. A second mini spaceship leaves the first at a speed of 0. 87 c with respect to the first. Calculate the speed of the second ship with respect to Earth if it is launched in the opposite direction, back towards Earth. • Vos = ? • Vms = 0. 71 c • Vom = - 0. 87 c

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. 87 c) 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. 50 c . 95 c

Calculating the length contraction: L = moving length Ls = 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 = 0. 0999 m or 9. 99 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

You are packing for a trip to another star, and on your journey you will be traveling at a speed of 0. 99 c. 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. Keeler’s 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 might see 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!

• This is in essence, time dilation: • Clocks that are moving, run slow. • Note: When a clock is “running slow” it shows a smaller time than it “should”

Consider a light clock • 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.

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: Stationary object on a moving spaceship. t = time (observer) ts = time if standing still (object) v = velocity c = speed of light

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. 95 c? t = 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. 95 c 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. 95 c. Upon his return, Speedo is shocked to discover that Goslo is now an old man while Speedo is not.

The Paradox From Speedo’s point of view, his brother Goslo was the one racing away and then back at 0. 95 c. 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 constant speed of 0. 50 c 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 not symmetric. So it should not be surprising that time flows differently for each. • Twin Paradox Video

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. 95 c) For Goslo: Speedo: t = 42 years Age = 42 yrs + 20 yrs Goslo is 62 years old ts = 13 years Age = 13 yrs + 20 yrs Speedo is 33 years old

E= 2 mc Probably the most famous scientific equation of all time, first derived by Einstein is the relationship E = mc 2. This tells us the energy corresponding to a mass, m, at rest. What this means is that when mass disappears, for example in a nuclear fission process, this amount of energy must appear in some other form. It also tells us the total energy of a particle of mass m sitting at rest.

E= 2 mc "It followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing -- a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to m c-squared, in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned above. This was demonstrated by Cockcroft and Walton in 1932, experimentally. "

What E = mc 2 doesn’t mean: Many people think that E = mc 2 means matter, when traveling at the speed of light transforms into pure energy. THIS IS NOT TRUE 1. Matter can’t travel at the speed of light. 2. The speed of light squared is not a velocity. All E = mc 2 means is that mass and energy are two sides of the same coin. They are the same thing.

Energy • Rest energy (also called Rest Mass) - The energy a particle has by simply being a particle (having mass). Also called mass-energy Erest = mc 2 • Kinetic Energy - Energy due to the movement of the particle • Total Energy - The total energy a particle has due to the fact it is a particle (rest energy) and the fact that it is moving (kinetic energy)

As a student, Einstein was a whiz in math & science, but he was lousy in grammar and home economics.

Example: A nickel has a mass of 5. 00 g. If this mass could be converted to electric energy, how long would it keep a 100. W light bulb lit? (remember 1 W = 1 J/sec) First: How much energy is equivalent to 5. 00 g (physics prefers kg!) Second: 100 W = 100. J/sec. We have ______ J, how many seconds is that? or 143, 000 years!

Fusion & Fission processes take advantage of a particle’s rest energy (E = mc 2) & change mass into energy.

Fusion processes in stars fuse hydrogen atoms together to form helium atoms according to the following (simplified) process: 41 H 4 He + energy • If one mole of 1 H = 1. 007824 g • & one mole of 4 He= 4. 002603 g, A) How much mass is lost in this process? Dmass = 4. 002603 – 4(1. 007824) = - 0. 028693 g Or 2. 8693 x 10 -5 kg

B) Where did this mass go/what did it turn in to? Energy, specifically light & heat C) How much energy was created when 2. 8693 x 10 -5 kg of mass turned into energy?