Notes - Special Relativity 1.) The problem that needs to be solved. - Special relativity is an interesting branch of physics. It often deals with looking at how the laws of physics pan out with regards to light movement. - In standard physics laws (Galilean) the velocity of a wave pulse measured by a moving observer would be the velocity of the wave +/- the velocity of the observer. For an observer toward the light source the light pulse would have a velocity of c + v (c = speed of light and v = velocity of observer), for an observer at rest the velocity would be c, and for an observer moving away from the source of its pulse its velocity would be c v. While this makes sense this leads to a problem. Many of the laws of physics are dependent on the speed of light, c, being finite and constant. According to the above example this isn t true. - The above example leads to 3 possible conclusions: 1.) The laws of physics depend on an observer s frame of reference. 2.) Only the Laws of Electro-Magnetism (Maxwell) are correct. 3.) Only the Laws of Mechanics (Newton) are correct. - In 1905 Einstein presented a paper called that answered this problem with two basic concepts: 1.) The laws of physics are the same in all frames of reference. 2.) The speed of light is the same as measured in all frames of reference. - For Einstein s ideas presented in his paper to be true he suggested that space and time ought to be considered the same, that motion at a significant fraction of the speed of light alters both space and time and that this motion alters our reality of simultaneity, time, length, addition of velocities and mass. - This body of work is what is now known as the Theory of Relativity (Special Relativity refers to frames of reference traveling at constant velocities). 1
2.) Simultaneity. - Simultaneity refers to events that occur at the same time in a frame of reference. When people aren t moving it makes sense that all observes perceive the event occurring at the same time. However, when one observer is moving fast (a significant fraction of the speed of light) simultaneity of events isn t necessarily occurring. That is, if you move really fast what one person sees as occurring simultaneously will not be seem as being simultaneously occurring to the other person. Ex. The diagram below shows a person on a rocket ship O moving at 0.5 c toward the right and passing an observer O at rest. Two bolts of lightning strike at point A and B, neither sees the bolts yet as the light has not reached either observer. - In the next diagram, the pulses of light are moving from A and B toward each other, and as O is moving right at 0.5c this observer moves half the distance the light pulses move as seen below. - The following diagram shows that the pulse from B has reached observer O. Therefore O states the lightning struck first at point B. 2
- The last image in the sequence occurs when light from A and B meet at observer O, who records the lightning strikes as simultaneous. - At some later time the light pulse from A will catch up to observer O and this observer will record the event of the lightning strike at A. The conclusion is that simultaneity is dependent upon the observer s frame of reference. 2.) Time Dilation - This involves a comparison of two frames of reference, with one moving at a significant fraction of the speed of light. - The theory of special relativity predicts that a clock in an at rest frame of reference with respect to an observer will read greater elapsed time than a clock in frame of reference which is in motion with respect to the same observer. - So, let s talk about what a clock actually means. To most students a clock measures seconds that are of equal duration by the motion of a hand in a circle or the changing of digits on a digital clock face. This is fine but there are other types of clocks that measuring aging; such as sub-atomic particles with lifetimes or half=lives can also be considered clocks as they measure increments of time passing. A light clock is one which, measures the time differential for a beam of light to leave a source, reflect off a mirror and return to the source (pictured below). Light clock 3
- The time elapsed on the above clock for the observer recording it is so the distance from the source to the mirror multiplied by two divided by the speed of light or. - An observer O measuring this time when at rest with respect to the clock will measure a different time than observer O who is in motion with respect to the clock. This is because, to the observer O, the light must travel a farther distance. The diagram below demonstrates this by placing the light clock on a ship travelling a significant speed of light. The three images below are 1.) the moment of the light pulse being emitted. 2.) the moment the light pulse hits the mirror. 3.) the moment the light pulse returns to the source. Velocity = v O frame of reference Height ct Distance vt - If it takes t seconds for the light to travel from the source to the mirror (images 1 and 2) for the observer O, then the ship travels to the right a distance equal to. As a result the light must travel the distance of the hypotenuse and since the speed of light is always c then the distance must be. According to Pythagoras theorem then. Expanding the squares gives Combining t 2 terms simplifies to Dividing by c 2 gives Factoring t 2 gives 1 Taking the square root and solving for t equals - Since must be less than, then will always be less than 1, causing the denominator to be less than 1, resulting in. 4
This proves that the time measured by observer O in a frame at rest with the clock t o will always be less than the time measured by an observer O who is not at rest with respect to the clock. Ex. 1 A sub atomic particle is created in the atmosphere, 6.4 10 m above the earth s surface, by absorption of light rays. The particle streams towards earth at 0.800 c. If these particles have a life time of 1.6 10 s when at rest, how long will the particle live as viewed from the particle s frame of reference and from the earth s frame of reference? Ex. 2 Two twins are separated from one another. One travels away from the earth at a high velocity, say 0.60 c and the other remains on the earth. If an observer on earth watched both twins for one year, how much would each twin age? 5
3.) Length Contraction - Time dilation is not the only effect of travelling at fractions of the speed of light. Another result is length contraction. To illustrate how this occurs, a thought experiment using a light clock is used, this time with the clock oriented so motion is in the direction of the light s path. Light clock - The light clock above considers the time required for the light to travel to the mirror and back (a distance of 2L o ) when the observer is in the same frame of reference as the clock. - So, time must be or if we solve for L o gives. - The figure of the light clock above considers the time required from an observer watching the light clock travel right at velocity v, a significant fraction of the speed of light (images are slightly offset to better illustrate the concept). The length measured for the clock is then L. As the mirror end of the clock then moves a (where t m is the time to the mirror), the total distance to the mirror becomes, which must equal, as the speed of light never changes. Conversely, on the return trip the light needs only travel, as the source has approached the returning pulse. On this portion of the trip,. These equations are solved for t m and t r respectively as shown below: The total time then in this frame is Simplification of this gives with Lv terms eliminating each other, 6
or Because and, these can both be substituted into the time dilation equation which gives: and after removing common factors, As before since v must be less than c the radical must always be less than 1. The length of an object in a moving frame of reference (L) is always measured as shorter than the length of the same object in a frame of reference at rest with respect to the object (L o ). Ex. 1 A high performance car is measured at rest to have a rest length of 5.00 m. The car is later measured when traveling past an observer O at 2.00 10 observer O measure for the length of the car?. What length will the driver and Ex. 2 Example 1 of time dilation considered a sub-atomic particle created in the atmosphere 6.4 10 m above the earth s surface. The particle streamed toward earth at 0.800 c. Will the particle reach the earth? 7
- It can be seen from the two frames of reference in example 1 of time dilation and example 2 from length contraction, that the effects of the increased life span of the moving particle are compensated by the contraction of the earth-particle length in the particle s frame of reference. This prevents the experiments from having different outcomes. There is no preferred frame of reference. 4.) Beyond Special Relativity - Special Relativity has many implications for classical physics. Consider what happens to the passage of time when an object attains the speed of light. Since the observer O of such an object will see the passage of time as then when v = c the radical becomes zero. As it is impossible to divide by zero the time t becomes infinite! - Therefore, for the observer O, time passes infinitely slowly for the object traveling at c. A similar argument can be put forward for the object s length. As the radical becomes zero, the length becomes zero for the observer. The result is that no object can travel at the speed of light. So what happens when a train travels at 0.800c and has a passenger moving inside at 0.400c? According to Galilean physics, the result as viewed from an observer O would be 1.200c. However, we just showed that you can t travel at the speed of light much less FASTER than the speed of light. Einstein used Lorentz s work to come up with another equation for the relativistic addition of velocities. We need another variable to describe the passengers velocity relative to the train (u ). The variable v is the velocity of the train and u is the velocity of the person as seen by an observer O at rest. - In this case the velocity as seen by O is found by doing the following: 8
........ 0.91 - It is valuable to see the effects of the relativistic addition of velocities if a high speed sub-atomic particle moving at 0.60c emits a photon (light wave). Here the velocity of the particle is v = 0.60c and the photon has a velocity u = c. The observer at rest will then determine the following velocity:...... - What is so important about this calculation is it shows that the second postulate of relativity is true, namely that the speed of light is the same for all observers as u = u = c. The photon is traveling at the speed of light for ALL frames of reference. - Another ramification of the Theory of Special Relativity comes from the theory s application to the law of conservation of momentum. If two particles of equal mass are moving towards each other at velocity v and v respectively and viewed from a frame of reference which is at rest, then their total momentum MUST equal zero, and in an inelastic collision remain at rest when combined to 2m as below: - In Galilean terms, the law of conservation of momentum is always obeyed even if the observer is moving with right hand mass. In that case the large mass 2m would simply seem to be moving after the collision at velocity v to the right, as the left-hand mass would have velocity 2v to the right before the collision. However, consider though, what happens if the observer were moving in the same frame of reference as the right-hand particle at a significant fraction of the speed of light. Then the observed velocity of the left hand particle is given by the formula for relativistic addition, namely but since u = v, this becomes - It is clear from before that v must always be less than c; therefore, the denominator must be less than 2. The velocity u must be less than v and therefore the momentum before the collision would 9
be p = mv and the momentum after would be p = mu. Since the law of conservation of momentum would be violated. This is also a contradiction to the first postulate, that the laws of physics remain constant in all reference frames! To compensate for this Einstein was able to use algebra to show that the mass of an object must INCREASE as it achieves greater and greater velocities. This is shown in the equation below: Where m o is the mass of an object as measured from the rest frame and m is the mass measured when in motion. - So it is as v approaches c, the term in the denominator approaches zero and m will become much greater than m o. Ex. 1 An electron has its mass measured when at rest to be 9.11 10 kg. It is accelerated to a speed of 2.85 10 from the electron s frame of reference?. What mass would an observer in the lab measure and what mass would be measured - This increase in the mass adds further support to the idea that the speed of light is unattainable. If an object moves at the speed of light (v = c), then the rest mass m o is divided by zero, giving the relativistic mass and infinite value. The result of this is that the kinetic energy of such an object found by must then be infinite as well. According to the work/energy theorem then, to achieve the speed of light with infinite mass requires an input of infinite work. Therefore only objects whose rest mass is zero (photons) can attain the speed of light. Einstein immediately saw that the addition of energy to an object could not go entirely into increasing the velocity, that some of the energy had to go into the mass. From this he concluded that mass is simply another expression of energy and this led to realize that kinetic energy can be converted into mass and vice versa. Einstein used calculus to show that energy E derived from a mass m to be. 10