( ) 2 + v t B = 2 D2 + L 2. = 2D c. t B. & t A. , L = v t B. D = c t A. + v t. = t A. = 2 c. c 2. ( v2 c ) = t. 1 v2. t B 2 = t A.

Transcription

1 LECTURE The Theory of Special Relativity [PSR] This principle seems innocuous enough, and it is the entirety of the theory of special relativity and in essence that s it however, there are consequences of this principle. These consequences seriously bend our notions of time, space, causality, and the nature of our universe. That is what we discuss in the remainder. 1) Constancy of the speed of light. [c] Discussed in previous section all observers that move at constant velocity with respect to each other (are not accelerating) observe the speed of light to have the same value, c. ) Time Dilation [TD] I introduce the Lightclock invented by a Swiss patent clerk to explore the nature of space and time. This device has a light pulse that travels a short distance perpendicular to the direction of travel. The round trip takes a time t = D/c where D is the separation of the detector and the mirror. We set this to 1 nanosecond just to have a number. Give two observers, Alice and ob, identical light clocks that are synchronized with extreme precision (Swiss Timing!). When at rest w.r.t. each other the clocks tick at exactly the same rate. Now set them in motion w.r.t. each other. Taken from one perspective (say Alice s) ob s light pulse now travels a longer distance. Utilizing the principle of special relativity that both observers see the light travel at the same speed, Alice observes that, though it takes her own clock to tick at t = D/c = 1 nanosecond, ob s appears to take t ob = *sqrt[d + L ]/c, where v is the speed of ob relative to Alice. Let s actually derive the result relating the two clocks rate from Alice s perspective. The distance L is how far ob moves in the time of one tick of his clock (as observed by Alice). Since he is traveling at speed v, we have that L = vt ob (t ob is the time observed for one tick as viewed by Alice). We can solve for L and insert it into our previous relation to get, t = D + L = c t = t A t = c ct A ( ) + vt t A 1 v c., L = v t + vt c = & t A = D c c 1 ct A D = ct A + vt = ( t A ) + vt c t t ( v c ) = t t A 1 v c = t A That s the result, Alice see ob s clock to run slow by that amount. Employing the lazy physicist principle (physicists are too lazy to continually write a big radical), the following definition is made, 1 1 v c and then t = t A. This factor, called the dilation factor, is a measure of how relativistic a situation is. Notice that when v = 0, when the clocks are at rest w.r.t. each other, = 1 and the clocks tick at the same rate. This factor gets bigger until you reach the speed of light where the factor becomes 1/0 = infinity. This means that if ob were flying by at the speed of light, Alice would see everything in his reference frame to not change in time! This can be seen in the gedankenexperiment where if ob were traveling at the same speed as light, the light would never reach back to the detector to trigger the clock, thus it would never appear to click according to Alice. This is a symmetric effect, Alice observes ob s clock to run slow and ob observes Alice s to run slow, but their own clocks run normally from their own perspective. And note, it is not just the clocks that run slow, everything in ob s

2 frame appears to run slow as seen by Alice ob appears to age less! So here is Einstein s beauty tip #1: if you want to stay looking young (and actually be young) just move at a high rate of speed with respect to your friends. Again, they each see their own clock to tick at the normal rate (1 ns say) yet when they look at the other clock moving by it runs slower (say ticks every ns). This effect has been measured experimentally in several different ways. There is no universal clock that ticks at the same rate for everyone sorry Newton. 3) Length Contraction [LC] If Alice and ob observe each other s clock to run slow then a direct result is that they must observe that the dimension along the direction of relative travel must shrink. That is, from Alice s perspective, ob s frame will appear shorter than what ob says his own frame appears (and vice versa). To see how this must occur consider the following scenario, Al (in frame A at position 1) has a twin Andy at position. They observe ob to move to the right at speed v from their perspective. From ob s perspective, he sees Al and Andy move to the left at the same speed, v. The question is, how do they each measure the distance between Al and Andy (at 1 and )? Examine the easiest way for ob to measure the distance. He knows that the frame A moves at speed v, and he can measure the time it takes for Al and then Andy to pass. y knowing the time difference between their passage, and the speed, he can measure the length. speed = (distance between 1 and )/(change in time) (distance between 1 and )=(change in time) x (speed), or as a formula L = t v. [Note there are four time measurements that can be discussed, A s frame s frame t A (time measured by Al as seen by Al) t A (time measured by Al as seen by ob) t (time measured by ob as seen by Al) t (time measured by ob as seen by ob) In this example t A = t A, and t =t. They must agree on the watch settings at the events even though they may disagree in the amount of time that passed between them.]

3 Al and Andy can measure the distance between them in a similar fashion: they know that ob is traveling at speed v to the right and they can measure the time that he passes each and (as long as their watches are synchronized) can find the distance between them. L A = t v. The speeds are the same but the times are different from our previous discussion of time dilation. Thus the lengths appear to be different, this effect is called Length Contraction. It occurs along the direction of travel only, perpendicular directions are unaffected. y how much do their measurements differ? Recall that from frame A s perspective, ob s clock runs slower than their own. Thus the time between the two events (ob passes Al and ob passes Andy) appears shorter for ob than it does for Al and Andy. Or, t < t and therefore L < L A. The formal relation is, L = L A 1 v = L A c. Since Al and Andy are at rest with respect to the length in question, say they measure it to be 10 m, then ob will measure it to be somewhat less than 10 meters. 4) Simultaneity What appear simultaneous in one frame (two events occurring at the same time) does not necessarily appear to be simultaneous in another frame in motion with the first. To demonstrate this we rely upon the previous result of length contraction and analyze one particular arrangement of observers and speeds. Consider the two frames described in the previous section moving at a rate such that the dilation effects are 1/. That is, the dilation factor = and each frame sees the other s clock to run as half as fast as their own and lengths in the other frame appear to be 1/ as long along the direction of travel. (y resorting to the definition of, you can quickly verify that the frames must be passing each other at 87% the speed of light.) Frame A: We arrange a very large number of observers in frame A to be 10 m apart. We will label their position as 1,,3,4, etc. In frame there is also a large number of observers (labeled 1,,3, etc.) spaced equally apart. It is arranged so that, from A s perspective, the observers in all appear to be 10 meters apart as well. At one particular instance the observers will be across from their numbered counter part (1-1, -, 3-3, etc.). At this instant all observers are instructed to set their watches to time t = 0. Thus at this point forward all watches in A and all watches in will remain synchronized in the frame A. Of course, the watches in A and will appear to tick at different rates so that watches in A and in will quickly diverge in their readings (will not appear synchronized). Frame : Now examine how this situation appears in the frame of. In order to have the observers appear to be 10 m apart in A the observers in must be 0 meters apart in their own frame (remember that the dilation factor is here). Only in this way can the observers appear to be 10 m apart according to A. How far apart do the observers in A appear to? Well they are moving at 87% the speed of light to the left, thus they appear 1/ the distance in their own frame 5 meters apart. The observers are not the same distance apart in A and in according to those in frame. 3

4 Notice now that it is impossible to have the observers line up at the same time (1 can not line up 1 at the same time that lines up with etc.). Thus the events that occurred simultaneously in frame A do not occur simultaneously in frame! The concept of simultaneous events is frame-dependent. 5) Addition of Velocities [AV] In ordinary physics if you watch a plane fly by at 300 m/s and someone with in the plane throws a ball forward at 50 m/s, from your perspective you would see the ball to travel at m/s = 350 m/s. However this changes for objects moving fast (how fast? Well to get a 10% effect you need to travel at 44% the speed of light). For if in the airplane a laser is shone forward, both you and a person in the ground see the light to travel at the same speed. Under relativity the way we add velocities is not so simple, or 1 x 10 8 m/s + 1 x 10 8 m/s does not equal x 10 8 m/s (1+1 is not equal to!). We will return to this relation later. 6) No material thing can travel at the speed of light. [Noc] In the previous section it was discussed that combining the observations of a moving object from two different frames (addition of velocities) no longer leads to a simple sum of the velocities (as was the case in classical physics). As an extreme example consider the following, a spaceship travels at (or infinitesimally close to) the speed of light and shoots a laser in the forward direction of its travel as seen in one frame. The principle of special relativity states that both the observer on board the space ship (who sees the light travel forward at c) and the Earth-based observer (who sees the ship travel at c) both see the light to travel at c. Thus in this case the Earthbased observer would find that c + c = c. Thus no matter what the speed of any projectile shot forward in the spaceship, the Earth-based observer sees it to travel at c. This can be seen also from the time dilation result, if the spaceship is traveling at c, the Earthbased observer sees everything within the ship to be frozen no time development. The light never appears to move to the right in the space ship because the ship is already traveling at c! Any scheme you can cook up to attempt to travel beyond the speed of light will never work. For e.g., if you travel at a spaceship traveling at % the speed of light and fire a projectile forward at 99% c, no frame of reference will ever see it to travel greater than c. [Noc] is a result of causality To see that the result that no object can travel faster than light is a result of our basic, comfortable, notion of causality, consider the following scenario. An execution is about to take place on board a spaceship. The condemned is strapped to a chair and next to him is a bomb. The execution is a three step process. First, the executioner presses the execute button, which simultaneously launches (from the same spot) a laser beam and a projectile towards the bomb (all velocities are in the positive x direction). The laser beam reaches the bomb first and arms it, without which the bomb can not detonate. Second the particle, traveling at a speed less than c yet is very large, strikes the detonation device, setting off the bomb. Call these three events 1,, and 3. The casual order for the whole affair is 13, if they do not occur in this order, no execution. Just to make the argument more dramatic, let s say the condemned figures a way to disarm the bomb, thus saving his self. However, he can only disarm the bomb once it is armed. In this reference frame he is saved and escapes. Now view these occurrences in another frame, one in which the prison ship is observed to travel in the positive x direction at a speed just less than c. If the only change in the world due to special relativity was the constancy of the speed of light and velocities added just as we do normally (meaning non-relativistically) then in this frame the projectile would be traveling faster than c and the laser would still be traveling at c. ut observe 4

5 now the causal order has changed. Event 1 occurs first (push the execute button), then event 3 since the projectile is traveling faster than the laser beam. The causal order is now 13. In this frame there is no causal link to the detonation of the bomb, if the bomb can not be detonated prior to it being armed, then it will not explode in this frame. To demonstrate vividly the logical inconsistency of this result, consider the prisoner. In this case, the bomb explodes before being armed and the prisoner can not disarm it prior to it being detonated. It would follow that the prisoner has been executed in this frame but lives to carry out more crime in the other frame. A logical inconsistency follows if the ship lands on the planet (corresponding to the second frame). Now in one frame the prisoner is both dead and alive. How can this be? There can not be two separate histories unfolding in both frames as they can always be merged. If this is still not clear just keep elaborating the story. Suppose the prisoner manages to destroy the ship and the planet if he lives. Now, according to those on the planet, the ship lands and all is ok but from the perspective on the ship the ship and planet no longer exist! The upshot here is that allowing objects to travel faster than light (while still maintaining the constancy of light) allows for a violation of cause and effect. If we demand causality to be maintained we must not allow objects to travel faster than light. Further gedankenexperiments will show that not only objects but any type of signal can not travel faster than light. Summary y devising simple gedankenexperiments we have demonstrated the following kinematical effects of special relativity. These all derive from the constancy of the speed of light, which stems from the principle of special relativity. [TD] Time dilation. (Time intervals may be measured to be different in different frames). [LC] Length contraction. (Distances may be measured to be different in different frames). [Sim] Lack of universal simultaneity. (Simultaneous events may not be simultaneous in other frames). [AV] Addition of velocities. (Velocities do not add simply in different frames). [Noc] No object, energy, or information can travel faster than light. In the following lectures we will explore how these effects arise in more sophisticated treatments of special relativity. Appendix Some facts that are good to remember. To get a sense of scale, use light time travel! - Light travels 1 foot (~0.3 m) in 1 nanosecond ( second). [Light-nanosecond] - Light takes approximately 1 second (1. s) to get to the Moon. [Light-second] - Light takes 8 minutes (500 sec) to get from the Sun to Earth. [8 Light-minutes] - Light takes 5. hours to get from the Sun to Pluto Diameter of solar system is roughly 10 Light hours. [10 Light-hours] - Light takes 4.3 years to get to the nearest star. [4.3 Light-years] - The diameter of the Milky Way Galaxy is about 100,000 ly. We are about 7,000 ly from the center of the galaxy. - The distance to the nearest galaxy is about,000,000 ly. - The furthest distance we can see is about 13,000,000,000 ly (1.3 x 10 9 ly). (or about 10,000,000,000,000,000,000,000,000 meters! (1. x10 6 m)) The speed of light in various units, c = 3 x 10 8 m/s (actually 99,79,458 m/s) c = 1 light-year/year = 1 ly/y = 365 light-days/year c = 4 light-hours/day = (60*60*4) light-seconds/day. c = 186,000 miles per second. c = 670,000,000 miles per hour. Light travels one foot in one nanosecond (10-9 s). 5