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Notes on Lecture 10 1 PHY3101 Detweiler January 30, 2008 Relativity Notes on lecture 10 The interval The interval between two events is the same when measured in any frame of reference: §interval2 = ¡c2¢t2 + ¢x2 + ¢y2 + ¢z2 The value of the interval is independent of which coordinate system is in use. De¯nition: Timelike interval We say that the interval between two events is timelike if the right hand side of the interval squared is negative. If the interval between two events is timelike then there is some frame of reference where the events occur at the same place, and there is no frame of reference where the events occur at the same time. And the event that occurs ¯rst in one frame of reference, occurs ¯rst in all frames of reference. The value of a timelike interval determines the proper time between the two events|that is the the time between the events as measured in a frame of reference where they occur at the same position. Example: Given two timelike-related events ¯nd a frame of reference in which the two events occur at the same place. We are looking for the speed v of the boosted frame of reference in which ¢x0 = 0. But ¢x ¡ v¢t ¢x0 = p so we require ¢x ¡ v¢t=c2 = 0 ) v = ¢x=¢t: 1 ¡ v2=c2 De¯nition: Spacelike interval We say that the interval between two events is spacelike if the right hand side of the interval squared is positive. If the interval between two events is spacelike then there is some frame of reference where the events occur at the same time, and there is no frame of reference where the events occur at the same place. The value of a spacelike interval is actually the proper distance between the two events|that is the distance as measured in a frame of reference where the two events occur at the same time. Example: Given two spacelike-related events ¯nd a frame of reference in which the two events occur at the same time. We are looking for the speed v of the boosted frame of reference in which ¢t0 = 0. But ¢t ¡ v¢x=c2 ¢t0 = p so we require ¢t ¡ v¢x=c2 = 0 ) v = c2¢t=¢x: 1 ¡ v2=c2 De¯nition: Light-like interval We say that the interval is light-like or null if the interval is zero. If the interval is light-like (or null) then a ray of light can go from one event to the other. If the interval between two events is light-like then there is no frame of reference where the events occur at either the same time or at the same place. And the event that occurs ¯rst in one frame of reference, occurs ¯rst in all frames of reference. 2 \Pythagorean Theorem" intuition does not work in a space-time diagram ct*#3 light ct" x' #2a* #2* ct' #1* x Twin paradox The traveling twin leaves the Earth at event #1, arrives at a distant planet and instan- taneously turns around at event #2 and ¯nally arrives back on Earth at event #3. While traveling from #1 to #2 the line ct0 marks the worldline of the traveling twin, and the proper time between events #1 and #2 is ¢t0 = t0 ¡t0 . In the Earth's frame of reference, the time (2;1) 2 1 p 0 2 2 interval between #1 and #2 is \time dilated" to be ¢t(2;1) = t2 ¡ t1 = ¢t(2;1)= 1 ¡ v =c . This is also the time interval between events #1 and #2a because event #2a is simultaneous with #2 in the Earth's frame of reference. 3 Faster than light allows time travel In the ¯rst ¯gure, notice that t = 0 on the x-axis and that x = 0 on the ct-axis. The line labeled ct0 is the world line of an object moving with a position given by x = vt where 4 0 0 v = 5 c, and this is the ct axis (where x = 0) for the \moving" frame of reference. Also the x0-axis is the line where t0 = 0 and also where t = vx=c2, which is consistent with the Lorentz transformations. x=0 x'=0 ct ct' light x' ct'=0 *#2 #1 * x *#−1 ct=0 Look at the location in spacetime of events #1 and #2. It is clear that x1 = 0 and t1 = 0 and also that x2 > 0 and that t2 > 0. We deduce that event #1 occurs before event #2 with this coordinate system. In this diagram it is more di±cult to understand the primed coordinates, but it should be clear that the x0 axis is the line where ct0 = 0: The Lorentz transformations then imply that this same line is described by t = vx=c2 in the unprimed coordinates. Also, it should be clear that the t0 axis is the line where x0 = 0. The Lorentz transformations then imply that this same line is described by x = vt in the unprimed coordinates. 4 x'=0 light ct ct' x ct=0 #1 * x' #−1* ct'=0 *#2 A spacetime diagram for the same events but for the primed coordinates is shown in the 0 0 second ¯gure. Event #1 is at the origin andp has coordinates x = 0 and tp= 0. The 0 2 2 0 2 2 2 coordinates for event #2 are x2 = (x2 ¡vt2)= 1 ¡ v =c and t2 = (t2 ¡vx2=c )= 1 ¡ v =c : 0 0 The ¯rst ¯gure shows qualitatively that t2 < 0, because event #2 lies below the line t = 0. In the primed coordinate system, notice that the unprimed x axis is the line where t = 0, which corresponds to the line t0 = vx0=c2. and that the unprimed t axis is the line where x0 = 0, which corresponds to the line x0 = vt0. Now imagine that it is possible to travel faster than light. In the ¯rst diagram, you could then start at event #1 and travel via warp-drive to #2. Having arrived at #2 you could quickly change to the primed coordinate system of the second ¯gure, and again using warp-drive then return to an event at the same place, in the unprimed coordinates, but just a little before #1 actually occurs. The net e®ect of this operation is to put you back in your starting place, but a little before you started | you would have gone back in time. If tachyons actually exist, and we if can control them in some way, then we could send information back into the past. If we can travel back into the past, or send information back into the past, then we might be able to create a logical paradox, by changing an event that we thought had already occurred. This might be a good reason to claim that \The laws of physics forbid sending information faster than the speed of light." 5 All evidence implies that this statement is true. However, it might be the case that information might be able to travel faster than the speed of light, but that we do not have free will (or at least enough free will) to go back and change the past. Most physicists have thought about such things when they were younger. But, found such ideas not a very fruitful way to increase their understanding. Automobile and garage paradox ct x=0 ct' back front x'=0 x' ct'=0 * * Doors are closed x ct=0 Above, is a spacetime diagram that helps \explain" problem #1 of homework set #4 | that was the problem with the 11 ft long automobile ¯tting into the 10 ft garage. The ¯gure is in the frame of reference of the garage. The two light vertical lines are the locations of the doors at either end of the garage. The two red lines are the front of the automobile (on the right) and the back of the automobile on the left. The front and the back of the automobile are both moving to the right at the same speed, so the red lines are parallel. The small blue star on the left marks where the back of the automobile enters the garage, and the door on the left can be closed. The small blue star on the right marks the event where the front of the automobile leaves the garage and the door on the right must be opened. 6 First look at the situation from the garage's point of view: Notice that the x-axis and the light grey horizontal line are both lines of \constant time" t in the garage's frame of reference. Also note that on the light grey line both the front and the back of the automobile are inside the garage at the same time t: Now, look at the situation from the automobile's point of view: Notice that the x0-axis and the light brown lines sloping up and to the right are lines of \constant time" t0 in the automobile's frame of reference. Look at the brown line with the blue star on the right that shows the front of the automobile leaving the garage. Now look at the brown line with the blue star on the left that shows the back of the automobile entering the garage. Here is the surprise: The front leaves before the back enters! | from the automobile's point of view. How did I come to that conclusion? The x0-axis is the light blue line where t0 = 0: The light brown lines are lines of constant t0.
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