Transcript March 1
Einstein’s postulates 1. The laws of nature are the same for everyone. 2. The speed of light in a vacuum is constant for all observers. Discussion You and Jackie are each on separate trains that will pass each other at high speed. Both of you decide to measure how fast the trains pass each other. How can you do this without leaving your seat? Discussion Now that you and Jackie have the relative speed of the trains, both of you decide to measure the length of the other’s train. How would you do this again without leaving your seat? Discussion How can you measure the length of your train without leaving your seat? Discussion Will the length of your train that you measure be the same as the length of your train that Jackie measures? Explain. Discussion How will your measurement of Jackie’s train compare with her own measurement of her train? Explain. Length contraction From your point of view, length of an object moving by you (or distance between objects moving by you) is shorter in the direction of motion than it would be if it were at rest. The faster the objects are moving, the shorter the lengths. Theory of Invariantness? Length and time measurements depend on the relative velocity of the observers in such a way as to make the speed of light constant for all observers in any reference frame. Discussion Suppose you have two identical space ships, one is at rest with respect to you, the other is moving at high velocity. As the ship passes you, you give each ship an identical push for the same duration of time. How do passengers on the moving ship perceive the duration of push you give them as compared to the duration of the push you gave the stationary ship? Discussion If the moving ship observers you giving the stationary ship a push for a longer duration, how does this change the acceleration they feel as compared to the acceleration they observe you giving the stationary ship? Explain. Discussion According to Newton’s second law F = ma. If the force F applied is the same in both cases and the acceleration is less in the case of the moving ship, what does this mean for the mass of the moving ship? Explain. Mass increase From your point of view, objects moving by you have a greater mass than they have while at rest with respect to you. The bigger the velocity, the greater the effective mass E = mc2 As the speed of an object approaches the speed of light, its mass tends toward infinity, resisting any further increase in speed. Kinetic energy added to a high speed object gets converted into mass. Special Theory of Relativity No matter can travel at or faster than the speed of light in a vacuum Length contraction Time dilation Mass increase Space fleet maneuvers Consider three space ships moving in formation. The flag ship is midway between the lead ship and the rear ship. The “go” order The flag ship now signals to the other two ships to fire their engines for exactly one minute. The signal travels at the speed of light to each ship which obeys the command. The flag ship simply adjusts its own speed to remain at the midpoint between the two ships. Because both ships are the same distance from the flag ship, both ships fire their engines at the same time. Because they are all moving together at the same speed, they all stop firing their engines at the same time and the formation is preserved. Eat lunch The flag ship then signals to the other two ships that it is time to eat lunch. From the moving ships perspective they are all at rest with the flagship at the midpoint. Thus both ships received the “lunch” command at the same time and lunch takes place on both ships simultaneously. The relativity of time For someone at rest with respect to the moving ships, the signals also travel at the speed of light, but during the time it takes to reach the back ship it moves forward, toward the signal and thus receives the command a little early. Also during the travel time for the light the lead ship has moved forward, away from the light and receives the signal a little late. Not simultaneous Thus, from someone at rest with respect to the moving ships, the last ship eats lunch first and the lead ship eats lunch last. Discussion After lunch, the flagship again gives an order for another engine burn. From the perspective on someone at rest which ship receives the order first? What happens to the formation of the ships? Desynchronized motion Again the rear ship receives the order first and fires its engine before the other two. Thus, the rear ship for a time is going faster than the other two ships and gets closer to the flag ship. The lead ship gets the message last and starts its engine burn later, allowing the other two ships to catch up. The formation length has shrunk After the lead rocket shuts down its engine, all ships will again be going the same velocity, but the distance between them is now smaller than it was before. Desynchronized time causes length contraction Discussion How will this appear to the captains of the ships flying in formation? Will they see their ships as being closer? Why or why not? Are the captains blind? The ships can consider themselves at rest before the second engine burn. Therefore, from their perspective all the ships fired their engines at the same time and the distance between them did not decrease. Discussion How could the ships get closer to each other without the captains knowing it? Consider doing another engine burn. The ships will get closer still. Eventually, even if the fool captains cannot see the difference won’t they feel it as the run into each other? Length contraction No. The ships will never run into each other. From the perspective of someone at rest the distance between the ships will decrease to 0 as the speed approaches the speed of light. But the lengths of the ships also decrease in size so that they never hit. Space is contracting It is not just objects that contract, but space itself is contracting. The spaceships contract because the space between their atoms is contracted. Width and height contraction? Can’t happen because it is logically inconsistent Example – firing a cannon A fist fight The force of a punch depends on how fast you can throw a punch (the speed) and the mass of your fist. A high speed fight Suppose two twin brothers get into a fight. Each is traveling on a train going in opposite directions with a constant velocity and each tries to punch the other as the trains pass. Time dilation again But it takes time to throw a punch. Each twin will consider himself at rest and he will see the other twin as moving in slow motion. Each will also conclude that the other’s slow motion punch will have little effect on them, and that their fast punch is going to knock the other’s fist back in his mouth. Who is right? Which twin wins the fight? Discussion Neither. The punches will hit with equal force and neither is able to shove the other’s fist back into his brother’s mouth. How can the other twin’s slow motion punch have as much force as their own fast punch? No magnetic fields The Special theory of Relativity shows that the magnetic field is just an electric field which arises due to the relative motions of charged particles. Discussion Given the results of the Special Theory of Relativity, why did Einstein conclude that Newton’s law of gravity needed to be revised?