The speed of the ball relative to O, which we call u, is found. u u' O'

Transcription

1 Einstein s Theory of relativity The theory of relativity has two parts. The Special Theory of Relativity relates the measurements of the same events made by two different observers moving relative to each other with constant velocity (inertial observers). The General Theory of Relativity includes the effects of gravity. The Special Theory of Relativity This is a very successful, well verified theory, discovered by Einstein based upon what were called the two postulates. The Postulates of Einstein s Special Relativity: (1) The laws of physics are the same for every inertial (nonaccelerating) observer. (2) The speed of light, c = 300, 000 km/sec, is absolute. That is, the speed of a pulse of light measured by an observer is independent of the speed of the source relative to the observer. Galilean Relativity: Galileo would agree with the first postulate, but not with the second. Say we have two observers, O is on the ground, and O is riding on a car moving with a speed v relative to O. Observer O would say that observer O is moving toward the left with the same speed v. An object, maybe a ball, is moving with a speed u relative to observer O as shown below. u u' O O' v The speed of the ball relative to O, which we call u, is found 1

2 in classical (Galilean) relativity from the formula u = u v This is the Galilean (everyday, common sense) speed subtraction formula. So if the object is a pulse of light, u = c, classical physics says c = c v, which is not equal to c, and disagrees with the second postulate. It turns out that this formula is valid only if both v and u are much smaller than c. Einstein s Speed Subtraction formula: Einstein found a different formula, u = u v 1 uv c 2. If u and v are much smaller than c, then uv/c 2 is much smaller than 1, and the denominator becomes approximately equal to 1, and then this formula gives that u u v, in agreement with the classical result. However, if the object is a pulse of light, so u = c, then u = c also, no matter what v is. So both observers measure the same speed c. The speed of light is very special! Every observer measures the same speed c, no matter how they are moving, even if they are accelerating, and even also if they are in gravitational fields. It is absolute. But how is this possible? After all, measuring the speed of something just involves measuring the distance traveled and the time elapsed, and then speed = distance/time Time contraction and length contraction: The answer is that the two observers do not agree about measurements of lengths and time intervals. Say there are two events that happen at the same place in the moving frame of 2

3 reference. Say that the traveler O goes to bed in his vehicle (event 1), and gets up (vent 2) a time t later as measured by his clocks. O' measures a length L' for the trip, L'= vδt' O' measures a time interval Δt' from event 1 to 2 event 1 event 2 O' goes to bed here O' gets up here v v O O measures a time interval Δt from event 1 to 2 lenght L of trip as measured by O, L= vδt Observer O would measure a different time t with his clocks, where t = t 1 v2 c 2. The square root is a number less than 1, so t < t. So, say an astronaut goes on trip and returns to Earth. The Earth time t > t. If v is much less than c, the square root is almost equal to 1, and the two times are essentially the same. But if v = 0.99c, then t < t, or t > 7.1 t, and if v = c, then t < t, or t > 707 t. So an astronaut goes on a trip that to him lasts for t = 1 year. When he returns to Earth, t = 707, and 707 Earth years have elapsed and all his friends have long since died! This amazing result has not been verified with astronauts, but it has been 3

4 verified with radioactive particles, than seem to have a much longer half life when they are moving than their sisters that have remained at rest. How does the astronaut explain that he measured a shorter time? To him the distance traveled is shorter, because L = v t, while L = v t, so L = L 1 v2 c 2, and L < L. So, to the astronaut the time was shorter because the length of the trip was shorter. Length and time measurements are relative. Simultaneity is relative: Observers moving relative to each other disagree on whether or not two events are simultaneous. Observer O is on the ground and observer O is standing at the middle of a train moving with speed v relative to O. As O passes in front of O two lightning bolts hit at points A and B on the train, and also make burns marks on points A and B on the track, as shown in the figure to the left. The flashes of light reach O at the same time. Observer O then verifies that the burn marks at points A and B were at the same distance from the point where he was standing. Therefore, since the distances were the same, the two flashes must have been simultaneous, as far as he is concerned. 4

5 But as the flashes traveled toward O, observer O was moving toward point B, so the flash coming from the front of the train would reach him before the other flash coming from the rear. Then O would verify that the burn marks at the front and rear of the train are at the same distance from him, since he was standing in the middle of the train. As far as O is concerned, the burn at the front of the train must have happened before the burn at the rear because, as far as he was concerned, the flashes each had to travel half the length of the train, but the flash from the front arrived first, so it must have happened earlier than the one at the rear. Both observers are right! Simultaneity is relative. Energy and momentum: In classical physics a particle of mass m moving with speed v has kinetic energy KE = 1 2 mv2, and momentum p = mv. But theses formulas are only approximately correct if v c. The correct formulas at all speeds are KE = m c 2 mc 2, and p = 1 v2 c 2 m v. 1 v2 c 2 A mass at rest, not moving, has energy E = mc 2, where c = m/s is the speed of light. This means that energy and mass are equivalent. In a process, we may start with an initial mass m i, and end up with a smaller final mass m f, then the difference m i m f has been converted to energy, and the amount of energy release is E = (m i m f )c 2. If we could 5

6 do this an annihilate 1 kg of mass the energy released would be E = Joules, which is enough to run a 1,000 Megawatt plant for seconds, which is almost three years. Note that a coal fired 1,000 Megawatt electric plant consumes around 10,000 tons of coal per day!! In chemical reactions the amounts of mass that are annihilated are immeasurably small, the energy released is reasonable, and it was thought that mass was conserved. Now we know that energy, not mas, is really conserved, but only with nuclear reactions did this become clear. The General Theory of Relativity Objects under gravity fall with the same acceleration independently of their mass. When a car accelerates forward with an acceleration a, all objects in the car that are not attached to anything, as measured from the frame of reference of the car, accelerate backwards with an acceleration a, independently of their mass. This fact led Einstein to his principle of equivalence, which he used to arrive at a theory of gravity called general relativity. This is a very mathematical theory in which space is not nothing, but has properties that determine how objects will move. One can say that space-time is curved. In a gravitationally curved space-time objects move in a straight line, a geodesic. The generalization of a straight line in a flat space is a geodesic. 6

7 This is a curve that is generated by translating a segment parallel to itself. On a sphere this follows a great circle. In flat geometry the angles inside a geodesic triangle (made of straight lines) always add up to 180. On a sphere, which is a space of positive curvature, the angles in a geodesic triangle always add up to more than 180, and two geodesics starting from a point eventually come together. On a saddle shaped surface, which is a surface of negative curvature, the angles in a geodesic triangle always add up to less than 180, and two geodesics starting from a point eventually move further apart. All this is expressed in the mathematically complicated language of differential geometry. Some of the consequences are that clocks placed in a stronger gravitational field, such as at the ground floor of a building, run slower than clocks placed in a weaker gravitational field, such as at the top of the building. A dramatic prediction of General Relativity is the possibility of black holes. This is a mathematical solution of Einstein s equations discovered by Karl Schwarzschild in 1916 for the gravitational field caused by a point mass M. It was not really 7

8 understood because the solution had a singularity at a particular distance from the mass, now called the Schwarzschild radius R s R s = 2GM c 2, which for the mass of the Sun would be only 2.95 km. The Schwarzschild radius was understood in David Finkelstein and Martin Kruskal explained that is is the distance of the event horizon. Any object that approaches the mass M to a distance smaller than R s will never be able to get out again, and will ultimately be driven toward the mass M where the object will be crushed by infinite gravitational forces. This represents a challenge that seems to require that gravity and quantum physics be reconciled, which has not been done up to now. Outside the event horizon time slows down as one approaches it. If an astronaut falls toward the event horizon, we as far away observers would see his clocks and his motion slowing down more and more, so as far we are concerned it would take an infinite amount of time for the astronaut to reach the event horizon. However, the astronaut wold measure a finite time in his clock, and he would actually pass through the event horizon, never to leave again. If the black hole is small the astronaut would be torn apart by tidal forces as he approaches the event horizon. But if ithe black hole is very large he would not feel much of anything, but would be disconnected from our world forever. Do black holes exist? There is evidence for very large concentrations of mass in small spaces that attract stars and stuff into a very hot rapidly rotating accretion disk as in the artist conception shown below. 8

9 An actual observation of something that fits the theory for the accretion disk around a black hole is shown in the figure below. 9