Why do we need a new theory?

Lecture 7: General Relativity We now begin our foray into the beautiful, yet sophisticated theory of General Relativity. At first we will explain the need for a new theory and then outline the ideas. We will not be able to go into the full details of the theory as that requires highly sophisticated mathematics, such as non-euclidean geometry and tensor calculus. At times we may display equations from the theory and explain their significance but will not expect you to get numerical results or understand the full implications of the equations. Our goal will be to display the geometric ideas back of the theory and tie it to what we have learned about special relativity. The ideas we learn here will allow us to discuss with more sophistication the ideas behind black holes and cosmology. Why do we need a new theory? First of all, as we pointed out before, SR is limited in validity to non-accelerating frames. To come up with a more general theory which can describe both relativistic scenarios for IRFs and frames which are non-inertial is key to have a complete description of physical phenomena. Second, and also very important has to do with the gravitational force as described by Newton s Universal Law of Gravitation (NUG). In all of our discussions of SR we never (or rarely) mentioned gravity. We explained the contradictions between Maxwell s equations and Galilean relativity and then modified the latter. But take a look at NUG for a moment, up to this point the definitive law of gravitation, F = G M M 1 2 ˆr 2 12 r 12 Notice that there is nothing within this equation that limits the speed at which the gravitational force is transmitted. In Maxwell s theory, light, or EM waves, transmitted the electric and magnetic forces at the speed of light. Under Newton the gravitational force is transmitted instantaneously. Einstein (and Minkowski) found that there is a logical inconsistency if gravity was transmitted instantaneously yet electromagnetic forces were transmitted only at c. A quick example will help make this clear. At planet A, a sensitive gravity detector is attached to a very large explosive charge which will destroy the planet. Spacestation B is 1 ly away. Dr. Evil sets out on a spaceship to B. When the ship reaches the station, thrusters start moving the station away from the planet. The detector on A picks up the change in gravitational attraction to the station and when it reaches a critical value the bomb explodes. (This example needs to be treated very carefully within general relativity since the energy used to push the station away will have an effect as well). So the spacetime diagrams in the planet s frame and Dr. Evil s frame would look like, As with the examples discussed previously, we see that causality is violated. In one frame Dr. Evil leaves the planet and causes the explosion, in the explosion occurs before Dr. Evil arrives to the space station, hence he could not have caused the explosion in this frame. 1

Another difference we can see right away. Under Newton, only masses can experience the gravitational force. However, we just learned that E= mc 2, the equivalence of mass and energy. So we see that energy sources (like heat energy) will now contribute to the gravitational force. So we see that there is a need to incorporate accelerating frames and gravity into the relativistic theory. We will see shortly that these two goals are closely related. In 1905 Einstein published the paper On the Electrodynamics of Moving Bodies (special relativity paper) and he knew at that time the limitation with it. From this period until 1915, when the full theory of General Relativity was published, Einstein worked on the bits and pieces that came to be the full theory. It was not an easy task. For the full theory required complex mathematics that physicists, for the most part, did not know at the time. The mathematical theories of non-euclidean geometry, tensor calculus, etc. were developed in the 1800s by many brilliant mathematicians. Yet up to this point there were not any physical applications of these theories. Fortunately, Einstein had some of these mathematicians as teachers, yet he still had to learn these ideas while developing his theory. Einstein worked on the general theory as a natural extension of the special theory. In the late 1800s there was evidence of a possible shortcoming of the Newtonian Law of Gravitation. It had been known by LeVerrier as early as 1859 that the advance of the perihelion of Mercury, the furthest point from the Sun for its elliptical orbit, did not quite agree with the theoretical result from NUG. Some suggested modifying NUG slightly but none of these ideas seemed to work exactly. In November of 1914, Einstein calculated the theoretical result from his partially created general theory and found that it agreed with the experimental result. Reiterating, as we approach this topic we will not be able to go into the full details nor do any serious quantitative analysis. We will explore the foundations of the theory with emphasis on the geometrical nature. As is the case in SR, the metric plays a central role in GR. We will spend some time exploring metrics of more complex spaces. To begin, let s highlight the path we will follow: 1) Gravity = Acceleration. Known as the Equivalence Principle, this will play a key role when we explain some of the effects of GR. We also see that this combines the tasks at hand (incorporating non-inertial frames and gravity) into one. 2) Curvature Acceleration. Here we will explore some metrics of curved spaces and show that two objects in a curved space will see each other accelerate if they move with respect to each other. The curvature tells objects how to move. 3) Energy Curvature This step is the key to GR and we will only be able to get a cursory look at how this works. The basic idea is that mass (more accurately energy from E = mc 2 ) curves spacetime. Diagramming these ideas, 2

In modern terms, there is no longer a force of gravity, there is a warping of spacetime which tells objects how to move through it. An object s energy curves spacetime, which in turn tells the object how to move. Gravity = Acceleration (The Equivalence Principle) So we begin with the first topic and examine some classical examples of non-inertial reference frames. First recall the definition of an inertial reference frame, An inertial reference frame is a frame in which Newton s First Law of Motion is valid. Alice stands on the train platform and Bob rides within a train car which accelerates past Alice. Within this car a mass hangs from a string. What do each observe. Alice observes the ball to be suspended at an angle due to the acceleration. Bob as well sees the ball hang at an angle. In Bob s frame there appears to be a force which does not have a source (not gravity, not electric). What causes this force that Bob sees? You may recall from your course on mechanics that this is termed a pseudoforce. Another example often discussed is the pseudoforce you experience when riding in a car which is going around a curve. You may think that some force is pushing you into the side of the car, but as you learned, this is nothing more than the car pushing on you as you want to travel in a straight line. A third example is seen in weather systems on Earth. You may have learned that weather systems (storms) circulate because of the Coriolis force, a pseudoforce due to the rotation of the Earth. The following set of four examples will bring us to our point. Example 1: Consider Bob in a sealed elevator which is free falling to the ground of a very tall building, (ignore air resistance and variation of Earth s gravity). Alice stands outside but can view into Bob s elevator. As the elevator falls, Bob holds out a ball and drops it. What do each observe? Alice observes: the elevator, Bob, and the ball to accelerate downwards at g = 9.8 m/s 2. Bob observes: a) There are no forces acting on the ball. The ball and himself are free floating. Equivalently; b) There is a pseudoforce acting upward which cancels the force of gravity. 3

Example 2: EPGY Special and General Relativity Bob reaches the ground floor (safely) and is next to Alice, both at respect to each other. Alice observes: The ball to accelerates downwards at g = 9.8 m/s 2. Bob observes: The ball to accelerates downwards at g = 9.8 m/s 2. Example 3: Now we relocate to outer space. Alice is free floating and observes Bob in his elevator equipped now with a rocket which accelerates him upwards. The rocket is tuned so that the elevator/spaceship accelerates at 9.8 m/s 2 from rest. Alice observes: The ball at rest (until the floor hits it and it then accelerates upward). Bob accelerates upwards. Bob observes: The ball accelerates down until it hits the ground. In this case Bob in no way can tell the difference between whether he is on the ground in the building or accelerating through space (examples 2 and 3). Example 4: Alice free floating in space and Bob at rest (or in motion at constant velocity) with respect to Alice. Alice observes: The ball at rest (or moving with constant velocity), does not accelerate. Bob observes: The ball at rest, does not accelerate. Examining example 2 and 3 we noted that Bob can not tell whether he is in a uniform gravitational field or in an accelerating reference frame. Examining example 1 and 4 we note that Bob in no way can determine whether he is free falling in a uniform gravitational field or whether he is in an IRF in deep space. The conclusion is that a (small, local) frame falling in a uniform gravitational field is an IRF! (Again, we need to limit this to his small elevator and a uniform field. These points will be brought up in the problem session). This argument (which Einstein says when he realized this it was one of the happiest days in his life) led Einstein in 1907 to the Equivalence Principle, which we now state. 4

The laws of physics are the same at each point in a uniform gravitational field as in a local reference frame undergoing uniform acceleration. Relate this to the first postulate of SR. The laws of physics are the same in all IRFs moving with respect to each other at constant velocity. Remember Einstein extended the Newtonian relativity principle to all physical laws. The same is proposed here. ALL physical laws are the same in a uniformly accelerating frame as they are in a uniform gravitational field. It becomes clear quickly that this leads to a new effect that can be (and has been) verified experimentally. Consider Bob in his rocket accelerating upwards. He fires a laser from one end of his elevator/rocket to the other. Now, since the Equivalence Principle says that what happens in the confines of the elevator as it accelerates is exactly what happens in a uniform gravitational field (where the acceleration due to gravity is the same as that of the elevator). In the accelerating frame (Bob s frame) the light appears to fall as it crosses the elevator. Hence, in the elevator sitting on the ground floor, the light should appear to fall as well! This result can not be explained via Newton s Theory of Gravitation since only massive objects can exert a gravitational force upon each other. Of course, we have greatly exaggerated the effect here, the amount that light falls is very small. In order to observe this effect you need a very massive body and a long distance for it to fall. Consider the Sun and light emanating from a far off star grazing the Sun and reaching Earth. Its path will be deflected slightly. In 1919 Eddington set out on an expedition to Africa to measure this effect during a total eclipse of the Sun, (in this way you can see the stars near the sun). The experiment was claimed a success and Einstein became a worldwide celebrity. (However it was later found that this experiment could not verify the theory within experimental error. Future expeditions provided more experimental evidence and today the result has been verified to rather high accuracy.) 5

Principle of Maximal Aging, geodesics. A local frame that is either free floating away from any massive bodies or is in free fall in a uniform gravitational field is an inertial reference frame, which we will now call a free falling frame (FFF). An object placed in one of these frames will not experience any net force upon it. Recall the principle of maximal aging; we considered the case of an object floating freely in deep space and stated, via the principle, that the proper time measured by this object between two events will be greater than any other proper time for a frame which connects the two events. Since this free falling frame is equivalent to the free floating frame, we see that this object will, as well, measure the greatest proper time between events as compared to other frames which connects the two events. To see that this is different than what we discussed in SR, consider the following example. Two objects are suspended high up in a uniform gravitational field with an acceleration due to gravity of 9.8 m/s 2. One of the objects is dropped and falls freely to the ground, say a distance of 1 km. The other object is released at the same time but is so equipped that it can travel at a constant velocity between the departure point and the ground. The two objects hit the ground at the same time. A simple calculation shows that it takes 14.3 s for the free falling object to hit and thus the other object must be moving at 70 m/s (about 150 MPH) to have the same arrival event. If we use the flat metric from before, (the interval s 2 = c 2 t 2 z 2 ), it should be clear that the object moving at constant velocity will have a longer proper time than the free falling object, (which has a curved worldline). However, an object in the free falling frame has no net force acting upon it (put it in an enclosed box), it is an inertial reference frame. If we examine the object moving at constant velocity there is a net force on it (again consider it to be in an enclosed box), there is a net force acting upon it and it falls to the floor of the box. Hence, this is not an inertial reference frame. By the Principle of Maximal Aging we would expect the free falling frame to have the longest proper time and not the one at constant velocity. What we have done here is to relegate the gravitational force to a pseudoforce, a force due solely to the non-inertial nature of this frame. Under this new view, there is no force of gravity, only the natural movements of objects through spacetime. Another conclusion from this example is that the flat 6

spacetime metric is no longer suitable to situations near gravitating bodies. A new metric is called for. These worldlines of longest proper time are special worldlines. They are the frames in which Newton s Laws are valid (IRFs). Since the first postulate of relativity demands that Newton s Laws remain valid in IRFs, these are the natural worldlines of objects moving through spacetime. (Much like objects at rest or traveling at constant velocity in Newtonian physics). These worldlines are actually specific mathematical entities, called geodesics. The formal definition will not concern us here but we will consider them to be free float or free falling frames (FFFs). Again, these are the natural trajectories for objects moving through spacetime, not disturbed by outside forces. We can use these trajectories to label or graph spacetime. For a two-dimensional space the geodesics are the shortest lines connecting two points. On the Earth, flying from San Francisco to Hong Kong the shortest path between these two points travels up near Alaska. This great circle route is the geodesic between these two points. As geodesics are lines of shortest distance in space, they are the lines of longest proper time in spacetime. Lines of shortest proper time are not unique, there are an infinite amount of them between two events. Consider the flat spacetime plot for a frame away from any gravitating bodies. The graph consists of many straight lines of constant position x. These are the (timelike) geodesics for objects at rest in this frame. Now consider another simple example. In deep space, five balls are thrown up in the z direction at different velocities. The worldlines for all of these objects are all straight lines (geodesics). We could use these lines to map spacetime (lines of angle ). Repeat this experiment on a platform high up in a uniform gravitational field. The worldlines of these objects are no longer straight but curve as they go up and then fall back down. (A rotated version is shown to highlight the parabolic motion). 7

Since these objects are free falling frames (consider again these to be elevators and place a ball inside of it) these lines are geodesics. From the previous example it should be clear that straight worldlines would be frames in which pseudoforces would be present. We see that the natural trajectories, geodesics, for spacetime in a uniform gravitational field are curved. The conclusion is that in the presence of a gravitational field we can no longer draw a spacetime diagram with straight lines. The flat SR interval is no longer valid in this situation. This leads us to the inescapable conclusion that spacetime near a gravitating body is no longer flat, but curved. The idea that we have been motivating (in a different order) is that curved space leads to acceleration, which is equivalent to gravity via the Equivalence Principle. In short, curved spacetime is equivalent to gravity. A simple example will make clear that a curved space leads to acceleration. Consider the surface of the Earth and place Alice and Bob in boats separated by some distance at the equator. Their natural tendency is to move north at constant speed, (their geodesic). As they both head off in the same direction they will observe each other to accelerate towards each other (restricting their vision to the surface of the Earth). It is the curvature of space that leads to the observed acceleration. They may ascribe this acceleration to some force, gravity, because they are unaware they live on a curved surface. Once they realize that they live on a curved surface they conclude that there is no such force but simply natural movement on this surface. This is an analogy between the Newtonian view of gravity and the General Relativistic view, where there is no force but natural movement through a curved space. To go on we will need to discuss curved spaces in a little more detail. We will return to a discussion of metrics for curved spaces. Afterwards, we will try to answer the question of what causes the curvature of spacetime. 8

Discussion questions: - Consider a reference frame near the Earth s surface that is in free fall but is extremely wide. Consider two objects on either end of this frame and their movement as the frame moves downwards. Is there anything which is observed which would make you believe that this is not an inertial reference frame? - Consider an IRF as introduced in the SR section; a lattice work of friends, each with a clock. How does this scenario need to be modified in the presence of a gravitating body? Consider both a uniform field and a field near a spherical massive body. (Near a spherical body consider how they should be placed in the radial direction). - What type of body would you need to produce a constant, uniform, gravitational field throughout all space. Do such objects occur commonly? Do they occur at all? 9