Lecture 7: General Relativity - Stanford...

EPGY Special and General Relativity

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Lecture 7: General Relativity

We now begin our foray into the beautiful, yet sophisticated theory of General Relativity. At first we willexplain the need for a new theory and then outline the ideas. We will not be able to go into the full details of the theoryas that requires highly sophisticated mathematics, such as non-Euclidean geometry and tensor calculus. At times wemay display equations from the theory and explain their significance but will not expect you to get numerical results orunderstand the full implications of the equations. Our goal will be to display the geometric ideas back of the theoryand tie it to what we have learned about special relativity. The ideas we learn here will allow us to discuss with moresophistication 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 andframes 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 byNewton’s Universal Law of Gravitation (NUG). In all of our discussions of SR we never (orrarely) mentioned gravity. We explained the contradictions between Maxwell’s equations andGalilean relativity and then modified the latter. But take a look at NUG for a moment, up to thispoint the definitive law of gravitation,

F =G

M1M

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r122

r̂12

Notice that there is nothing within this equation that limits the speed at which the gravitationalforce is transmitted. In Maxwell’s theory, light, or EM waves, transmitted the electric andmagnetic forces at the speed of light. Under Newton the gravitational force is transmittedinstantaneously. Einstein (and Minkowski) found that there is a logical inconsistency if gravitywas 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, thrustersstart moving the station away from the planet. The detector on A picks up the change in gravitational attraction to thestation and when it reaches a critical value the bomb explodes. (This example needs to be treated very carefully withingeneral 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 planetand causes the explosion, in the explosion occurs before Dr. Evil arrives to the space station, hence he could not havecaused the explosion in this frame.


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Another difference we can see right away. Under Newton, only masses can experience thegravitational force. However, we just learned that E=mc2, the equivalence of mass and energy. Sowe 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 therelativistic theory. We will see shortly that these two goals are closely related.

In 1905 Einstein published the paper “On the Electrodynamics of Moving Bodies” (specialrelativity paper) and he knew at that time the limitation with it. From this period until 1915, whenthe full theory of General Relativity was published, Einstein worked on the bits and pieces thatcame to be the full theory. It was not an easy task. For the full theory required complexmathematics that physicists, for the most part, did not know at the time. The mathematical theoriesof non-Euclidean geometry, tensor calculus, etc. were developed in the 1800s by many brilliantmathematicians. 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 theseideas while developing his theory. Einstein worked on the general theory as a natural extension ofthe special theory.

In the late 1800s there was evidence of a possible shortcoming of the Newtonian Law ofGravitation. It had been known by LeVerrier as early as 1859 that the advance of the perihelion ofMercury, the furthest point from the Sun for its elliptical orbit, did not quite agree with thetheoretical result from NUG. Some suggested modifying NUG slightly but none of these ideasseemed to work exactly. In November of 1914, Einstein calculated the theoretical result from hispartially 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 doany serious quantitative analysis. We will explore the foundations of the theory with emphasis onthe geometrical nature. As is the case in SR, the metric plays a central role in GR. We will spendsome 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 someof the effects of GR. We also see that this combines the tasks at hand (incorporatingnon-inertial frames and gravity) into one.

2) Curvature Acceleration.

Here we will explore some metrics of curved spaces and show that two objects in acurved space will see each other accelerate if they move with respect to each other. Thecurvature ‘tells’ objects how to move.

3) Energy CurvatureThis step is the key to GR and we will only be able to get a cursory look at how thisworks. The basic idea is that mass (more accurately energy from E = mc2) curvesspacetime.

Diagramming these ideas,


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In modern terms, there is no longer a force of gravity, there is a warping of spacetimewhich tells objects how to move through it. An object’s energy curves spacetime, which in turntells the object how to move.

Gravity = Acceleration (The Equivalence Principle)

So we begin with the first topic and examine some classical examples of non-inertialreference 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 rideswithin a train car which accelerates past Alice. Within thiscar a mass hangs from a string. What do each observe. Aliceobserves the ball to be suspended at an angle due to theacceleration. Bob as well sees the ball hang at an angle. InBob’s frame there appears to be a force which does not havea source (not gravity, not electric). What causes this forcethat Bob sees? You may recall from your course onmechanics that this is termed a pseudoforce. Anotherexample 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 youinto the side of the car, but as you learned, this is nothing more than the car pushing on you as youwant to travel in a straight line. A third example is seen in weather systems on Earth. You mayhave learned that weather systems (storms) circulate because of the Coriolis force, a pseudoforcedue 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 ofEarth’s gravity). Alice stands outside but can view into Bob’selevator. 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 acceleratedownwards at g = 9.8 m/s2.

Bob observes: a) There are no forces acting on the ball. The ball andhimself are free floating.

Equivalently;b) There is a pseudoforce acting upward which cancels the

force of gravity.


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Example 2:

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/s2.Bob observes: The ball to accelerates downwards at g = 9.8 m/s2.

Example 3:

Now we relocate to outer space. Alice is free floating andobserves Bob in his elevator equipped now with a rocket whichaccelerates him upwards. The rocket is tuned so that theelevator/spaceship accelerates at 9.8 m/s2 from rest.

Alice observes: The ball at rest (until the floor hits it and it thenaccelerates 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 betweenwhether he is on the ground in the building or accelerating throughspace (examples 2 and 3).

Example 4:

Alice free floating in space and Bob at rest (or in motion at constantvelocity) with respect to Alice.

Alice observes: The ball at rest (or moving with constant velocity), does notaccelerate.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 uniformgravitational field or in an accelerating reference frame. Examining example 1 and 4 we note thatBob in no way can determine whether he is free falling in a uniform gravitational field or whetherhe is in an IRF in deep space. The conclusion is that a (small, local) frame falling in a uniformgravitational 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 whenhe realized this it was one of the happiest days in his life) led Einstein in 1907 to the EquivalencePrinciple, which we now state.


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“The laws of physics are the same at each point in a uniform gravitationalfield 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 movingwith respect to each other at constant velocity. Remember Einstein extended the Newtonianrelativity principle to all physical laws. The same is proposed here. ALL physical laws are thesame 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) verifiedexperimentally. Consider Bob in his rocket accelerating upwards. He fires a laser from one end ofhis elevator/rocket to the other.

Now, since the Equivalence Principle says that what happens in the confines of the elevatoras it accelerates is exactly what happens in a uniform gravitational field (where the accelerationdue to gravity is the same as that of the elevator). In the accelerating frame (Bob’s frame) the lightappears to fall as it crosses the elevator. Hence, in the elevator sitting on the ground floor, the lightshould appear to fall as well! This result can not be explained via Newton’s Theory of Gravitationsince only massive objects can exert a gravitational force upon each other. Of course, we havegreatly 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 tofall. 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 measurethis effect during a total eclipse of the Sun, (in this way you can see the stars near the sun). Theexperiment was claimed a success and Einstein became a worldwide celebrity. (However it waslater found that this experiment could not verify the theory within experimental error. Futureexpeditions provided more experimental evidence and today the result has been verified to ratherhigh accuracy.)


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Principle of Maximal Aging, geodesics.

A local frame that is either free floating away from any massive bodies or is in free fall in auniform gravitational field is an inertial reference frame, which we will now call a free fallingframe (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 deepspace and stated, via the principle, that the proper time measured by this object between two eventswill be greater than any other proper time for a frame which connects the two events. Since thisfree 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 thetwo 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/s2. One of the objects is dropped and falls freely to theground, say a distance of 1 km. The other object is released at the same time but is soequipped that it can travel at a constant velocity between the departure point and theground. The two objects hit the ground at the same time. A simple calculation shows that ittakes 14.3 s for the free falling object to hit and thus the other object must be moving at 70m/s (about 150 MPH) to have the same arrival event.If we use the flat metric from before, (the interval s2 = c2t2 – z2), 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. Ifwe 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 uponit and it falls to the floor of the box. Hence, this is not an inertial referenceframe. By the Principle of Maximal Aging we would expect the free fallingframe 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 apseudoforce, 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


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spacetime metric is no longer suitable to situations near gravitating bodies. A new metric is calledfor.

These worldlines of longest proper time are special worldlines. They are the frames inwhich Newton’s Laws are valid (IRFs). Since the first postulate of relativity demands thatNewton’s Laws remain valid in IRFs, these are the natural worldlines of objects moving throughspacetime. (Much like objects at rest or traveling at constant velocity in Newtonian physics). Theseworldlines are actually specific mathematical entities, called geodesics. The formal definition willnot 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 outsideforces. We can use these trajectories to label or graph spacetime.

For a two-dimensional space the geodesics are the shortest lines connecting two points. Onthe Earth, flying from San Francisco to Hong Kong the shortest path between these two pointstravels up near Alaska. This great circle route is the geodesic between these two points. Asgeodesics are lines of shortest distance in space, they are the lines of longest proper time inspacetime. Lines of shortest proper time are not unique, there are an infinite amount of thembetween two events.

Consider the flat spacetime plot for a frame away from any gravitating bodies. The graph consists of manystraight 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 differentvelocities. The worldlines for all of these objects are all straight lines (geodesics). We could use these lines to mapspacetime (lines of angle ).

Repeat this experiment on a platform high up in a uniform gravitational field. The worldlines of these objects are nolonger straight but curve as they go up and then fall back down. (A rotated version is shown to highlight the parabolicmotion).


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Since these objects are free falling frames (consider again these to be elevators and place a ballinside of it) these lines are geodesics. From the previous example it should be clear that straightworldlines would be frames in which pseudoforces would be present. We see that the naturaltrajectories, geodesics, for spacetime in a uniform gravitational field are curved. The conclusion isthat in the presence of a gravitational field we can no longer draw a spacetime diagram withstraight lines. The flat SR interval is no longer valid in this situation. This leads us to theinescapable 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 toacceleration, which is equivalent to gravity via the Equivalence Principle. In short, curvedspacetime is equivalent to gravity.

A simple example will make clear that a curved space leads to acceleration. Consider thesurface 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 offin the same direction they will observe each other to accelerate towards each other (restricting theirvision to the surface of the Earth). It is the curvature of space that leads to the observedacceleration.

They may ascribe this acceleration to some force, ‘gravity’, because they are unaware theylive on a curved surface. Once they realize that they live on a curved surface they conclude thatthere is no such force but simply natural movement on this surface. This is an analogy between theNewtonian view of gravity and the General Relativistic view, where there is no force but naturalmovement through a curved space.

To go on we will need to discuss curved spaces in a little more detail. We will return to adiscussion of metrics for curved spaces. Afterwards, we will try to answer the question of whatcauses the curvature of spacetime.


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Discussion questions:

- Consider a reference frame near the Earth’s surface that is in free fall but is extremely wide. Consider twoobjects on either end of this frame and their movement as the frame moves downwards. Is there anythingwhich 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 doesthis scenario need to be modified in the presence of a gravitating body? Consider both a uniform field and afield near a spherical massive body. (Near a spherical body consider how they should be placed in theradial direction).

- What type of body would you need to produce a constant, uniform, gravitational field throughout allspace. Do such objects occur commonly? Do they occur at all?

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