Introduction to General Relativity

Problems with Newtonian Gravity

Newton was fully aware of the conceptual difficulties of  his action-at-a-distance theory of gravity. In a letter to Richard Bentley Newton wrote: So, clearly, Newton believed that something had to convey gravitational influence from one body to another. When later it became clear that influences travel at finite speeds it was reasonable to suppose this true of gravity also. But Newton's law of gravity did not incorporate the finite travel time of gravitational influences. If  right now the sun were to be destroyed by a passing black hole we would not feel the gravitational effects until about 8 minutes had elapsed. Because Newton's law did not include such retardation effects, and permitted violations of special relativity, it was clear that  Newton's law had to be an approximation to the correct law of gravity.

Special Relativity

In 1905 Albert Einstein introduced his theory of special relativity. With this theory Einstein sought to make the laws of motion consistent with James Clerk Maxwell's (1831-1879) laws of electromagnetism. Those laws predicted that light in vacuum traveled at a speed c (about 300,000 km/s) that was independent of the motion of the observer of the light and of the light source. Newton's law of motion, however, predicted that the speed of light should depend upon the motion of the observer. Einstein basically sided with Maxwell! Special relativity makes two postulates: A non-accelerating observer is said to be in an inertial frame of reference.

If you conduct an experiment in a moving vehicle (provided it is moving at a constant velocity relative to the ground) the experiment will give exactly the same result as one conducted in a laboratory at rest relative to the ground. This is why we can drink a can of soda just as well in a vehicle moving at a constant velocity as in one that is at rest relative to the ground. The first postulate says that there is no experiment we can do that can determine whether it is we who are moving, or the ground, or both. The most that any observer can do is to determine their speed relative to something. The earth goes around the sun at a relative speed of  30 km/s. But this value is just the speed relative to the sun. The earth has also a speed relative to the galactic center. Einstein proposed that there is no absolute meaning to the phrase: the earth's speed through space.

The second postulate says that the speed of light is always observed to be the same however we, or the source, might be moving. It is a universal invariant.

The consequence of Einstein's two postulates are radical: time and space become intertwined in surprising ways. Events that may be simultaneous for one observer can occur at different times for another. This leads to length contraction and time dilation, the slowing down of time in a moving frame. Every observer has her own personal time, caller proper time. That is the time measured by a clock at the observer's location. Two observers, initially the same age as given by their proper times, could have different ages when they met again after traveling along different spacetime paths.

Spacetime

It is extremely useful to think of space as filled with points, each labeled so that we can distinguish one point from another.  These labels are called coordinates.  We are free to use whatever labels we like. But, in practice, we use numbers as the labels because numbers are easy to manipulate.
A coordinate system is a set of rules that tell you how the labels are related to each other and to the points they label.  For example, in a Cartesian coordinate system the symbols x, y and z represent three numbers that label the points in 3-dimensional space.  These numbers can be arbitrarily assigned. We can, for example, choose any point in space to be the origin, that is to have the label (0,0,0), of the coordinate system; and we are free to use any coordinate system that's convenient. In astronomy it is often convenient to use a coordinate system called spherical polar coordinates, with each point labeled by the three numbers  p = (r, q, f), as shown in the figure below. The labeling of points is arbitrary, and has no intrinsic significance. But, the distance dl between two nearby points is an intrinsic geometrical property of the two points. The distance is independent of how the points are labeled. It is therefore independent of the coordinate system. An event is a given place at a given time. Einstein, and others, suggested that we should think of space and time as a single entity called spacetime. An event is a point p in spacetime. To keep track of events we label each by four numbers: p = (t,x,y,z), where t represents the time coordinate and x, y and z represent the space coordinates (assuming a Cartesian coordinate system). A good way to visualize ideas about spacetime is to use spacetime diagrams. Usually, we draw the time axis vertically with a single spatial coordinate, x say, horizontally. But sometimes we represent space as a two-dimensional plane. (As far as we know space is really 3-dimensional, as shown in the figure above, but it is impossible to fully visualize a 4-dimensional world.) In the figure below we represent space as a two-dimensional plane in x and y, with time increasing upwards. As a particle moves in space and time (that is, in spacetime) it traces a path called a worldline. The plane represents the present which, from the viewpoint of the particle, is of course always changing. Suppose at some event (that is, a given place at a given time), labeled O in the figure, there is a flash of light. From the viewpoint of an observer at O a spherical shell of light will expand away from event O at the speed of light. However, in spacetime this spherical shell looks like a hyper-cone. If we restrict space to a 2-dimensional plane in x and y, then the set of all paths followed by the light rays from O will form not a hyper-cone but a cone.

The angle a light ray (and therefore the cone) makes with the time axis is determined by the speed of light. If we choose our units so that the speed of light, denoted by c, is equal to 1 unit, this angle will be 45 degrees. Because nothing can travel faster than light the wordline of any material object must make an angle with respect to the time axis that, necessarily, is smaller than that of a light ray.

All points that lie within the forward cone are accessible, in principle, from O by a messenger from O traveling at a speed less than that of light. These points define the future of O. Similarly, the points within the backward pointing light cone define the past of O. They are all the points from which a messenger could have reached O without exceeding the speed of light. All other points are neither in the future nor the past of event O! This is so because it is not possible to send a messenger from O to any of these points, for the messenger would have to travel at a speed greater than that of light, which by assumption is impossible.

Just as we can define the distance between two points in space, dl, we can also define the distance between two nearby events in spacetime, ds. And like ordinary spatial distance, ds (which is called the spacetime interval) is invariant; that is, it is measured to be the same by all observers and its value does not depend on how we choose to label the events. The spacetime interval is defined to be where c is the speed of light and dt is the difference between the time labels at the two nearby events. The difference dt is called the coordinate time difference to remind us of the fact that its value depends on how we choose to label the events, in contrast to the interval ds which does not. Notice something peculiar about the interval. Its square can be positive, zero or negative! It turns out that this is the only way we can get an invariant length in spacetime.

If  ds2 > 0
the two nearby events are said to separated by a timelike interval. This means that a messenger could travel from one event to the other at less than the speed of light.  In other words, a messenger could set off from one spatial point, at a given time, and arrive at the nearby spatial point at the time which together with the spatial point define the second event.

If  ds2 < 0
the two nearby events are said to be separated by a spacelike interval. This means that no messenger could travel from one event to the other because to do so would require the messenger to travel at superluminal speeds (speeds greater than that of light).

Finally, if  ds2 = 0
the two events are said to be separated by a lightlike, or null, interval. That is, a messenger could connect these two events if  the messenger were to travel at exactly the speed of light. That is, you could start from one point, at a given time, and arrive at the other just in time, if your speed were exactly that of light. It follows that two null separated events can be connected by a light ray. Indeed, worldlines that are null intervals are precisely the paths followed by light rays in spacetime.

Consider a spacetime interval that has dl = 0. In this case What does this mean? This means that the two nearby events are actually at the same spatial point, but are separated in time by an amount dt. In this case the coordinate time difference dt coincides with the proper time between the two events; dt is the proper time that has elapsed between the two events because it is the time measured by a clock at that point. The elapsed proper time is just ds/c, that is, the spacetime interval divided by the speed of light.

Now consider setting dt = 0. In this case

This shows that when two events have the same time, that is, when they are simultaneous the spatial distance between them is invariant, that is, measured to be the same by all observers. This distance is called the proper distance between the two points.

Proper time and proper distance are very important concepts because they give us a way of mapping out spacetime so that all observers throughout spacetime would agree on the mapping. This is particularly important in cosmology, where we have to think about the entire universe in some sensible way.

As noted above the proper time is the time measured by a clock at an observer's location. So far we have considered the proper time of a special class of observers, namely, those who are in a fixed spatial location relative to a coordinate system; that is, those for whom dl = 0. But what about the proper time of observers who are moving relative to the coordinate system. How do we compute their proper times? Let's start with the formula for the spacetime interval By dividing throughout by c2 and pulling out a factor of dt2 on the right-hand side we can  re-write the interval as But, by definition, v = dl/dt is the observer's speed, relative to the coordinate system; so we can write We assume observers can travel only along timelike intervals; therefore, their speed v is always less than the speed of light, c. So in the above expression (v/c)2 is always less than 1. We now take the square root of that expression to obtain: The spacetime interval between two nearby points along the worldline of our moving observer is, by definition, just ds. The latter has the dimensions of distance; but when we divide it by the speed of light we convert ds to a quantity dt = ds/c that has the dimensions of time. Notice that dt = ds/c is always less than or equal to dt. It is only equal to dt when our observer is at rest in the coordinate system. This is the case we considered earlier, when we identified ds/c as the proper time of an observer fixed relative to the coordinate system (that is, dl = 0).

By continuity, we conclude that, in fact, dt = ds/c is the proper time not only for observers at rest in the coordinate system but also for moving observers, provided that they move only along timelike worldlines. This then gives us the rule for computing the proper time of any observer, whether at rest, or in motion, relative to our (arbitrarily chosen) coordinate system: the elapsed proper time between any two nearby events, along the worldline of an observer (regardless of their state of motion) is just the spacetime interval between the nearby points divided by the speed of light:

By adding up (that is, integrating) all the small elapsed proper times dt along the worldline we can compute the total elapsed proper time measured by a clock attached to any observer. The formula that relates dt to dt shows that the former is always less than or equal to the latter: the clocks of moving observers run slower relative to a clock that is stationary in the coordinate system. This is the famous time dilation effect of relativity theory.

Observers that are fixed in space (dl = 0) have particularly simple worldlines; this makes it easy to compute their elapsed proper times.  For more complicated worldlines the calculation is not quite as easy because we then have to worry about displacements both in coordinate time dt as well as in coordinate space dl.

The principle of equivalence

In Newton's theory two kinds of mass appear: inertial mass in his law of motion and gravitational mass in his law of gravity. In Newton's theory there is no physical reason why these masses should be related to each other. Therefore, the fact that, for all objects for which this has been checked, the inertial and gravitational masses are equal in value, to a very high precision, is an astonishing mystery in Newton's theory, that begs to be explained.

In characteristic fashion, Einstein turned this observation on its head. He hypothesized that these two kinds of mass are, in fact, one and the same and he sought to deduce the remarkable consequences of this hypothesis. Einstein was very good at using simple, but profound, physical reasoning to get to the heart of things. His discussion of free-falling elevators is a classic example, that we shall now repeat.

You are in an elevator that is at rest relative to the earth's gravitational field. The gravitational force on your body, called your weight, pushes you down onto the floor of the elevator. However, because you are neither going through the floor nor being thrown into the air it follows that the floor must be pushing up on you with exactly the same force. You experience this reaction force as your weight. 

Suddenly, disaster strikes; the elevator cables snap. Undaunted by the imminent termination of your worldline, as a brave seeker of truth, you decide to take stock of what's happening in the elevator. The most striking thing is that you have become weightless. Because of the equivalence of inertial and gravitational mass all objects fall freely with the same acceleration. The floor of the elevator is accelerating towards the center of the earth as fast as you are. This has two consequences: it cannot impart any force on your body and you remain at rest relative to the elevator. You take a pen from your pocket and let go of it. The pen appears to remain suspended in thin air. Again, this is because the pen is in free fall and is accelerating towards the earth at the same rate as the elevator.

What this shows is that gravity can be made to vanish merely by going to a frame of reference that is in free fall. If gravity can be so easily banished, Einstein reasoned that what we call the force of gravity may be an illusion; perhaps, gravity is not a force at all, but is somehow related to free motion in spacetime. Einstein went further: since gravity has been transformed away within the elevator all experiments conducted therein should give the same results as experiments carried out in a region far away from gravitational influences. Einstein summarized the results of his reasoning in his Principle of Equivalence, which can be stated thus:

That is, there is no experiment we can perform that will tell us whether we are in a free falling reference frame (like the elevator above) or in a reference frame far away in space. The consequences of this profound hypothesis are quite remarkable.

(By the way, note the use of the word local. Local means in this context a region of space that is sufficiently small so that the gravitational field can be considered uniform. If the region is too large we would notice the effects of the non-uniform gravitational field. Then the equivalence principle would not apply. Think about what would happen to two particles in a large falling elevator: the particles are falling towards the center of the earth along radial trajectories; therefore, we would see the particles come progressively closer as the elevator approaches the center of the earth. This would tell us that we were near a massive object, like the earth, rather than at a place far away in space. The equivalence principle would not be valid.)

Consider the figure below. In a rocket in free space (somewhere far away from stars and planets) a laser beam is emitted from one side of the rocket to a light detector on the opposite side. The astronaut sees the laser beam travel in a straight line from one side to the other. Now consider the same rocket in free fall near a planet. According to the equivalence principle the astronaut will again see the laser beam travel in a straight line across the cabin. So far, nothing strange! But now consider the same experiment viewed from the vantage point of someone who is at rest relative to the planet. The stationary observer also sees the event: laser beam hitting the light detector. (Events can't be changed just by changing your point of view!) But, by the time the light beam has crossed the cabin, the rocket and the light detector will have fallen a small distance. So the observer who is stationary with respect to the planet will see the laser beam follow a curved path.

Since the stationary observer believes herself to be in a gravitational field (because she feels her weight) she will conclude that gravity bends light. Einstein assumed that light, nonetheless, travels in as straight a line as possible. The fact that light's natural motion is curved could be understood if the spacetime through which it traveled were itself curved. Similar reasoning led Einstein to the further conclusion that clocks in a gravitational field run slower than clocks far from gravitational influences. Einstein sought to explain these effects as consequences of natural (that is, unforced) motion in curved spacetime.

After ten years of arduous intellectual searching, in 1915, Einstein succeeded, finally, in translating his profound physical intuition about Nature into a rigorous mathematical theory of free motion in curved spacetimes. Thus was born the general theory of relativity. Einstein's equations

connect matter and energy (the right-hand side) with the geometry of spacetime (the left-hand side). Each superscript stands for one of the 4 coordinates of spacetime; so what looks like one equation is actually 4 x 4 = 16 equations. But since some are repeated there are really 10 equations. Contrast this with the single gravitational law of Newton! That alone gives a hint of the complexity of these equations. Indeed, they are amongst the most difficult equations in science. Happily, however, some exact solutions have been found. Below we discuss one such exact solution, the first, found in 1916 by Karl Schwarzchild.

The Schwarzchild Solution

A solution of Einstein's equations is a formula that allows us to compute the spacetime interval between any two nearby points, given an assumed distribution of matter and energy. Given this formula we can calculate the free motion of objects in the spacetime geometry the formula describes, using the fact that freely moving particles travel on worldlines that are as straight as possible, that is, on geodesics. A well-known example of a curved geometry is the surface of the earth; it is a 2-dimensional space whose geodesics are great circles. These are circles whose centers coincide with the center of the earth. Aircraft fly along great circles because the latter are the shortest lines between any two points on the earth's surface.

The German scientist Karl Schwarzchild found the first exact solution of Einstein's equations just months after the publication of these equations which, given their complexity, was a pretty impressive feat. The metric (as the expression for spacetime intervals is called) found by Schwarzchild is

    (1) ds2 = c2(1-2MG/c2r)dt2 - dr2/(1-2MG/c2r) - r2(dq2+sin2qdf2).

It describes the spacetime geometry surrounding a spherically symmetric object of mass M, situated at the spatial coordinate r = 0. This metric gives an excellent description of the spacetime geometry around objects like the sun and the earth, that are, to a good approximation, spherical. This metric is the basis of the three classic tests of Einstein's theory of general relativity: 1) the perihelion advance of Mercury, 2) the bending of starlight by the sun and 3) the slowing down of clocks by gravity. A detailed understanding of the first two phenomena requires familiarity with differential equations, and shall therefore not be considered here. To understand the third requires nothing more than basic algebra; so let's take a look at it.

Clocks run more slowly in the presence of gravity. To see this imagine two clocks (these could be two atoms of hydrogen) fixed in space near a large mass M; one clock placed at the radial coordinate r = r1 and the other at r = r2. What we want to do is to compare the elapsed proper times of the two clocks. How do we compute their proper times? Remember the rule: to get the proper time difference set the spatial coordinate differences to zero in the metric, that is, in the formula for the spacetime interval and then divide the interval by the speed of light.  Since each clock remains fixed in space the elapsed time measured by each clock will be a proper time interval. That's what we want to compute.

If we follow our rule and set dr = dq = df = 0, and then divide by c, in equation (1), we obtain

as the elapsed proper time dt on a clock fixed at location r. We see that dt is not the proper elapsed time. Now apply equation (2) to our two clocks at r = r1 and r = r2 and compare the two proper time intervals, for the same coordinate time difference dt. We find It should be clear from equation (3) that if r1 < r2 the numerator will be smaller than the denominator; that is, the clock that is closer to the mass M measures a shorter elapsed proper time than the clock that is further out. The closer clock runs slower. Notice, also, a very strange thing: as the closer clock (clock number 1) is moved ever closer its elapsed proper time gets progressively shorter. At some point r1 will be equal to the critical radius At that radius the elapsed proper time is zero; the clock appears to have stopped! The critical radius is called the Schwarzchild radius. It divides the spacetime around the mass M into two regions, an outer region r > rc and an inner realm r < rc. Later we shall learn that rc is the coordinate radius of the event horizon of a black hole.

To finish off we shall apply equation (3) to the earth. We'll set r1 = R, the earth's radius, and r2 = R+h, where the height h is taken to be much smaller than R. We can get a simpler (approximate) expression for the ratio of proper times by noting the that for the earth 2MG << c2R. (<< means very much less than.) With this approximation and using the formula (1+x)n = 1 + nx, which is valid for x << 1, we obtain, from equation (3),

in which we have kept only terms linear in 1/R, since the rest are much smaller and therefore can be neglected. The expression in the square brackets is approximately h/R2 and with a minor rearrangement we arrive at If we interpret the proper time interval dt2 as one complete period of oscillation of light waves at position r = R+h, then we can compute by how much the frequency of a light wave at the ground differs from one a height h above it, due to the slow-clock effect. The light wave at ground level has only advanced by an amount dt1, which is not yet a complete cycle. The proper duration dt1 of the complete cycle for the light wave at ground level is longer than dt2 by an amount dt2/dt1. Therefore, dt1 = dt2 (dt2/dt1), or (1/dt1)/(1/dt2) = dt1/dt2. By definition, the frequencies of the light waves are n1 = 1/dt1 and n2 = 1/dt2, so our final answer is So if a light ray leaves the ground and rises to a height h it will be found to have a lower frequency n1 than the frequency n2 of a similar light ray that is already at height h.

This effect has actually been observed. In 1960 the scientists R.V. Pound and G.A. Rebka (Phys. Rev. Letters 4, 337 (1960)) shot the 14,400 electron-Volt gamma rays from radioactive iron (Fe57) up the 21.6 meter tower at Harvard University, and tried to absorb the gamma rays in similar iron nuclei at the top of the tower. But since the frequency of the gamma rays is predicted to be (slightly) lower than the natural frequency of the iron the gamma rays were absorbed less efficiently than normal. Then Pound and Rebka introduced an inspired trick (based on an effect discovered shortly before by Mossbauer): they changed the natural frequency of the iron absorber by moving the iron nuclei upwards, at just the right speed, thus causing a lowering of the natural frequency due to the Doppler effect. The gamma rays were then readily absorbed by the moving  nuclei. The scientists determined that the frequency of the rising gamma rays was less than the natural frequency of the stationary iron nuclei, at the top of the tower, by a fractional amount equal to 2.56 x 10-15, in excellent agreement with the prediction from equation (7) of 2.46 x 10-15. The Pound-Rebka experiment is one of the most beautiful of 20th century science.