Introduction to Relativity

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 or the light source. 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, that is, a constant speed in a fixed direction) the experiment will give exactly the same result as one conducted at rest. This is why we can drink a can of soda just as well in a vehicle moving at a constant velocity as we can when we are at rest relative to the ground. 

The first postulate says, in effect, that it is impossible to determine whether it is we who are moving, or the ground, or both. The most we can do is to determine our speed relative to something. The Earth goes around the Sun at a relative speed of  30 km/s. But this value is the speed relative to the Sun. The Earth also moves relative to the galactic center. Einstein proposed that there is no absolute meaning to the phrase: the Earth's speed through space. All we can ever say is the Earth's speed relative to something. That something could be, for example, the Earth's speed relative to the cosmic microwave background.

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. Every observer has her own personal time, called proper time. It 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 meet again after traveling along different paths through space and time. 

Example: Time dilation. To see how these postulates alter our view of time and space consider the appearance of a laser, on board an aircraft, which is fired vertically downwards towards a light detector, also on the aircraft. From the point of view of someone on the aircraft, which we suppose is traveling horizontally at a speed v relative to the ground, the laser light travels a vertical distance d at the speed of light c in a time t. So we have

d = ct

From the point of view of someone on the ground, by the time the light reaches the light detector the aircraft has moved a horizontal distance 

vT 

in a time T as measured by the ground observer. In that time T the light will have traveled a distance 

cT

at the same invariant speed c along the hypotenuse of a triangle. From Pythagorus' theorem we have

(cT)2 = (vT)2 + d2 = (vT)2 + (ct)2

which, when rearranged yields

t = T[1 - (v/c)2]1/2

We see that the two observers do not agree on how long it takes the light to reach the light detector. The aircraft's clock runs slowly, as judged by the observer on the ground. This is a simple example of the time dilation effect, which is discussed more fully later. 

It is important to note that the time dilation effect is symmetric: the observer on the aircraft, who sees the ground go past at the speed -v, will determine that it is the ground clock that slows down, not his! This seems paradoxical. The paradox is resolved by noting that events that are simultaneous for one observer need not be simultaneous for another. But this is best understood by thinking in terms of space-time. 

Space-Time

It is extremely useful to think of space  and time as a dense set of 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.  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 space-time. An event is a point p in space-time. 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, if we are using a Cartesian coordinate system, or p = (t,r,q,f), if we are using spherical polar coordinates. A good way to visualize space-time is to use space-time diagrams. Usually, for a particular observer, we draw the time axis vertically with a single spatial coordinate, x, 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 for us to 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 space-time) it traces a path called a world-line. The plane represents the present which, from the viewpoint of the particle, is of course always changing. It is important to remember that each observer has her own time and space axes. Another observer, moving relative to the first, would have a time axis that is tilted with respect to the time axis of the first observer; that is, tilted with respect to the time axis shown in Fig. 2. 

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 space-time this spherical shell looks like a hyper-cone, called the light cone. If we restrict space to a 2-dimensional plane in x and y, then the set of all world-lines traced out by the light rays from O form a cone. 

The angle a light ray (and therefore the cone) makes with the time axis, of a given observer, 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 word-line 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 events that lie within the forward light cone are accessible, in principle, from O by a messenger from O traveling at a speed less than that of light. The set of 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 according to Einstein's theory is impossible.

  Where is the Past?

It is important to understand the distinction between events, which are true elements of reality, and the arbitrary labeling of events in terms of space and time coordinates, which can be done in infinitely many different ways. Moreover, according to the view of space and time as a single indivisible entity called spacetime events do not somehow cease to be as we perceive time to progress towards the future. What we call the past is every bit as real as what we call the present. According to Einstein:

"The distinction between past, present and future is only an illusion, even if a stubborn one."

When is Now?

Time is not absolute. Therefore, we should not be surprised if observers disagree about "Now". Let's consider a nice example, due to Prof. Paul Davies. Two sisters Ann and Betty decide to do the following.

Ann stays on Earth, while Betty goes off on a trip to a star that is 8 light years away at 0.8 the speed of light and immediately returns to Earth at the same speed. From Ann's point of view it takes 10 years for Betty's spaceship to reach the star and another 10 years for Betty to return to Earth. Thus if Betty sets off in 2000, Ann and Betty will not meet again until 2020 as measured by Earth time. 

However, because of the time dilation effect, Betty's clock runs slower by a factor 0.6 relative to Ann's time. (This factor comes from putting v/c = 0.8 in [1 - (v/c)2]1/2.) Ann therefore deduces that Betty arrives at the star when Betty's clock reads 2006, while the clock on Earth reads 2010. Betty does in fact arrive at the star in 2006 according to her clock. Note, however, that Ann will not be able to actually verify Betty's 2006 arrival date at the star until 2018 Earth time when the light from Betty's spacecraft reaches the Earth after traveling for 8 years according to Ann's reckoning! 

But the time dilation effect is symmetric for inertial observers (that is, for non-accelerating observers)! Therefore, Betty reckons that it is Earth time that is slowed down by the factor of 0.6. So when Betty arrives at the star in 2006, according to her clock only 0.6*6 = 3.6 years will have elapsed on Earth. Therefore, for Betty, her arrival at the star is simultaneous with 2003.6 back on Earth, not 2010! 

To summarize: according to Ann 2010 Earth time is simultaneous with Betty's arrival at the star, while for Betty it is 2003.6 Earth time that is simultaneous with that same event. Ann's "now" and Betty's "now" do not agree. This is why both can infer that each other's time is slowed relative to their own without contradiction. However, when they meet again Betty is definitely younger by 8 years than she would have been had she stayed on Earth. 

But how can this be if time dilation is a symmetric effect? The answer is that Ann and Betty's situation are not quite symmetrical. While Betty is moving at a constant speed relative to Earth both she and Ann are inertial observers and the time dilation effect is indeed perfectly symmetrical. But at the star, Betty must decelerate and then accelerate to return to Earth. She is then no longer an inertial observer. It is this period of deceleration and acceleration that breaks the symmetry between Ann and Betty and which permanently warps Betty's time relative to Ann's.

There is one more puzzle. In spite of Ann's deduction, it is not clear why Betty should necessarily arrive at the star in 2006. The puzzle is resolved because of another relativistic effect: length contraction. Ann reckons the star to be 8 light years away. But not Betty! For Betty the distance between the Earth and the star has shrunk by the same relativistic factor of 0.6; therefore the star is only 0.6*8 = 4.8 light years away! Since Betty approaches the star at 0.8 times the speed of light she will get there in 4.8/0.8 = 6 years, that is, in 2006 in agreement with Ann's deduction and Betty's clock.  In order to avoid paradox in relativity it is necessary to include both length contraction and time dilation effects. Only then does one get a consistent picture.

Faster Than Light, Tachyons and Causal Paradoxes

According to relativity the speed of light presents an impenetrable barrier in the sense that anything traveling slower than the speed of light can never reach or exceed it. 

But could there exist objects whose speeds already exceed that of light? In principle, yes, since relativity does not explicitly forbid their existence. Such objects are called tachyons. To be consistent with relativity, however, tachyonic objects must have some very odd properties:

We can see how the last point arises by considering the diagram below

In 2000 Betty sets off to visit a nearby star. Unfortunately, the severe acceleration of Betty's spaceship as she hurtles away from Earth causes a tachyonic shockwave to propagate to another star that is also 8 light years from Earth. Let's suppose that the shockwave arrives at the other star in 2006, Betty's time; that is, coincident with Betty's arrival at the first star. And, alas, the shockwave destroys the second star.  

This shockwave propagates into Betty's future, from 2000 to 2006. Nothing odd about that. But the star's destruction is coincident with 1997.2 on Earth. Therefore, from Ann's perspective the shockwave travels from 2000 to 1997.2; that is, backwards in time! 

Even stranger, the destruction of the star occurred before Betty set off on her journey! So in Ann's frame of reference, the star's destruction (the effect) occurred prior to the shockwave (the cause). Since the star is 8 light years away, according to Ann, the light heralding the star's demise will not reach her, on Earth, until 1997.2 + 8 = 2005.2. We have engineered a causality violation in Ann's frame of reference: that is, we have arranged for the effect to precede its cause in at least one frame of reference.

Suppose a civilization near the star witnesses its destruction. They send an emergency tachyonic message to Earth that reaches Earth, say, in 1998 before Betty's departure, warning her to forgo her journey and thereby avert the accidental destruction of the star by the shockwave. 

Betty acts on this message and abandons her trip; no shockwave occurs and therefore the star is not destroyed. But then since the star is not destroyed no warning message is sent, Betty does not abandon her trip and consequently the star is destroyed inadvertently. We have arrived at a logical contradiction!

Because of such paradoxes, some scientists, notably Stephen Hawkins, have conjectured that the universe enforces a chronology protection:   The laws of the universe are such that it limits what can be done so as to avoid paradox. This does not necessarily imply that faster than light objects do not exist; it merely implies that if they do they must behave in such a way as to avoid paradox. How that can be achieved, however, is far from clear.

Just as we can define the distance between two points in space, dl, we can also define the distance between two nearby events in space-time. The important fact about the space-time distance, called the space-time interval, is that it 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 space-time interval ds is defined by 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 space-time. (In some books a different sign convention is used: the negative sign is attached to the spatial part of the interval. Here I've decided to adopt the convention used in modern books on relativity.)

If ds2 < 0
the two nearby events are said to separated by a time-like 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 coordinates define the second event.

If ds2 > 0
the two nearby events are said to be separated by a space-like 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 light-like, 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, world-lines that are null intervals are precisely the paths followed by light rays in space-time.

Consider a space-time 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 |ds2|1/2/c, that is, the square-root of the absolute value of the square of the space-time interval divided by the speed of light. The vertical lines |..| mean absolute value, that is, the value ignoring the sign.

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 space-time so that all observers throughout space-time 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 space-time 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 time-like 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 space-time interval between two nearby points along the world-line of our moving observer is, by definition, just |ds2|1/2. Notice, when v = 0, dt = |ds2|1/2/c reduces to the proper time of an observer at rest relative to the coordinate system. By continuity, we conclude that

dt = |ds2|1/2/c

is the proper time not only for observers at rest in the coordinate system but it is the proper time for all observers, however they move,  

provided that they move only along time-like world-lines. By integrating (that is, adding up) all the proper time intervals dt along a time-like world-line 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 for time-like world-lines. The clocks of moving observers run slower relative to a clock that is stationary in the coordinate system. This is the time dilation effect of relativity theory that we have already considered in the Ann/Betty example.

Observers that are fixed in space (dl = 0), relative to a coordinate system, have particularly simple world-lines; this makes it easy to compute their elapsed proper times.  For more complicated world-lines 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. However, if you know how the speed v(t) varies with time, as measured by an observer who is at rest relative to the coordinate system, then you can compute the proper time by integration:

(b) t = òdt (1- v2(t)/c2)1/2

Here's a nice problem to try: (You need a little integral calculus!) Assuming that the speed of a spaceship varies like

v(t) = c sin(2pt/T),

where t is the proper time on Earth, compute the elapsed proper time on the spaceship, assuming that the total elapsed proper time on Earth is T. Note that quantity (1- v2(t)/c2)1/2 must always be positive and that the return journey (which lasts an Earth time of T/2) is just the mirror image of the outward journey. 

 

General Relavity

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.

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. Following Einstein, we shall now engage in a bit of simple physical reasoning whose conclusions are quite extraordinary.

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 we conclude 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 world-line as a sentient being 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: The floor 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 and you.

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 somehow is related to free motion in space-time. 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 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 are elements of reality and therefore can't be changed just by changing your point of view! Both observers agree about the existence of the event "laser beam hits light detector." 

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. That's the only way to maintain consistency between the two observers.

Since the stationary observer believes herself to be in a gravitational field (because she feels her weight) she concludes that gravity causes the light to bend. 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 space-time through which light 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 space-time.

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 space-times. Thus was born the general theory of relativity. Einstein's equations

connect matter and energy (the right-hand side) with the geometry of space-time (the left-hand side). Each superscript stands for one of the 4 coordinates of space-time; so what looks like one equation is actually 4 x 4 = 16 equations. But since some are repeated there are really only 10 independent 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 Schwarzschild Solution

A solution of Einstein's equations is a formula that allows us to compute the space-time 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 space-time geometry the formula describes, using the fact that freely moving particles travel on world-lines 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 Schwarzschild found the first exact solution of Einstein's equations just months after the publication of these equations which, given their complexity, was an impressive feat. Even more remarkable is the fact that he did this whilst a soldier at the front, during the First World War! The metric (the distance relation) found by Schwarzschild is

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

It describes the space-time geometry surrounding a spherically symmetric object of mass M, situated at the spatial coordinate r = 0. Notice that when M = 0, that is, when there is no mass the metric reduces to that given earlier (see equation (a)). This metric gives an excellent description of the space-time geometry around objects like the Sun and the Earth, that are, to a good approximation, spherical. It also shows that both time as well as space are warped by the presence of the mass. 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 space-time interval) and then divide |ds2|/1/2 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, after setting the spatial components dr = dq = df = 0 in equation (1), we obtain

as the elapsed proper time dt given by a clock fixed at coordinate location r. We see that, unlike the situation in flat space-time (equation (a)), dt is not the proper elapsed time at that location. But if dt is not the proper elapsed time what is it? Equation (2) shows that when r is very large, that is, when we are far away from the mass, dt = dt. So we see that dt is the proper time of an observer far away from the mass. 

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 (or, if you like, for the same proper time difference of an observer far away from the mass). 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 becomes equal to the critical radius and the elapsed proper time is zero; the clock stops, as seen from the point of view of a distant observer! The critical radius is called the Schwarzschild radius. It divides the space-time 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 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 advanced by an amount dt1, which is not yet a complete cycle. The proper duration dt1 of one 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.

The Pound-Rebka Experiment. This effect has actually been observed. In 1960 the scientists R.V. Pound and G.A. Rebka (Phys. Rev. Letters 4, 337 (1960)) shot 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.