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General Relativity

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

Background Reading: J. P. McEvoy and O. Zarate, Introducing Stephen Hawking. Totem Books. pp. 9 - 46.

Special and General Relativity

The special theory of relativity was a first step for Einstein. The fuller development of his goal of relativizing physics came with his general theory of relativity. That theory was completed in its most important elements in November of 1915. By many measures, the special theory was a smaller achievement. Its final creative phase took Einstein some 5 to 6 weeks. Of all the new theories of 20th century physics it is usually regarded as the most conservative. Had Einstein not published the theory in 1905, we have good reason to think that it would have emerged in one form or another. Both Lorentz and Poincaré had developed the essential equations; they just put a different interpretation on them than did Einstein.

The general theory of relativity took seven years of work by Einstein, the final two to three being years of intense and exhausting labor. No one else was even close to Einstein's ideas. Had he not worked on them, they would most probably not have emerged then. We may not even have them today. In some ways, Einstein's theory is conservative. It is the last "classical" field theory in the sense that "classical" can mean "non-quantum." In another sense, it is anything but conservative. The theory is quite different from any theory before or after. It treats a force by means of geometry and eventually leads to startling notions: black holes, other universes and the bridges to them and even the possibility of time travel. All other theories of forces have been readily swept into quantum theory. General relativity has resisted and the problem of bringing general relativity and quantum theory together remains one of the most difficult, outstanding puzzles of modern physics.

In a Nutshell: Gravitation is Curvature of Spacetime

Before we start to delve into the theory in greater detail, we should just state its basic idea. The theory is based on a single, luminous, dominant idea.

In Newton's classical account of gravitation, the earth wants to move inertially, that is, uniformly in a straight line. A gravitational force from the sun deflects it and causes it to move in an elliptical orbit around the sun.

In Einstein's theory, the presence of the sun disturbs--curves--the very fabric of space and time. The earth then merely moves inertially in this new disturbed spacetime. It follows an inertial trajectory, but that trajectory has been distorted so that it ends up as an ellipse in the space around the sun; or, more precisely, a helical trajectory winding around the sun's worldline in spacetime.

General relativity combines the two major theoretical transitions that we have seen so far. These two transitions are depicted in the table below. The first is represented in the vertical direction by the transition from space to spacetime. We learned from Minkowski that special relativity can be developed as the geometry of a spacetime. The analogy is quite close. The trajectories of bodies in inertial motion are straight lines in spacetime in the sense that they are curves of greatest proper time, that is, timelike geodesics. That makes them the analogs of the straight lines of Euclidean geometry, which are also called geodesics, the curves of shortest distance.

The second transition is represented in the horizontal direction in the table. It is the transition from flat to curved geometry. In the context of ordinary spatial geometry, that transition takes us from the venerable geometry of Euclid to the geometry of curved surfaces of the nineteenth centry. In the context of spacetime theories, that same transition takes us from the geometry of a flat spacetime, the Minkowski spacetime of special relativity, to the geometry of the curved spacetimes of general relativity. The central idea of Einstein's general theory of relativity is that this curvature of spacetime is what we traditionally know as gravitation.

This makes learning Einstein's general theory of relativity much easier, for we have already done much of the ground work. The mathematics needed to develop the theory is just the mathematics of curved spaces, but with the one addition shown: it is transported from space to spacetime.

There is a great deal more that could be said--and some of it will be. Einstein himself gave a rather detailed account of the theory as generalizing the principle of relativity to accelerated motion. In first approaching the theory, I will say little about that. I will take you along a different pathway that avoids many of the unnecessary pitfalls of Einstein's account. The problem is that, in retrospect, it is very far from clear just how that generalization was brought about or even if it was done at all. So let's concentrate on curvature. The royal road to curvature is geodesic deviation.

Geodesic Deviation: a Refresher

Let us recall how geodesic deviation allows us to detect the positive curvature of a spherical surface.

A number of observers all start at the equator of a sphere. They proceed in the same direction, due North. As they proceed, the paths converge, eventually crossing at the North Pole. Here is the familiar view of the surface of the two dimensional sphere embedded in a three dimensional space is shown in the first figure.

How would this appear to someone trapped in the surface, without the higher viewpoint of the third dimension? They would map out the trajectories as shown. We detect the positive curvature of the surface in the convergence of the paths of the travelers. Had they diverged, we would have diagnosed negative curvature.

Free Fall inside the Earth...

Can we find similar effects in spacetime? Then we would have found curvature. So what we seek is a sheet of spacetime in which we find converging or diverging curves. As we shall see, that will be easy to find. A collection of masses in free fall in a gravitational field will provide exactly the sort of curves we need.

To get us started, we will take the simplest case as far as the curvature is concerned, although the set up physically is a bit messier.

Imagine that we drill a hole through to the center of the earth and out to the other side. It will be 6400 miles long. We evacuate the resulting tube and cap it so that bodies dropped in the hole can fall without any air resistance at all.

One of the oddities of gravity is that this period of 84 minutes (=42 minutes there + 42 minutes back) is fixed, no matter where the ball may first be released. Imagine, for example, that it is released from rest halfway between the surface and center. It would take the same 42 minutes to cross the center and come momentarily to rest at the corresponding point on the other side of the center; and then another 42 minutes to make the trip back to its starting point.

Here's an animation that shows balls starting at different places in the hole. Imagine that the balls are so small that they pass by one another without interference. (That is hard to draw, so the animation just shows them passing through each other.)

Now let's plot the motions through time over the 42 minutes needed for a ball to fall past the center and come to rest at the other side:

If you compare this spacetime diagram to the earlier figure of the travelers on the earth's surface, you will see that they agree in the essential aspect. They both show converging trajectories, the hallmark of positive curvature. This allows us to interpret the gravitational motions in a novel way.

The temptation is to call this convergence "geodesic deviation." We need to be a little cautious here since the trajectories in the spacetime are not necessarily geodesics, that is, curves of shortest distance. They are not in Newtonian theory. In relativity, both special and general, they are timelike geodesics. That is, they are curves of greatest proper time, which is the analog of the straight lines of Euclidean geometry, the curves of shortest distance. So we can yield to the temptation and, in so doing, arrive at the essential idea of Einstein's theory.

...Reinterpreted

Here's how we pass to the essential idea of Einstein's theory.

Newton's theory: These motions are due the force of gravity deflecting the bodies from the trajectories they want to follow into the oscillations we see.

Reinterpreted theory: the sheet of space-time displayed in the spacetime diagram is instrinsically curved. The trajectories followed by the bodies in free fall are simply the straightest lines of this new curved geometry.

(We'll call this a sheet of space-hyphen-time to indicate that the sheet has one spatial dimension and one temporal dimension.)

This turns out to be an especially simple case as far as curvature is concerned in two ways.

Second, the magnitude of the curvature does not depend on the mass or size of the earth; it depends only on the mass density of the earth. (This is not obvious. An easy calculation in Newtonian theory can show it, however.)

These last two points are important enough to be stated in a relation that is close to (but not quite) one that holds very generally:

In this formula Newtonian "mass density" has been replaced by the vaguer "matter density" in anticipation of what will transpire in general relativity, where the density of matter is a more complicated quantity that embraces energy and momentum densities as well as stresses.

The analysis can be generalized. We considered just one space-time sheet, the one swept out through time by the hole we imagined drilled through the earth. Nothing in the analysis depended upon where we drilled the hole. We could have drilled many holes. Each would sweep out a different surface in spacetime to which this analysis would apply. In general, there are three independent spatial directions we could have chosen, correspondingly to the three axes of a three dimensional space. Finding the curvature in the three resulting sheets would be enough to fix the curvature in all possible sheets generated by holes we may dig.

Uniqueness of Free Fall

We have reinterpreted gravitational accelerations as manifestations of an intrinsic curvature of spacetime. So far, we have actually posited nothing new, physically beyond "geometrizing away gravitational forces." Everything said so far could be carried through in Newton's theory of gravity without affecting any of the observationally testable predictions of the theory. We have just re-packaged an old theory in a very unfamiliar way.

One special fact about gravity makes it an especially apt redescription. There is a uniqueness in free fall trajectories that is peculiar to gravity. If we drop a one pound ball in the tube, it will take 42 minutes to pass to the other side of the earth. The same is true of a two pound ball; or a three pound ball; or a ball of any mass. They all take 42 minutes to pass to the other side of the earth. While they do it, they follow exactly the same trajectory. So if we release a one, two and three pound ball at the same moment, they will remain together as they traverse the hole to the other side. This is the uniqueness of free fall.

It is just the latest version of the result Galileo made famous when he wrote of dropping objects of different mass from a tower and noting that they would fall alike.

In Newtonian theory, the result is given more complicated expression. The quantity that measures how a gravitational force will act on a body in some gravitational field is its gravitational mass. The quantity that measures how much a given body will accelerate when acted on by a force is the body's inertial mass. It is an unexplained coincidence in Newtonian theory that these two masses are equal. The result is the uniqueness of free fall. A two pound mass feels twice the gravitational force than does a one pound mass in the same gravitational field, since it has twice the gravitational mass. But the two pound mass is still accelerated by the same amount, since it has twice the inertial mass and so resists acceleration twice as much.

Notice that if electric forces were pulling the balls through the tube, this uniqueness of fall would fail. There is no coupling of inertial mass and electric charge. So if we drop one body which carries twice the charge of a second, there is no assurance that the inertial mass is also doubled; and so no assurance that the two will fall alike.

This remarkable result of the uniqueness of free fall is what makes the reinterpretation very comfortable. We can think of the spacetime sheet as having a natural spacetime geometry revealed to us by masses. That geometry is largely independent of the masses. For all masses--big and small--reveal the same trajectories. The masses are more like probes exploring an independently existing structure.

Finally Einstein's reinterpretation eradicates an awkwardness of Newtonian theory. That theory had to posit that increases in gravitational mass in bodies are perfectly and exactly compensated by corresponding increases in inertial mass, so that the uniqueness of free fall can be preserved. Einstein's redescription does away with that coincidence and even the very idea of distinct inertial and gravitational masses. In his theory, bodies now just have mass, or, in the light of special relativity, mass-energy. For Einstein the primitive notion is the geometrical structure of spacetime with the curved trajectories traced out by all freely falling bodies, independently of their mass.

Gravity Above the Surface of the Earth

So far, we have dealt with an especially simple case in which the curvature of the space-time sheet is everywhere the same. More generally curvature in spacetime will vary from event to event and, even at one event, it will be different according to the particular space-time sheet considered. This is simply a re-expression in curvature language of the more familiar fact that gravitation varies from place to place and acts differently in different directions.

We can explore this variability by considering masses falling under the action of gravity above the surface of the earth. (As before, we will ignore the rotation of the earth.)

Masses Distributed Vertically

To begin, consider masses 100 miles above the surface, stacked up 1 mile apart as shown and initially at rest with respect to earth.

Now let them fall freely. The masses closer to the earth will feel a slightly stronger pull of gravity, so they will fall slightly faster. It is easy to compute the effect. In the course of 18.3 seconds, the masses will fall roughly one mile. A mass that is one mile closer , however, will fall 1.6 feet more than a mass starting one mile higher.

If we plot these motions through time on a spacetime diagram, we recover a familiar figure.

This is just a space-time sheet showing diverging trajectories; that is, this particular space-time sheet has negative curvature.

Masses Distributed Horizontally

We will get a different outcome if we consider masses aligned horizontally. As before they are 100 miles above the surface of the earth and spaced one mile apart, but now they are distributed horizontally.

In this case, each mass will feel the same gravitational force. However those forces will pull in a slightly different direction for each mass. The forces are all directed towards the center of the earth. If the masses start from rest and go into free fall, after 18.3 seconds they will have fallen one mile. Since they are being pulled by forces that converge to one point, the center of the earth, the masses will have converged slightly in the course of falling. It turns out that each mass will be 0.8 feet closer to its neighboring masses as a result of the motion.

These motions can be plotted on a spacetime diagram, from which we recover a familiar figure.

This is just a space-time sheet showing converging trajectories; that is, this particular spacetime sheet has positive curvature.

To sum up, we can identify three space-time sheets passing through an event 100 miles above the surface of the earth in which free fall motions are plotted. In the sheet spatially oriented in the up-down direction, we find a negative curvature. The remaining two sheets are spatially oriented east-west and north-south. In each of those we found a positive curvature.

Summed Curvature and Matter Density

By considering masses in free fall within a tube bored through the earth, we saw a connection between the curvature of the space-time sheet and the matter density. Since we know that matter produces gravity and that gravity is now to be represented by a curvature of spacetime, you might suppose that this is a general relation. Could the spacetime curvature just be proportional to matter density everywhere? That clearly doesn't work. Above the surface of the earth, there is no matter density, but there certainly is gravity and, as we have just seen, curvature of the space-time sheets as well.

So we need a weaker relationship between curvature of the space-time sheets and matter density. That relationship turns out to be easy to see if we just tabulate the cases we've seen so far. The curvature revealing deviations in the space-time sheets, as mapped out by masses in free fall, will be measure by the convergence or divergence of masses one mile apart when they fall one mile.

The table suggests the correct result. For the case of space-time sheets above the earth where there is no matter density, the curvature revealing deviations in each is non-zero. But their sum is zero. We define the summed curvature to be the sum of the curvatures of the space-time sheets for the three different spatial directions. Then we can write the connection between curvature and matter density as:

This relation amounts to a natural relaxing of the too stringent condition that curvature must be proportional to matter density. For with the relaxed condition, we can still have curvature in the individual space-time sheets at events where the matter density vanishes. For example, we can have negative curvature in the sheets aligned with the up-down spatial direction. But to keep the summed curvature zero, we must have positive curvature in sheets aligned in the other directions.

From Curvature in Space-Time Sheets to Space-Space Sheets

All our considerations so far apply equally to Newton's theory of gravity as to Einstein's new theory. They have dealt only with curvature of space-time sheets, that is, in two dimensional surfaces in spacetime that are spacelike in one direction and timelike in another.

As the figure shows, the spacetime will also have space-space sheets. These are just the ordinary two dimensional slices of three dimensional space. Within the context of Newton's theory, reinterpreted as a theory of spacetime curvature, curvature does not extend to them.

That they should be treated differently makes sense in the Newtonian context. For there space and time are treated very differently.

It is quite another matter when we move to relativity theory. The core innovation of Einstein's special theory of relativity was a mixing together of space and time, manifested most vividly as the relativity of simultaneity. In a Minkowski spacetime, there are many ways to slice up the spacetime into spaces that persist through time. So, in the relativistic context, it is no longer so natural to have one rule for space-time sheets and another for space-space sheets.

Where Einstein's general theory of relativity deviates sharply from Newton's is that Einstein requires the curvature associated with gravity to extend from space-time sheets to space-space sheets as well; and for all to be governed by the same relationship. This difference can be summarized as:

This table summarizes the core ideas but avoids a lot of very messy technical and mathematical issues. Let us just consider Einstein's theory.

What ought to represent matter density? In Newtonian mechanics, that would just be mass density. We learned from special relativity that mass is not such a simply quantity. The real concept is mass-energy and what complicates things further is that the amount of mass-energy a body has will vary with the frame of reference. These are just the beginning of a series of complications.

It turns out that an adequate representation of the matter density at an event in spacetime requires a catalog of a lot of information: energy density, moment densities, energy fluxes, momentum fluxes and all the various forms of stress that may also be present. The synopsis of all this information in a 4x4 table is known as the stress energy tensor.

There is a similar problem in determining precisely which quantity should represent the summed curvature. There is a single 4x4x4x4 table, known as the Riemann curvature tensor, that represents all the curvature information pertaining to the different sheets in spacetime. Somehow we need to extract an appropriate sum of curvature quantities from it. There are several ways to do this.

Deciding which was the right way proved to be a special stumbling block for Einstein. The final answer, however, became so strongly connected with Einstein's theory that it is now named after him. It is called the Einstein tensor. As with the stress-energy tensor, the Einstein tensor can be written down as a 4x4 table of numbers computed from the numbers in the 4x4x4x4 table of the Riemann curvature tensor.

The precise mathematical expression of the idea of a connection between summed spacetime curvature and matter density is known as

Einstein's gravitational field equations

EINSTEIN TENSOR equals STRESS-ENERGY TENSOR

These are the core equations of Einstein theory and the crowning glory of Einstein's discovery. In one set of equations they embrace the entirety of gravitational phenomena as well as the geometry of space.

These equations decide which spacetimes--that is, which universes--are admissible according to Einstein's theory. The admissible ones will be those spacetimes in which the spacetime curvature and matter density are related appropriately.

While this is easy to say, the mathematical difficulty of finding a spacetime that satisfies Einstein's equations is immense. Success is so hard that we usually celebrate it by naming the spacetime after the person who first shows that it satisfies the gravitational field equations.

Note that we cannot turn this around. If we have a spacetime in which the stress energy tensor is zero, so that the Einstein tensor is zero, it does not now follow that the curvature is also zero. We have already seen that one can have a non zero curvature that yields a zero summed curvature.

The Geometry of Space Near a Massive Object

Einstein's theory allows that the geometry of space can become curved as well in the vicinity of very massive objects. That is true for the space we know that is close to both the great masses of the earth and sun. However the deviation from flatness in these spaces is so slight that no ordinary measurement can detect it.

For this reason, we believed for millennia that our space is exactly Euclidean, whereas it is only very nearly so. The deviation of spatial geometry from the Euclidean becomes more noticeable once we consider very intense gravitational fields or the enormous distances of cosmology.

To get a sense of just how close our local geometry is to Euclidean, let us estimate the disturbance to it due to the presence of the sun. Consider a huge circle around the sun that roughly coincides with our earth's orbit. Euclidean geometry tells us that the circumference of this circle is 2π x radius of the orbit.

Imagine that we now approach the sun one mile at a time and draw a new circle centered on the sun at each step. The Euclidean result tells us that for each mile we come closer to the sun, the circumference of the circle is diminished by 2π miles.

That is the Euclidean result. Because of the presence of the sun, space around the sun is not exactly Euclidean. According to general relativity, for each mile that we come closer to the sun, the circle does not lose 2π miles in circumference; it loses only (0.99999999)x2π miles.

If we tried to build a model out of paper or plastic that had this property, it could not lie flat in the Euclidean space of our model builder's room. Instead as we added the portions of the surface that lie closer to the sun, those portions would pop out of the surface. That popping out is a kind of embedding diagram and one of the most frequently built models in the context of general relativity.

The model captures an important geometrical fact about the space around our sun--that it is no longer exactly Euclidean. However it is misleading in two ways.

First, since it is an embedding diagram, we should not be misled into assigning any physical reality to the higher dimensioned space in which the surface is modeled. It is introduced solely for our ease of visualization.

Second, a common way of encapsulating Einstein's theory is to roll marbles across the model and suggest that gravitational attraction somehow comes from the resulting deflection of the marble's roll. From the discussion above, you can see why that is misleading. The gravitational deflection of ordinary objects falling in the vicinity of the sun is due to the curvature of the space-time sheets. What the model shows is the curvature of the space-space sheets and that curvature is so small as to have negligible effects on the motions of ordinary objects

Causal Structure

One of the consequences of Einstein's theory will have special importance to us. Gravity is a curvature of spacetime that affects all free fall motions. Light propagating is one of those motions. So just as massive bodies like planets and comets are deflected toward the sun, so also in light.

One of the characteristics of a Minkowski spacetime and the more general spacetimes of Einstein's theory spacetime is that it has a light cone structure that is usually taken to map out the fastest trajectories for causal interactions. Since gravity affects light, it will also affect this causal structure. The effect of gravitation is to tip the light cones in the direction of the gravitational attraction.

This can have some very interesting consequences, such as new regions of spacetime causally isolated from our region. These arise in the theory of black holes and we will see more of them later.

The Three Tests

Shortly after Einstein complete his theory, he announced three empirical tests that he believed established the theory. Two had yet to be done. They were:

Mercury

According to Newton's theory, planets orbit the sun along elliptical paths. Here's a picture of the orbital motion according to Newton's theory; and an animation:

Einstein's theory predicted the same, but added that the axis of the ellipses of the planetary orbits would advance very slightly. That means the axis would rotate slowly in the same direction as the planet's motion. In Mercury's case, the advance would be about 43 seconds of arc per century. This amount of advance is really very small. To see this, note that there are 60 minutes in one degree and 60 seconds in one minute. So 43 seconds of arc is very much less than a single degree. It would be impossible to use a sharp pencil and a big sheet of paper to draw two intersecting straight lines that intersect at 43 seconds of arc. They would be so close that they would appear like one line. Yet this is the extra advance Einstein's theory predicts over the time of 100 years.

Here is a picture of this advance, with the size of the advance greatly exaggerated, and an animation:

That so called "anomalous" advance had already been observed but no final explanation had been agreed on for it. When Einstein discovered that his theory predicted this elusive 43 seconds of arc, it might well have been the greatest scientific moment of his life. He recalled having heart palpitations, being unable to sleep and a sense that something inside snapped.

Of course the matter was more complicated than the above gloss suggests. Even in Newtonian theory, the ellipse of Mercury's orbit was expected to move by over 400 seconds per century due to the perturbations of the other planets. That means that the gravitational attraction of the other planets pulls Mercury off the simple elliptical orbit computed in their absence. Adding in the effects of these perturbations, Newtonian theory could account for all but about 40 seconds of the motion of the axis of Mercury's orbit. Until Einstein was able to explain it exactly with his general theory of relativity in late 1915, this small discrepancy did not seem to be very worrisome. It was only afterwards that explaining it became a sine qua non for any new gravitation theory.

Light bending.

According to Einstein's theory, light, just like any other form of matter, is affected by gravity. That is, light also "falls" in a gravitational field. Just as a comet's trajectory is deflected by the sun when is passes nearby, a ray of starlight grazing the sun would also be deflected. Einstein computed that the deflection would be about 1.75 seconds of arc. The deflection had two components. Half of the deflection is due to the curvature of space near the sun. The other half arises merely from the light falling towards the sun. This deflection was verified by expeditions in 1919 that took photos of the stars near the sun at the time of a solar eclipse.

What complicates the measurement is that one gets half of Einstein's predicted deflection in Newtonian theory. One merely needs to assume that light is a form of matter that falls in a gravitational field in Newtonian theory, just as every other form of matter falls. That is sufficient to give half the deflection of Einstein's theory.

A minor variation on this effect arises if the deflecting body is massive enough to bring together the light that passes on either side of it from a luminous body behind it. Then the deflecting body acts a kind of lens, focusing the light. In the figure, the observer would see two images of the same object. In the case of perfect alignment, the observer would see a ring of duplicated images. This effect, known as "gravitational lensing," has only recently been observed. While Einstein did not discuss the effect in his publications, it turns out that he had computed it in a private notebook in 1913.

Here's a spectular image of gravitational lensing:

Dowloaded from http://hubblesite.org/newscenter/archive/releases/1995/14/image/a/format/web_print/ February 15, 2007.

Red Shift

According to Einstein's theory, time runs slower closer to massive bodies. That means that natural clocks in the sun run slower than the same clocks on earth. Of course there are no ordinary clocks in the sun. But there is something much better. Excited atoms emit light in very specific frequencies and our measuring the frequency of that light is akin to our measuring the frequency of ticking of a clock. Any slowing of those atomic clocks would result in a change in the frequency of light emitted from the sun.

Einstein's theory predicts a very small degree of slowing of clocks in the sun. It manifests in the light from the sun being slightly reddened for observers watching from far afield on the earth. The red shift for light from the sun is merely 0.00002%, which proved extremely difficult to detect. The effect was found later in the light from stars far more massive than the sun. The figure shows light climbing out of the stronger gravitational field of the sun towards the earth.

Einstein's Pathway to General Relativity

The pathway to general relativity mapped out above was not Einstein's. His own was more tortured and fallible. The theory emerged after seven years of struggles with many things: strong hunches on how the theory should look, vivid thought experiments to support the hunches, lengthy explorations into new mathematics, errors and confusions that thoroughly derailed him and final insight that answered to exhausted desperation.

It is impractical to review all these considerations. However some are so famous and so characteristic of Einstein, that they must be mentioned.

The Starting Point

There Can be no Adequate Theory of Gravitation in Special Relativity.

Einstein came to this startling conclusion in 1907, while still a clerk in a patent office and only two years after he had published his special theory of relativity. He had been commissioned to write a review article on relativity theory. His efforts to show how gravitation theory could be modified to fit with special relativity failed. The stumbling block was an old result due to Galileo. All bodies in free fall descend with the same acceleration. Einstein could not recover this result in the theories he tried.

One problem was that the fall of bodies in the theories he tried was affected by their sideways motion. So a body moving horizontally would fall vertically at a different rate from one without the horizontal motion.

Gravitation and Acceleration

The decisive insight came to Einstein in the form of a simple thought experiment. If one is in a closed box (in special relativity or Newtonian theory) and the box is accelerated uniformly in a straight line, everything would accelerate in one direction inside the box. In Newtonian terms, they are left behind by the acceleration due to their inertia. This is the same effect that presses you back in the seat of a car when it accelerated from a stop light. They are sometimes called "inertial effects."

How are we to interpret these inertial effects? Einstein noticed masses undergoing inertial effects behave formally exactly like bodies in a homogeneous gravitational field. So, he asked, what if the state of space inside the box just is a gravitational field? Then he would have found a far reaching link between gravitation and acceleration. Einstein liked this a lot since he wanted to extend the principle of relativity to acceleration. It looked like this reinterpretation would help. An observer inside the box need not presume that the box is really accelerating. It could be at rest with a homogeneous gravitational field inside. So Einstein asserted:

Principle of Equivalence

The inertial effects inside a uniformly accelerated box in gravitation free space are equivalent to those of a homogeneous gravitational field; more tersely, uniform acceleration creates a homogeneous gravitational field.

Learning About Gravitation

The immediate benefit of this new principle was that it let Einstein learn a lot about gravitation. He knew exactly what would happen to processes in a uniformly accelerated box; he now knew these things must happen to the same processes in a gravitational field. Two conclusions were especially important. He knew that a light beam traversing the box across the direction of acceleration would be bent; so a gravitational field must bend a light beam. He knew that clocks at the bottom of the box run slower than those at the top; so he inferred that clocks deeper in a gravitational field must run slower.

The Rotating Disk

If one has a circular disk at rest in some inertial reference system in special relativity, the geometry of its surface is Euclidean. That is quite obvious, but it will be useful to spell out what that means in terms of the outcomes of measuring operations. If the disk is ten feet in diameter, then it means that we can lay 10 foot long rulers across a diameter. Euclidean geometry tells us that the circumference is π x 10 feet, which is about 31 feet. That means that we can traverse the full circumference of the disk by laying 31 rulers round the outer rim of the disk.

Thus we measure the circumference of the rotating disk to be greater than 31 feet, the Euclidean value. In other words, we find that the geometry of the disk is not Euclidean. The circumference of the disk is more than the Euclidean value of π times its diamater.

The significance of this thought experiment was great for Einstein. Through his principle of equivalence, Einstein had found that linear acceleration produces a gravitational field. Now he found that another sort of acceleration, rotation, produces geometry that is not Euclidean.

Assembling the Pieces

Einstein had all this in place by the summer of 1912. He knew that gravitation could bend light and slow clocks. He expected that the final theory would somehow involve accelerations in a new way and that such accelerations came with a breakdown of Euclidean geometry. He also knew that the natural arena in which to conduct relativity theory is Minkowski's spacetime.

To us, the final step does not seem like such a great leap. Assemble the pieces and infer that gravitation is a curvature of spacetime! All that is needed is nice mathematical clothing to dress this idea. For Einstein in 1912 it was far from easy. He first needed the assistance of his mathematician friend Marcel Grossmann to find his way in the new and difficult mathematics the theory required.

Then he took a series of wrong turnings and ended up with the wrong gravitational field equations--not the celebrated Einstein equations that appear in all the modern textbooks. It required three years of painful work first to recognize that something had gone wrong and then to find the final equations. What made the last phase especially urgent was the fact that David Hilbert, the greatest mathematician of the era, had also gotten interested in his theory and had started to formulate the gravitational field equations in a mathematically more elegant formulation.

What You Should Know

• The difference between special and general relativity.
• How geodesic deviation reveals curvature.
• How free fall motion in a gravitational field can be reinterpreted as a curvature of a spacetime sheet.
• The connection between summed curvature and matter density.
• The difference between the curvature of a space-time sheet and space-space and how each reveals itself to us.
• The three famous tests of general relativity.
• The ideas that helped Einstein find general relativity.

Copyright John D. Norton. February 2001; January 2, 2007, February 15, August 23, 2008.