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Gravity Near a Massive Body

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

In the last chapter, we learned the barest elements of Einstein general theory of relativity. We now need to understand what those elements entail for gravity. The first place to start is the most familiar, the gravitational effects arising near a massive object like our earth or sun. These were the first applications of Einstein's new theory.

The Geometry of Space

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.

This representation of how the geometry of space is affected by the presence of the sun is quite adequate. It is limited to intrinsic curvature. However another representation is possible. We can capture the deviations from Euclidean flatness in the space through extrinsic curvature. To do it, we imagine that the surface lives in a higher dimensioned Euclidean space and that its intrinsic curvature comes about from its extrinsic curvature in that space. We will pursue this since it leads to one of the most familiar images associated with general relativity in the popular literature.

To make things concrete, let us imagine that we will build a real model out of some flexible sheeting material whose extrinsic curvature will display the geometry of space around the sun.

To build the model, we take the largest circle and lay it down as a ring on a flat, Euclidean bench surface. We then move inward one unit distance in radius to the next circle. We lay down a second smaller ring with the appropriate circumference and connect the two by the flexible sheeting of the model. In the case of Euclidean geometry, the circumference of the second circle will be reduced 2π units of distance, matching the behavior of Euclidean circles on the table top. Thus it will lie flat on the table, to produce the two circles shown below. The two circles would be joined by a flat ring of the sheeting that comprises the model.

In the case of the non-Euclidean geometry, when we move inward one unit of distance, we arrive at circle whose circumference has lost less than 2π units of distance. As a result, it will have a greater circumference than the circles of the table surface one unit of radius closer to the center. That means that it will not lie flat on the Euclidean surface of the table. If we tried to lie it flat on the table surface, we would have to crumple the sheeting connecting the two circles. Since we want to avoid crumpling the sheeting, we would have to let it pop up above the table, as shown below.

The effect is greatly exaggerated in the figure. The amount the surface would pop up would be undetectible, if it accuractely modeled the deviations from Euclidean space in our part of space. However the qualitative effect is correct. If we were to model of the geometry of space very close to a very massive body, then the effect would be more marked.

We continue building our model by adding more circles nearer the sun, connected by sheeting to the circles farther out. What would result is more popping out of the sheet, until we have a familiar funnel shape. The figure below inverts the funnel in keeping with the standard presentation of the model in the literature.

The model is an embedding diagram of the geometry of space near our sun. That is, it recovers the intrinsic curvature of a two-dimensional sheet of space-space by the extrinsic curvature of a model surface bending into a higher dimensional, Euclidean space. The model captures an important geometrical fact about the space around our sun--that it is no longer exactly Euclidean. It is one of the most frequently built models in the context of general relativity.

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. In fact the diagram is a step backwards in that it is a return to the old way of visualizing curvature as a bending of a surface into a higher dimensioned space. While it might be a useful aid to visualization, it is factually false. There is, as far as we know, no higher dimensioned space into which the surface bends.

http://en.wikipedia.org/wiki/File:Spacetime_curvature.png

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.

Here's a contemporary account from Simon Newcomb's authoritative The Elements of the Four Inner Planets and the Fundamental Constants of Astronomy: Supplement to the American Ephemeris and Nautical Almanax for 1897. Washington: Government Printing Office, 1895, p. 184.

Note that Newcomb allows that the anomalous motion of Mercury could be accommodated if Newton's law of gravitation was not exactly an inverse square law. That is, he considers the possibility that the force of gravity does not dilute in inverse proportion with (distance)² but with (distance)^{2.00000016120}. We might wonder if this is an admission that no hypothesis within the existing system is expected to accommodate the anomaly so that an alteration of fundamental law has to be contemplated. Or, more likely, is it just a working astronomer noting the simplest way to develop a rule that will allow prediction of planetry motion?

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. The deflection is measured as the change in apparent direction of the star from the earth; that is, it is measured as the angle between the direction in which we see the star and in which we expected to see the star. Einstein computed that the deflection would be about 1.75 seconds of arc. The deflection has 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.

From the New York Times, November 10, 1919. Full article.

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 spectacular 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, informally speaking, 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.

What You Should Know

• The difference between the curvature of a space-time sheet and space-space and how each reveals itself to us.
• What the rubber sheet-like embedding diagram shows (and does not show).
• The three famous tests of general relativity.

Copyright John D. Norton. February 2001; January 2, 2007, February 15, August 23, October 16, 27, 2008; February 5, July 20, 2010, February 25, 2013.