HPS 0410 Einstein for Everyone

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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.

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. In fact the diagram is a step backwards in that it is 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.

 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. The model is often described as a rubber membrane model and the picture is of a massive object sitting on a rubber membrane distorts the membrane. Just about the only thing right in the model is that the surface of the membrane is similar to the surface of the embedding diagram. Almost everything else is misleading and has to be imagined away. There is no gravity outside the membrane, for example, pulling the mass down so it distorts the membrane. Most importantly, there is no curvature of the space-time sheets of spacetime represented, even though that curvature is responsible for the familiar gravitational effects.
 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.

 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. One of Eddington's eclipse photos Here's the account of light deflection, told in reverse order. You might find it clearer. First, let's consider the spatial slice of spacetime that holds the earth and the star at a time when the sun is in a different part of the sky from the star. Then the portion of that spatial slice between the star and earth will be Euclidean. Second, let's consider what happens to the spatial slice after the position of earth and sun have changed so that the sun lies between the star and earth. The spatial slice is no longer exactly Euclidean. Its geometry is now slightly curved and is represented by the funnel shaped embedding diagram given above. A straight line in the spatial slice from the star to the earth will bow out around the sun, where the "bow out" is judged in relation to Euclidean expectations. This bowing out gives half the deflection observed in 1919. Third, let's put back in the time element and consider a pulse of light traveling from the star to earth. Its trajectory will be in spacetime, but we can project the trajectory down to the spatial slice and follow its progress there. You might expect that it would just follow the "bowed out" straight of the non-Euclidean geometry. But it doesn't. As it propagates, its motion deviates from the bowed out straight line of the spatial slice and it moves slightly in the direction of the sun. This second effect is the extra deflection that gives the other half of the deflection found in 1919. It turns out that this second half of the amount of deflection is the same deflection predicted by Newtonian theory for an object traveling at the speed of light that passes the sun and falls towards it. Because of this agreement in the magnitude of the deflection, it is easy to mash the two together and say that the light "falls" towards the sun. The quotes are there since I'm mixing notions from two theories.

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.