Celebrating Einstein

“The General Theory of Relativity”

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D.  The Law of Gravity

Einstein spent a good deal of time over several years, with the help of a number of collaborators, trying to work out a mathematical law that would account for gravity in terms of spacetime curvature, and publishing various speculations and findings along the way.  Other physicists read these publications and took the ideas in new directions on their own.  But in 1915, Einstein had arrived at a spacetime-curvature law and explained it in a series of papers that year and the next.

Two clues were particularly important in Einstein's choice of a curvature law.  One was the known accuracy of Isaac Newton's law of gravity.  Much more than "what goes up must come down" (which Newton's law doesn't imply), Newton found that the acceleration in one object's motion produced by another object's gravitational attraction is proportional to the attracting object's mass divided by the square of the distance between the objects:

attracted object's acceleration =

Acceleration, a speed divided by a time, is measured in units other than mass divided by distance squared; the "" in the equation is a universal conversion ratio.  Since this law had been proven accurate in many experiments and astronomical observations, Einstein's curvature law had to give equally accurate results when applied to the same situations.

Einstein also hoped that his new law would explain something that Newton's law couldn't.  If you apply Newton's law to the motions of planets in the solar system, you find that the sun's gravity is the main influence on each planet, while the attractions of the planet's own satellites, and the other planets, provide a smaller but still significant influence.  Just these attractions account for every feature of planetary motion that astronomers had observed by the early 20th century—except for one.

Newton's law implied that, if the sun alone were the only object attracting the planet Mercury, then Mercury should move in the same orbit over and over indefinitely, with its closest approach to the sun occurring at the same place every time, much like the orbit shown in Figure 3:

Figure 3
Figure 3.

while the additional attractions from the other planets should disturb Mercury's orbit to make it something like this one in Figure 4:

Figure 4
Figure 4.

The effect is exaggerated in Figure 4 to make it visible:  Mercury's orbit isn't as elliptical as the orbit shown, and Mercury actually has to go around the sun many more times for its closest approach to the sun to change position by the amount pictured.

Mercury's closest approach to the sun shifts around even faster—if only slightly faster—than the other planets' attractions can account for (Figure 5):

Figure 5
Figure 5.

so if we subtract the other planets' attraction, we're left, not with the eternal orbit of Figure 3, but with a slightly shifting orbit—something that Newton's law doesn't explain (Figure 6):

Figure 6
Figure 6.

These last two figures also exaggerate.  The actual and residual shifts are much smaller:  in 100 years, Mercury's closest approach to the sun moves around by an angle of 2.78 milliradians (574 arcseconds), which is roughly a fifth of a milliradian (43 arcseconds) more than the shift caused by other planets' gravity.  One-fifth of a milliradian is the angular width of one-fifth of a millimeter seen from one meter away, or a similar small width seen at arm's length.  This angle is small, but measurable, and was well documented by centuries of astronomical observations.  Newton's law of gravity was utterly unable to explain it.

The equation Einstein eventually proposed took several features of spacetime and gravity into account.  First, to account for gravity's operation in vacuum, Einstein realized that it should be possible for spacetime to be curved even when there's nothing in it.  Second, spacetime's curvature should be affected by the presence of matter.  Third, as we noted earlier, the curvature would have to produce paths in spacetime that look like the ones you'd expect from Newton's law under conditions for which Newton's law is known to be accurate.

These requirements didn't define a unique law of curvature, though they did limit the possible choices.  Einstein thus made use of another reasonable hypothesis.  A gravitational field can be thought of as having energy.  Where gravity is stronger, the gravitational field's energy is denser.  Since no natural process produces energy from nothing or turns energy into nothing, accurate mathematical descriptions of energy can only imply that energy changes its form or location, never its total amount; at any place where the density of energy changes, a corresponding flow of momentum moves some energy in or out.  Einstein found an equation that implied spacetime curvature would have these properties when the spacetime did not contain matter.  He then assumed that the same equation would describe the energy and momentum within any region of spacetime, even if it did contain matter.  The equation was 

G = 8 πκ T/c4

G = the sum of certain components of spacetime curvature,
T = corresponding features of the energy, momentum, and stresses within matter,
κ = the same universal converstion ratio found in Newton's gravitational law,
c = the speed of light in vacuum.

Since G represents only the sum of certain features of the curvature of spacetime, there can still be curvature even in the empty space around matter—for example, approximately what we find in outer space around the earth.  In empty space where there is no matter, the energy, momentum, and stresses of matter all equal zero, and Einstein's equation implies that the sum G equals 0.  If G represented all the features of spacetime curvature individually, the absence of matter at any point would imply the absence of gravity at that point.  Gravity in empty space does exist, though, embodied in other features of spacetime curvature not included in G.  Einstein's equation thus conforms to the first feature of gravity noted above.  The second feature is accounted for as well, since the remaining aspects of spacetime curvature that G does represent are directly proportional to the energy, stresses, and momentum flow of any matter occupying the spacetime.  While G and T can represent extremely complex spacetime curvatures and matter distributions, the relation between them—their proportionality—is a simple one.

Although it's not immediately obvious, Einstein's equation turns out to conform to Newton's law as well.  When you define a matter distribution whose density and internal stresses are not extremely high, and whose motions are slow compared to the speed of light, the straightest possible lines in the surrounding spacetime are very nearly the orbits that objects would have if Newton's law were 100% accurate.

In one respect, though, the orbits aren't quite the same.  Einstein found that, if his equation were correct, any planet whose orbit isn't circular should move like the one illustrated in Figure 6 if there were no other planets to influence its motion.  A planet in Mercury's situation should gradually shift its nearest approach to the sun a fifth of a milliradian per century faster than the rate that Newton's law could account for.  Einstein considered this fact one of the strongest indications that his equation G = 8 πκ T/c4 was on the right track.     (.....continued)

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