Page 166 - Intro to Tensor Calculus

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The solutions given by equations (1.5.190) tells us that φ 0 varies slowly with time. For less than 1, the
elliptical motion is affected by this change in φ 0 . It causes the semi-major axis of the ellipse to slowly rotate
at a rate given by dφ 0 . Using the following values for the planet Mercury
dt
2
G =6.67(10 −8 )dynecm /g 2
33
M =1.99(10 )g
12
a =5.78(10 )cm
=0.206
10
c =3(10 )cm/sec (1.5.191)
2GM 5
β ≈ =2.95(10 )cm
c 2
p 19 2
2
h ≈ GMa(1 − )= 2.71(10 )cm /sec
1/2
dφ GM −1
≈ sec Kepler’s third law
dt a 3
we calculate the slow rate of rotation of the semi-major axis to be approximately

2
1/2
dφ 0 dφ 0 dφ 3 dφ GM GM −14
= ≈ βA ≈ 3 =6.628(10 ) rad/sec
dt dφ dt 2 dt ch a 3 (1.5.192)
=43.01 seconds of arc per century.
This slow variation in Mercury’s semi-major axis has been observed and measured and is in agreement with
the above value. Newtonian mechanics could not account for the changes in Mercury’s semi-major axis, but
Einstein’s theory of relativity does give this prediction. The resulting solution of equation (1.5.188) can be
viewed as being caused by the curvature of the space-time continuum.
The contracted curvature tensor G ij set equal to zero is just one of many conditions that can be assumed
in order to arrive at a metric for the space-time continuum. Any assumption on the value of G ij relates to
imposing some kind of curvature on the space. Within the large expanse of our universe only our imaginations
limit us as to how space, time and matter interact. You can also imagine the existence of other tensor metrics
in higher dimensional spaces where the geodesics within the space-time continuum give rise to the motion
of other physical quantities.
This short introduction to relativity is concluded with a quote from the NASA News@hg.nasa.gov news
release, spring 1998, Release:98-51. “An international team of NASA and university researchers has found
the first direct evidence of a phenomenon predicted 80 years ago using Einstein’s theory of general relativity–
that the Earth is dragging space and time around itself as it rotates.”The news release explains that the
effect is known as frame dragging and goes on to say “Frame dragging is like what happens if a bowling
ball spins in a thick fluid such as molasses. As the ball spins, it pulls the molasses around itself. Anything
stuck in the molasses will also move around the ball. Similarly, as the Earth rotates it pulls space-time in
its vicinity around itself. This will shift the orbits of satellites near the Earth.”This research is reported in
the journal Science.