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Introduction to General Relativity. Note
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Advance of the perihelion of Mercury
In the 19th century it was discovered that interplantary perturbations
cannot account fully for the turning rate of the Mercury's orbit. About
43 arcseconds per century remained unexplained. The general theory of
relativity exactly accounts for this descrepancy.
The newtonian equation for the trajectory of a planet,
u'' + u = M/J2, (u = 1/r) has a periodic elliptic solution with
a period T = 2π (angular!). The corresponding relativistic equation has
an additional term, u'' + u = M/J2 +3Mu2, which causes the
perihelion to advance by dφ = 6π M2/J2 per revolution.
Bending of light
General relativity predicts apparent bending of light rays passing
through gravitational fields. The bending was first observed in 1919 by
Sir Arthur Stanley Eddington during a total eclipse when stellar images
near the occulted disk of the Sun appeared displaced by some arcseconds
from their usual locations in the sky.
Again, extended massive objects such as galaxies may act as
gravitational lenses, providing more than one optical path for light
emanating from a source far behind the lens and thus producing multiple
images. Such multiple images, typically of quasars, had been discovered
by the early 1980s.
In the newtonian theory the light rays travel along straight lines
described by and equation u'' + u = 0. The corresponding relativistic
equation has the same additional term, u'' + u = 3Mu2, which causes
the light trajectory to deflect by Δφ = 4M/b, where b is the
closest approach of the light ray to the centrum.
Exercises
- Show that a light ray can travel around a massive star in a circular
orbit much like a planet. Calculate the radius (in Schwarzschild
coordinates) of this orbit. (Answer: r = 3/2 rg)
- Calculate the bending angle of the light ray that just skirts the edge
of the Sun (what Sir Arthur presumably must have measured).
M© ≈ 2.0×1030 kg,
R© ≈ 7.0×108 m.
(Answer: δφ = 1.75")
- (Non-obligatory) Newtonian gravitation theory can be made covariant
(the scalar theory) by a suitable modification of the equation of motion
of a test point particle
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dpa = -ηacΦ,cpbdxb + pbΦ,bdxa,
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where Φ is a scalar potential related to the energy momentum tensor
by
- Is this theory in agreement with the Galileo's Piza experiment?
- Does this theory predict the bending of starlight near the sun?
- Does this theory predict the advance of the planet's perihelion?
Copyleft
©
2003 D.V.Fedorov
(fedorov@ifa.au.dk)