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General relativity
• Special relativity applies only to observers moving with
constant velocity.
• Can we generalize relativity to accelerated observers ?
• No problem for Einstein, just a few more thought
experiments …
Ch. 11.1
The forces from acceleration and gravity
cannot be distinguished
In an accelerating spaceship
a stone drops just like Newton’s apple
A stone thrown sideways follows an arc
(again just like Newton)
Light follows an arc, too
(that’s Einstein)
Einstein’s logic
1) Acceleration and gravity
Starlight deflected
by Sun’s gravity
cannot be distinguished.
2) Acceleration deflects light.
3) Therefore, gravity should
deflect light.
Confirmed by experiment:
Stars can be detected close to
the Sun during a solar eclipse.
Their light is indeed deflected
by the Sun’s gravity.
Eddington’s Eclipse Expedition 1919
• British astronomer Eddington
traveled to Principe Island in
the Gulf of Guinea to observe
deflection of starlight during
a solar eclipse.
• After months of drought it was
pouring rain on the day of the
eclipse. The clouds parted just
in time to see the eclipse.
• The photographs revealed an
apparent deflection of stars
away from the Sun, as predicted by Einstein.
• Einstein was instantly famous.
• The Sun acts like
a lens deflecting
the light from
stars behind it.
• A cluster of galaxies can also
act as lens by
deflecting light
from galaxies
behind it.
• The mass of the
galaxy cluster is
obtained from
faint arcs produced by a lens.
Gravitational lensing simulation
When a black hole with the mass of Saturn is placed in front
of the two central towers , gravitational lensing bends them
into arcs. This explains the faint arcs in the Hubble image.
Curved space-time
• The bending of light by the gravity of the Sun
brings up the question: What is a straight line?
• Laser beams are our best method to produce a
straight line.
• If light does not go straight, what else would ?
• We conclude that space itself must be curved.
It does not allow a straight line. The path of a
light beam is the straightest possible path.
A light clock slows down in curved space
In curved space, light takes a detour compared to flat space
(dotted line). The detour increases the time to complete one
cycle of a light clock (Lect. 14, Slide 3). The clock slows down.
Gravitational time dilation
• Gravity warps both space and time!
• Clocks run more slowly in a gravitational force field.
• In a GPS satellite the clock speeds up during launch
into orbit , where gravity is reduced. The satellite
clock needs to be set slightly slow before launch to
compensate (by a factor of 4.5 10-10).
• This effect is much larger than the time dilation due
to special relativity (Lect. 14, Slide 7), which acts in
the opposite direction.
Curved space explains the planet’s orbits
• In general relativity, gravity is a warp in space.
• The orbits of the planets and comets can be explained
by their movement around the dimple in space that is
created by the mass of the Sun.
Einstein’s equations of gravity
• Einstein’s equations of gravity are rather complicated.
Even Einstein needed help from expert mathematicians.
• Einstein’s equations in words (by John Wheeler):
Matter tells space how to curve,
and space tells matter how to move.
• General relativity connects space and matter, concepts
that seem to have nothing in common.
• General relativity is reduced to special relativity in the
absence of gravity or acceleration. It includes the results
of special relativity on length contraction, time dilation.