1. 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

2. The forces from acceleration and gravity cannot be distinguished

3. In an accelerating spaceship a stone drops just like Newton’s apple

4. A stone thrown sideways follows an arc (again just like Newton)

5. Light follows an arc, too (that’s Einstein)

6. Starlight deflected by Sun’s gravity
Moon
Star’s real position
Star’s apparent position
Starlight deflected by Sun’s gravity
Einstein’s logic
1) Acceleration and 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.

7. 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 predic- ted by Einstein.
Einstein was instantly famous.

8. Gravitational lensing
The Sun acts like a lens deflecting the light from stars behind it.
A cluster of ga-laxies can also act as lens by deflecting light from galaxies behind it.
The mass of the galaxy cluster is obtained from faint arcs pro-duced by a lens.

9. 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.

10. 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.

11. A light clock slows down in curved space
Mirror
Mirror
Light beam
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.

12. 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.

13. 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.

14. 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.