1. Relativity
H7: General relativity

2. Special relativity is concerned with inertial frames that are not accelerating
But what happens when the inertial frame is accelerating
This is the subject of General Relativity

3. What effect does mass have?
Gravity: tendency of massive bodies to attract each other
Inertia: resistance of a body against changes of its current state of motion

4. Is gravity and inertia the same thing ?
No. They are completely different physical concepts.
There is no a priori reason, why they should be identical. In fact, for the electromagnetic force (Coulomb force), the source (the charge Q) and inertia (m) are indeed different.
But for gravity they appear to be identical
 Equivalence Principle

5. The equivalence principle
Einstein’s principle of equivalence states
No experiment can be performed that could distinguish between a uniform gravitational field and an equivalent uniform acceleration

6. Eötvös experiment

7. Result of the Eötvös experiment
Gravitational and inertial mass are identical to one part in a billion
modern experiments: identical to one part in a hundred billion

8. What effect does mass have?
Source of gravity
Inertia

9. Principle of Equivalence

10. Implications of General Relativity
Gravitational mass and inertial mass are not just proportional, but completely equivalent
A clock in the presence of gravity runs more slowly than one where gravity is negligible
The frequencies of radiation emitted by atoms in a strong gravitational field are shifted to lower frequencies
This has been detected in the spectral lines emitted by atoms in massive stars

11. More Implications of General Relativity
A gravitational field may be “transformed away” at any point if we choose an appropriate accelerated frame of reference – a freely falling frame
Einstein specified a certain quantity, the curvature of time-space, that describes the gravitational effect at every point

12. Testing General Relativity
General Relativity predicts that a light ray passing near the Sun should be deflected by the curved space-time created by the Sun’s mass
The prediction was confirmed by astronomers during a total solar eclipse in 1919

13. Bending light rays
If a rocket ship is undergoing constant acceleration and a flash light is shown on one side of the rocket ship towards the other side, the light will not hit the opposite side of the ship at the same height as the window.

14. Bending light rays
The light will leave point A traveling at the speed of light in a purely horizontal direction.
When it reaches the other wall it will not reach the same height. According to Einstein’s principle of equivalence light will also bend in a gravitational field…

15. Gravitational time dilation
Time slows down in a strong gravitational field
clocks undergoing acceleration will run slow compared to non-accelerating clocks
the greater the gravitational field, the greater the time dilation

16. Gravitational time dilation
Clocks on the ground floor of a tall building will run slower than those in the upper floors; if you want to keep (relatively) young, find a job in the basement - or become a miner!

17. Gravitational time dilation
Special relativity
General relativity
G = Gravitational constant, M is the mass and R is the radius

18. Minkowski’s spacetime
As we have seen, time intervals, lengths, and simultaneity is relative and depend on the relative velocity of the observer.
velocity connects time and space
Let’s stop separating space and time, let’s rather talk about spacetime
spacetime is 4 dimensional, 3 spatial + 1 time dimension but is space and time really the same thing ?

19. Minkowski diagram
time
space
ct x
light: x=ct

20. World lines — slowly moving
ct x
world line
of a particle

21. World lines — fast moving
ct x
world line
of a particle

22. Faster than speed of light ?
ct x

23. geometrical interval (x1,y1) y1 s2= y2 + x2 y y y2 (x2,y2) x x1

24. Spacetime interval (x1,t1) ct1 s2= (ct)2 – x2 ct ct ct2

25. Spacetime interval
– sign: difference between space and time
s2 is invariant under Lorentz transformation
for particle moving at speed of light: x = ct  s2= 0  light like (null) distance

26. Character of spacetime intervals
s2>0  ct > x
spatial distance can be traveled by speed of light
there exist an inertial frame, in which the two events happen at the same position
but they never happen simultaneously  time like distance
s2<0  ct < x
spatial distance cannot be traveled by speed of light
there exist an inertial frame, in which the two events happen simultaneously
but they never happen at the same place  space like distance

27. Warping of spacetime
Gravitation can be explained by the curvature of spacetime. As an object travels in straight line in curved space its path will curve towards a massive object. No force is needed to explain the change of path, the curvature of spacetime is enough to explain the motion. Pretty clever.

28. Warping of spacetime
These diagrams shows the change in path of a light wave close to a massive body

29. Newtonian gravity
What velocity is required to leave the gravitational field of a planet or star?
Example: Earth
Radius: R = 6470 km = 6.47106 m
Mass: M = 5.97 1024 kg
escape velocity: vesc = 11.1 km/s

30. Newtonian gravity
What velocity is required to leave the gravitational field of a planet or star? Example: Sun
Radius: R = km = 7108 m
Mass: M = 21030 kg
escape velocity: vesc = 617 km/s

31. Newtonian gravity
What velocity is required to leave the gravitational field of a planet or star? Example: a solar mass White Dwarf
Radius: R = 5000 km = 5106 m
Mass: M = 21030 kg
escape velocity: vesc = 7300 km/s

32. Newtonian gravity
What velocity is required to leave the gravitational field of a planet or star? Example: a solar mass neutron star
Radius: R = 10 km = 104 m
Mass: M = 21030 kg
escape velocity: vesc = km/s  ½ c

33. Newtonian gravity Example: for a solar mass
Can an object be so small that even light cannot escape ?
RS: “Schwarzschild Radius”
Example: for a solar mass
Mass: M = 21030 kg
Schwarzschild Radius: RS = 3 km

34. Some definitions ...
The Schwarzschild radius RS of an object of mass M is the radius, at which the escape speed is equal to the speed of light.
The event horizon is a sphere of radius RS. Nothing within the event horizon, not even light, can escape to the world outside the event horizon.
A Black Hole is an object whose radius is smaller than its event horizon.

35. Sizes of objects

36. Let’s do it within the context of general relativity — spacetime
spacetime distance (flat space):
Fourth coordinate: ct
time coordinate has different sign than spatial coordinates
time
space

37. Let’s do it within the context of general relativity — spacetime
spacetime distance (curved space of a point mass):
time
space

38. What happens if R  RS time space
R > RS: everything o.k.: time: +, space:  but gravitational time dilation and length contraction
R  RS: time  0 space  
R < RS: signs change!! time: , space: +  “space passes”, everything falls to the center  infinite density at the center, singularity
time
space

39. Structure of a Black Hole

40. What happens to an astronaut who falls into a black hole?
Far outside: nothing special
Falling in: long before the astronaut reaches the event horizon, he/she is torn apart by tidal forces
For an outside observer:
astronaut becomes more and more redshifted
The astronaut’s clock goes slower and slower
An outside observer never sees the astronaut crossing the event horizon.

41. What happens, if an astronaut falls into a black hole?
For the astronaut:
He/she reaches and crosses the event horizon in a finite time.
Nothing special happens while crossing the event horizon (except some highly distorted pictures of the local environment)
After crossing the event horizon, the astronaut has 10 microseconds to enjoy the view before he/she reaches the singularity at the center.

42. Doppler effect (for sound)
The pitch of an approaching car is higher than that of a car moving away.

43. Doppler effect (for light)
The light of an approaching source is shifted to the blue,
the light of a receding source is shifted to the red.

44. Doppler effect blue shift red shift
The light of an approaching source is shifted to the blue,
the light of a receding source is shifted to the red.
blue shift
red shift

45. Doppler effect
redshift:
z=0: not moving
z=2: v=0.8c
z=: v=c

46. Doppler effect The formula for normal Doppler effect is
This is modified for general relativity