Transcript Intro to general relativity

Relativity
H7: General relativity
1
 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
2
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
3
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
4
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
5
Eötvös experiment
6
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
7
What effect does mass
have?
 Source of gravity
F G
M mgravity
r
2
 Inertia
F  minertia  a
8
Principle of Equivalence
F  minertial  a  G
M mgravity
r
2
 mgravity 
M
  G 2
 a  
r
 minertial 
9
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
10
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
11
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
12
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.
13
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…
14
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
15
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!
16
Gravitational time dilation
 Special relativity
t    t0
 
 General relativity
t    t0

1
v2
1 2
c
1
2GM 2
1
Rc 2
 G = Gravitational constant, M is the mass and R is
the radius
17
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 ?
18
Minkowski diagram
time
ct
light: x=ct
space
x
19
World lines — slowly
moving
world line
of a particle
ct
x
20
World lines — fast moving
world line
of a particle
ct
x
21
Faster than speed of light
?
ct
x
22
geometrical interval
y1
y
(x1,y1)
y
y2
s2= y2 + x2
x
x1
(x2,y2)
x x2
23
Spacetime interval
ct1
(x1,t1)
c ct
t
ct2
x1
s2= (ct)2 – x2
x
x
(x2,t2)
x2
24
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
25
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
26
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.
27
Warping of spacetime
 These diagrams shows the change in
path of a light wave close to a massive
body
28
Newtonian gravity
 What velocity is required to leave the
gravitational field of a planet or star?
vesc
2G M

R
 Example: Earth
 Radius: R = 6470 km = 6.47106 m
 Mass: M = 5.97 1024 kg
 escape velocity: vesc = 11.1 km/s
29
Newtonian gravity
 What velocity is required to leave the
gravitational field of a planet or star?
vesc
2G M

R
Example: Sun
 Radius: R = 700 000 km = 7108 m
 Mass: M = 21030 kg
 escape velocity: vesc = 617 km/s
30
Newtonian gravity
 What velocity is required to leave the
gravitational field of a planet or star?
2G M
vesc 
R
Example: a solar mass White Dwarf
 Radius: R = 5000 km = 5106 m
 Mass: M = 21030 kg
 escape velocity: vesc = 7300 km/s
31
Newtonian gravity
 What velocity is required to leave the
gravitational field of a planet or star?
2G M
vesc 
R
Example: a solar mass neutron star
 Radius: R = 10 km = 104 m
 Mass: M = 21030 kg
 escape velocity: vesc = 163 000 km/s  ½ c
32
Newtonian gravity
Can an object be so small that even light cannot
escape ?
2G M
RS  2
c
RS: “Schwarzschild Radius”
Example: for a solar mass
 Mass: M = 21030 kg
 Schwarzschild Radius: RS = 3 km
33
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.
34
Sizes of objects
35
Let’s do it within the context
of general relativity —
spacetime
 spacetime distance (flat space):
s  c t  R
2
2
time
2
2
space
 Fourth coordinate: ct
 time coordinate has different sign than spatial
coordinates
36
Let’s do it within the context
of general relativity —
spacetime
 spacetime distance (curved space of a
point mass):
RGM
11

2

2
S  2 2 2 2
ss  11 2  c ctt 

R
2 GM
1

 cR R 
1  RS c/2 RR
22
time
space
37
What happens if
R  RS
RS  2 2
1

s  1 
R 2
 c t 
R 
1  RS / R

2
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
38
Structure of a Black Hole
39
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.
40
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.
41
Doppler effect (for sound)
 The pitch of an approaching car is higher
than that of a car moving away.
42
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.
43
Doppler effect
 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
44
Doppler effect
redshift:
 z=0: not
moving
 z=2: v=0.8c
 z=: v=c
45
Doppler effect
 The formula for normal Doppler effect is
 v0 
f   f 1  
v

 This is modified for general relativity
f gh
 2
f
c
46