PH5011 General Relativity

PH5011 General Relativity

Dr HongSheng Zhao shortened/expanded from notes of MD

[email protected]

Martinmas 2012/2013

0.1 Summation convention 0 General issues

2

pairwise indices imply sum

0.2 Indices

dimension of coordinate space

Apart from a few exceptions, upper and lower indices

are to be distinguished thoroughly

1.  Basis (not examined) intro. tensor and Coordinates transformation (exam).

2.  Tensor operations all examinable. 3.  Mechanics classical NOT exam. 4.  Mechanics in curved space NOT exam. 5.  Special Rela. NOT exam. 6.  General Rela. (Einstein Eq.) exam. 7.  Application of GR Examinable: FRW (p1-6), Schwarzschild (p1-4), Tutorials (1,2,3). Adv. (1p, for intuition)

To Exam Or Not To Exam

3

1.1 Basis and coordinates 1 Curvilinear coordinates

set of basis vectors spans tangent space at ⤿�

in general, the basis vectors depend on

described by set of coordinates location

infinitesimal displacement in space on variation of coordinate given by line element

≡ basis vector related to coordinate

coordinate line given by for all

tangent vector at

3

1 Curvilinear coordinates 1.1 Basis and coordinates

4

1 Curvilinear coordinates

Example A: Cartesian coordinates (I)

1 Curvilinear coordinates 1.1 Basis and coordinates

5


Example B: Constant, non-orthogonal system (I)

1.2 Reciprocal basis

Kronecker-delta

orthonormal basis orthogonal basis

6


construction:

orthogonality normalization for

for

for

1 Curvilinear coordinates 1.2 Reciprocal basis

7


Special case: 3 dimensions

Example A: Cartesian coordinates (II)


8


⤿�

Example B: Constant, non-orthogonal system (II)


9


⤿�

1.3 Metric 1 Curvilinear coordinates

10

⤿�

coefficients of metric tensor (→ 1.5)

as matrix

symmetry:

1 Curvilinear coordinates 1.3 Metric

11

Examples A+B: Cartesian & non-orthogonal constant basis (III)

⤿�

⤿�


12

length of curve given by

1.3 Metric

parametric representation of curve

1 Curvilinear coordinates 1.3 Metric

13

Example: Length of equator in spherical coordinates

in

one only needs to consider :

⤿�

use parameter along the azimuth

one full turn for and

⤿�


equivalent to the condition for the inverse matrix

which fulfill ,

With the reciprocal basis ,

one defines reciprocal components of the metric tensor

14


metric tensor

orthonormality condition

“lowers index”

“raises index”

⤿�

15

1.4 Vector fields 1 Curvilinear coordinates

mathematics: vector field physics: vector (field)

covariant components contravariant components

(→ 1.6)

“raising/lowering indices”

vector components defined by means of basis vectors

16

1.5 Tensor fields 1 Curvilinear coordinates

tensor is multi-dimensional generalization of vector

mathematics: tensor field physics: tensor (field)

behaves like a vector with respect to each of the vector spaces

17

product of vector spaces

tensor of rank 2 square matrix tensor of rank 1 tensor of rank 0

tensor of rank 3 cube

vector scalar

........

rank of tensor


18

1.5 Tensor fields

basis vectors apply to each of the vector spaces

⤿�


mixed components

1.5 Tensor fields 1 Curvilinear coordinates

19

Example: Rank-2 tensor

⤿�

Coincidentally, with the matrix product

For Cartesian coordinates:

1.6 Coordinate transformations 1 Curvilinear coordinates

20

consider different set of coordinates

(chain rule)

different coordinate systems describe same locations

⤿�

⤿�


21

covariant contravariant

derivatives differentials

components transform like coordinate { }

1.6 Coordinate transformations

vector fields

tensor fields

⤿�

⤿�

1.6 Coordinate transformations 1 Curvilinear coordinates

22


Proof: are covariant components of a tensor

⤿�

1.7 Affine connection 1 Curvilinear coordinates

in general, basis vectors depend on the coordinates

derivative of basis vector written in basis

23

affine connection (Christoffel symbol)

derivative of reciprocal basis vector:

⤿�

1 Curvilinear coordinates 1.7 Affine connection


24

Example C: Spherical coordinates (IV)

⤿�

⤿�



25

Example C: Spherical coordinates (IV) [continued]

⤿�

⤿�

⤿�



given that

the Christoffel symbols can be expressed by means of the components of the metric tensor and their derivatives

26


27


Proof:

⤿� (I)

(II)

(III)

(II) + (III) - (I) :

⤿�

2 Tensor analysis

derivative:

28

vector field both the vector components and the basis vectors depend on the coordinates

define covariant derivative of a contravariant vector component

as so that

2.1 Covariant derivative

2 Tensor analysis 2.1 Covariant derivative

derivatives transform as

⤿� can be considered the covariant components

of the vector

29

covariant components of a vector

(gradient)

form components of a tensor, not



30

for each upper

lower index , add { { }

takes place of in or where

covariant derivatives of tensor components

31


Covariant derivative of 2nd-rank tensor

⤿�

2.2 Riemann tensor 2 Tensor analysis

with the Riemann (curvature) tensor

(not intended to be memorized)

order of 2nd covariant derivatives of vector is not commutative

with

⤿�

and

, but

32

2 Tensor analysis

33

2.2 Riemann tensor

Riemann tensor has two pairs of indices and is

antisymmetric in the indices of each pair

[ [

symmetric in exchanging the pairs

Moreover,

(1st Bianchi identity)

(2nd Bianchi identity)

2 Tensor analysis

34

2.2 Riemann tensor

Proof:

The scalar product of two vectors is a scalar

⤿�

On the other hand

⤿�

(Riemann curvature tensor is antisymmetric in first two indices)

2 Tensor analysis

35

2.3 Einstein tensor

must relate to Riemann tensor

matches required conditions ⤿�

only a single non-vanishing contraction (up to a sign)

(Ricci tensor)

with next-level contraction

(Ricci scalar)

2nd-rank curvature tensor fulfilling

3 Review: Classical Mechanics 3.1 Principle of stationary action

action

Fermat’s principle (optics) Feynman’s path integral (QM)

(Hamilton’s) principle of stationary action

for : kinetic energy potential energy

Mechanical system completely described by

(Lagrangian) coordinate velocity time

36

(Euler-) Lagrange equations ⤿�

Example: 1D harmonic oscillator (I)

⤿�

⤿�

⤿�

3 Classical mechanics 3.1 Principle of stationary action

37

(geodesic equation, assume = s )

4.1 Principle of stationary paths

47

stationary path between two points (e.g. path length is locally shortest)

Christoffel symbols (affine connection)

4 Intro: Mech. in curved space

Path length ds = G d , stationary path means

48

4 Mechanics in curved space 4.1 Stationary paths

Define

⤿�

Constant L factored out of derivatives. Write derivative as dot, if we define t = s =

⤿�

⤿�

L resembles Lagrangian for a free particle of mass m in curved space

with and

⤿�

(Euler-Lagrange equations)

⤿�

⤿�

↳

} ⤿�

44

4 Mechanics in curved space 4.1 Stationary paths

4 Mechanics in curved space

49

4.1 Stationary paths

Eq. of motion along geodesics, = s, or in shorthand:

based on Newton’s law purely space geometry

(geodesic equation)


4.2 Geodesics as parallel transport

• moving along geodesics means to keep the same direction • geodesics form “straight lines”

46

= tangent unit vector to a curve

i.e. is geodesic if unit tangent vector is parallelly transported

if all do not depend on ⤿�

4.3 Conserved momentum pk dpk/d=0 if the metric g independent of qk


50

(geodesic equation)

5 Review: Special Relativity

54

5.1 Minkowski space

“inertial system” force-free particles move uniformly

all reference frames moving uniformly with respect to an inertial system are inertial system themselves

“reference frame” defines coordinate origin and motion

“event” described by time and location

laws of physics assume the same form in all inertial systems

55

5 Special Relativity 5.1 Minkowski space

describes distance in four-dimensional space ⤿�depend on reference frame both and whereas

homogeneity and isotropy of space and time

⤿� invariance of

⤿�

⤿� along light rays: for all reference frames

invariance of speed of light

56

5 Special Relativity 5.1 Minkowski space

Latin indices Greek indices

use 4-dimensional vectors

flat three-dimensional space described by cartesian coordinates

⤿�

photons trace null geodesics between events defines light cone 45° opening angle in

5.2 Light cone 5 Special Relativity

57

or: “causality and the finite speed of light”

instantaneous knowledge of interaction non-relativistic theories:

light cone widens, all events get into causal contact

:

invariance of categorization holds irrespective of coordinate system and reference frame ⤿�

outside light cone ‘elsewhere’, no causal connection

inside light cone massive particles move on time-like geodesics

5.3 Proper time 5 Special Relativity

58

time shown on clock

⤿� proper time

invariance of ⤿�

⤿�

(moving clock observed “t” appears big ) so that

along worldline of clock with attached rest frame

5 Special Relativity

59

5.4 Relativistic mechanics define 4-velocity as

⤿�

⤿�

⤿�(as anticipated for inertial system)

known: free particle moves along geodesic

⤿� [ all ]

5 Special Relativity

( ) non-relativistic limit

60

⤿�let�

5.4 Relativistic mechanics

(matches invariance of ) ansatz:

relativistic action

5.4 Relativistic mechanics 5 Special Relativity

conjugate momentum

61

energy

⤿�

(relativistic Hamilton-Jacobi equation)

with

⤿�

(sign in spatial part due to in metric)

components of stress tensor

provides relation between the forces and the cross-sections these are exerted on

force area of cross-section normal to cross-section

5.5 Energy-momentum tensor 5 Special Relativity

62

for fluid in thermodynamic equilibrium: (no shear stresses)

pressure

energy-momentum tensor

in fluid rest frame: mass density

complement to energy density momentum density

stress

5 Special Relativity 5.5 Energy-momentum tensor

63

non-relativistic limit:

(continuity equation) (↔ Newton’s law)

6 General Relativity

64

6.1 Principles experiments cannot distinguish between:

•  virtual forces present in non-inertial frames •  true forces

gravitation can be described by space-time metric

⤿�

gravitation becomes property of space-time with particles moving on geodesics ⤿�

local free-falling frame is an inertial frame, where free particles are on straight lines and

→ Einstein’s field equations only remaining issue: relation between and Newton’s law

6 General Relativity 6.1 Principles

The laws of physics are the same for all observers, irrespective of their motion

Physical laws take the same covariant form in all coordinate systems

We live in a 4-dimensional curved metric space-time

Particles move along geodesics

The laws of Special Relativity apply locally for all non-accelerated (inertial) observers

The curvature follows the energy-momentum tensor as described by Einstein’s field equations

General Relativity summarized in 6 points

65

66


6.2 Einstein’s field equations independence on choice of coordinates

formulate theory by means of tensor fields ⤿�

if non-relativistic limit reproduces Newton’s law, this is not necessarily the only possible theory,

but the most simple one that conforms to the principles ⤿�

⤿� ?

(energy-momentum tensor) matter is completely described by 2nd-rank tensor

description of curvature by 2nd-rank tensor (Einstein tensor)

6 General Relativity 6.2 Einstein’s field equations

67

Einstein’s field equations:

non-relativistic limit ( ):

dominating

,

⤿�


Newton:

⤿�

⤿�

⤿�with

⤿�

68


[note: Einstein’s orignal sign convention for the Ricci tensor differs from ours] 69


theories modifying the law of gravity provide alternative models

6.3 Cosmological constant

70

negligible correction, unless huge length scales are considered

modified Einstein tensor

also fulfills

(dark) “vacuum” energy ?? effective repulsion

measurements suggest

Solar neighbourhood baryonic matter in the Universe

71


6.4 Time and distance Laws of physics — described by tensors — do not depend on coordinates

coordinates do not have immediate physical meaning ⤿�

⤿� What is the time and distance?

can be locally transformed to

are not completely arbitrary

⤿� matrix with eigenvalues of

corresponding to 1 time-like and 3 space-like coordinates have signs

⤿�

6 General Relativity 6.4 Time and distance

cannot define spatial distance by means of for neighbouring events at the same time

⤿�

in general, the relation between the proper time interval and

depends on the location

time interval between two events at the same location given by

⤿� proper time

72

coordinate transformation can always provide (at cost of time-dependent )

(synchronized reference frame) everywhere

coordinate line of (i.e. ) is geodesic

6 General Relativity 6.5 Synchronisation

(with regard to time coordinate, but measured depends on location) global synchronisation possible ⤿�if

76

6.5 Synchronisation (e.g. FRW cosmology)

Challenge: prove this

110

7 GR Applications

7. Satellites: GPS orbit Earth ~ stars orbit BH Beepers on sat. are Doppler/Gravitational-shifted, time delayed

112

7 Consequences 7. Satellite navigation

GPS satellites perform two orbits per sidereal day

GPS clocks are shipped with “factory offset” to compensate

in total, GPS clock appears to run faster by

⤿� ,

Doppler shift (transverse motion) per day

gravitational potential

per day

per day

7 Consequences 7.1 Relativistic Kepler problem

87

Perihelion shift of the planets in the Solar system semi-major axis

a [AU]

orbital period P

[yr]

eccentricity ε

perihelion shift

per century

Mercury ☿ 0.39 0.25 0.206 43˝ Venus ♀ 0.72 0.62 0.0068 8.6˝ Earth ♁ 1 1 0.0167 3.8˝ Mars ♂ 1.5 1.88 0.0933 1.4˝

Jupiter ♃ 5.2 11.9 0.048 0.06˝ Saturn ♄ 9.5 29.5 0.056 0.01˝ Uranus ♅ 19 84 0.046 0.002˝

Neptune ♆ 30 165 0.010 0.0008˝ (essentially inversely proportional to a5/2)

7 Consequences 7.1 Relativistic Kepler problem

86

90

7 Consequences 7.2 Bending of light

asymptotics

⤿� total deflection ⤿�

Deflection of light by gravity (1915)

α = 4GM c2ξ 1.″7 measurable at Solar limb: α =

bending angle

92

93


"The present eclipse expeditions may for the first time demonstrate theweight of light; or they may confirm Einstein's weird theory of non-Euclidean space; orthey may lead to a result of yet

more far-reaching consequences -- no deflection."

"The generalized relativity theory is a most profound theory of Nature,embracing almost all the phenomena of physics."

(Sir) Arthur Stanley Eddington Negative of one of the photographic plates

taken by the British expedition to Sobral (Brazil) during the total Solar Eclipse of 29 May 1919©

The Royal Society

The British expeditions to Sobral (Brazil) and the island of Principe to observe the total Solar Eclipse of 29 May 1919

95


99

Notes about gravitational microlensing dated to 1912 on two pages of Einstein’s scratch notebook

I−

I+

ξ η

side view


Images by a gravitational lens

96

⤿�

with (angular Einstein radius)

(two images) ⤿�

6˝

(animation by Daniel Kubas, ESO)

98


bending of light of stars due to intervening foreground stars

image distortion leads to observable transient brightening

images cannot be resolved ⤿�

within the Milky Way

⤿�

The chance is one in a million ! B. Paczyński 1986, ApJ 304, 1


100

First reported microlensing event

MACHO LMC#1

Nature 365, 621 (October 1993)


101

Astronomy & Geophysics Vol. 47

(June 2006)


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A Sample of Advanced Material: Geodesics around Black Hole Metric

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PH5011 General Relativity