ph5011 general relativity dr hongsheng zhao shortened/expanded from notes of md [email protected]...
TRANSCRIPT
![Page 1: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/1.jpg)
PH5011
General Relativity
Dr HongSheng Zhao shortened/expanded from notes of MD
Martinmas 2012/2013
![Page 2: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/2.jpg)
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
![Page 3: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/3.jpg)
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
![Page 4: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/4.jpg)
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 coordinategiven by line element
≡ basis vector related to coordinate
coordinate line given by for all
tangent vector at
3
![Page 5: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/5.jpg)
1 Curvilinear coordinates1.1 Basis and coordinates
4
1 Curvilinear coordinates
Example A: Cartesian coordinates (I)
![Page 6: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/6.jpg)
1 Curvilinear coordinates1.1 Basis and coordinates
5
1 Curvilinear coordinates
Example B: Constant, non-orthogonal system (I)
![Page 7: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/7.jpg)
1.2 Reciprocal basis
Kronecker-delta
orthonormal basis
orthogonal basis
6
1 Curvilinear coordinates
construction:
orthogonality
normalization for
for
for
![Page 8: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/8.jpg)
1 Curvilinear coordinates1.2 Reciprocal basis
7
1 Curvilinear coordinates
Special case: 3 dimensions
![Page 9: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/9.jpg)
Example A: Cartesian coordinates (II)
1 Curvilinear coordinates
8
1 Curvilinear coordinates1.2 Reciprocal basis
⤿
![Page 10: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/10.jpg)
Example B: Constant, non-orthogonal system (II)
1 Curvilinear coordinates
9
1 Curvilinear coordinates1.2 Reciprocal basis
⤿
![Page 11: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/11.jpg)
1.3 Metric1 Curvilinear coordinates
10
⤿
coefficients of metric tensor (→ 1.5)
as matrix
symmetry:
![Page 12: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/12.jpg)
1 Curvilinear coordinates1.3 Metric
11
Examples A+B: Cartesian & non-orthogonal constant basis (III)
⤿
⤿
![Page 13: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/13.jpg)
1 Curvilinear coordinates
12
length of curve given by
1.3 Metric
parametric representation of curve
![Page 14: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/14.jpg)
1 Curvilinear coordinates1.3 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
⤿
![Page 15: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/15.jpg)
1.3 Metric1 Curvilinear coordinates
equivalent to the condition for the inverse matrix
which fulfill ,
With the reciprocal basis ,
one defines reciprocal components of the metric tensor
14
![Page 16: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/16.jpg)
1.3 Metric1 Curvilinear coordinates
metric tensor
orthonormality condition
“lowers index”
“raises index”
⤿
15
![Page 17: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/17.jpg)
1.4 Vector fields1 Curvilinear coordinates
mathematics: vector fieldphysics: vector (field)
covariant components
contravariant components(→ 1.6)
“raising/lowering indices”
vector components defined by means of basis vectors
16
![Page 18: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/18.jpg)
1.5 Tensor fields1 Curvilinear coordinates
tensor is multi-dimensional generalization of vector
mathematics: tensor fieldphysics: tensor (field)
behaves like a vector with respect to each of the vector spaces
17
product of vector spaces
tensor of rank 2 square matrixtensor of rank 1 tensor of rank 0
tensor of rank 3 cube
vectorscalar
........
rank of tensor
![Page 19: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/19.jpg)
1 Curvilinear coordinates
18
1.5 Tensor fields
basis vectors apply to each of the vector spaces
⤿
covariant components
contravariant components
mixed components
![Page 20: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/20.jpg)
1.5 Tensor fields1 Curvilinear coordinates
19
Example: Rank-2 tensor
⤿
Coincidentally, with the matrix product
For Cartesian coordinates:
![Page 21: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/21.jpg)
1.6 Coordinate transformations1 Curvilinear coordinates
20
consider different set of coordinates
(chain rule)
different coordinate systems describe same locations
⤿
⤿
![Page 22: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/22.jpg)
1 Curvilinear coordinates
21
covariantcontravariant
derivativesdifferentials
components transform like coordinate {}
1.6 Coordinate transformations
vector fields
tensor fields
⤿
⤿
![Page 23: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/23.jpg)
1.6 Coordinate transformations1 Curvilinear coordinates
22
1 Curvilinear coordinates
Proof: are covariant components of a tensor
⤿
![Page 24: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/24.jpg)
1.7 Affine connection1 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:
⤿
![Page 25: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/25.jpg)
1 Curvilinear coordinates1.7 Affine connection
1 Curvilinear coordinates
24
Example C: Spherical coordinates (IV)
⤿
⤿
![Page 26: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/26.jpg)
1 Curvilinear coordinates1.7 Affine connection
1 Curvilinear coordinates
25
Example C: Spherical coordinates (IV) [continued]
⤿
⤿
⤿
![Page 27: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/27.jpg)
1 Curvilinear coordinates1.7 Affine connection
1 Curvilinear coordinates
given that
the Christoffel symbols can be expressed by means
of the components of the metric tensor and their derivatives
26
![Page 28: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/28.jpg)
1 Curvilinear coordinates
27
1 Curvilinear coordinates1.7 Affine connection
Proof:
⤿ (I)
(II)
(III)
(II) + (III) - (I) :
⤿
![Page 29: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/29.jpg)
2 Tensor analysis
derivative:
28
vector field
both the vector components and the basis vectorsdepend on the coordinates
define covariant derivative of a contravariant vector component
as so that
2.1 Covariant derivative
![Page 30: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/30.jpg)
2 Tensor analysis2.1 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
![Page 31: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/31.jpg)
2 Tensor analysis2.1 Covariant derivative
covariant componentscontravariant components
30
for eachupper
lowerindex , add {{ }
takes place of in orwhere
covariant derivatives of tensor components
![Page 32: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/32.jpg)
31
2 Tensor analysis2.1 Covariant derivative
Covariant derivative of 2nd-rank tensor
⤿
![Page 33: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/33.jpg)
2.2 Riemann tensor2 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
![Page 34: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/34.jpg)
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)
![Page 35: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/35.jpg)
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)
![Page 36: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/36.jpg)
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
![Page 37: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/37.jpg)
3 Review: Classical Mechanics3.1 Principle of stationary action
action
Fermat’s principle (optics)Feynman’s path integral (QM)
(Hamilton’s) principle of stationary action
for : kinetic energypotential energy
Mechanical system completely described by
(Lagrangian)coordinatevelocitytime
36
(Euler-) Lagrange equations⤿
![Page 38: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/38.jpg)
Example: 1D harmonic oscillator (I)
⤿
⤿
⤿
3 Classical mechanics3.1 Principle of stationary action
37
![Page 39: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/39.jpg)
(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
![Page 40: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/40.jpg)
Path length ds = G d⤿ , stationary path means
48
4 Mechanics in curved space4.1 Stationary paths
Define
⤿
Constant L factored out of derivatives. Write derivative as dot, if we define t = s = ⤿
⤿
⤿
![Page 41: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/41.jpg)
L resembles Lagrangian for a free particle of mass m in curved space
with and
⤿
(Euler-Lagrange equations)
⤿
⤿
↳
}⤿
44
4 Mechanics in curved space4.1 Stationary paths
![Page 42: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/42.jpg)
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
![Page 43: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/43.jpg)
(geodesic equation)
4 Mechanics in curved space
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
![Page 44: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/44.jpg)
if all do not depend on ⤿
4.3 Conserved momentum pk dpk/d⤿=0 if the metric g independent of qk
4 Mechanics in curved space
50
(geodesic equation)
![Page 45: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/45.jpg)
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
![Page 46: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/46.jpg)
55
5 Special Relativity5.1 Minkowski space
describes distance in four-dimensional space⤿depend on reference frameboth and whereas
homogeneity and isotropy of space and time
⤿ invariance of
⤿
⤿ along light rays:
for all reference frames
invariance of speed of light
![Page 47: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/47.jpg)
56
5 Special Relativity5.1 Minkowski space
Latin indices
Greek indices
use 4-dimensional vectors
flat three-dimensional space described by cartesian coordinates
⤿
![Page 48: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/48.jpg)
photons trace null geodesics between eventsdefines light cone 45° opening angle in
5.2 Light cone5 Special Relativity
57
or: “causality and the finite speed of light”
instantaneous knowledge of interactionnon-relativistic theories:
light cone widens, all events get into causal contact
:
invariance ofcategorization holds irrespective of coordinate system and reference frame⤿
outside light cone‘elsewhere’, no causal connection
inside light conemassive particles move on time-like geodesics
![Page 49: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/49.jpg)
5.3 Proper time5 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
![Page 50: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/50.jpg)
5 Special Relativity
59
5.4 Relativistic mechanicsdefine 4-velocity as
⤿
⤿
⤿(as anticipated for inertial system)
known: free particle moves along geodesic
⤿ [ all ]
![Page 51: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/51.jpg)
5 Special Relativity
( ) non-relativistic limit
60
⤿⤿⤿⤿
5.4 Relativistic mechanics
(matches invariance of )ansatz:
relativistic action
![Page 52: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/52.jpg)
5.4 Relativistic mechanics5 Special Relativity
conjugate momentum
61
energy
⤿
(relativistic Hamilton-Jacobi equation)
with
⤿
(sign in spatial part due to in metric)
![Page 53: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/53.jpg)
components of stress tensor
provides relation between the forces and the cross-sections these are exerted on
forcearea of cross-sectionnormal to cross-section
5.5 Energy-momentum tensor5 Special Relativity
62
for fluid in thermodynamic equilibrium:(no shear stresses)
pressure
energy-momentumtensor
in fluid rest frame:
mass density
complement to
energy densitymomentum density
stress
![Page 54: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/54.jpg)
5 Special Relativity5.5 Energy-momentum tensor
63
non-relativistic limit:
(continuity equation)
(↔ Newton’s law)
![Page 55: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/55.jpg)
6 General Relativity
64
6.1 Principlesexperiments cannot distinguish between:
• virtual forces present in non-inertial frames• true forces
gravitation can be described byspace-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 equationsonly remaining issue: relation between and Newton’s law
![Page 56: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/56.jpg)
6 General Relativity6.1 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 locallyfor all non-accelerated (inertial) observers
The curvature follows the energy-momentum tensoras described by Einstein’s field equations
General Relativity summarized in 6 points
65
![Page 57: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/57.jpg)
66
6 General Relativity
6.2 Einstein’s field equations
independence on choice of coordinatesformulate 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)
![Page 58: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/58.jpg)
6 General Relativity6.2 Einstein’s field equations
67
Einstein’s field equations:
non-relativistic limit ( ):
dominating
,
⤿
![Page 59: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/59.jpg)
6 General Relativity6.2 Einstein’s field equations
Newton:
⤿
⤿
⤿with
⤿
68
![Page 60: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/60.jpg)
6 General Relativity6.2 Einstein’s field equations
[note: Einstein’s orignal sign convention for the Ricci tensor differs from ours]69
![Page 61: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/61.jpg)
6 General Relativity
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 neighbourhoodbaryonic matter in the Universe
![Page 62: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/62.jpg)
71
6 General Relativity
6.4 Time and distanceLaws 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 coordinateshave signs
⤿
![Page 63: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/63.jpg)
6 General Relativity6.4 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 locationgiven by
⤿ proper time
72
![Page 64: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/64.jpg)
coordinate transformation can always provide(at cost of time-dependent )
(synchronized reference frame)everywhere
coordinate line of (i.e. ) is geodesic
6 General Relativity6.5 Synchronisation
(with regard to time coordinate, but measured depends on location)global synchronisation possible⤿if
76
6.5 Synchronisation (e.g. FRW cosmology)
![Page 65: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/65.jpg)
![Page 66: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/66.jpg)
![Page 67: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/67.jpg)
![Page 68: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/68.jpg)
![Page 69: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/69.jpg)
![Page 70: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/70.jpg)
Challenge: prove this
![Page 71: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/71.jpg)
110
7 GR Applications
7. Satellites: GPS orbit Earth ~ stars orbit BHBeepers on sat. are Doppler/Gravitational-shifted, time delayed
![Page 72: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/72.jpg)
112
7 Consequences7. 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
![Page 73: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/73.jpg)
7 Consequences7.1 Relativistic Kepler problem
87
Perihelion shift of the planets in the Solar system
semi-major semi-major axisaxis
a a [AU][AU]
orbital orbital periodperiod
P P [yr][yr]
eccentriciteccentricityyεε
perihelion perihelion shiftshift
per centuryper 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)
![Page 74: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/74.jpg)
7 Consequences7.1 Relativistic Kepler problem
86
![Page 75: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/75.jpg)
90
7 Consequences7.2 Bending of light
asymptotics
⤿ total deflection ⤿
Deflection of light by gravity (1915)
α =4GMc2ξ1.″7
measurable at Solar limb: α =
bending angle
![Page 76: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/76.jpg)
92
![Page 77: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/77.jpg)
93
![Page 78: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/78.jpg)
7 Consequences7.2 Bending of light
"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
![Page 79: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/79.jpg)
7 Consequences7.2 Bending of light
99
Notes about gravitational microlensing dated to 1912on two pages of Einstein’s scratch notebook
![Page 80: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/80.jpg)
I−
I+
ξη
side view
7 Consequences7.2 Bending of light
Images by a gravitational lens
96
⤿
with (angular Einstein radius)
(two images)⤿
6˝
![Page 81: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/81.jpg)
(animation by Daniel Kubas, ESO)
98
7 Consequences7.2 Bending of light
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
⤿
![Page 82: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/82.jpg)
The chance isone in a million !
B. Paczyński 1986, ApJ 304, 1
7 Consequences7.2 Bending of light
100
![Page 83: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/83.jpg)
First reported microlensing event
MACHO LMC#1
Nature 365, 621 (October 1993)
7 Consequences7.2 Bending of light
101
![Page 84: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/84.jpg)
Astronomy & Geophysics Vol. 47
(June 2006)
7 Consequences7.2 Bending of light
102
![Page 85: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/85.jpg)
![Page 86: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/86.jpg)
![Page 87: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/87.jpg)
![Page 88: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/88.jpg)
![Page 89: PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013](https://reader033.vdocuments.us/reader033/viewer/2022061513/56649e155503460f94aff24d/html5/thumbnails/89.jpg)
A Sample of Advanced Material: Geodesics around Black Hole Metric