General relativity Dr. Luigi E. Masciovecchio
[email protected] first published on http://mio.discoremoto.alice.it/luigimasciovecchio/, October 2011 available as notebook and PDF on http://sites.google.com/site/luigimasciovecchio/ Print#"Revision ", IntegerPart#Date#''' Revision 2013, 7, 30, 8, 0, 59
A) INTRODUCTION Dear Colleagues, This is my personal Mathematica notebook on Albert Einstein's genial general theory of relativity. This document wasn't originally intended for publication, but a few formulas and tricks are maybe of interest to you, so here they are. The code seems to work well and I added some comments to make it more understandable. This is not an introduction to this field, so use it at your own risk! The main point about this work is to show how to do the typical mathematics of general relativity easily and rigorously with Mathematica. In addition, I "streamlined" a little bit the derivation of some classical results (perihelion advance, bending of light etc.). As main textbook I have chosen the excellent and brilliantly instructive "A short course in general relativity" by James Foster and J.David Nightingale. Mathematica together with the packages Tensorial and GeneralRelativity have been used by David Park to do all the derivations, examples and exercises of this textbook. Most of the present notebook is actually a rewrite of Park's very fine original work. Once again, the combination of a good textbook and Mathematica provides a fun, easy and mathematical rigorous learning environment that stimulates greatly understanding and own experiments with the formulas. Don't miss it!
*** General relativity is a metric theory of gravitation. At its core are Einstein's field equations, which describe the relation between the geometry of a four-dimensional, pseudo-Riemannian manifold representing spacetime, and the energy-momentum contained in that spacetime. First published by Albert Einstein in 1915 as a tensor equation, the Einstein's field equations equate spacetime curvature (expressed by the Einstein tensor) with the energy and momentum within that spacetime (expressed by the energy-momentum-stress tensor). General relativity's predictions have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data. (Wikipedia, 2011) General relativity is a geometric theory and incorporates special relativity in the sense that locally the spacetime of the general theory is like that of the special theory. So it's important for the sake of conceptual cleanness to derive in your course first special relativity from the basic geometrical spacetime symmetries without using the postulate of constant speed of light or any other "unneeded physics" (see for example Jean-Marc Lévy-Leblond, "One more derivation of the Lorentz transformation", American Journal of Physics 44, 271-277 (1976); visit http://o.castera.free.fr for more information). Valuable web resources on general relativity: • David Park, Mathematica notebooks (2005) based on "A short course in general relativity" (Foster/Nightingale)
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• See books.google.com or the Springer editor web site for a preview of the above-mentioned textbook. • Florian Schrack "Gravitation - Theorien, Effekte und Simulation am Computer" (2002) • Gerard ’t Hooft "Introduction to General Relativity" (2007) • Matt Visser "Math 464: Notes on Differential Geometry" (2009) • Matt Visser "Math 465: Notes on General Relativity and Cosmology" (2009) • Norbert Dragon "Geometrie der Relativitätstheorie" (2011) • Sean Carroll "Lecture Notes on General Relativity" (1997) • Tom Marsh "Notes for PX436, General Relativity" (2009) • Clifford M. Will "The Confrontation between General Relativity and Experiment", Living Rev. Relativity, 9, (2006) • Neil Ashby "Relativity in the Global Positioning System", Living Rev. Relativity, 6, (2003) • Wikipedia: "General relativity", "Allgemeine Relativitätstheorie" and links • General relativity video courses (Charles Bailyn, Alexander Maloney, Lenny Susskind) Note: Ê Mathematica by Wolfram Research is a (fabulous) computer algebra system. Ê A notebook is an interactive Mathematica document (extension .nb). Ê Tensorial 3.0 (R. Cabrera, D. Park, J.-F. Gouyet, August 2005) is a general-purpose tensor calculus package for Mathematica Version 4.1 or later. Ê TGeneralRelativity1`GeneralRelativity` (D. Park, 29 January 2005) is a subpackage for the Tensorial package that adds routines useful in special and general relativity. (This also automatically loads the regular Tensorial package.) Print#"This system is:"' "ProductIDName", "ProductVersion" s. $ProductInformation ReadList#"ver", String'##2'' $MachineType, $ProcessorType, $ByteOrdering, $SystemCharacterEncoding This system is:
Mathematica, 5.2 for Microsoft Windows +June 20, 2005/ Windows 98 #versione 4.10.1998'
PC, x86, 1, WindowsANSI
B) HELP (Extracted from the Tensorial package help.)
{x,G,g,*} are the standard set of tensor labels used in all Tensorial derivative routines. They tell the routines which labels will be considered to represent the coordinates x, Kronecker G, metric tensor g and Christoffel symbol *. DeclareBaseIndices[{index..}] declares the base indices for the underlying linear space. DeclareIndexFlavor[{flavorname,flavorform}...] will add the index flavors to the IndexFlavors list and establish the Format for displaying indices with the given flavor name.
ToArrayValues[baseindices][expr] will convert the expression to a vector, matrix or array by expansion and substitution of any stored values.
EvaluateDotProducts[e,g,metricsimplify:True][expr] expands Dot products of vectors expressed in a given basis e using the metric tensor g. Metric simplification is performed if the default argument metricsimplify is True.
LinearBreakout[f1,f2,...][v1,v2,...][expr] will break out the linear terms of any expressions within expr that
have heads matching the patterns fi over variables matching the patterns vj.
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SetMetricValues[g,metricmatrix,flavor:Identity]
creates value definitions for the up and down forms of the
metric tensor using the label g and a metric matrix.
CoordinateToTensors[{r,T,I...},coord:x,flavor:Identity][expr] will convert the coordinate symbols in the
expression to the corresponding indexed tensors. The optional arguments coord and flavor give the coordinate label and index flavor to use. Their default values are x and plain.
SetChristoffelValueRules[xu[i,metricmatrix,*,simplification:Identity] calculates and stores substitution rules for the Christoffel values of *udd[i,j,k] and *ddd[i,j,k] from the values of metricmatrix and the xu[i] vector pattern.
SelectedTensorRules[label,pattern] will select the rules for label whose right hand sides are nonzero and whose left
hand sides match the pattern.
SimplifyTensorSum[expr] will check that all terms in a tensor sum have valid indices,that the free indices are the same in all terms,and will simplify the sum by matching dummy indices in all terms that have the same index structure.
ExpandCovariantD[{x,G,g,*},a][expr] will expand first order covariant derivatives of tensors using x as the label for the coordinates, G as the label for the Kronecker, g as the label for the metric tensor and * as the label for Christoffel symbols. The introduced dummy index will be a.
MapLevelParts[function,{topposition,levelpositions}][expr] will map the function onto the selected level
positions in an expression. The function is applied to them as a group and they are replaced with a single new expression. Other parts not specified on the list are left unchanged.
C) PHYSICAL CONSTANTS Some physical constants as given by Mathematica. Print#"Miscellaneous`PhysicalConstants`:"' Miscellaneous`PhysicalConstants` Miscellaneous`Units` SpeedOfLight, GravitationalConstant, CosmicBackgroundTemperature, HubbleConstant HubbleConstant1 , AgeOfUniverse s HubbleConstant1 , Convert#AgeOfUniverse, Year' "Earth:", EarthMass, EarthRadius, "Sun:", Convert#SolarSchwarzschildRadius SpeedOfLight2 s +2 GravitationalConstant/, Kilogram', SolarRadius, SolarSchwarzschildRadius Miscellaneous`PhysicalConstants`:
299792458 Meter 6.673 1011 Meter2 Newton 3.2 1018 cccccccccc , cccccccccccccccccccccccccccccccc cccccccccccccccc ccccccccccccccccccccc , 2.726 Kelvin, cccccccccccccccc cccccccccc cccccccccccccccccccccccccccccccc 2 Second
Kilogram
Second
3.125 1017 Second, 1.504, 1.49036 1010 Year Earth:, 5.9742 1024 Kilogram, 6378140 Meter, Sun:, 1.9888 1030 Kilogram, 6.9599 108 Meter, 2953.25 Meter Print#"Gravitational constant\nG ", Convert#GravitationalConstant, Kilogram1 Meter3 Second2 ', " ", Convert#GravitationalConstant, Gram1 Centimeter3 Second2 '' Convert#8 S GravitationalConstant s SpeedOfLight2 , Meter s Kilogram'; Print# "Einsteinsche Gravitationskonstante in SexlsUrbantke S.69\nN 8 S G c2 1 Convert#%, Gram Centimeter '' 4 Print#"coupling constant in FostersNightingale p.113\nN 8 S G c ", 2 %% s SpeedOfLight '
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", %, "
",
3
Gravitational constant 6.673 1011 Meter3 G cccccccccccccccccccccccccccccccc ccccccccccccccccccc Kilogram Second2
6.673 108 Centimeter3 cccccccccccccccccccccccccccccccc cccccccccccccccc 2cccccccccccccccc Gram Second
Einsteinsche Gravitationskonstante in SexlsUrbantke S.69 1.86603 1026 Meter 1.86603 1027 Centimeter N 8 S G c2 cccccccccccccccccccccccccccccccc cccccccccccccccccccccc cccccccccccccccccccccccccccccccc cccccccccccccccc cccccccccccccccc ccccc Kilogram Gram coupling constant in FostersNightingale p.113 2.07624 1043 Second2 N 8 S G c4 cccccccccccccccccccccccccccccccc cccccccccccccccc cccccccccccc Kilogram Meter
I will use the CODATA 2010 values. (See http://physics.nist.gov/ for updates.) Print#"CODATA 2010: G 4 Print#"N 8 S G c CODATA 2010: G
N
8 S G c4
6.673 84+80/1011 m3 kg1 s2 "' ", NumberForm# 8S 6.67384 1011 s +2997924584 /, 7', " m1 kg1 s2 "'
6.673 84+80/1011 m3 kg1 s2
2.076504 1043 m1 kg1 s2
D) OWN (?) CONSIDERATIONS Special relativity teaches us how spacetime dictates the behavior of matter-energy and general relativity teaches us how matter-energy influences the behavior of spacetime. We could say that this two entities, spacetime and matter-energy, are in some kind of interaction. Starting from a heuristic principle that states that entities who can interact can not be completely different "in essence", we could tentatively postulate a symmetry between spacetime and matter-energy, implying the possibility of a transformation of spacetime into matter-energy and vice-versa. So it's maybe sensible to ask: • How much spacetime can we get from a given quantity of matter-energy or vice-versa? What is the conversion factor O between (geometrized) spacetime and matter-energy (1 m4 C O · 1 J)? Is O a universal constant? • What are the observable signatures of spacetime matter-energy transformations? • How "expands" newly created spacetime in some finite region into the rest of the Universe? How works the local "collapse" of the universe caused by the destruction of a finite piece of spacetime? • How works the spacetime - matter-energy - transformation at a fundamental level?
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Since wild speculations don't cost a thing, we can go further and postulate the existence of a substance called "Essenz" in which matter-energy and spacetime are not separated entities and which constitutes all of the Universe at some point. We can assume that the Big Bang represents the moment of the evolution of the Universe where the Essenz undergoes a phase transition separating into the two components spacetime and matter-energy. Since then we have "matter-energy acting on the stage of spacetime" and we can define (at most locally) a metric to measure space and time. We said that the Essenz undergoes a phase transition at some point: this means that this substance is not static. But the time coordinate that we need to catalogue events in the pre-BigBang era has to be interpreted as an intrinsic parameter of the Essenz. This intrinsic time parameter must not necessarily be a measurable quantity (if there is no metric) but may defines only an order relation between events, the evolution of the Essenz proceeding by "leaps" much like today quantum systems evolves (e.g. successive decays in a radioactive series). Perhaps this analogy is not accidental and points to some connection between quantum mechanics and spacetime physics! Well, as I said, wild speculations don't cost a thing...
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E) CALCULATIONS FROM James Foster, J.David Nightingale A SHORT COURSE IN GENERAL RELATIVITY (3.ed., 2006) with Mathematica code by David Park (2005, for the 2. ed. [1995]) partially modified, corrected and simplified by Luigi E. Masciovecchio (2011) Utilization note: Every of the following Mathematica subsections should be evaluated by its own! The initialization code for a subsection ends with a horizontal line. I present here only my limited set of calculations from the textbook, for a complete (!) and extensively commented set see the huge work by David Park.
Chapter 1: Vector and tensor fields
1.0 Introduction p. 7
1.1 Coordinate systems in Euclidean space p. 7 - 13 (nonsuffix notation)
1.2 Suffix notation p. 13 - 19
1.3 Tangents and gradients p. 19 - 23
1.4 Coordinate transformations in Euclidean space p. 23 - 27
1.5 Tensor fields in Euclidean space p. 27 - 30
1.6 Surfaces in Euclidean space p. 30 - 35
1.7 Manifolds p. 35-37
coming soon...
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1.8 Tensor Fields on manifolds p. 38 - 43 "We can create new tensors from old tensors by a number of methods."
coming soon...
1.9 Metric properties p. 43 - 46 (pseudo-Riemannian manifolds)
coming soon...
1.10 What and where are the bases? p. 46 - 49
coming soon...
Chapter 2: The spacetime of general relativity and paths of particles
2.0 Introduction p. 53 - 56
2.1 Geodesics p. 56 - 64
2.2 Parallel
2.3 Absolute
2.4 Geodesic
2.5 The
2.6 Newton's laws of motion p. 86 - 87
2.7 Gravitational potential and the geodesic p. 87 - 89
2.8 Newton's law of universal gravitation p. 89 - 90
vectors along a curve p. 64 - 71
and covariant differentiation p. 71 - 79
coordinates p. 79 - 81
spacetime of general relativity p. 82 - 85
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2.9 A rotating reference system p. 90 - 93
Chapter 3: Field equations and curvature
3.0 Introduction p. 97
3.1 The stress tensor and fluid motion p. 97 - 102
3.2 The curvature tensor and related tensors p. 102 - 105
3.3 Curvature and parallel transport p. 105 - 110
3.4 Geodesic deviation p. 110 - 112
3.5 EINSTEIN's field equations p. 112 - 114 3.6 Einstein's equation compared with Poisson's equation p. 115 - 116
3.7 The Schwarzschild solution p. 116 - 119
Chapter 4: Physics in the vicinity of a massive object
4.0 Introduction p. 123
4.1 Length and time p. 124
4.2 Radar sounding (Shapiro-Effekt) p. 129
4.3 Spectral Shift p. 131
Addendum: The Hafele-Keating experiment (Heuristische ex post Machbarkeitsstudie des Hafele-Keating-Experiments)
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4.4 General particle motion (Including photons) p. 136
4.5 Perihelion advance p. 144
4.6 Bending of light p. 146
4.7 Geodesic effect p. 149
coming soon...
4.8 Black holes p. 152
coming soon...
4.9 Other coordinate systems p. 157
coming soon...
4.10 Rotating objects; the Kerr solution p. 167
coming soon...
Chapter 5: Gravitational radiation coming soon...
Chapter 6: Elements of cosmology coming soon...
Appendices coming soon...
"Nur wer nicht sucht, ist vor Irrtum sicher." Albert Einstein (1879-1955)
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