Quantum Field Theory III

In the present volume, we concentrate on the classical aspects of gauge theory
related to curvature. These have to be supplemented by the crucial, but elusive
quantization procedure. The quantization of the Maxwell–Dirac system leads to
quantum electrodynamics (see Vol. II). The quantization of both the full Standard
Model in elementary particle physics and the quantization of gravitation will be
studied in the volumes to come.
One cannot grasp modern physics without understanding gauge theory,
which tells us that the fundamental interactions in nature are based on
parallel transport, and in which forces are described by curvature, which
measures the path-dependence of the parallel transport.
Gauge theory is the result of a fascinating long-term development in both mathematics
and physics. Gauge transformations correspond to a change of potentials,
and physical quantities measured in experiments are invariants under gauge transformations.
Let us briefly discuss this.
Gauss discovered that the curvature of a two-dimensional su***ce is an intrinsic
property of the su***ce. This means that the Gaussian curvature of the su***ce can
be determined by using measurements on the su***ce (e.g., on the earth) without
using the surrounding three-dimensional space. The precise formulation is provided
by Gauss’ theorema egregium (the egregious theorem). Bernhard Riemann (1826–
1866) and ´Elie Cartan (1859–1951) formulated far-reaching generalizations of the
theorema egregium which lie at the heart of
• modern differential geometry (the curvature of general fiber bundles), and
• modern physics (gauge theories).
Interestingly enough, in this way,
• Einstein’s theory of general relativity (the curvature of the four-dimensional
space-time manifold), and
• the Standard Model in elementary particle physics (the curvature of a specific
fiber bundle with the symmetry group U(1) × SU(2) × SU(3))
can be traced back to Gauss’ theorema egregium.
In classical mechanics, a large class of forces can be described by the differentiation
of potentials. This simplifies the solution of Newton’s equation of motion
and leads to the concept of potential energy together with energy conservation (for
the sum of kinetic and potential energy). In the 1860s, Maxwell determined that
the computation of electromagnetic fields can be substantially simplified by introducing
potentials for both the electric and the magnetic field (the electromagnetic
four-potential).
Gauge theory generalizes this by describing forces (interactions) by the
differentiation of generalized potentials (also called connections).
The point is that gauge transformations change the generalized potentials, but not
the essential physical effects.
Physical quantities, which can be measured in experiments, have to be invariant
under gauge transformations.
Parallel to this physical situation, inmathematics the Riemann curvature tensor can
be described by the differentiation of the Christoffel symbols (also called connection
coefficients or geometric potentials). The notion of the Riemann curvature tensor
was introduced by Riemann in order to generalize Gauss’ theorema egregium to
higher dimensions. In 1915, Einstein discovered that the Riemann curvature tensor
of a four-dimensional space-time manifold can be used to describe gravitation in
the framework of the theory of general relativity.
The basic idea of gauge theory is the transport of physical information
along curves (also called parallel transport).
This generalizes the parallel transport of vectors in the three-dimensional Euclidean
space of our intuition.
In 1917, it was discovered by Levi-Civita that the study of curved manifolds
in differential geometry can be based on the notion of parallel transport of
tangent vectors (velocity vectors).
In particular, curvature can be measured intrinsically by transporting a tangent
vector along a closed path. This idea was further developed by ´Elie Cartan in
the 1920s (the method of moving frames) and by Ehresmann in the 1950s (the
connection of both principal fiber bundles and their associated vector bundles).
The very close relation between
• gauge theory in modern physics (the transport of local SU(2)-phase factors investigated
by Yang and Mills in 1954), and
• the formulation of differential geometry in terms of fiber bundles in modern
mathematics
was only noticed by physicists in 1975 (see T. Wu and C. Yang, Concept of nonintegrable
phase factors and global formulation of gauge fields, Phys. Rev. D12
(1975), 3845–3857).
The present Volume III on gauge theory and the following Volume IV on quantum
mathematics form a unified whole. The two volumes cover the following topics:

Volume III: Gauge Theory
Part I: The Euclidean Manifold as a Paradigm
Chapter 1: The Euclidean Space E3 (Hilbert Space and Lie Algebra Structure)
Chapter 2: Algebras and Duality (Tensor Algebra, Grassmann Algebra, Clifford
Algebra, Lie Algebra)
Chapter 3: Representations of Symmetries in Mathematics and Physics
Chapter 4: The Euclidean Manifold E3
Chapter 5: The Lie Group U(1) as a Paradigm in Harmonic Analysis and Geometry
Chapter 6: Infinitesimal Rotations and Constraints in Physics
Chapter 7: Rotations, Quaternions, the Universal Covering Group, and the Electron
Spin
Chapter 8: Changing Observers – A Glance at Invariant Theory Based on the
Principle of the Correct Index Picture
Chapter 9: Applications of Invariant Theory to the Rotation Group
Chapter 10: Temperature Fields on the Euclidean Manifold E3
Chapter 11: Velocity Vector Fields on the Euclidean Manifold E3
Chapter 12: Covector Fields on the Euclidean Manifold E3 and Cartan’s Exterior
Differential – the Beauty of Differential Forms
Part II: Ariadne’s Thread in Gauge Theory
Chapter 13: The Commutative Weyl U(1)-Gauge Theory and the Electromagnetic
Field
Chapter 14: Symmetry Breaking
Chapter 15: The Noncommutative Yang–Mills SU(N)-Gauge Theory
Chapter 16: Cocycles and Observers
Chapter 17: The Axiomatic Geometric Approach to Vector Bundles and Principal
Bundles
Part III: Einstein’s Theory of Special Relativity
Chapter 18: Inertial Systems and Einstein’s Principle of Special Relativity
Chapter 19: The Relativistic Invariance of the Maxwell Equations
Chapter 20: The Relativistic Invariance of the Dirac Equations and the Electron
Spin
Part IV: Ariadne’s Thread in Cohomology
Chapter 21: Exact Sequences
Chapter 22: Electrical Circuits as a Paradigm in Homology and Cohomology
Chapter 23: The Electromagnetic Field and the de Rham Cohomology.

Volume IV: Quantum Mathematics
Part I: The Hydrogen Atom as a Paradigm
Chapter 1: The Non-Relativistic Hydrogen Atom via Lie Algebra, Gauss’s Hypergeometric
Functions, von Neuman’s Functional Analytic Approach, the Weyl–
Kodaira Theory, Gelfand’s Generalized Eigenfunctions, and Supersymmetry
Chapter 2: The Dirac Equation and the Relativistic Hydrogen Atom via the Clifford
Algebra of the Minkowski Space
Part II: The Four Fundamental Forces in the Universe
Chapter 3: Relativistic Invariance and the Energy–Momentum Tensor in Classical
Field Theories
Chapter 4: The Standard Model for Electroweak and Strong Interaction in Particle
Physics
Chapter 5: Gravitation, Einstein’s Theory of General Relativity, and the Standard
Model in Cosmology
Part III: Lowest-Order Radiative Corrections in Quantum Electrodynamics (QED)
Chapter 6: Dimensional Regularization for the Feynman Propagators in QED
(Quantum Electrodynamics)
Chapter 7: The Electron in an External Electromagnetic Field (Renormalization
of Electron Mass and Electron Charge)
Chapter 8: The Lamb Shift
Part IV: Conformal Symmetry
Chapter 9: Conformal Transformations According to Gauss, Riemann, and Lichtenstein
Chapter 10: Compact Riemann Su***ces
Chapter 11: Minimal Su***ces
Chapter 12: Strings and the Graviton
Chapter 13: Complex Function Theory and Conformal Quantum Field Theory
Part V: Models in Quantum Field Theory
Part VI: Distributions and the Epstein–Glaser Approach to Perturbative Quantum
Field Theory
Part VII: Nets of Operator Algebras and the Haag–Kastler Approach to Quantum
Field Theory
Part VIII: Symmetry and Quantization – the BRST Approach to Quantum Field
Theory
Part IX: Topology, Quantization, and the Global Structure of Physical Fields
Part X: Quantum Information.

Readers who want to understand modern differential geometry and modern physics
as quickly as possible should glance at the Prologue of the present volume and at
Chaps. 13 through 17 on Ariadne’s thread in gauge theory.
Cohomology plays a fundamental role in modern mathematics and physics.
It turns out that cohomology and homology have their roots in the rules for
electrical circuits formulated by Kirchhoff in 1847.
This helps to explain why the Maxwell equations in electrodynamics are closely
related to cohomology, namely, de Rham cohomology based on Cartan’s calculus
for differential forms and the corresponding Hodge duality on the Minkowski space.
Since the Standard Model in particle physics is obtained from the Maxwell equations
by replacing the commutative gauge group U(1) with the noncommutative gauge
group U(1) × SU(2) × SU(3), it should come as no great surprise that de Rham
cohomology also plays a key role in the Standard Model in particle physics via
the theory of characteristic classes (e.g., Chern classes which were invented by
Shing-Shen Chern in 1945 in order to generalize the Gauss–Bonnet theorem for
two-dimensional manifolds to higher dimensions).
It is our goal to show that the gauge-theoretical formulation of modern physics
is closely related to important long-term developments in mathematics pioneered by
Gauss, Riemann, Poincar´e and Hilbert, as well as Grassmann, Lie, Klein, Cayley,
´Elie Cartan and Weyl. The prototype of a gauge theory in physics is Maxwell’s
theory of electromagnetism. The Standard Model in particle physics is based on the
principle of local symmetry. In contrast to Maxwell’s theory of electromagnetism,
the gauge group of the Standard Model in particle physics is a noncommutative
Lie group. This generates additional interaction forces which are mathematically
described by Lie brackets.
We also emphasize the methods of invariant theory. In terms of physics, different
observers measure different values in their experiments. However, physics does
not depend on the choice of observers. Therefore, one needs both an invariant approach
and the passage to coordinate systems which correspond to the observers, as
emphasized by Einstein in the theory of general relativity and by Dirac in quantum
mechanics. The appropriate mathematical tool is provided by invariant theory.
Acknowledgments. In 2003, J¨urgen Tolksdorf initiated a series of four International
Workshops on the state of the art in quantum field theory and the search
for a unified theory concerning the four fundamental interactions in nature. I am
very grateful to Felix Finster, Olaf M¨uller, Marc Nardmann, and J¨urgen Tolksdorf
for organizing the workshop Quantum Field Theory and Gravity, Regensburg, 2010.
The following three volumes contain survey articles written by leading experts:
F. Finster, O. M¨uller, M. Nardmann, J. Tolksdorf, and E. Zeidler (Eds.),
Quantum Field Theory and Gravity: Conceptual and Mathematical Advances
in the Search for a Unified Framework, Birkh¨auser, Basel (to appear).
B. Fauser, J. Tolksdorf, and E. Zeidler (Eds.), Quantum Field Theory –
Competitive Methods, Birkh¨auser, Basel, 2008.
B. Fauser, J. Tolksdorf, and E. Zeidler (Eds.), Quantum Gravitation:
Mathematical Models and Experimental Bounds, Birkh¨auser, Basel, 2006.
These three volumes are recommended as supplements to the material contained
in the present monograph.

Prologue
Geometry is the knowledge of what eternally exists.
Plato of Athen (428–348 B.C.)
He who understands geometry may understand anything in this world.
Galileo Galilei (1564–1642)
The way of people to the laws of nature are not less admirable than the
laws themselves.
Johannes Kepler (1571–1630)
In humbleness, we have to admit that if ‘number’ is a product of our
imagination, ‘space’ has a reality outside of our imagination, to which a
priori we cannot assign its laws.
Gauss (1777–1855) in a letter to Bessel, 1840
This prologue should help the reader to understand the sophisticated historical
development of gauge theory in mathematics and physics.We will not follow a strict
logical route. This will be done later on. At this point, we are going to emphasize the
basic ideas. It is our goal to show the reader how the methods of modern differential
geometry work in the case of Einstein’s theory of general relativity, which describes
the gravitational force in nature. In particular, we want to show how
• the language of physicists created by Einstein and used in most physics textbooks
(based on the use of local space-time coordinates) and
• the language of mathematicians used in modern textbooks on differential geometry
(based on the invariant – i.e., coordinate-free – formulation)
are related to each other. This should help physicists to enter modern differential
geometry. One cannot grasp modern physics without understanding gauge field
theory which tells us the following crucial facts:
• interactions in nature are based on the parallel transport of physical information;
• forces are described by curvature which measures the path-dependence of the
parallel transport.
Here, we will discuss the following points:
• an interview with the Nobel prize laureate Chen Ning Yang (born 1922) on the
history of modern gauge theory,
• Einstein’s theory of general relativity on gravitation,
• changing observers in the universe and tensor calculus,
• the Riemann curvature tensor and the beauty of Gauss’ theorema egregium,
• two fundamental variational principles in general relativity,
• symmetry and Felix Klein’s invariance principle in geometry (a glance at the
history of invariant theory in the 19th century),1
• Einstein’s principle of general relativity and invariants – the geometrization of
physics (the paradigm of higher-dimensional cartography),
• gauge transformations:
– Einstein’s gauge transformation in the theory of both special relativity and
general relativity (change of the observer),
– Dirac’s unitary gauge transformations in the Hilbert space approach to quantum
mechanics (change of the observer by changing the measurement device),
– Yang’s gauge transformation by changing the local phase factor of the wave
function,
– the U(1)-gauge transformation in classical electrodynamics and quantum electrodynamics,
– the U(1) × SU(2) gauge transformations in electroweak interaction,
– the SU(3) gauge transformations in strong interaction (quantum chromodynamics),
– the U(1) × SU(2) × SU(3) gauge transformations in the Standard Model in
particle physics,
– the conformal gauge transformations in string theory,
– ´Elie Cartan’s gauge transformations in his method of moving frames (change
of the frame),
• construction of invariants by the universal index killing principle,
• Lie’s intrinsic tangent vectors,
• ´Elie Cartan’s algebraization of calculus and infinitesimals,
• Riemann’s invariant sectional curvature and the geometric meaning of Riemann’s
curvature tensor,
• Levi-Civita’s parallel transport and the geometric meaning of the Riemann curvature
tensor,
• two fundamental approaches in differential geometry:
– Gauss’ method of symmetric tensors, and
– Cartan’s method of antisymmetric tensors,
• Yang’s matrix trick (the relation between the Einstein equations in general relativity
and the Maxwell–Yang–Mills equations), and Cartan’s calculus for matrices
with differential forms as entries,
• Cartan’s structural equations:
– local structural equations,
– global structural equations,
• partial covariant derivative and the classical Ricci calculus,
• the Lie structure behind curvature,
• the generalized Riemann curvature tensor in modern mathematics and physics,
• parallel transport of physical information and curvature,
• the modern language of fiber bundles in mathematics and physics,
• summary of typical applications,
• perspectives (instantons and gauge theory, conformal symmetry and twistors,
the Seiberg–Witten equations and the quark confinement, the Donaldson theory
for 4-dimensional manifolds, Morse theory and Floer homology, quantum cohomology,
J-holomorphic curves, Frobenius manifolds, Ricci flow and the Poincar´e
conjecture).
The classical formulas (0.13) and (0.14) on page 11 for defining the Riemann curvature
tensor via Christoffel symbols for the metric tensor are clumsy. The development
of modern differential geometry was essentially influenced by the desire of
mathematicians to get insight into the true structure of curvature. This led to a
better understanding of curvature and to far-reaching generalizations which proved
to be useful in modern physics. The basic paper in mathematics is due to:
C. Ehresmann, Les connexions infinit´esimales dans un espace fibr´e differentiable
(in French) (The infinitesimal connections in a differentiable fiber
bundle), Colloque de Topologie, Bruxelles, 1950, pp. 29–55.
Charles Ehresmann (1905–1979) based his theory on ´Elie Cartan’s work created in
the 1920s.2 The first textbook on modern differential geometry was written by:
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols.
1, 2, Wiley, New York, 1963.
We also recommend:
Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis,
Manifolds, and Physics. Vol. 1: Basics; Vol. 2: 92 Applications, Elsevier,
Amsterdam, 1996.
T. Frankel, The Geometry of Physics, Cambridge University Press, Cambridge,
2004.
S. Novikov and T. Taimanov, Geometric Structures and Fields, Amer.
Math. Soc., Providence, Rhode Island, 2006.
As an introduction to the theory of general relativity based on the use of local
coordinates, we recommend the classical Lecture Notes by
P. Dirac, General Theory of Relativity, Princeton University Press, 1996
(70 pages)
together with
Ø. Grøn and S. Hervik, Einstein’s Theory of General Relativity: with Modern
Applications in Cosmology, Springer, New York, 2007.
Both the invariant formulation and the formulation in terms of local coordinates is
discussed in great detail in the classic textbook by
C. Misner, K. Thorne, and A. Wheeler, Gravitation, Freeman, San Francisco,
California, 1973.
For the sophisticated mathematical problem of solving the initial-value problem for
the Einstein equations on the gravitational field, we recommend:
P. Cru´sciel and H. Friedrich, The Einstein Equations and the Large Scale
Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General
Relativity, Birkh¨auser, Boston, 2004.
Y. Choquet–Bruhat, General Relativity and the Einstein Equations, Oxford
University Press, 2008.
As a comprehensive modern textbook, we recommend:
T. Padmanabhan, Gravitation: Foundations and Frontiers, Cambridge
University Press, 2010.
The nature of dark matter is one of the great open problems in physics. We refer
to:
G. Bertone (Ed.), Particle Dark Matter, Cambridge University Press, 2010.

An Interview with Chen Ning Yang on the History of Modern
Gauge Theory
To begin with, let us quote some parts of an interview given by the physicist Chen
Ning Yang answering the questions of Dianzhou Zhang:
Zhang: Chen Ning Yang (born 1922 in Hefei, China), one of the twentieth
century’s great theoretical physicists, shared the Nobel prize in physics
with Tsung-Dao Lee in 1957 for their joint contribution to parity nonconservation
in weak interaction. Mathematicians, however, know Yang
best for the Yang–Mills gauge field theory and the Yang–Baxter equation.
After Einstein and Dirac, Yang is perhaps the twentieth-century physicist
who has had the greatest impact on the development of mathematics . . .
While a student in Kunming (China) and Chicago, Yang was impressed
with the fact that gauge invariance determined all electromagnetic interactions.
This was known from the works in the years 1918–1929 of Weyl,
Fock, and London, and through later review papers by Pauli. But by the
1940s and the early 1950s, it played only a minor and technical role in
physics. In Chicago, Yang tried to generalize the concept of gauge invariance
to non-Abelian groups (the gauge group for electromagnetism being
the Abelian group U(1)). In analogy with Maxwell’s equations he tried
Fαβ = ∂αAβ − ∂βAα,
where Aα are matrices (α, β = 0, 1, 2, 3). As Yang pointed out later on,
“This led to a mess, and I had to give up.”
In 1954, as a visiting physicist at Brookhaven National Laboratory on Long
Island, New York, Yang returned once again to the idea of generalizing
gauge invariance. His officemate was Robert Mills, who was about to finish
his Ph.D. degree at Columbia University, New York City. Yang introduced
the idea of non-Abelian gauge field to Mills, and they decided to add a
quadratic term:
Fαβ = ∂αAβ − ∂βAα + AαAβ −AβAα. (0.1)
That cleared up the “mess” and led to a beautiful new field theory.
Zhang: Did you study gauge field theory continuously after 1954?
Yang: Yes, I did . . . In the late 1960s, I began a new formulation of gauge
field theory through the approach of non-integrable phase factors. It happened
that one semester I was teaching general relativity, and I noticed
that the formula (0.1) in gauge field theory and the formula
      (0.2)
with α, β, γ, δ = 0, 1, 2, 3 for the Riemann curvature tensor in Riemannian
geometry are not just similar – they are, in fact, the same if one makes
the right identification of symbols.6 It is hard to describe the thrill I felt
at understanding the point.
Zhang: Is that the first time that you realized the relation between gauge
theory and differential geometry?
Yang: I had noticed the similarity between Levi-Civita’s parallel displacement
and non-integrable phase factors in gauge fields. But the exact relationship
was appreciated by me only when I realized that the formula (0.1)
in gauge field theory and the Riemann formula (0.2) are the same. With
an appreciation of the geometrical meaning of gauge theory, I consulted
Jim Simons, a distinguished geometer, who was then the chairman of the
Mathematics Department at Stony Brooke (Long Island, New York). He
said gauge theory must be related to connections on fiber bundles. I then
that is, self-dual connections. I would be inaccurate to say after studying
mathematics for thirty years, I felt prepared to return to physics.”
Yang: In 1975, impressed with the fact that gauge fields are connections
on fiber bundles, I drove to the house of Shing-Shen Chern (1911–2004)
in El Cerrito near Berkeley (California) . . . I said I found it amazing
that gauge theory are exactly connections on fiber bundles, which the
mathematicians developed without reference to the physical world. I added
“This is both thrilling and puzzling, since you mathematicians dreamed
up these concepts out of nowhere.” Chern immediately protested “No, no.
These concepts were not dreamed up. They were natural and real.”
Zhang: The Yang–Baxter equation
A(u)B(u + v)A(v) = B(v)A(u + v)B(u)
appearing in statistical mechanics is just a simple equation for matrix functions.
Why does it have such great importance?
Yang: In the simplest situation, the Yang–Baxter equation has the form
ABA = BAB.
This is the fundamental equation of Artin (1898–1962) for the braid group.
The braid group is, of course, a record of the history of permutations. It
is not difficult to understand that the history of permutations is relevant
to many problems in mathematics and physics. Looking at the developments
of the last six or seven years, I got the feeling that the Yang–Baxter
equation is the next pervasive algebraic equation after the Jacobi identity
[A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0.
The study of the Jacobi identity has, of course, led to the whole of Lie
algebra and its relationship to Lie groups that govern symmetry in nature.
Zhang: Yang–Mills theory and the Yang–Baxter equation both figure
prominently in today’s score mathematics. One can see this by the Fields
medals awarded in 1986 and 1990. Simon Donaldson was awarded a Fields
medal at the International Congress of Mathematicians held in Berkeley
in 1986. Sir Michael Atiyah spoke on Simon Donaldson’s work: “Together
with the important work of Michael Freedman (another Fields medal winner
in 1986), Donaldson’s result implied that there exist ‘exotic’ fourdimensional
spaces which are topologically but not differentially equivalent
to the standard Euclidean four-dimensional space R4 . . . Donaldson’s
results are derived from the Yang–Mills equations of theoretical physics
which are nonlinear generalizations of Maxwell’s equations. In the Euclidean
case the solution to the Yang–Mills equations giving the absolute
minimum are of special interest and called instantons.”
There were four Fields medalists in 1990: Vladimir Drinfeld, Vaughan
Jones, Shigefumi Mori, and Edward Witten. The work of three of them
was related to the Yang–Mills equations
−D ∗ F = ∗J, DF = 0
and/or the Yang–Baxter equation (see Sect. 15.4).
tried to understand fiber-bundle theory from such books as Steenrod’s
“The Topology of Fiber Bundles,” Princeton University Press, 1951, but
I learned nothing. The language of modern mathematics is too cold and
abstract for a physicist.
Zhang: I suppose only mathematicians appreciate the mathematical language
of today.
Yang: I can tell you a relevant story. About ten years ago, I gave a talk
on physics in Seoul, South Korea. I joked “There exist only two kinds
of modern mathematics books: one which you cannot read beyond the
first page and one which you cannot read beyond the first sentence. The
Mathematical Intelligencer later reprinted this joke of mine. But I suspect
many mathematicians themselves agree with me.
Zhang: When did you understand bundle theory?
Yang: In early 1975, I invited Jim Simons to give us a series of luncheon
lectures on differential forms and bundle theory. He kindly accepted the
invitation, and we learned about de Rham’s theorem, differential forms,
patching and so on . . .
Zhang: Simon’s lecture helped Wu and Yang to write a famous paper
in 1975.7 In this paper, they analyzed the intrinsic meaning of electromagnetism,
emphasizing especially its global topological aspects. They
discussed the mathematical meaning of the Aharonov–Bohm experiment
and of the Dirac magnetic monopole. They exhibited a dictionary on the
translation of terminologies used in mathematics and physics. Half a year
later, Isadore Singer of the Massachusetts Institute of Technology (MIT,
Cambridge, Massachusetts) visited Stony Brooke and discussed these matters
with Yang at length. Singer had been an undergraduate student in
physics and a graduate student in mathematics in the 1940s. He wrote in
1985:“Thirty years later I found myself lecturing on gauge theories, beginning
with the Wu and Yang dictionary and ending with instantons,
(i) We should mention Drinfeld’s pioneering work with Yuri Manin on
the construction of instantons. These are solutions to the Yang–Mills
equations which can be thought of as having particle-like properties
of localization and size. Drinfeld’s interest in physics continued with
his investigation of the Yang–Baxter equation.
(ii) Jones opened a whole new direction upon realizing that under certain
conditions solutions of the Yang–Baxter equation could be used for
constructing invariants of links . . . The theory of quantum groups
(i.e., deformations of classical Lie groups based on non-commutative
Hopf algebras) was devised by Jimbo and Drinfeld to produce solutions
of Yang–Baxter equations.
(iii) Witten described in these terms the invariants of Donaldson and Floer
(extending the earlier ideas of Atiyah) and generalized the Jones polynomials
to the case of an arbitrary ambient three-dimensional manifold.
We note with amusement that there were complaints that the plenary lectures
at the International Congress of Mathematicians in Kyoto, 1990, were
heavily slanted toward the topics of mathematical physics: “Everywhere
we heard quantum group, quantum group, quantum group!” . . .
Yang: Many theoretical physicists are, in some ways, antagonistic to mathematics,
or at least have a tendency to downplay the value of mathematics.
I do not agree with these attitudes. I have written:8 “Perhaps of my father’s
influence, I appreciate mathematics. I appreciate the value judgement of
the mathematician, and I admire the beauty and power of mathematics:
there are ingenuity and intricacy in tactical maneuvers, and breathtaking
sweeps in strategic campaigns. And, of course, miracle of miracles, some
concepts in mathematics turn out to provide the fundamental structures
that govern the physical universe!”
In the present volume, we will show that the Yang–Mills equations generalize the
Maxwell equations in electromagnetism.

Einstein’s Theory of General Relativity on Gravitation
We set


This completes the general theory of relativity as a logical structure. The
postulate of relativity in its most general form, which makes the space-time
coordinates meaningless parameters, leads necessarily to a certain form of
gravitational theory which explains the motion of the Perihelion of the
planet Mercury.
Anyone who has really grasped the general theory of relativity, will be
captured by its beauty. It is a triumph of the general differential calculus,
which was created by Gauss (1777–1855), Riemann (1826–1866), Christoffel
(1829–1900), Ricci-Curbastro (1853–1925), Bianchi (1856–1928), and
Levi-Civita (1873–1941).
Albert Einstein, 1915
The two fundamental Einstein equations. In 1915, motivated by the study
of classical differential geometry, Einstein based his theory of general relativity on
the Riemann curvature tensor of the four-dimensional space-time manifold M4.
The points P of M4 are called space-time points or events. Einstein’s fundamental
equations read as follows:
(i) The equation of motion for the gravitational field:
                                 (0.3)
(ii) The equation of motion for the trajectories  of celestial
bodies (e.g., planets, the sun, stars, or galaxies) and light rays:

This equation generalizes Newton’s classical equation of motion.

(P) In terms of physics, Einstein postulated that: Physics does not depend on the
choice of observers. This is Einstein’s principle of general relativity.
(M) In terms of mathematics, Einstein’s principle of general relativity is realized
by the use of tensor calculus introduced in the second half of the 19th century.

Gauss posed the following fundamental question:
Is it possible to compute the Gaussian curvature K of a 2-dimensional
su***ce by only using measurements on the su***ce?
After a long fight, Gauss found that the answer is “yes”! He discovered the following
sophisticated formula:
K(P) =R1221(P)/g(P)                    (0.19)
where g := g11g22 − g12g12. This is the famous theorema egregium. Let us discuss
this.
The Gaussian curvature K is an intrinsic property of the 2-dimensional
su***ce; it depends on the components gαβ of the metric tensor and their
first and second partial derivatives with respect to the local coordinates.
In fact, Gauss did not explicitly use the Riemann curvature tensor, but in terms
of the modern terminology, his key formula can be written as (0.19). Concerning
cartography, the theorema egregium tells us in rigorous terms that it is impossible
to introduce geographic charts which are length preserving after rescaling. Indeed,
one can show that length preserving maps preserve the components of the metric
tensor. In turn, such maps preserve the Gaussian curvature. Finally, note that the
Gaussian curvature of the sphere is positive, but the Gaussian curvature of the
Euclidean plane vanishes.
Gauss’ theorema egregium had an enormous impact on the development of
modern differential geometry and modern physics culminating in the principle
“force equals curvature.” This principle is basic for both Einstein’s
theory of general relativity on gravitation and the Standard Model in elementary
particle physics.
In order to understand the intuitive meaning of both the components Rαβ of the
Ricci tensor and the scalar curvature R on the 2-dimensional su***ce M2, observe
that the components of the Riemann curvature tensor read as
Rαβγδ = K(gαδgβγ − gαγgβδ), α, β, γ, δ = 1, 2.
Since gαβ = gβα, we get the following symmetry properties
Rαβγδ = −Rβαγδ = −Rαβδγ = Rγδαβ
for all indices α, β, γ, δ = 1, 2. Therefore, the 2*2*2*2 = 16 components of the Riemann
curvature tensor reduce to one essential component, namely,
R1221 = K(g11g22 − g12g12).
In fact, we have R1221 = −R2121 = −R1212 = −R2112. The remaining 12 components
vanish identically. For example, it follows from Rαβγδ = −Rβαγδ that
R1112 = 0. In order to simplify notation, let us introduce an orthogonal local coordinate
system, that is, we have the special case where g12 = g21 = 0.

In his famous 1854 lecture on the foundations of geometry, Riemann described
the Riemann curvature tensor only in intuitive terms. In his 1861 paper, Riemann
published the precise analytic formula of the Riemann curvature tensor for the first
time.20 In the textbook by M. Spivak, A Comprehensive Introduction to Differential
Geometry, Vol. 2, Publish or Perish, Boston, one finds seven variants of the proof
of Riemann’s solution of the Paris Academy problem.
Riemann died in 1866 at the age of 40. His collected works fill only one volume.
But his ideas, revealing deep connections between analysis, topology, and geometry,
profoundly influenced the mathematics and physics of the 20th century. This is
described in the beautiful book by K. Maurin, The Riemann Legacy: Riemannian
Ideas in Mathematics and Physics of the 20th Century, Kluwer, Dordrecht.
The importance of conformal maps. Conformal mappings are essential for
both classifying Riemann su***ces and proving the existence of minimal su***ces
with prescribed boundary curves (the problem of Plateau (1801–1883) on soap
bubbles spanned by a metallic frame).
Conformal mappings play also a fundamental role in modern physics,
namely, in string theory and conformal quantum field theory.
The point is that the principle of critical action in string theory is invariant under
conformal mappings (which represent the gauge transformations in string theory).
In 2-dimensional conformal quantum field theory, the conformal symmetry strongly
restricts the structure of possible correlation functions (i.e., Green’s functions). Two
Riemann su***ces M and N are called conformally equivalent iff there exists a
conformal diffeomorphism
χ :M→N.
Let dimRMg denote the real dimension of the space of all compact Riemann su***ces
of genus g modulo conformal equivalence. By considering the description of
Mg by real parameters called moduli, Riemann suggested that
dimRMg = 6g −6 if g = 2, 3, . . . , dimRM1 = ∞, dimRM0 = 0. (0.22)
This was the beginning of the sophisticated theory of moduli spaces which describe
the set of given geometric (or algebraic) structures up to equivalence via symmetry
groups. The rigorous proof of theorem (0.22) can be given in the setting of
Teichm¨uller spaces.
In what follows, we will pass back to the 4-dimensional space-time manifold
M4 used in Einstein’s theory of general relativity.

Symmetry and Klein’s Invariance Principle in Geometry
Felix Klein (1849–1925) emphasized the importance of invariants in geometry.
Sophus Lie (1842–1899) discovered the importance of the linearization
principle due to Newton (1643–1727) and Leibniz (1646–1716) for constructing
invariants in differential geometry via Lie algebras and Lie
groups.
´Elie Cartan (1859–1951) combined the methods of Gauss (1777–1855) and
Riemann (1826–1866) in order to describe curvature based on the ideas
due to Klein and Lie.
Folklore
Klein’s Erlangen program and gauge theory in physics. In the 19th century,
numerous new geometries emerged in mathematics (e.g., non-Euclidean geometry
and projective geometry). Missing was a general principle for classifying geometries.
In 1869, the young German mathematician Felix Klein (1849–1925) and the young
Norwegian mathematician Sophus Lie (1842–1899) met each other in Berlin and
became close friends. Klein and Lie extensively discussed the classification problem
for geometry. They agreed that symmetry groups play a distinguished role. In his
1872 Erlangen program, Felix Klein formulated the following general principle:
Geometry is the invariant theory of transformation groups.
In physics, gauge theory corresponds to a special case of this principle:
Gauge theory studies the invariants of physical fields under both space-time
transformations and gauge transformations.
The main goal of gauge theory is the formulation of
• variational principles (principle of critical action) and
• partial differential equations (Euler–Lagrange equations)
which are invariant under both space-time transformations and gauge transformations.
Such invariant variational principles and differential equations appear in:
(a) electrodynamics (the Maxwell equations),
(b) the Standard Model in elementary particle physics,
(c) the theory of general relativity (e.g., the Standard Model in cosmology).
In this connection, our main goal is
to create a differential calculus which respects both space-time transformations
and gauge transformations.
It was the beautiful idea of ´Elie Cartan to combine curvature in differential geometry
with local symmetry. Nowadays we know that precisely this idea is basic for modern
physics, too.
A glance at the history of invariant theory. Invariant theory was created
in the 19th century by George Boole (1815–1864), James Sylvester (1814–1897),
and Arthur Cayley (1821–1895). Hermann Weyl wrote:
The theory of invariants came into existence about the middle of the nineteenth
century somewhat like Minerva: a grown-up virgin, mailed in the
shining armor of algebra, she sprang forth from Cayley’s Jovian head.
Her Athens over which she ruled and which she served as a tutelary and
beneficent goddess was projective geometry.
Cayley was a master in doing long computations and in inventing algorithms. A
brief history of invariant theory can be found in the introduction of Peter Olver’s
book: Classical Invariant Theory, Cambridge University Press, 1999. We also refer
to Felix Klein’s famous book: Development of Mathematics in the 19th Century,
Math. Sci. Press, New York, 1979.
The goal of invariant theory. We are given a mathematical object O and
a symmetry group G which transforms the object O. The final goal is to construct
G-invariants of O. That is, we are looking for quantities which are assigned to O and
which are invariant under the action of the symmetry group G. Moreover, we are
interested in determining a complete system of invariants. By definition, a system
of G invariants of O is called complete iff it uniquely determines the object O up
to symmetry operations contained in the group G.

Einstein’s Principle of General Relativity and Invariants –
the Geometrization of Physics
Einstein emphasized the importance of invariants in physics.
Folklore
In the theory of general relativity, transformations of local space-time coordinates
are called gauge transformations. Using this term, one can say that
Einstein wanted to construct his theory of general relativity in such a way
that it is gauge invariant.
In other words, starting with his philosophical principle of general relativity, Einstein
was looking for a mathematical approach which describes invariants in terms
of local coordinates. The prototype of such an approach is given by cartography.
Cartography as a paradigm. In cartography, parts of the su***ce of earth are
described by local geographic charts collected in a geographic atlas. The Euclidean
coordinates of each chart are called local coordinates of earth. Obviously, geometric
properties of the su***ce of earth do not depend on the choice of the geographic
charts, for example, the distance of two points on the su***ce of earth does not
depend on the choice of local coordinates. Geometric properties are invariants with
respect to the possible choices of local coordinates.
Intuitively spoken, Einstein looked for higher-dimensional cartography.
His friend – the mathematician Marcel Grossmann (1878–1936) – told him that
Riemann generalized Gauss’ theory of cartography to higher dimensions and that
there exists a well-developed calculus for higher-dimensional manifolds, namely,
the Ricci calculus due to Gregorio Ricci-Curbastro (1853–1925). By the help of
Grossmann, Einstein studied the Ricci calculus and he applied it to his theory of
gravitation.
The geometrization of physics. Geometry is a mathematical model for describing
both invariant geometric properties and their representation by local coordinates.
In ancient times, one only considered invariant geometric properties. The
description of geometric properties by coordinates dates back to Ren´e Descartes
(1596–1650). In 1667 Descartes published his “Discours de la m´ethode” which contains,
among a detailed philosophical investigation and its application to the sciences,
the foundation of analytic geometry (e.g., the use of Cartesian coordinates).26
Einstein geometrized gravitation in his 1915 theory of general relativity. Quantum
mechanics was geometrized by Dirac, as a unitary geometry of Hilbert spaces.
In the introduction to his book “The Principles of Quantum Mechanics,” Clarendon
Press, Oxford, 1930, the young Dirac (1902–1984) wrote:
The important things in the world appear as invariants . . . The things we
are immediately aware of are the relations of these invariants to a certain
frame of reference . . . The growth of the use of transformation theory, as
applied first to relativity and later to the quantum theory, is the essence
of the new method in theoretical physics.
Finally, note that the Standard Model in particle physics starts from a classical
field theory which is closely related to the geometry of specific fiber bundles.

Symmetry and Klein’s Invariance Principle in Geometry
Felix Klein (1849–1925) emphasized the importance of invariants in geometry.
Sophus Lie (1842–1899) discovered the importance of the linearization
principle due to Newton (1643–1727) and Leibniz (1646–1716) for constructing
invariants in differential geometry via Lie algebras and Lie
groups.
´Elie Cartan (1859–1951) combined the methods of Gauss (1777–1855) and
Riemann (1826–1866) in order to describe curvature based on the ideas
due to Klein and Lie.
Folklore
Klein’s Erlangen program and gauge theory in physics. In the 19th century,
numerous new geometries emerged in mathematics (e.g., non-Euclidean geometry
and projective geometry). Missing was a general principle for classifying geometries.
In 1869, the young German mathematician Felix Klein (1849–1925) and the young
Norwegian mathematician Sophus Lie (1842–1899) met each other in Berlin and
became close friends. Klein and Lie extensively discussed the classification problem
for geometry. They agreed that symmetry groups play a distinguished role. In his
1872 Erlangen program, Felix Klein formulated the following general principle:
Geometry is the invariant theory of transformation groups.
In physics, gauge theory corresponds to a special case of this principle:
Gauge theory studies the invariants of physical fields under both space-time
transformations and gauge transformations.
The main goal of gauge theory is the formulation of
• variational principles (principle of critical action) and
• partial differential equations (Euler–Lagrange equations)
which are invariant under both space-time transformations and gauge transformations.
Such invariant variational principles and differential equations appear in:
(a) electrodynamics (the Maxwell equations),
(b) the Standard Model in elementary particle physics,
(c) the theory of general relativity (e.g., the Standard Model in cosmology).
In this connection, our main goal is
to create a differential calculus which respects both space-time transformations
and gauge transformations.
It was the beautiful idea of ´Elie Cartan to combine curvature in differential geometry
with local symmetry. Nowadays we know that precisely this idea is basic for modern
physics, too.
A glance at the history of invariant theory. Invariant theory was created
in the 19th century by George Boole (1815–1864), James Sylvester (1814–1897),
and Arthur Cayley (1821–1895). Hermann Weyl wrote:
The theory of invariants came into existence about the middle of the nineteenth
century somewhat like Minerva: a grown-up virgin, mailed in the
shining armor of algebra, she sprang forth from Cayley’s Jovian head.
Her Athens over which she ruled and which she served as a tutelary and
beneficent goddess was projective geometry.
Cayley was a master in doing long computations and in inventing algorithms. A
brief history of invariant theory can be found in the introduction of Peter Olver’s
book: Classical Invariant Theory, Cambridge University Press, 1999. We also refer
to Felix Klein’s famous book: Development of Mathematics in the 19th Century,
Math. Sci. Press, New York, 1979.
The goal of invariant theory. We are given a mathematical object O and
a symmetry group G which transforms the object O. The final goal is to construct
G-invariants of O. That is, we are looking for quantities which are assigned to O and
which are invariant under the action of the symmetry group G. Moreover, we are
interested in determining a complete system of invariants. By definition, a system
of G invariants of O is called complete iff it uniquely determines the object O up
to symmetry operations contained in the group G.

James Sylvester (1814–1897) said in 1864:
As all roads lead to Rome so I find in my own case at least that all algebraic
inquiries, sooner or later, end at the Capitol of modern algebra over whose
shining portal is inscribed the Theory of Invariants.
Invariant theory is essential for modern physics. In the present volume we will
encounter invariant theory again and again.
Einstein’s Principle of General Relativity and Invariants –
the Geometrization of Physics
Einstein emphasized the importance of invariants in physics.
Folklore
The components  of the Riemann curvature tensor depend on the choice of
local space-time coordinates, that is, they depend on the choice of the
observer. Recall that Einstein’s principle of general relativity tells us that:
Physics is independent of the choice of the observer.
This means that proper physical quantities have to be independent of the choice of
local coordinates.

In the theory of general relativity, transformations of local space-time coordinates
are called gauge transformations. Using this term, one can say that
Einstein wanted to construct his theory of general relativity in such a way
that it is gauge invariant.
In other words, starting with his philosophical principle of general relativity, Einstein
was looking for a mathematical approach which describes invariants in terms
of local coordinates. The prototype of such an approach is given by cartography.
Cartography as a paradigm. In cartography, parts of the su***ce of earth are
described by local geographic charts collected in a geographic atlas. The Euclidean
coordinates of each chart are called local coordinates of earth. Obviously, geometric
properties of the su***ce of earth do not depend on the choice of the geographic
charts, for example, the distance of two points on the su***ce of earth does not
depend on the choice of local coordinates. Geometric properties are invariants with
respect to the possible choices of local coordinates.
Intuitively spoken, Einstein looked for higher-dimensional cartography.
His friend – the mathematician Marcel Grossmann (1878–1936) – told him that
Riemann generalized Gauss’ theory of cartography to higher dimensions and that
there exists a well-developed calculus for higher-dimensional manifolds, namely,
the Ricci calculus due to Gregorio Ricci-Curbastro (1853–1925). By the help of
Grossmann, Einstein studied the Ricci calculus and he applied it to his theory of
gravitation.
The geometrization of physics. Geometry is a mathematical model for describing
both invariant geometric properties and their representation by local coordinates.
In ancient times, one only considered invariant geometric properties. The
description of geometric properties by coordinates dates back to Ren´e Descartes
(1596–1650). In 1667 Descartes published his “Discours de la m´ethode” which contains,
among a detailed philosophical investigation and its application to the sciences,
the foundation of analytic geometry (e.g., the use of Cartesian coordinates).26
Einstein geometrized gravitation in his 1915 theory of general relativity. Quantum
mechanics was geometrized by Dirac, as a unitary geometry of Hilbert spaces.
In the introduction to his book “The Principles of Quantum Mechanics,” Clarendon
Press, Oxford, 1930, the young Dirac (1902–1984) wrote:
The important things in the world appear as invariants . . . The things we
are immediately aware of are the relations of these invariants to a certain
frame of reference . . . The growth of the use of transformation theory, as
applied first to relativity and later to the quantum theory, is the essence
of the new method in theoretical physics.
Finally, note that the Standard Model in particle physics starts from a classical
field theory which is closely related to the geometry of specific fiber bundles.

Riemann curvature tensor

From Wikipedia, the free encyclopedia


[th]General relativity[/th]

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In the mathematical field ofdifferential geometry, theRiemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to expresscurvature of Riemannian manifolds. It associates atensor to each point of aRiemannian manifold (i.e., it is a tensor field), that measures the extent to which the metric tensor is not locally isometric to a Euclidean space. The curvature tensor can also be defined for anypseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.
The curvature tensor is given in terms of the Levi-Civita connection  by the following formula:

where [u,v] is the Lie bracket of vector fields. For each pair of tangent vectors u, v, R(u,v) is a linear transformation of the tangent space of the manifold. It is linear in u and v, and so defines a tensor. Occasionally, the curvature tensor is defined with the opposite sign.
If  and  are coordinate vector fields then  and therefore the formula simplifies to

The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). The linear transformation  is also called the curvature transformation or endomorphism.
The curvature formula can also be expressed in terms of the second covariant derivative defined as:^[1]

which is linear in u and v. Then:

Thus in the general case of non-coordinate vectors u and v, the curvature tensor measures the noncommutativity of the second covariant derivative.

[/ltr][/size]

1 Geometrical meaning



• 1.1 Informally
  
• 1.2 Formally
  



2 Coordinate expression

3 Symmetries and identities

4 Special cases

5 See also

6 Notes

7 References

[size][ltr]

Geometrical meaning[edit]

Informally[edit]

Imagine walking around the bounding white line of a tennis court with a stick held out in front of you. When you reach the first corner of the court, you turn to follow the white line, but you keep the stick held out in the same direction, which means you are now holding the stick out to your side. You do the same when you reach each corner of the court. When you get back to where you started, you are holding the stick out in exactly the same direction as you were when you started (no surprise there).
Now imagine you are standing on the equator of the earth, facing north with the stick held out in front of you. You walk north up along a line of longitude until you get to the north pole. At that point you turn right, ninety degrees, but you keep the stick held out in the same direction, which means you are now holding the stick out to your left. You keep walking (south obviously – whichever way you set off from the north pole, it's south) until you get to the equator. There, you turn right again (and so now you have to hold the stick pointing out behind you) and walk along the equator until you get back to where you started from. But here is the thing: the stick is pointing back along the equator from where you just came, not north up to the pole how it was when you started!
The reason for the difference is that the su***ce of the earth is curved, whereas the su***ce of a tennis court is flat, but it is not quite that simple. Imagine that the tennis court is slightly humped along its centre-line so that it is like part of the su***ce of a cylinder. If you walk around the court again, the stick still points in the same direction as it did when you started. The reason is that the humped tennis court has extrinsic curvature but nointrinsic curvature. The su***ce of the earth, however, has both extrinsic and intrinsic curvature.
The Riemann curvature tensor is a way to capture a measure of the intrinsic curvature. When you write it down in terms of its components (like writing down the components of a vector), it consists of a multi-dimensional array of sums and products of partial derivatives (some of those partial derivatives can be thought of as akin to capturing the curvature imposed upon someone walking in straight lines on a curved su***ce).

Formally[edit]

When a vector in a Euclidean space is parallel transported around a loop, it will again point in the initial direction after returning to its original position. However, this property does not hold in the general case. The Riemann curvature tensor directly measures the failure of this in a generalRiemannian manifold. This failure is known as the non-holonomy of the manifold.
Let x_t be a curve in a Riemannian manifold M. Denote by τ_x[sub]t[/sub] : T_x[sub]0[/sub]M → T_x[sub]t[/sub]Mthe parallel transport map along x_t. The parallel transport maps are related to the covariant derivative by

for each vector field Y defined along the curve.
Suppose that X and Y are a pair of commuting vector fields. Each of these fields generates a pair of one-parameter groups of diffeomorphisms in a neighborhood of x₀. Denote by τ_tX and τ_tY, respectively, the parallel transports along the flows of X and Y for time t. Parallel transport of a vector Z ∈ T_x[sub]0[/sub]M around the quadrilateral with sides tY, sX, −tY, −sX is given by

This measures the failure of parallel transport to return Z to its original position in the tangent space T_x[sub]0[/sub]M. Shrinking the loop by sending s, t → 0 gives the infinitesimal description of this deviation:

where R is the Riemann curvature tensor.

Coordinate expression[edit]

Converting to the tensor index notation, the Riemann curvature tensor is given by

where  are the coordinate vector fields. The above expression can be written using Christoffel symbols:

(see also the list of formulas in Riemannian geometry).
The Riemann curvature tensor is also the commutator of the covariant derivative of an arbitrary covector  with itself:^[2][3]

since the connection  is torsionless, which means that the torsion tensor  vanishes.
This formula is often called the Ricci identity.^[4] This is the classical method used by Ricci and Levi-Civita to obtain an expression for the Riemann curvature tensor.^[5] In this way, the tensor character of the set of quantities is proved.
This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows

This formula also applies to tensor densities without alteration, because for the Levi-Civita (not generic) connection one gets:^[4]

It is sometimes convenient to also define the purely covariant version by

Symmetries and identities[edit]

The Riemann curvature tensor has the following symmetries:

The last identity was discovered by Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it is equivalent to the Bianchi identity below. (Also, if there is nonzero torsion, the first Bianchi identity becomes a differential identity of the torsion tensor.) These three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has  independent components.
Yet another useful identity follows from these three:

On a Riemannian manifold one has the covariant derivative  and theBianchi identity (often called the second Bianchi identity or differential Bianchi identity) takes the form:

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:
Skew symmetryInterchange symmetryFirst Bianchi identityThis is often writtenwhere the brackets denote the antisymmetric part on the indicated indices. This is equivalent to the previous version of the identity because the Riemann tensor is already skew on its last two indices.Second Bianchi identityThe semi-colon denotes a covariant derivative. Equivalently,again using the antisymmetry on the last two indices of R.
The algebraic symmetries are also equivalent to saying that R belongs to the image of the Young symmetrizer corresponding to the partition 2+2.

Special cases[edit]

Su***ces
For a two-dimensional su***ce, the Bianchi identities imply that the Riemann tensor can be expressed as

where  is the metric tensor and  is a function called the Gaussian curvature and a, b, c and d take values either 1 or 2. The Riemann tensor has only one functionally independent component. The Gaussian curvature coincides with the sectional curvature of the su***ce. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvaturetensor of the su***ce is simply given by
Space forms
A Riemannian manifold is a space form if its sectional curvature is equal to a constant K. The Riemann tensor of a space form is given by

Conversely, except in dimension 2, if the curvature of a Riemannian manifold has this form for some function K, then the Bianchi identities imply that K is constant and thus that the manifold is (locally) a space form.

See also[edit]

[/ltr][/size]

• Introduction to mathematics of general relativity
  
• Decomposition of the Riemann curvature tensor
  
• Curvature of Riemannian manifolds
  

[size][ltr]

Notes[edit]

[/ltr][/size][list=references]
[*]Jump up^ Lawson, H. Blaine, Jr.; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton U Press. p. 154. ISBN 0-691-08542-0.
[*]Jump up^ Synge J.L., Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. pp. 83; 107. ISBN 978-0-486-63612-2.
[*]Jump up^ P. A. M. Dirac (1996). General Theory of Relativity. Princeton University Press. ISBN 0-691-01146-X.
[*]^ Jump up to:^a b Lovelock, David; Rund, Hanno (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover. p. 84,109. ISBN 978-0-486-65840-7.
[*]Jump up^ Ricci, Gregorio; Levi-Civita, Tullio (March 1900), "Méthodes de calcul différentiel absolu et leurs applications", Mathematische Annalen(Springer) 54 (1–2): 125–201, doi:10.1007/BF01454201
[/list]
[size][ltr]

References[edit]

[/ltr][/size]

• Besse, A.L. (1987), Einstein manifolds, Springer
  
• Kobayashi, S.; Nomizu, K. (1963), Foundations of differential geometry, vol. 1, Interscience
  
• Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973),Gravitation, W. H. Freeman, ISBN 0-7167-0344-0
  

[th] [show] v t e Various notions of curvature defined indifferential geometry[/th]

[th] [show] v t e Theories of gravitation[/th]

[th] [show] v t e Tensors[/th]

General relativity

From Wikipedia, the free encyclopedia


[ltr]For a generally accessible and less technical introduction to the topic, see Introduction to general relativity.[/ltr]




[th]General relativity[/th]

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[ltr]
General relativity, or thegeneral theory of relativity, is the geometrictheory of gravitationpublished by Albert Einsteinin 1916^[1] and the current description of gravitation inmodern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of spaceand time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentumof whatever matter andradiation are present. The relation is specified by theEinstein field equations, a system of partial differential equations.
Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences includegravitational time dilation,gravitational lensing, thegravitational redshift of light, and the gravitational time delay. The predictions of general relativity 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. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws ofquantum physics to produce a complete and self-consistent theory ofquantum gravity.
Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes; for example, microquasars and active galactic nuclei result from the presence of stellar black holes and black holes of a much more massive type, respectively. The bending of light by gravity can lead to the phenomenon of gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waves, which have since been observed indirectly; a direct measurement is the aim of projects such asLIGO and NASA/ESA Laser Interferometer Space Antenna and variouspulsar timing arrays. In addition, general relativity is the basis of currentcosmological models of a consistently expanding universe.[/ltr]

• 1 History
  
• 2 From classical mechanics to general relativity
  
  
  
  • 2.1 Geometry of Newtonian gravity
    
  • 2.2 Relativistic generalization
    
  • 2.3 Einstein's equations
    
  
  
  
• 3 Definition and basic applications
  
  
  
  • 3.1 Definition and basic properties
    
  • 3.2 Model-building
    
  
  
  
• 4 Consequences of Einstein's theory
  
  
  
  • 4.1 Gravitational time dilation and frequency shift
    
  • 4.2 Light deflection and gravitational time delay
    
  • 4.3 Gravitational waves
    
  • 4.4 Orbital effects and the relativity of direction
    
  
  
  
• 5 Astrophysical applications
  
  
  
  • 5.1 Gravitational lensing
    
  • 5.2 Gravitational wave astronomy
    
  • 5.3 Black holes and other compact objects
    
  • 5.4 Cosmology
    
  
  
  
• 6 Advanced concepts
  
  
  
  • 6.1 Causal structure and global geometry
    
  • 6.2 Horizons
    
  • 6.3 Singularities
    
  • 6.4 Evolution equations
    
  • 6.5 Global and quasi-local quantities
    
  
  
  
• 7 Relationship with quantum theory
  
  
  
  • 7.1 Quantum field theory in curved spacetime
    
  • 7.2 Quantum gravity
    
  
  
  
• 8 Current status
  
• 9 See also
  
• 10 Notes
  
• 11 References
  
• 12 Further reading
  
• 13 External links
  
  

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History[edit]

Main articles: History of general relativity and Classical theories of gravitation[/ltr]


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Soon after publishing thespecial theory of relativity in 1905, Einstein started thinking about how to incorporategravity into his new relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as theEinstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, and form the core of Einstein's general theory of relativity.^[2]
The Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But as early as 1916, the astrophysicist Karl Schwarzschildfound the first non-trivial exact solution to the Einstein field equations, the so-called Schwarzschild metric. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, which eventually resulted in the Reissner–Nordström solution, now associated with electrically charged black holes.^[3] In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—thecosmological constant—to reproduce that "observation".^[4] By 1929, however, the work of Hubble and others had shown that our universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot and dense earlier state.^[5] Einstein later declared the cosmological constant the biggest blunder of his life.^[6]
During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein himself had shown in 1915 how his theory explained the anomalous perihelion advance of the planetMercury without any arbitrary parameters ("fudge factors").^[7] Similarly, a 1919 expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of May 29, 1919,^[8] *** Einstein instantly famous.^[9] Yet the theory entered the mainstream of theoretical physics and astrophysics only with the developments between approximately 1960 and 1975, now known as the golden age of general relativity.^[10] Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations.^[11] Ever more precise solar system tests confirmed the theory's predictive power,^[12] and relativistic cosmology, too, became amenable to direct observational tests.^[13]

From classical mechanics to general relativity[edit]

General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity.^[14]

Geometry of Newtonian gravity[edit]

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At the base of classical mechanicsis the notion that a body's motion can be described as a combination of free (or inertial) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's secondlaw of motion, which states that the net force acting on a body is equal to that body's (inertial) massmultiplied by its acceleration.^[15]The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics, straight world lines in curved spacetime.^[16]
Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and *** allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space, as well as a time coordinate. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvösand its successors (see Eötvös experiment), there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties.^[17] A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard an accelerating rocket generating a force equal to gravity.^[18]
Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specificconnection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However, spacetime as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable. From this, one can deduce that spacetime is curved. The result is a geometric formulation of Newtonian gravity using only covariantconcepts, i.e. a description which is valid in any desired coordinate system.^[19] In this geometric description, tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.^[20]

Relativistic generalization[edit]

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As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of(special) relativistic mechanics.^[21] In the language of symmetry: where gravity can be neglected, physics is Lorentz invariantas in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is thePoincaré group which also includes translations and rotations.) The differences between the two become significant when we are dealing with speeds approaching the speed of light, and with high-energy phenomena.^[22]
With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see the image on the left). The light-cones define a causal structure: for each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer-independent.^[23]In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the space–time's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines aconformal structure.^[24]
Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.^[25]
A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for thegravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. below). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.^[26] The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity.^[27]
The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- orpseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).^[28]

Einstein's equations[edit]

Main articles: Einstein field equations and Mathematics of general relativity
Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor, which includes bothenergy and momentum densities as well as stress (that is, pressure and shear).^[29] Using the equivalence principle, this tensor is readily generalized to curved space-time. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equationfor gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy–momentum corresponds to the statement that the energy–momentum tensor is divergence-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-manifold counterparts, covariant derivatives studied indifferential geometry. With this additional condition—the covariant divergence of the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero— the simplest set of equations are what are called Einstein's (field) equations:[/ltr]

Einstein's field equations

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On the left-hand side is the Einstein tensor, a specific divergence-free combination of the Ricci tensor  and the metric. Where  is symmetric. In particular,

is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as

On the right-hand side,  is the energy–momentum tensor. All tensors are written in abstract index notation.^[30] Matching the theory's prediction to observational results for planetary orbits (or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics), the proportionality constant can be fixed as κ = 8πG/c⁴, with G thegravitational constant and c the speed of light.^[31] When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations,

There are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Brans–Dicke theory, teleparallelism, andEinstein–Cartan theory.^[32]

Definition and basic applications[edit]

See also: Mathematics of general relativity and Physical theories modified by general relativity
The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building.

Definition and basic properties[edit]

General relativity is a metric theory of gravitation. At its core are Einstein's 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.^[33] Phenomena that inclassical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow.^[34] The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler, spacetime tells matter how to move; matter tells spacetime how to curve.^[35]
While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases. For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.^[36]
As it is constructed using tensors, general relativity exhibits general covariance: its laws—and further laws formulated within the general relativistic framework—take on the same form in all coordinate systems.^[37]Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers.^[38] Locally, as expressed in theequivalence principle, spacetime is Minkowskian, and the laws of physics exhibit local Lorentz invariance.^[39]

Model-building[edit]

The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.^[40]
Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.^[41] Nevertheless, a number of exact solutionsare known, although only a few have direct physical applications.^[42] The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner–Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe,^[43] and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes, each describing an expanding cosmos.^[44] Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub-NUT solution (a model universe that ishomogeneous, but anisotropic), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture).^[45]
Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes.^[46] In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities. Approximate solutions may also be found by perturbation theories such as linearized gravity^[47]and its generalization, the post-Newtonian expansion, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.^[48] An extension of this expansion is theparametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.^[49]

Consequences of Einstein's theory[edit]

General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of the ninety years of research that followed Einstein's initial publication.

Gravitational time dilation and frequency shift[edit]

Main article: Gravitational time dilation[/ltr]


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Assuming that the equivalence principle holds,^[50] gravity influences the passage of time. Light sent down into a gravity wellis blueshifted, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) isredshifted; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.^[51]
Gravitational redshift has been measured in the laboratory^[52] and using astronomical observations.^[53] Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks,^[54] while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS).^[55] Tests in stronger gravitational fields are provided by the observation of binary pulsars.^[56] All results are in agreement with general relativity.^[57] However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.^[58]

Light deflection and gravitational time delay[edit]

Main articles: Kepler problem in general relativity, Gravitational lensand Shapiro delay[/ltr]


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General relativity predicts that the path of light is bent in a gravitational field; light passing a massive body is deflected towards that body. This effect has been confirmed by observing the light of stars or distant quasars being deflected as it passes the Sun.^[59]
This and related predictions follow from the fact that light follows what is called a light-like or null geodesic—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance oflightspeed in special relativity.^[60] As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion),^[61] several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending theuniversality of free fall to light,^[62] the angle of deflection resulting from such calculations is only half the value given by general relativity.^[63]
Closely related to light deflection is the gravitational time delay (or Shapiro delay), the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction.^[64] In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.^[65]
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Global and quasi-local quantities[edit]

Main article: Mass in general relativity

The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.^[155]

Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)^[156] or suitable symmetries (Komar mass).^[157] If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the so-called Bondi mass at null infinity.^[158] Just as in classical physics, it can be shown that these masses are positive.^[159]Corresponding global definitions exist for momentum and angular momentum.^[160] There have also been a number of attempts to definequasi-local quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about isolated systems, such as a more precise formulation of the hoop conjecture.^[161]

Relationship with quantum theory[edit]

If general relativity is considered one of the two pillars of modern physics,quantum theory, the basis of understanding matter from elementary particles to solid state physics, is the other.^[162] However, it is still an open question as to how the concepts of quantum theory can be reconciled with those of general relativity.

Quantum field theory in curved spacetime[edit]

Main article: Quantum field theory in curved spacetime

Ordinary quantum field theories, which form the basis of modernelementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth.^[163] In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.^[164] Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as Hawking radiation, leading to the possibility that they evaporate over time.^[165] As briefly mentionedabove, this radiation plays an important role for the thermodynamics of black holes.^[166]

Quantum gravity[edit]

Main article: Quantum gravity

See also: String theory, Canonical general relativity, Loop quantum gravity, Causal Dynamical Triangulations and Causal sets

The demand for consistency between a quantum description of matter and a geometric description of spacetime,^[167] as well as the appearance ofsingularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.^[168] Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.^[169]



Attempts to generalize ordinary quantum field theories, used inelementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems. At low energies, this approach proves successful, in that it results in an acceptableeffective (quantum) field theory of gravity.^[170] At very high energies, however, the result are models devoid of all predictive power ("non-renormalizability").^[171]



One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute one-dimensional extended objects.^[172]The theory promises to be aunified description of all particles and interactions, including gravity;^[173] the price to pay is unusual features such as six extra dimensions of space in addition to the usual three.^[174] In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity andsupersymmetry known assupergravity^[175] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.^[176]

Another approach starts with the canonical quantization procedures of quantum theory. Using the initial-value-formulation of general relativity (cf.evolution equations above), the result is the Wheeler–deWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be ill-defined.^[177] However, with the introduction of what are now known asAshtekar variables,^[178] this leads to a promising model known as loop quantum gravity. Space is represented by a web-like structure called aspin network, evolving over time in discrete steps.^[179]

Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,^[180] there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being dynamical triangulations,^[181] causal sets,^[182] twistor models^[183] or the path-integral based models of quantum cosmology.^[184]

All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.^[185]

Current status[edit]

General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong indications the theory is incomplete.^[186] The problem of quantum gravity and the question of the reality of spacetime singularities remain open.^[187] Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.^[188] Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations,^[189] and increasingly powerful computer simulations (such as those describing merging black holes) are run.^[190] The race for the first direct detection of gravitational waves continues,^[191] in the hope of creating opportunities to test the theory's validity for much stronger gravitational fields than has been possible to date.^[192] More than ninety years after its publication, general relativity remains a highly active area of research.^[193]

See also[edit]

Center of mass (relativistic) Contributors to general relativity Derivations of the Lorentz transformations Ehrenfest paradox Einstein–Hilbert action

Introduction to mathematics of general relativity Relativity priority dispute Ricci calculus

Tests of general relativity Timeline of gravitational physics and relativity Two-body problem in general relativity

Introduction to mathematics of general relativity

From Wikipedia, the free encyclopedia



[ltr]This article is a non-technical introduction to the subject. For the main encyclopedia article, see Mathematics of general relativity.[/ltr]


[th]General relativity[/th]

Introduction Mathematical formulation Resources  · Tests

Fundamental concepts[show]

Phenomena[show]

Equations[show]

Advanced theories[show]

Solutions[show]

Scientists[show]

Spacetime[show]

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The mathematics of general relativity are very complex. In Newton's theories of motions, an object's length and the rate of passage of time remain constant as it changes speed. As a result, many problems in Newtonian mechanics can be solved with algebra alone. In relativity, on the other hand, length, and the passage of time change as an object's speed approaches the speed of light. The additional variables greatly complicate calculations of an object's motion. As a result, relativity requires the use of vectors,tensors, pseudotensors,curvilinear coordinates and many other complicated mathematical concepts.
For an introduction based on the specific physical example of particles orbiting a large mass in circular orbits, see Newtonian motivations for general relativity for a nonrelativistic treatment and Theoretical motivation for general relativity for a fully relativistic treatment.[/ltr]

• 1 Vectors and tensors
  
  
  
  • 1.1 Vectors
    
  • 1.2 Tensors
    
  • 1.3 Applications
    
  • 1.4 Dimensions
    
  • 1.5 Coordinate transformation
    
  
  
  
• 2 Oblique axes
  
• 3 Nontensors
  
• 4 Curvilinear coordinates and curved spacetime
  
• 5 Parallel transport
  
  
  
  • 5.1 The interval in a high-dimensional space
    
  • 5.2 The relation between neighboring contravariant vectors: Christoffel symbols
    
  • 5.3 Christoffel symbol of the second kind
    
  • 5.4 The constancy of the length of the parallel displaced vector
    
  • 5.5 The covariant derivative
    
  
  
  
• 6 Geodesics
  
• 7 Curvature tensor
  
• 8 See also
  
• 9 Notes
  
• 10 References
  
• 11 Related information
  
  

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Vectors and tensors[edit]

Main articles: Euclidean vector and Tensor

Vectors[edit]

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In mathematics, physics, andengineering, a Euclidean vector(sometimes called a geometric^[1]or spatial vector,^[2] or – as here – simply a vector) is a geometric object that has both a magnitude(or length) and direction. A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "one who carries".^[3] The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. Many algebraic operations onreal numbers such as addition, subtraction, multiplication, and negationhave close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity.

Tensors[edit]

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A tensor extends the concept of a vector to additional dimensions. A scalar, that is, a simple set of numbers without direction, would be shown on a graph as a point, a zero-dimensional object. A vector, which has a magnitude and direction, would appear on a graph as a line, which is a one-dimensional object. A tensor extends this concept to additional dimensions. A two dimensional tensor would be called a second order tensor. This can be viewed as a set of related vectors, moving in multiple directions on a plane.

Applications[edit]

Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such asvelocity, the magnitude of which is speed. For example, the velocity 5 meters per second upward could be represented by the vector (0,5) (in 2 dimensions with the positive y axis as 'up'). Another quantity represented by a vector is force, since it has a magnitude and direction. Vectors also describe many other physical quantities, such as displacement,acceleration, momentum, and angular momentum. Other physical vectors, such as the electric and magnetic field, are represented as a system of vectors at each point of a physical space; that is, a vector field.
Tensors also have extensive applications in physics:[/ltr]

• Electromagnetic tensor (or Faraday's tensor) in electromagnetism
  
• Finite deformation tensors for describing deformations and strain tensor for strain in continuum mechanics
  
• Permittivity and electric susceptibility are tensors in anisotropic media
  
• Stress–energy tensor in general relativity, used to representmomentum fluxes
  
• Spherical tensor operators are the eigenfunctions of the quantumangular momentum operator in spherical coordinates
  
• Diffusion tensors, the basis of diffusion tensor imaging, represent rates of diffusion in biologic environments
  

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Dimensions[edit]

In general relativity, four-dimensional vectors, or four-vectors, are required. These four dimensions are length, height, width and time. A "point" in this context would be an event, as it has both a location and a time. Similar to vectors, tensors in relativity require four dimensions. One example is the Riemann curvature tensor.

Coordinate transformation[edit]

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[ltr]
In physics, as well as mathematics, a vector is often identified with a tuple, or list of numbers, which depend on some auxiliary coordinate system orreference frame. When the coordinates are transformed, for example by rotation or stretching, then the components of the vector also transform. The vector itself has not changed, but the reference frame has, so the components of the vector (or measurements taken with respect to the reference frame) must change to compensate.
The vector is called covariant or contravariant depending on how the transformation of the vector's components is related to the transformation of coordinates.[/ltr]

• Contravariant vectors are "regular vectors" with units of distance (such as a displacement) or distance times some other unit (such as velocity or acceleration). For example, in changing units from meters to millimeters, a displacement of 1 m becomes 1000 mm.
  
• Covariant vectors, on the other hand, have units of one-over-distance (typically such as gradient). For example, in changing again from meters to millimeters, a gradient of 1 K/m becomes 0.001 K/mm.
  

[ltr]
Coordinate transformation is important because relativity states that there is no one correct reference point in the universe. On earth, we use dimensions like north, east, and elevation, which are used throughout the entire planet. There is no such system for space. Without a clear reference grid, it becomes more accurate to describe the four dimensions as towards/away, left/right, up/down and past/future. As an example event, take the signing of the Declaration of Independence. To a modern observer on Mt Rainier looking east, the event is ahead, to the right, below, and in the past. However, to an observer in medieval England looking north, the event is behind, to the left, neither up nor down, and in the future. The event itself has not changed, the location of the observer has.

Oblique axes[edit]

[/ltr]

Main article: Metric tensor

[ltr]An oblique coordinate system is one in which the axes are not necessarilyorthogonal to each other; that is, they meet at angles other than right angles.

Nontensors[edit]

[/ltr]
See also: Pseudotensor
[ltr]A nontensor is a tensor-like quantity that behaves like a tensor in the raising and lowering of indices, but that does not transform like a tensor under a coordinate transformation. For example, Christoffel symbolscannot be tensors themselves if the coordinates don't change in a linear way.

Curvilinear coordinates and curved spacetime[edit]

[/ltr]


[ltr]
Curvilinear coordinates are coordinates in which the angles between axes can change from point-to-point. This means that rather than having a grid of straight lines, the grid instead has curvature.
A good example of this is the su***ce of the Earth. While maps frequently portray north, south, east and west as a simple square grid, that is not in fact the case. Instead, the longitude lines running north and south are curved and meet at the north pole. This is because the Earth is not flat, but instead round.
In general relativity, gravity has curvature effects on the four dimensions of the universe. A common analogy is placing a heavy object on a stretched out rubber sheet, causing the sheet to bend downward. This curves the coordinate system around the object, much like an object in the universe curves the coordinate system it sits in. The mathematics here are conceptually more complex than on Earth, as it results in 4 dimensions of curved coordinates instead of 3 as used to describe a curved 2D su***ce.

Parallel transport[edit]

Main article: Parallel transport[/ltr]

测地线[编辑]





[ltr]
测地线又称大地线或短程线，数学上可视作直线在弯曲空间中的推广；在有度规定义存在之时，测地线可以定义为空间中两点的局域最短路径。测地线（geodesic）的名字来自对于地球尺寸与形状的大地测量学（geodesy）。[/ltr]

• 1 三维空间中的曲面
  
  
  
  • 1.1 相关定理及推论
    
  
  
  
• 2 微分几何的测地线
  
  
  
  • 2.1 唯一性及存在性
    
  • 2.2 局部最短性
    
  
  
  
• 3 度量几何的测地线
  
• 4 参考
  
  
  

[ltr]

三维空间中的曲面[编辑]

在大地线上，各点的主曲率方向均与该点上曲面法线相合。它在圆球面上为大圆弧，在平面上就是直线。在大地测量中，通常用大地线来代替法截线，作为研究和计算椭球面上各种问题。测地线是在一个曲面上，每一点处测地曲率均为零的曲线。

相关定理及推论[编辑]

曲面上非直线的曲线是测地线的充分必要条件是除了曲率为零的点以外，曲线的主法线重合于曲面的法线。
如果两曲面沿一曲线相切，并且此曲线是其中一个曲面的测地线，那么它也是另一个曲面的测地线。
过曲面上任一点，给定一个曲面的切方向，则存在唯一一条测地线切于此方向。
在适当的小范围内联结任意两点的测地线是最短线，所以测地线又称为短程线。

微分几何的测地线[编辑]

在一个黎曼流形上，一条曲线若符合常微分方程

就称之为测地线。其中是上的列维-奇维塔联络。方程左边为曲线在流形上的加速度向量，所以方程是说测地线是在流形上加速度为零的曲线，也因此测地线必定是等速曲线。
以上方程用局部座标表示为

其中是的黎曼度量的克里斯托费尔符号。

唯一性及存在性[编辑]

给定流形上一点及点上一个非零的切向量，因测地线方程是二阶常微分方程，柯西－利普希茨定理指出存在区间，使得方程在此区间上存在唯一解

满足初值条件,。但因为方程是非线性的，故未必在实数线上存在解。
从上述方程解的唯一性，可知若两条测地线经过同一点，且在此点上有相同的切向量，则这两条测地线是同一条测地线中的两部份。
设是一条测地线，。如果对起点及起点的切向量改变得足够细微，则存在新的测地线符合新的初值条件，且仍然定义在上。这个结果用严格语言叙述为：
给定测地线。在切丛中存在的一个邻域，使得对任何，都存在测地线满足初值条件。
从这结果可以得出，如果是定义在有界开区间上的测地线，对它的起点和此点上的切向量改变得足够细微的话，则存在一条新的测地线满足新的初值条件，并且定义在接近整条上。^[1]
如果对于任意初始条件,都存在一条定义在整条实数线上的测地线，则称是测地完备的。霍普夫－里诺定理指出，若是一个完备的度量空间，则是测地完备的。（上两点间的度量，是连接此两点的所有曲线的长度的最大下界。）

局部最短性[编辑]

在黎曼流形上连接两点之间的等速曲线，若其长度等于两点间的距离，即这曲线是两点间最短的曲线，那么这曲线必定是测地线。然而，连接两点间的测地线未必最短。比如在单位球面上，一条长度大于的测地线，不是连接这条线的两端点间的最短曲线。因为球面上的测地线都是大圆的弧，若测地线长度大于，那么测地线所在大圆上的另一条弧，其长度会小于，是连接这两点的最短测地线。
连接两点间最短测地线，也未必唯一。比如单位球面上两个对径点（即球面和一条直径的两个交点）之间，有无数条最短测地线相连。然而，流形上任何一点都存在一个邻域，使得该点和邻域上其他点之间，都有唯一的最短测地线相连（不计测地线的速度）。因此流形上任何测地线都是局部最短的。
对流形上一点，一条从出发的单位速的测地线，考虑所有的使得，即是说是一条最短测地线。这集合可以是或。若是前者，称是沿着的割点，那么对所有,是从点到的唯一最短测地线；若是后者，则对所有,都是点到的唯一最短测地线。沿着全部从出发的测地线的割点组成的集合，称为的割迹。

度量几何的测地线[编辑]

一般的度量空间中，测地线是从区间的映射，使得对任何，都存在区间，使得包含在中一个开邻域，并且对任何有

换言之，是连接其上任何两点的一条最短路线。^[2]
如果一个度量空间任何两点都有测地线相连，称为测地度量空间。
度量空间上的测地线的性质，和微分几何有些不同：
两条测地线即使有部分线段重合，却未必属于同一条测地线。例如在上定义度量

设是从（0,0）到（1,0）再到（1,1）的两条线段所组成，而是从（0,0）到（2,0）的线段。这两条都是测地线，且在（0,0）到（1,0）一段重合，但明显不属同一条测地线，因为这两条线过了点（1,0）之后就分开。
一个测地度量空间中，在一点上未必存在一个邻域，使得该点其邻域其他点都有唯一的测地线。在上例的度量空间中，两点间如果两个座标都不同，则有无限多条测地线连接两点。例如从（0,0）到（2,1），以下都是连接这两点的最短测地线：任取一数，

就是先向右走到，再向上走到，再向右走到（2,1）。在任何一点的任何邻域中，和该点两个座标都不同的点有无数个，所以从该点到这些点之间，最短测地线都不是唯一。

参考[编辑]

[/ltr]

http://physics.stackexchange.com/questions/13870/gauge-symmetry-is-not-a-symmetry/29205#29205

Gauge symmetry is not a symmetry?

Gauge symmetry is not a symmetry?


I have read before in one of Seiberg's articles something like, that gauge symmetry is not a symmetry but a redundancy in our description, by introducing fake degrees of freedom to facilitate calculations.

Regarding this I have a few questions:

Why is it called a symmetry if it is not a symmetry? what about Noether theorem in this case? and the gauge groups U(1)...etc?
Does that mean, in principle, that one can gauge any theory (just by introducing the proper fake degrees of freedom)?
Are there analogs or other examples to this idea, of introducing fake degrees of freedom to facilitate the calculations or to build interactions, in classical physics? Is it like introducing the fictitious force if one insists on using Newton's 2nd law in a noninertial frame of reference?

按照现在人们对于量子场论的认识，所有的相互作用都可以从规范
原理导出来。每一个规范理论都对应一个规范对称群，这个群的元素对应于
一定的规范变换；比如，量子电动力学可以由一个规范对称群为U(1)的规范
理论来描述，其中规范场就是电磁场，U(1)群的群元素满足乘法交换性，所
以它是可交换的Abelian群。1954年，C．N．Yang并HR．Mills把规范场和规范变换
推广到群元素不满足乘法交换性的non．Abelian群，这就导致non．Abelian规范
场。non—Abelian规范场在诞生之初面临两大难题，一个是如何对场进行量子
化，另一个是如何赋予规范粒子以质量。对前者的研究一直贯穿着上个世纪
六十年代，其中Feynman、Fadeev、Popov$I]DeWitt等人做出了重要的贡献，其
成果就是所谓的路径积分量子化方法。对后者的研究一直伴随着对弱相互作
用的规范理论的处理，这个问题的解决得益于由对称性自发破缺(spontaneous
symmetry breaking)导致的Higgs机i昔lJ(Higgs Mechanism)[161。借助于这个机制，
以S．Glashow、S．Weinberg$[IA．Salam为代表的物理学家们最终建立了统一电磁
相互作用和弱相互作用的规范理论模型，臣PGlashow．Weinberg．Salam模型【17】。
该模型预言两个带电的(W士)和一个中性的(Zo)有质量的中间规范玻色子，它们
已经在实验上被发现；同时为了保证对称性自发破缺，Higgs机制还要求至少有
一个标量粒子—碰ggs粒子，但是到目前为止，还没有令人信服的实验上的证据
表明存在该粒子。