Astronomy

Compact Objects in Astrophysics
White Dwarfs,Neutron Stars and Black Holes

Preface
In astronomy, a compact star (sometimes called a compact object) is a star that is
a white dwarf, a neutron star or a black hole. Our Galaxy is populated by billions of
white dwarfs, a few hundred million neutron stars and probably by a few hundred
thousand black holes. Of all these objects, only a very tiny fraction has been detected
so far by astronomical instruments, just a few thousand white dwarfs, about 2000
neutron stars, and only a few dozen black holes. Of all these objects, only black holes
can appreciably grow in mass. Its is one of the great successes of the last 15 years that
it could be shown that practically every center of galaxies harbors a supermassive
black hole with a mass in the range of one million to a few billion solar masses. The
visible Universe therefore contains at least 100 billion supermassive black holes.
Only about 100,000 of these objects have now been detected as quasars and only
about 50 as mass centers of nearby galaxies. Black holes of varying mass are also
thought to be the driver behind gamma bursters.
Compact stars form the endpoint of stellar evolution. A star shines and thus loses
its nuclear energy reservoir in a finite time. When a star has exhausted all its energy
(which is called a stellar death), the gas pressure of the hot interior can no longer
support the weight of the star and the star collapses to a denser state – a compact
star. One could see the compact stars, such as the white dwarf and the neutron star,
as a solid state as opposed to the gaseous interior of all other stars. In contrast to
this, the interior of a black hole is very enigmatic. Its su***ce is formed by a kind of
semipermeable membrane forbidding any classical emission from its su***ce. The
very source of the gravitational field of black holes is a kind of curvature singularity,
which is hidden behind this membrane. It is expected that quantum effects will
smooth these singular mass currents in the center of a rotating Black hole.
A normal star is a fully Newtonian object, in the sense that its gravitational
field is a mere solution of the Poisson equation. Gravity of compact objects, on the
other hand, must rely on the concepts of space and time. The classical textbook
by Shapiro and Teukolsky [15] on the theory of white dwarfs, neutron stars and
black holes handles many aspects on these objects. In the last 20 years, however,
a great deal of observational data and theoretical insights into the physics of compact
objects force us to a more complicated approach for modelling. Just to mention one
example: though the Tolman–Oppenheimer–Volkoff equation is still the basis for
the calculation of the interior structure of neutron stars, the inclusion of rotation for
these objects leads to a nontrivial set of partial differential equations for handling
the gravitational field of rapidly rotating neutron stars. For this reason, the author of
this book has decided to base the description of gravity on the general framework
which is nowadays used in numerical computations when Einstein’s equations are
involved.
The concept of this book therefore relies heavily on the concepts of modern gravity.
For this reason, Chap. 2 gives an overview of the modern description of gravity.
This does not, however, preclude any study of classical textbooks on Einstein’s
theory of gravity.
Compact objects such as white dwarfs and neutron stars have extremely high
densities that cannot be created in terrestrial laboratories and involve phases of
matter that are not yet well understood. In these lectures we will work out the associated
highly relativistic phenomena theoretically and observationally. One theoretical
focus is understanding the interplay between magnetic and thermal processes for
strongly magnetic neutron stars. In addition, just like their stellar precursors, many
compact objects occur in binary systems. We will study the origin and evolution of
compact X-ray binaries using data from RXTE as well as ASCA and ROSAT and
other X-ray data. With the successful launch of Chandra and XMM–Newton, X-ray
astronomy is in a key position to conduct new high-resolution imaging and spectral
studies of compact objects in both binaries and AGN.
High-energy gamma-ray bursts are being detected with regularity now, but their
nature remains a mystery. Researchers are actively involved in modelling these bursts
and identifying tests and consequences of suggested mechanisms for a wide array
of data sets. Cosmic gamma-ray bursts are important for their own intrinsic physics
as well as for providing a probe of cosmology. We still do not know the nature of
the tremendous explosions that in about one minute release a few percent of a solar
mass of rest energy in the form of gamma-rays. However, several clues point to an
association with the explosions of massive stars, and current models assume that
a gamma-ray burst is triggered by the formation of a black hole.
The study of compact objects probes physics at extreme conditions of density,
temperature, and magnetic fields. The mass–radius relation for neutron stars, for
example, probes the equation of state at supranuclear densities and may reveal in
the future the existence of quark matter in one of the color-superconducting phases.
Accurate neutron star masses can be measured for some binaries, especially those
including radio pulsars; measuring radii is more difficult, but may be possible through
studies of gravitational redshifts, neutron star cooling or the dynamics of gas near
the innermost stable circular orbit predicted by general relativity.
Different models for the composition and equation of state of neutron-star matter
produce neutron-star models with different properties which might then be detected
in observations. Particularly important in this context is the possibility of constraining
the form of particle interactions in high-density matter or of finding evidence for
the occurrence of phase transitions in the stellar interior or of exotic states of matter
(strange stars being an extreme example).
Important stellar evolution questions are being addressed concerning the evolutionary
pathways to each of the endpoints for compact objects. Binary star systems
can undergo complex mass transfer evolutionary phases. In particular, considerable
insight has been gained into how close binary systems containing compact ob-
jects are formed from primordial binaries in the Galaxy and via dynamical capture
processes in globular star clusters. Once an accreting compact binary forms, many
questions remain about the accretion process itself. For example, largely through observational
work conducted with the Rossi X-ray Timing Explorer Satellite (RXTE),
astronomers have found that accreting neutron stars often flicker quasiperiodically
at frequencies ranging from a few hertz to more than one kilohertz. The cause of this
flickering is poorly understood, but may involve effects of strong field gravity in the
accretion disk or oscillations of the neutron star.
One exciting fact is that compact objects offer the ultimate strong-field tests
of general relativity through the gravitational radiation emitted when black holes
form. The recent detection of a double pulsar system opens up a new window on
testing relativistic gravity by using compact objects. Together with black holes, these
neutron stars will provide the deepest insight into the structure of relativistic gravity.
These systems are sources of gravitational waves. The existence and ubiquity of
gravitational waves is an unambiguous prediction of Einstein’s theory of general
relativity. Although gravitational radiation has not yet been unambiguously and
directly detected, there is already significant indirect evidence for its existence.
Most notably, observations of binary pulsars, which are thought to consist of two
neutron stars orbiting rather tightly and rapidly around each other, have revealed
a gradual in-spiral at exactly the rate which would be predicted by general relativity.
Heidelberg, December 2006 Max Camenzind


Contents
1 Compact Objects in Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Why is Newtonian Gravity Obsolete? . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Einstein was Skeptical about the Existence of Black Holes . . . . . . . . 3
1.3 Subrahmanyan Chandrasekhar and Compact Objects . . . . . . . . . . . . . 4
1.4 Classes of Compact Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 White Dwarfs and Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 Compact X-Ray Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.3 Radio Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Supermassive Black Holes in Galactic Centers . . . . . . . . . . . . . . . . . . 16
1.6 Gamma-Ray Bursters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Gravity of Compact Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 Geometric Concepts and General Relativity . . . . . . . . . . . . . . . . . . . . 27
2.2 The Basic Principles of General Relativity . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Einstein’s Equivalence Principle and Metricity . . . . . . . . . . . . 29
2.2.2 Metric Theories of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Basic Calculus on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 Tensors and Forms on Manifolds . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 The Metric Field and Pseudo-Riemannian Manifolds . . . . . . 42
2.3.3 The Calculus of Forms on Lorentzian Manifolds . . . . . . . . . . 44
2.4 Affine Connection and Covariant Derivative . . . . . . . . . . . . . . . . . . . . 47
2.4.1 Affine Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.2 Covariant Derivative of Vector Fields . . . . . . . . . . . . . . . . . . . 47
2.4.3 Covariant Derivative for Tensor Fields. . . . . . . . . . . . . . . . . . . 48
2.4.4 Parallel Transport and Metric Connection . . . . . . . . . . . . . . . . 50
2.4.5 Metric Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.6 Divergence of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5 Curvature of Pseudo-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . 56
2.5.1 Mathematical Definition of Torsion and Curvature. . . . . . . . . 57
2.5.2 Bianchi Identities for Metric Connection . . . . . . . . . . . . . . . . . 58
2.5.3 Ricci, Weyl and Einstein Tensor . . . . . . . . . . . . . . . . . . . . . . . . 60
2.5.4 Cartan’s Structure Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Gravity is a Lorentzian Connection on Spacetime . . . . . . . . . . . . . . . . 65
2.6.1 The Four Key Principles of General Relativity . . . . . . . . . . . . 65
2.6.2 The Hilbert Action and Einstein’s Field Equations . . . . . . . . . 68
2.6.3 On the Cosmological Constant . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.6.4 Limits of General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.7 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.7.1 The Geodesic Deviation – Relativistic Tidal Forces . . . . . . . . 73
2.7.2 Gravity Wave Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.7.3 The Nature of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . 76
2.7.4 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.7.5 Gravitational Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.7.6 The Quadrupole Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.8 3+1 Split of Einstein’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.8.1 Induced Spatial Metric and Extrinsic Curvature . . . . . . . . . . . 92
2.8.2 Hypersu***ce Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.8.3 Split of Affine Connection and Curvature . . . . . . . . . . . . . . . . 95
2.8.4 Split of Einstein’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.8.5 Black Hole Simulations and Gravitational Waves. . . . . . . . . . 100
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3 Matter Models for Compact Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.1 General Relativistic Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.1.1 Relativistic Plasma Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.1.2 On Numerics of Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 110
3.2 The Boltzmann Equation in GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.2.1 The Geodesics Spray on the Cotangent Bundle . . . . . . . . . . . 113
3.2.2 Particle Number Current and Energy–Momentum Tensor . . . 116
3.2.3 The Relativistic Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 117
3.2.4 Liouville Operator in 3+1 Split . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.2.5 Transformation into the Local Rest Frame . . . . . . . . . . . . . . . 119
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4 Relativistic Stellar Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.1 Spacetime of Relativistic Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2 Derivation of the TOV Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2.1 The Curvature of Static Spacetimes . . . . . . . . . . . . . . . . . . . . . 125
4.2.2 Matter in the Interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.2.3 The Exterior Schwarzschild Solution . . . . . . . . . . . . . . . . . . . . 130
4.2.4 Stable Branches for Degenerate Stars . . . . . . . . . . . . . . . . . . . 131
4.2.5 Metric for Relativistic Stars. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.3 A Variational Principle for the Stellar Structure. . . . . . . . . . . . . . . . . . 132
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1 Observations of Isolated White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . 138
5.1.1 Sirius B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.1.2 Field White Dwarfs and Classification . . . . . . . . . . . . . . . . . . . 139
5.1.3 White Dwarfs in Globular Clusters . . . . . . . . . . . . . . . . . . . . . 143
5.1.4 Magnetic White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.1.5 Ultracool White Dwarfs as Cosmochronometers . . . . . . . . . . 145
5.2 What is Inside a White Dwarf? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.3 Equation of State below the Neutron Drip Density . . . . . . . . . . . . . . . 153
5.4 Structure of White Dwarfs and the Chandrasekhar Mass . . . . . . . . . . 159
5.4.1 Polytropic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.4.2 Beyond the Chandrasekhar Treatment . . . . . . . . . . . . . . . . . . . 162
5.4.3 Comparison with Observations . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.5 The Relativistic Instability of White Dwarf Stars . . . . . . . . . . . . . . . . 167
5.5.1 Necessary Condition for Stability . . . . . . . . . . . . . . . . . . . . . . . 168
5.5.2 The Total Energy in the Post-Newtonian Limit . . . . . . . . . . . . 169
5.5.3 GR White Dwarf Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.6 Cooling White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.6.1 Structure of the Su***ce Layers . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.6.2 Cooling Curves and Crystallization . . . . . . . . . . . . . . . . . . . . . 177
5.6.3 Testing WD Crystallization Theory . . . . . . . . . . . . . . . . . . . . . 179
5.7 White Dwarfs in Binary Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.1 The Structure of a Neutron Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.2 Equations of State beyond Neutron Drip . . . . . . . . . . . . . . . . . . . . . . . 189
6.2.1 From Neutron Drip to Saturation . . . . . . . . . . . . . . . . . . . . . . . 190
6.2.2 Nuclear EoS for Dense Neutron Matter . . . . . . . . . . . . . . . . . . 199
6.2.3 Relativistic Mean Field Theory above Saturation . . . . . . . . . . 206
6.2.4 Analytical Fits to EoS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
6.3 Neutron Star Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.3.1 Hadronic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.3.2 Quark Matter Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.3.3 Grand Canonical Potential for Quark Matter . . . . . . . . . . . . . . 231
6.3.4 Strange Quark Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
6.3.5 The Structure of Massive Neutron Stars . . . . . . . . . . . . . . . . . 242
6.4 Neutron Stars in Close Binary Systems . . . . . . . . . . . . . . . . . . . . . . . . 244
6.4.1 Post-Newtonian Potentials for Many-Body Systems . . . . . . . 244
6.4.2 Periastron Shift in Two-Body Systems. . . . . . . . . . . . . . . . . . . 248
6.4.3 The Shapiro Time Delay in a Binary System . . . . . . . . . . . . . 250
6.4.4 Decay of Binary Orbits due to Gravitational Radiation . . . . . 251
6.5 Masses of Neutron Stars from Radio Pulsar Timing . . . . . . . . . . . . . . 255
6.5.1 What is Pulsar Timing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
6.5.2 The Timing Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
6.5.3 Timing of the Binary System PSR B1913+16 . . . . . . . . . . . . . 263
6.5.4 Masses of Companion Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
6.5.5 The Double Pulsar System PSR 0737-3039A+B . . . . . . . . . . 265
6.6 Neutron Stars in our Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
6.6.1 100 Million Neutron Stars in the Galaxy . . . . . . . . . . . . . . . . . 269
6.6.2 Thermal Emission from Isolated Neutron Stars . . . . . . . . . . . 272
6.6.3 Rotation-Powered Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
6.6.4 Accretion-Powered Neutron Stars
and the Mass–Radius Relation . . . . . . . . . . . . . . . . . . . . . . . . . 294
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
7 Rapidly Rotating Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
7.1 Spacetime of Stationary and Axisymmetric Rotating Bodies . . . . . . . 308
7.1.1 Physical Interpretation of the Metric . . . . . . . . . . . . . . . . . . . . 309
7.1.2 Geodetic and Lense–Thirring Precession. . . . . . . . . . . . . . . . . 312
7.1.3 On General 3+1 Split of Spacetime . . . . . . . . . . . . . . . . . . . . . 315
7.2 Einstein’s Field Equations for Rotating Objects . . . . . . . . . . . . . . . . . 317
7.2.1 Ricci Tensors of Time-Slices. . . . . . . . . . . . . . . . . . . . . . . . . . . 318
7.2.2 Extrinsic Curvature and 4D Ricci Tensors . . . . . . . . . . . . . . . . 319
7.2.3 3+1 Split of Einstein’s Equations . . . . . . . . . . . . . . . . . . . . . . . 320
7.3 Stellar Structure Equations in Isotropic Gauge . . . . . . . . . . . . . . . . . . 321
7.3.1 The Isotropic Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
7.3.2 Structure Equations for Rotating Stars . . . . . . . . . . . . . . . . . . . 322
7.3.3 Mechanical Equilibrium and Effective Potential . . . . . . . . . . . 324
7.3.4 Stellar Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
7.4 The Slow-Rotation Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
7.5 Numerical Integration of the Stellar Structure Equations . . . . . . . . . . 335
7.5.1 Comparison of Numerical Codes . . . . . . . . . . . . . . . . . . . . . . . 337
7.5.2 Properties of Rotating Equilibrium Stellar Structures . . . . . . 338
7.6 Towards Analytical Vacuum Solutions for Rotating Neutron Stars . . 342
7.6.1 Weyl–Papapetrou Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
7.6.2 Ernst Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
7.6.3 Manko’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
7.7 On Oscillation and Formation of Rotating Neutron Stars . . . . . . . . . . 350
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
8 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
8.1 The Schwarzschild Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
8.1.1 Tortoise Coordinates and Null Cones . . . . . . . . . . . . . . . . . . . . 356
8.1.2 Roads towards Black Hole Formation . . . . . . . . . . . . . . . . . . . 358
8.1.3 The Kruskal Extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
8.1.4 Penrose Diagram – the Conformal Structure of Infinity . . . . . 363
8.2 Geodetic Motions in Schwarzschild Spacetime . . . . . . . . . . . . . . . . . . 369
8.2.1 A Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
8.2.2 The Effective Potential for Equatorial Motion . . . . . . . . . . . . 371
8.2.3 Orbital Equation and Bound Orbits
in Schwarzschild Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
8.3 The Kerr Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
8.3.1 Kerr Black Hole in Boyer–Lindquist Coordinates . . . . . . . . . 379
8.3.2 A Short Derivation of the Kerr Solution . . . . . . . . . . . . . . . . . 379
8.3.3 The Weyl–Papapetrou Form of the Kerr Metric . . . . . . . . . . . 384
8.3.4 Uniqueness of the Kerr Solution . . . . . . . . . . . . . . . . . . . . . . . . 385
8.3.5 Global Properties of the Kerr Metric . . . . . . . . . . . . . . . . . . . . 386
8.3.6 On the Conformal Structure of the Kerr Solution . . . . . . . . . . 393
8.3.7 Ernst’s Equations for the Kerr Geometry . . . . . . . . . . . . . . . . . 394
8.3.8 The Kerr–Schild Metric and Two-Black-Hole States . . . . . . . 395
8.4 Rotational Energy and the Four Laws of Black Hole Evolution. . . . . 399
8.4.1 Su***ce Gravity and Angular Velocity of the Horizon . . . . . . 400
8.4.2 First Law of Black Hole Dynamics . . . . . . . . . . . . . . . . . . . . . 402
8.4.3 Rotational Energy of Astrophysical Black Holes . . . . . . . . . . 405
8.4.4 On the Second and Third Laws of Black Hole Dynamics . . . 406
8.5 Time Evolution of Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
8.5.1 Quasistationary Evolution of Accreting Black Holes . . . . . . . 408
8.5.2 Merging of Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
8.6 Geodesics in the Kerr Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
8.6.1 Direct Integration of Geodesics Equations . . . . . . . . . . . . . . . 414
8.6.2 Geodesics in the Equatorial Plane . . . . . . . . . . . . . . . . . . . . . . 416
8.6.3 Geodesics Including Lateral Motion . . . . . . . . . . . . . . . . . . . . 424
8.6.4 Null Geodesics and Ray-Tracing in Kerr Geometry . . . . . . . . 431
8.7 Dark Energy Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
8.7.1 Why Dark energy Stars? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
8.7.2 Structure of Gravastars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
8.7.3 The Necessity of an Anisotropic Crust. . . . . . . . . . . . . . . . . . . 445
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
9 Astrophysical Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
9.1 Classes of Astrophysical Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . 450
9.2 Measuring Black Hole Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
9.2.1 BHs in X-Ray Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
9.2.2 Intermediate-Mass Black Holes . . . . . . . . . . . . . . . . . . . . . . . . 456
9.2.3 Supermassive Black Holes in Nearby Galaxies . . . . . . . . . . . . 456
9.2.4 Black Holes in Quasars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
9.3 Estimating Black Hole Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
9.3.1 Black Hole Spin and Radio Galaxies . . . . . . . . . . . . . . . . . . . . 471
9.3.2 Spectral Fitting of Accretion Disks . . . . . . . . . . . . . . . . . . . . . 471
9.3.3 Relativistic Iron Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
9.3.4 Quasiperiodic Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
9.4 Black Holes and Galaxy Formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
9.5 Black Hole Magnetospheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
9.5.1 The 3+1 Formalism for Maxwell’s Equations . . . . . . . . . . . . . 474
9.5.2 Plasma Equations in the 3+1 Split . . . . . . . . . . . . . . . . . . . . . . 478
9.5.3 Time Evolution of Magnetic and Current Flux
in Turbulent Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
9.5.4 Stationary Magnetospheres on Kerr Black Holes . . . . . . . . . . 486
9.5.5 Relaxation of Black Hole Magnetospheres
and the Blandford–Znajek Process . . . . . . . . . . . . . . . . . . . . . . 499
9.6 Magnetic Spin-Down of Rotating Black Holes . . . . . . . . . . . . . . . . . . 509
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
10 Physics of Accretion Flows around Compact Objects . . . . . . . . . . . . . . . 513
10.1 Angular Momentum Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
10.2 Magnetohydrodynamics for Accretion Disks . . . . . . . . . . . . . . . . . . . . 517
10.2.1 Equations of Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . 517
10.2.2 Time and Space Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 523
10.2.3 MRI Driven Turbulence in Disks . . . . . . . . . . . . . . . . . . . . . . . 525
10.2.4 Two-Temperature Plasmas and Radiation Pressure
in Accretion Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
10.3 States of Turbulent Accretion Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
10.3.1 Turbulent Angular Momentum Transport
in Accretion Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
10.3.2 Truncated Accretion and Standard Disk Models in 1D . . . . . 540
10.3.3 Standard Thin Disk Solutions (SSD) . . . . . . . . . . . . . . . . . . . . 545
10.3.4 Advection-Dominated Flows (ADAF) . . . . . . . . . . . . . . . . . . . 551
10.3.5 Super-Eddington Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
10.3.6 Unified Models of Disk Accretion . . . . . . . . . . . . . . . . . . . . . . 553
10.3.7 Fundamental Time-Scales for Accreting Black Holes . . . . . . 555
10.4 Relativistic MHD – Turbulent Accretion onto Black Holes . . . . . . . . 558
10.4.1 From SRMHD to GRMHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
10.4.2 The Equations for GRMHD . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
10.4.3 Nonradiative Accretion onto Rotating Black Holes . . . . . . . . 563
10.5 Jets and the Ergosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
10.5.1 Jets as Outflows from the Ergospheric Region . . . . . . . . . . . . 566
10.5.2 From the Ergosphere to the Cluster Gas . . . . . . . . . . . . . . . . . 572
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
11 Epilogue and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
Astrophysical Constants and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
SLy4 Equation of State for Neutron Star Matter . . . . . . . . . . . . . . . . . . . . . . . 591
3+1 Split of Spacetime Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
C.1 Gauss Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
C.2 Codazzi–Mainardi Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
3+1 Split of Rotating Neutron Star Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 599
D.1 The 3+1 Split of the Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
D.2 The Curvature of Time Slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
Equations of GRMHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
E.1 Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
E.2 Conservative Formulation of GRMHD . . . . . . . . . . . . . . . . . . . . . . . . . 607
E.3 Numerical Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

1 Compact Objects in Astrophysics
As a class of astronomical objects, compact objects include white dwarfs, neutron
stars and black holes. As the endpoint states of stellar evolution, they form today
fundamental constituents of galaxies. In the form of supermassive black holes, these
objects also live in practically every center of a galaxy. Our Galaxy harbors a black
hole of 3.8 million solar masses, but the center of M87 in the Virgo cluster encloses
a black hole of three billion solar masses. These supermassive black holes are the
most extreme objects found in the Universe.
While neutron stars and stellar mass black holes mainly entered astrophysical
research by means of their radio and X-ray emission, white dwarfs had already been
detected 100 years ago by their optical emission.
1.1 Why is Newtonian Gravity Obsolete?
The gravitational collapse of normal matter produces some of the most exotic objects
in the Universe – neutron stars and black holes. Proving that these objects exist in
Nature occupied theoretical and observational astrophysicists for much of the 20th
century. Most of the detailed debate centered around understanding the possible
final states of massive stars. On his now famous sea voyage from India to England in
1930, Subrahmanyan Chandrasekhar considered the structure of white dwarf stars –
compact stellar remnants in which gravitational forces are balanced by electron
degeneracy pressure. He realized that, if the white dwarf was sufficiently massive,
the degenerate electrons will become relativistic thereby rendering the star susceptible
to further gravitational collapse. Although hotly debated by Arthur Eddington,
Chandrasekhar correctly deduced that a white dwarf would undergo gravitational
collapse if its mass exceeded MCh 1.4 M (where M is the mass of the Sun),
a limit now known as the Chandrasekhar limit.
Once gravity overwhelms electron degeneracy pressure, neutron degeneracy
pressure is the last, best hope for averting total gravitational collapse. Objects in
which gravitational forces are balanced by neutron degeneracy pressure are called
neutron stars. Although there was initial hope that nuclear forces would always be
sufficient to resist gravity, the upper limit to the mass of a neutron star is now believed
to be in the range (1.5–2.2) M . Uncertainties arising from the equation of
state at supranuclear densities continue to plague our determination of this critical
mass, but an absolute upper limit of 3 M arises from very general considerations,
i.e. the validity of general relativity and the principle of causality. Above this mass,
it is thought that complete gravitational collapse cannot be avoided. In particular,
Hawking’s singularity theorems show that the formation of a spacetime singularity
is unavoidable (irrespective of the mass/energy distribution) once the object is
contained within the light-trapping su***ce. The result is a black hole, i.e. a region
of spacetime bounded by an event horizon and, at its heart, possessing a spacetime
singularity.
While the above considerations now have a firm theoretical base, observational
astrophysics was, and continues to be, critically important in guiding our understanding
of such extreme objects. In the case of both neutron stars and black holes,
the very existence of these objects was only widely accepted when compelling observational
evidence was forthcoming. For neutron stars, the pivotal observation
was the discovery of radio pulsars by Jocelyn Bell and Anthony Hewish via radio
observations taken from Cambridge. Black holes gained wide acceptance after it
was demonstrated that the X-ray emitting compact object in the binary star system
Cygnus X-1 did, in fact, possess a mass in excess of the maximum possible neutron
star mass. This made it the first of the so-called Galactic Black Hole Candidates
(GBHCs), a class that has now grown to include some two dozen objects.
We now know of another class of black holes – the supermassive black holes,
with masses in the range of (106 –1010 ) M , that reside at the dynamical centers of
most, if not all, galaxies. Today, by far the strongest case for a supermassive black
hole can be made for our own Galaxy. Modern high-resolution, infrared imaging
reveals that the stars in the central-most regions of our Galaxy are orbiting an unseen
mass of three million solar masses. Furthermore, studies of the orbital dynamics
(which now include measured accelerations as well as velocities) constrain the
central mass to be extremely compact. According to conventional physics, the only
long-lived object with these properties is a supermassive black hole. Alternatives,
such as a compact cluster of neutron stars, would suffer a dynamical collapse on
much shorter time-scales.
Having established beyond reasonable doubt that black holes exist, it is obviously
interesting to perform detailed observational studies of them. The regions in the
immediate vicinity of a black hole bear witness to complex interactions between
matter moving at relativistic velocities, electromagnetic fields, and the black hole
spacetime itself. Given that the apparent angular scales of even the biggest black
hole event horizons are 10−6 arcsec, direct imaging studies of these regions will
not be possible for many years. In the meantime, we must study these regions using
more indirect methods, chief among which are spectroscopic methods.
As we will detail in this book, Nature has provided us with a well-understood and
extremely useful spectral diagnostic of matter in the near vicinity of astrophysical
black holes. In essence, relatively cold matter in the near vicinity of an astrophysical
black hole will inevitably find itself irradiated by a spectrum of hard X-rays. The
result can be a spectrum of fluorescent emission lines, the most prominent being
the Kα line of iron at an energy of 6.4 keV (depending upon the ionization state of
the iron). Ever since the launch of the Advanced Satellite for Cosmology and Astrophysics
(ASCA) in February 1993, X-ray astrophysicists have had the capability
to identify this emission line and measure its spectral profile. Figure 1.1 shows the
iron line in the X-ray emissions originating near the supermassive black hole in the
galaxy MCG–6–30–15. Bearing in mind that the line is intrinsically narrow with
a rest-frame energy of 6.4 keV, it can be seen that the line has been dramatically
broadened and skewed to low energies. It is now widely accepted that the line originates
from material that is just a few gravitational radii from the black hole, and
possesses a profile that is shaped by (relativistic) Doppler shifts and gravitational
redshift effects. Investigating these spectral features in X-ray luminous black hole
systems has given us the clearest window to date on the physics that occurs in the
immediate vicinity of astrophysical black holes.
1.2 Einstein was Skeptical about the Existence of Black Holes
A black hole is a region of space whose attractive gravitational force is so intense
that no matter, light, or communication of any kind can escape. A black hole would
thus appear black from the outside. However, gas around a black hole can be very
bright. It is believed that black holes form from the collapse of stars. As long as they
are emitting heat and light into space, stars are able to support themselves against
their own inward gravity with the outward pressure generated by heat from nuclear
reactions in their deep interiors. Every star, however, must eventually exhaust its
nuclear fuel. When it does so, its unbalanced self gravitational attraction causes it
to collapse. According to theory, if a burned-out star has a mass larger than about
twice the mass of our Sun (as a protoneutron star), no amount of additional pressure
can stave off total gravitational collapse. The star collapses to form a black hole. For
a nonrotating collapsed star, the size of the resulting black hole is proportional to the
mass of the parent star; a black hole with a mass three times that of our Sun would
have a diameter of about 20 km. The possibility that stars could collapse to form
black holes was first theoretically discovered in l939 by J. Robert Oppenheimer
and H. Snyder [318], who were manipulating the equations of Einstein’s general
relativity. The first black hole believed to be discovered in the physical world, as
opposed to the mathematical world of pencil and paper, was Cygnus X-1, about
7000 lightyears from Earth. Cygnus X-1 was found in 1970. Since then, a few
dozens excellent black hole candidates have been identified. Many astronomers and
astrophysicists believe that massive black holes, with sizes up to 10 billion times that
of our Sun, inhabit the centers of energetic galaxies and quasars and are responsible
for their enormous energy release. Ironically, Einstein himself did not believe in the
existence of black holes, even though they were predicted by his theory.
1.3 Subrahmanyan Chandrasekhar and Compact Objects
Subrahmanyan Chandrasekhar was born in Lahore (then in British India) and studied
physics at the Presidency College, Madras. In 1930, he became a research student of
R.H. Fowler at Cambridge University and earned his PhD in 1933. He developed the
theory of white dwarf stars, showing that quantum mechanical degeneracy pressure
cannot stabilize a massive star. He showed that a star of a mass greater than 1.4 times
that of the Sun (now known as the Chandrasekhar limit) had to end his life by
collapsing into an object of enormous density such as a black hole. In 1937, he
joined the University of Chicago and the Yerkes Observatory. He investigated and
wrote important books on stellar structure and evolution, dynamical properties of star
clusters and galaxies, radiative transfer of energy, hydrodynamic and hydromagnetic
stability, the stability of ellipsoidal figures of equilibrium, and the mathematical
theory of black holes. He also worked in relativistic astrophysics, and his last book
was Newton’s Principia for the Common Reader. In 1952, he received the Catherine
Wolfe Bruce gold medal, for lifetime contributions to astronomy. He was awarded
the Royal Medal of the Royal Society in 1962, and he edited the Astrophysical
Journal for nearly 20 years. Chandrasekhar shared the 1983 Nobel Prize in physics
with W.A. Fowler for his studies of the physical processes of importance to the
structure and evolution of stars.
Chandrasekhar left Bombay on a boat on 31st July 1930. On the voyage, after
overcoming his seasickness, he remembered Fowler’s paper and decided to combine
it with his knowledge of special relativity theory. To his great surprise, he found
that this combination predicted that white dwarfs could only exist up to a certain
limiting mass which depended chiefly on fundamental constants such as h, G and
the mass of the hydrogen atom; the mass was about 1.45 times the mass of the Sun.
England’s two leading astrophysicists, Eddington and Milne, could not believe this
result, and neither of them would recommend Chandra’s paper for publication by the
Royal Society. So Chandra sent it to the Astrophysical Journal in America, which
published it in March 1931.
Of course, Eddington was wrong. But his resistance to Chandra’s mass limit was
understandable: his life’s work had been to show that every star, whatever its mass,
had a stable configuration. It was generally (and correctly) believed that white dwarfs
were the end stage of stellar evolution, after their energy source was exhausted. Why
should there be a limit to the mass of a star in its old age? Chandra appealed to
physicists he knew – Rosenfeld, Bohr, Pauli. Unanimously, they decided that there
was no flaw in his argument. But it took decades before the Chandrasekhar limit was
accepted by the astrophysics community.
The paradox that normal stars can exist with any mass whereas white dwarfs
can only exist up to 1.45 solar masses is now understood. Stars, in their evolution,
go through a giant stage in which their radius may be hundreds of times larger than
originally. In this stage, the atoms at the su***ce are not strongly held by gravity, while
there is strong radiation pressure from the inside. Some atoms, especially hydrogen,
are blown off and the star gradually loses mass. Theory shows that stars up to eight
solar masses lose mass in this manner, ending up below the Chandrasekhar limit.
None of this was known in 1935.
The limit also affects stars heavier than eight solar masses. Matter in the central
core of stars evolves to iron by successive nuclear reactions. At this point, no further
nuclear energy can be obtained, just as in white dwarfs. When the iron core grows
to the Chandrasekhar mass, it collapses by gravitation into a neutron star, and the
rest of the star is expelled, giving a type II supernova. Some white dwarfs accrete
matter from the outside, and when their mass has grown to the Chandrasekhar limit,
they also become supernovae, in this case type Ia. Chandra’s theory is basic to much
modern astrophysics.
1.4 Classes of Compact Objects
The study of compact stars begins with the discovery of white dwarfs and the
successful description of their properties by the Fermi–Dirac statistics, assuming
that they are held up against gravitational collapse by the degeneracy pressure of
the electrons, an idea first proposed by Fowler in 1926 [160]. A maximum mass for
white dwarfs was found to exist in 1930 by the seminal work of Chandrasekhar due
to relativistic effects [113]. In 1932 Chadwick discovered the neutron. Immediately,
the ideas formulated by Fowler for the electrons were generalized to neutrons. The
existence of a new class of compact stars, with a large core of degenerate neutrons,
was predicted – the neutron stars (NS). The first NS model calculations were achieved
by Oppenheimer and Volkoff [317] and Tolman [394] in 1939, describing the matter
in such a star as an ideal degenerate neutron gas. Their calculations also showed
the existence of a maximum mass, like in the case of white dwarfs, above which
the star is not stable and collapses into a black hole. They found a maximum stable
mass of 0.75 M [317]. Only nearly 30 years later, in 1967, was the first neutron
star observed – in fact, a strange object pulsating in the radio range (radio pulsar),
which was however quickly identified as a fast rotating neutron star. Already in
1964, black holes have been proposed as the ultimate energy source for quasars.
In the meantime, the existence of black holes has been established in a huge mass
range, from about three solar masses to 10 billion solar masses in the centers of huge
elliptical galaxies.
In 1974, the pulsar PSR 1913+16 was observed for the first time in a binary
system by Hulse and Taylor. This allowed a precise measurement of its mass which
was found to be 1.44 M . Hence, this mass measurement ruled out the simple picture
of an ideal gas of neutrons for the interior of this star. It shows that the interactions
between the nucleons must be taken into account.
Shortly after the introduction of the quark model for nucleons, theoreticians
speculated about the possible existence of quark matter inside neutron stars. Gerlach
demonstrated in his PhD thesis with Wheeler in 1968 [170] that a third family of
compact stars could exist in Nature, besides white dwarfs and neutron stars. He
derived general conditions on the equation of state for such a new form of stars
to exist, in particular that a strong softening in the equation of state, like in phase
transitions, has to occur in neutron stars. Some astrophysicists even argue that the
very ground state of matter is in fact strange quark matter (composed of u, d and s
quarks). Such objects have now been studied since the mid-1980s and are referred
to as strange stars [27, 186].
A study of compact objects – white dwarfs, neutron stars, and black holes –
begins when normal stellar evolution ends. All these objects differ from normal stars
in at least two aspects:
– They are not burning nuclear fuel, and they cannot support themselves against
gravitational collapse by means of thermal pressure. Instead white dwarfs are supported
by the pressure of the degenerate electrons, and neutron stars are largely
supported by the pressure of the degenerate neutrons and quarks. Only black
holes represent completely collapsed stars, assembled by mere self-gravitating
forces. These objects can be considered as a kind of soliton solution of Einstein’s
equations.
– The second characteristic property of compact stars is their compact size. They
are much smaller than normal stars and therefore have much stronger su***ce
gravitational fields.
– Often compact objects carry strong magnetic fields, much stronger than found
in normal stars.
1.4.1 White Dwarfs and Neutron Stars
White dwarfs are stars of about one solar mass with a characteristic radius of
5000 km, corresponding to a mean density of 106 gcm−3 . They are no longer burning
nuclear fuel, but are steadily cooling away their internal heat. In 1926, only three
white dwarfs were firmly detected. In that year, Dirac formulated the Fermi–Dirac
statistics, which was used by Fowler [160] in the same year, in a pioneering paper on
compact stars – to explain the puzzling nature of white dwarf stars. He identified the
pressure holding up the stars from gravitational collapse with the electron degeneracy
pressure.
Actual models of white dwarf stars, taking into account the special relativistic
effects in the degenerate electron equation of state were then constructed in 1930
by Chandrasekhar [113]. He made the fundamental discovery of a maximum mass
of 1.4 M for white dwarfs – the exact value somewhat depends on the chemical
composition.
The prediction of the existence of neutron stars as a possible endpoint of stellar
evolution was independent of observations. Following the discovery of the neutron
by Chadwick, it was realized by many people that at very high densities electrons
would react with protons to form neutrons via inverse beta decay. Neutron stars had
been found at the end of the 1960s as radio pulsars and in the beginning of the 1970s
as X-ray stars. A firm upper limit for the mass of neutron stars was then seen as
evidence for the existence of even more exotic objects – black holes. At the time
of the discovery of Cyg X-1 by Uhuru (1970) the value of this upper limit was,
however, the subject of great debate.
1.4.2 Compact X-Ray Sources
A new era in astronomy was opened up in the 1960s by means of the launch of
various rockets (Giacconi 1962). They discovered Sco X-1 in the energy band of
1–10 keV. At the end of one decade, about 20 X-ray sources had been identified. One
of the strongest sources, Cyg X-1, was also found to vary in time. Already at that
time, gas accretion in a close binary system was seen to be the source of this X-ray
emission. But, for example, Prendergast and Burbidge [332] argued that gas flowing
onto a compact star in a binary system would have too much angular momentum
to flow radially inwards. They suggested that the gas would form a disk around the
compact star, with approximately Keplerian angular momentum. There should exist
a small inward drift velocity. The notion of an accretion disk was born (Fig. 1.5).
A comprehensive and up-to-date survey on compact stellar X-ray sources written by
leading experts in the field can be found in the book [6]. This book covers the details
of recent developments in X-ray astronomy and multiwavelength observations, as
well as some theoretical issues for these objects.
Modes of Accretion
Over the years, Cyg X-1 has been found to show two pronounced X-ray states
:
– a soft or high state: Here a pronounced black-body (BB) spectrum is visible with
a temperature of about 1 keV, and the luminosity is high;
– a hard or low state: the BB disappears and the X-rays are emitted in the hard
X-ray region up to 150 keV.
The existence of these two states is generally interpreted as evidence for two different
modes of accretion. The high energies of the photons is seen as evidence for the
existence of a hot plasma with electron temperatures of 109 K in the neighborhood
of the accretion disk. Soft photons from the optically thick disk would then be
up-scattered by Compton processes. Repeated Compton scattering can explain the
power-law form of the observed spectra in the hard state. The very location of the
hot plasma is still under debate. But already in 1977 Liang and Price introduced the
concept of a hot dissipative corona above the accretion disk following the example
of the solar corona (Liang and Price [254]). The energy could be dissipated by MHD
waves, or jets. In recent years, this transition between high state and low state is seen
as a transition from an optically thick disk flow to an optically thin disk flow. This
latter mode of accretion always exists within the marginal stable orbit, since matter
has to flow with the speed of light through the horizon and has to be supersonic
when entering the horizon (for causality reasons). In this way, the inner accretion
onto a black hole has to be hot (i.e. high sound speed near the speed of light). When
the accretion rate is high, then the soft flux is dominant and cooling of the corona
is efficient. On the other hand, when the accretion rate is very low, the inner disk
is probably in a very hot state cooling by Comptonization of soft photons from the
outer disk; this would correspond to the low state (hard) spectrum.
The existence of these luminosity states is not only generic for black hole systems.
Also neutron stars in low-mass X-ray binary systems (LMBXBs) are found to dispose
such luminosity states. The difference is the missing hard tail in the soft state in the
case of neutron stars, while the soft spectrum of Cyg X-1 has a pronounced hard
excess extending to at least a few hundred keV and probably into the MeV region.
1.4.3 Radio Pulsars
Pulsars are the lighthouses of the Galaxy – rapidly spinning neutron stars whose
strong magnetic fields produce conical beams of electromagnetic radiation that sweep
past the Earth with each rotation of the star, producing the eponymous pulses that are
observed primarily at radio wavelengths (Fig. 1.7). Pulsars were discovered, albeit
accidentally, by Jocelyn Bell at Cambridge in 1967. The apparently sporadic bursts
of radio emission appeared during the course of a survey to investigate the effects of
interplanetary scintillation of radio sources. Working as a graduate student in a team
lead by Anthony Hewish, Bell soon realized that the emission always occurred at the
same position in the celestial sphere indicating that the source was not of terrestrial
origin. Subsequent observations with greater time resolution showed the emission
to be a train of pulses with a precise repetition period of 1337 ms. The Cambridge
team published their discovery the following and, soon afterwards, announced the
discovery of three more pulsars found from subsequent inspection of the remaining
survey data.
Hewish was awarded the 1974 Nobel Prize in physics for his “decisive role in the
discovery of pulsars” and his pioneering work in radio astronomy. Bell’s key role in
the discovery has been widely recognized: among other awards, she has received the
Michelson Medal of the Franklin Institute in Philadelphia (jointly with Hewish), the
Tinsley Prize of the American Astronomical Society, and the Herschel Medal from
the Royal Astronomical Society.
Chinese astronomers in 1054 AD. Using the rotating neutron star model, Thomas
Gold of Cornell University, USA, was able to show that the Crab pulsar is the
dominant energy supply to its surrounding nebula. The connection between pulsars
and rotating neutron stars is now universally accepted.
Like other neutron stars, radio pulsars are born in the supernova explosions that
accompany the collapse of massive stars. The nascent pulsars are born rotating at
up to about one hundred times per second. It is this stored rotational kinetic energy
that powers the pulsar, so like a spinning top the pulsar gradually slows down,
reaching spin periods of about a second within a few million years. Eventually,
within about 100 million years, the pulsar is spinning too slowly to maintain its radio
emission, and it fades from view. Some old pulsars that have binary companions can
be “recycled,” or spun back up to fast rotation periods by mass transfer from their
companions. Because the resulting millisecond pulsars have relatively low magnetic
field strengths and hence low energy-loss rates, they can continue to spin rapidly for
times that are long compared to the age of the galaxy.
In over 35 years since the discovery, pulsars have proved to be exciting objects
to study and, presently, over 1500 are known. Most of these are normal in
the sense that their pulse periods are of order one second and, with few exceptions,
are observed to increase secularly at the rate of about one complete period
in 1,000,000,000,000,000! This is naturally explained as the gradual spin-down of
the neutron star as it radiates energy at the expense of its rotational kinetic energy.
A small fraction of the observed sample are the so-called millisecond pulsars which
have much shorter periods (< 20 ms) and rates of slowdown of typically only one
period in 10,000,000,000,000,000,000, proving to be extremely accurate clocks.
In addition, some pulsars are known to be members of binary systems in which
the companion is another neutron star, a white dwarf, or even a main sequence
star.
Just over 1500 radio pulsars are now known, all in our own Galaxy except
for a few pulsars detected in the Magellanic Clouds. They are studied because
neutron stars are intrinsically interesting astronomical objects, but also because
the study of pulsars is deeply intertwined with many different branches of both
astronomy and physics. Pulsars are, for example, very useful astrophysical probes.
For example, a sharp radio pulse emitted by a pulsar is delayed and broadened during
its propagation through the dispersive, turbulent interstellar medium, in a way that
depends on the frequency and polarization of the signal as well as the properties
of the medium. Multifrequency studies of pulsar signals have been used to map the
distribution and turbulence structure of ionized material in the Galaxy, as well as the
average Galactic magnetic field.
Pulsars are also, by virtue of their very regular, clock-like pulses, useful probes
of the gravitational environments in which they are found. The Doppler shifts of
the signals from pulsars in binaries can be used to study the binary properties, just
as spectral lines are used with normal stars. Some very close pulsar binaries have
orbits that are substantially deformed from Keplerian ellipses by general relativistic
effects; in these systems, very precise tests of “post-Keplerian” gravity theory have
been possible.
The first binary pulsar was discovered by Russell Hulse and Joseph Taylor in
1974, during a survey for new pulsars done at the Arecibo Observatory as part
of Hulse’s PhD thesis work. Follow-up observations showed that the pulsar is in
a high eccentricity (e = 0.6), short period (7.8 hours) orbit with another star,
which is almost certainly a second neutron star. It was immediately realized that
the high velocities and strong gravitational fields in this binary make this object
an extraordinary laboratory for studying fundamental physics. In 1993, Hulse and
Taylor were awarded the Nobel Prize in physics for the discovery of a new type of
pulsar, a discovery that has opened up new possibilities for the study of gravitation.
The most famous application of pulsar timing techniques has been to tests of
experimental gravitation. In most cases, binary orbits are well approximated as
Keplerian ellipses. The high velocities ( 0.001 c) and strong gravitational fields in
some binary pulsar systems cause relativistic deviations from Keplerian motion to be
significant. Five relativistic corrections have been measured: the advance of the angle
of periastron of the elliptical orbit (as is seen in the orbit of Mercury); the combined
effect of the transverse Doppler shift and the changing gravitational redshift as
the eccentric orbit carries the pulsars closer and further from its companion; two
parameters describing the Shapiro time delay of the pulsar signal as it propagates
through the gravitational potential well of the companion; and the decay of the binary
orbit due to gravitational radiation back reaction. The measurement of any two of
these effects allows the amplitude of the other three to be predicted, *** possible
very precise tests of general relativity and alternative gravity theories. This was the
subject of the 1993 Nobel Prize in physics, discussed above.
Observed pulsar radio luminosities, together with the small source size, imply
extraordinarily high brightness temperatures – as high as 1031 K. To avoid implausibly
high particle energies, coherent radiation processes are invoked. A maser-like
mechanism, involving particles bunched in momentum space, is attractive, if only
because maser action has been observed elsewhere in astrophysics, but models with
coherent emission from bunches of particles have also been widely discussed, with
a bunch of N particles localized in physical space radiating power proportional to
N2 . Coherence by bunching is seen in terrestrial lightning flashes. In detail, severe
problems remain in understanding pulsar emission by either the maser or bunching
models, and no consensus has emerged.
The emission mechanism itself also remains uncertain. Charged particles gyrating
around magnetic field lines produce synchrotron radiation. In the strong magnetic
fields of the pulsar magnetosphere, a particle will quickly radiate away its components
of momentum that are perpendicular to the field lines, so will be confined to
the lowest Landau level. Roughly, the charged particle can be pictured as a bead
on a wire, along which the bead is free to move. As a particle moves out along
a curved field line, it will produce synchrotron-like radiation that is conventionally
called “curvature radiation.” Coherent curvature radiation is currently the most
widely accepted model for pulsar radio emission, but many other possible models
have been discussed, including models based on relativistic plasma instabilities that
are variants of the mechanisms proposed for type III solar radio bursts.
general relativity were already in place; at issue was the exact form of the curvature
term in the Einstein equation. A precise relativistic model of the Sun’s gravitational
field was not needed – Einstein used a simple polynomial approximation. Late in
the year 1915, he succeeded, and the 43 second lag was explained.
A few weeks later, Einstein, working in Berlin, received a paper from Karl
Schwarzschild, an astronomer who, though no longer young, was serving in the
German army in Russia. Hospitalized by an illness that soon proved fatal, Schwarzschild
had time to discover the desired precise relativistic model, and Schwarzschild
spacetime replaced the Newtonian model as the best description of the gravitational
field of an isolated spherically symmetrical star. But only a few theorists were familiar
with relativity, and significant experimental tests were not possible in Earth-borne
laboratories at that time.
In 1963, the British-educated New Zealand physicist Roy Kerr, working at the
University of Texas, adopted a shrewd strategy: Bearing in mind that Schwarzschild
spacetime has Petrov type D, he did not aim directly at the elusive rotating model, but
instead examined an algebraically simple class of type D metric tensors. The longsought
metric appeared. Kerr’s minimal one-and-a-half page announcement of his
discovery [220] was followed two years later by elaborate detailed calculations [221].
Black holes with masses of a million to a few billion times the mass of the Sun
are now believed to be the engines that power nuclear activity in galaxies (Fig. 1.10).
Active nuclei range from faint, compact radio sources like that in M31 to quasars like
3C 273 that are brighter than the whole galaxy in which they live. Some nuclei fire
jets of energetic particles millions of lightyears into space. Almost all astronomers
believe that this enormous outpouring of energy comes from the death throes of
stars and gas that are falling into the central black hole. This is a very successful
explanation of the observations, but until recently, it was seriously incomplete: we
had no direct evidence that supermassive black holes exist.
The Hubble Space Telescope provides the best evidence to date of supermassive
black holes that lurk in the center of some galaxies. The Space Telescope Imaging
Spectrograph (STIS) revealed large orbiting velocities around the nucleus of these
galaxies, suggesting a huge mass inside a very small region.
Since the mass of black holes can only grow with time, at least some fraction
of nearby galaxies should host such supermassive black holes – like dead quasars
with insufficient fuel to trigger the activity in real quasars. For the past 20 years,
astronomers have looked for supermassive black holes by measuring rotation and
random velocities of stars and gas near galactic centers. If the velocities are large
enough, as in the Sombrero Galaxy, then they imply more mass than we see in stars.
The most probable explanation is a black hole. About 50 have been found as of the
year 2005 (Fig. 1.9). Their masses are in the range expected for nuclear engines,
are at great distances, beyond our own Milky Way Galaxy, VLA observations have
revealed the size of the fireball and the speed of its expansion. The May 8, 1997,
GRB, for example, was only a tenth of a lightyear across when first detected and
expanded at very nearly the speed of light.
The VLA’s ability to locate GRBs in the sky with pinpoint precision has helped
astronomers at other observatories to locate GRB afterglows that they otherwise
might have missed. With the image shown for the GRB of March 29, 1998, the
position determined by the VLA was provided to optical and infrared observers,
who had failed to find the object, but then, armed with the precise information on its
location, found it on images they had already made.
After three decades of mystery, astronomers now know that GRBs, the most
violent events in the current Universe, occur in galaxies far from Earth. In addition,
the VLA has provided strong evidence that these tremendous explosions occur in
dusty areas of those galaxies, where it is likely that young stars are located within the
clouds of dust and gas from which they formed. This evidence supports the theory
that GRBs result from a hypernova, the explosive death of a very massive star that
collapses and forms a black hole.
Radio telescopes are the only instruments presently capable of measuring the
size of a GRB fireball. In addition, while GRB afterglows fade quickly at other
wavelengths, the VLA has been able to follow an afterglow for more than a year,
tracking changes in its intensity and other characteristics. These observations indicate
the extraordinary importance of radio astronomy for providing information that can
be gained in no other way about one of the frontier areas of astrophysics.
Gamma-ray bursts (GRBs) are brief gamma-ray flashes detected with spacebased
detectors in the range 0.1–100 MeV, with typical photon fluxes of 0.01–
100 photons cm−2 s−1 and durations of 0.1–1000 seconds (Fig. 1.13). Their origin
is clearly outside the Solar System, and more than a thousand events have been
recorded so far. Before there was any firm evidence on the isotropy of classical
gamma-ray bursts, the most plausible interpretations involved magnetospheric events
on neutron stars (NS) within our Galaxy. However, the remarkable isotropy of these
events discovered by the BATSE experiment on the NASA Compton Gamma Ray
Observatory (together with the flatter than Newtonian counts) clearly shifts the odds
substantially in favor of a cosmological interpretation.
If gamma-ray bursts came from objects in our Galaxy, one would expect to
see more of them from the Galactic equator, where most other Galactic objects are
found. However, BATSE found that equal numbers of gamma-ray bursts come from
all directions (Fig. 1.15).
In principle, the isotropy could be interpreted in terms of either (i) a cosmological
distribution similar to that of the distant galaxies and clusters, i.e. hundreds and
thousands of Mpcs, (ii) a distribution in an “extended halo” of our galaxy, which
is so large that the small dipole moment associated with our off-center location is
not noticeable (i.e. greater than= 200 kpc), or (iii) a “galactic disk” distribution,
where objects are sufficiently faint that they are detectable only out to distances
smaller than the width of the disk (few kpc). The “galactic disk” model has difficulty
in explaining the large number of events (a few per day) occurring within a few
kpc, and the dipole and quadrupole moment of the spatial distribution appear to
rule out such an origin. The “extended halo” option may satisfy (just) the dipole
and quadrupole observational restrictions, but the physical origin of the bursts and
the number of sources at such large distances is not straightforward to explain.
On the other hand, the “cosmological” interpretation does have at least two rather
plausible energy sources: either NS-NS (or NS-black-hole) binary mergers (e.g.
binary pulsars merging under the effect of gravitational wave energy losses), or else
“failed supernova” events (where a star undergoes core collapse to a NS but with
much reduced optical display). Either of these should occur with a frequency of
10−5 per galaxy per year, and produces 1050 –1051 ergs, detectable out to redshifts
of order unity, so that the typical frequency and fluence is easily explained. More
importantly, the discovery in February 1997 of GRB afterglows and counterparts
gives strong support to the cosmological origin.
Fig. 1.16. Schematics of GRB models. The collapse towards a black hole produces large
amounts of electron–positron pairs which escape as relativistic jets in polar directions. Initially,
the plasma is optically thick, and only at distances of the order of astronomical units, can
gamma-rays escape. These jets drive a shock-front which is visible in the afterglows
One model for GRBs is a binary pair of neutron stars, with their orbital separation
ground down by billions of years of gravitational radiation, finally merging to form
a black hole. In the process of merging the neutron stars tear each other apart,
forming an accretion disk and jets (Fig. 1.16). A straightforward prediction of
cosmological models is that, if GRB are standard candles, one would expect the
weaker fluence bursts (which presumably are farther) to have longer durations due
to cosmological time dilation. Such an effect has been recently reported. However,
the duration of the burst can depend on intrinsic properties of the source, so that this
cosmological signature may be smeared out by details of the source physics. For
cosmological GRBs, a very interesting prediction is that they should be accompanied
by gravitational wave bursts of energy comparable to a solar rest mass. These would
be detectable at the rate of several per year with coincidence measurements from two
advanced versions of the proposed LIGO or VIRGO detectors. Such measurements
might also distinguish between failed supernova events or compact binary mergers,
through their wave profile. One may also obtain valuable information concerning
early star formation, through limits on the typical redshift derived from the counts
of events as a function of the fluence, and it may be possible to derive limits on the
GRB luminosity distribution.

2 Gravity of Compact Objects
The gravity of compact objects requires a description of gravitational fields much
beyond the Newtonian picture. In this chapter, we give a short overview for the
most important concepts and methods of general relativity. This does not replace
a thorough study of Einstein’s theory. This marvellous theory is explained in many
classical textbooks; see for example the books by Misner, Thorne and Wheeler [10],
Schutz [14], Carroll [2], or Straumann [18]. Since recent research on compact objects
goes much beyond a simple stationary description of gravitational fields, we also give
a short introduction to the concepts of the 3+1 split of Einstein’s equations, which is
now the basis of numerical treatments of Einstein’s field equations. Simulations for
the merging of two black holes or two neutron stars are based on these techniques.
The 3+1 technique is now a very powerful method, which can also be implemented
in deriving, e.g. the field equations for rapidly rotating compact objects, such as
neutron stars and black holes.
2.1 Geometric Concepts and General Relativity
In 1915 Albert Einstein published a geometrical theory of gravitation [141]: the
general theory of relativity. He presented a fundamentally new description of gravity
in the sense that the relative acceleration of particles is not viewed as a consequence
of gravitational forces, but results from the curvature of the spacetime in which the
particles are moving. As long as no nongravitational forces act on a particle, it is
always moving on a “straight line.” If we consider curved manifolds, there is still
a concept of straight lines which are called geodesics, but these will not necessarily
have the properties we intuitively associate with straight lines from our experience
in flat Euclidean geometry. It is, for example, a well known fact that two distinct
straight lines in two-dimensional flat geometry will intersect each other exactly once,
unless they are parallel, in which case they do not intersect each other at all. These
ideas result from the fifth Euclidean postulate of geometry, which plays a special
role in the formulation of geometry.
It is a well known fact that one needs to impose it separately from the first
four Euclidean postulates in order to obtain flat Euclidean geometry. It was not
realized until the work of Gauss, Lobachevsky, Bolyai and Riemann in the 19th
century that the omission of the fifth postulate leads to an entirely new class of non-
Euclidean geometries in curved manifolds. A fundamental feature of non-Euclidean
geometry is that straight lines in curved manifolds can intersect each other more
than once and correspondingly diverge from and converge towards each other several
times.
In order to illustrate how these properties give rise to effects we commonly
associate with forces such as gravitation, we consider two observers on the Earth’s
su***ce, say one in Heidelberg, Germany, and one in Vienna, Austria. We assume
that these two observers start moving due south in “straight lines” as for example
guided by an idealized compass exactly pointing towards the south pole. If we follow
their separate paths we will discover exactly the ideas outlined above. As long as
both observers are in the northern hemisphere the proper distance between them
will increase and reach a maximum when they reach the equator. From then on
they will gradually approach each other and their paths will inevitably cross at the
south pole.
In the framework of Newtonian physics, the observers will attribute the relative
acceleration of their positions to the action of a force. It is clear, however,
that no force is acting in the east–west direction on either observer at any stage
of their journey. In a geometric description, the relative movement of the observers
finds a qualitatively new interpretation in terms of the curvature of the
manifold they are moving in, the curvature of the Earth’s su***ce. With the development
of general relativity, Einstein provided the exact mathematical foundation
for applying these ideas to the forces of gravitation in four-dimensional spacetime.
One may ask why such a geometrical interpretation has only been developed
for gravitation. Or in other words: which feature distinguishes gravitation from the
other three fundamental interactions? The answer lies in the gravitational charge, the
mass. It is a common observation that the gravitational mass mG which determines
the coupling of a particle to the gravitational field is virtually identical to the inertial
mass mI which describes the particle’s kinematic reaction to an external force. High
precision experiments have been undertaken to measure the difference between
these two types of masses. All these results are compatible with the assumption that
the masses are indeed equal. The mass will therefore drop out of the Newtonian
equations governing the dynamics of a particle subject exclusively to gravitational
forces m a = GmM/r2 , where a is the acceleration of the particle, G the gravitational
constant, M the mass of an external source and r the distance from this source.
The particle mass m can be factored out so that the movement of the particle is
described in purely kinematic terms. The redundancy of the concept of a gravitational
force is naturally incorporated into a geometric theory of gravity such as general
relativity.
It is important to note that this behavior distinguishes gravity from the other
fundamental interactions which are associated with different types of charges, such
as electric charge in the case of electromagnetic interaction. It is not obvious how
and whether it is possible to obtain similar geometric formulations for the electromagnetic,
weak and strong interaction. The unification of these three fundamental
forces with gravity in the framework of quantum theory is one of the important areas
of ongoing research.
2.2 The Basic Principles of General Relativity
2.2.1 Einstein’s Equivalence Principle and Metricity
The principle of equivalence has historically played an important role in the development
of gravitation theory. Newton regarded this principle as such a cornerstone
of mechanics that he devoted the opening paragraph of the Principia to it. In 1907,
Einstein used the principle as a basic element of general relativity. We now regard
the principle of equivalence as the foundation, not of Newtonian gravity or of GR,
but of the broader idea that spacetime is curved. One elementary equivalence principle
is the kind Newton had in mind when he stated that the property of a body
called “mass” is proportional to the “weight,” and is known as the weak equivalence
principle (WEP). An alternative statement of WEP is that the trajectory of a freely
falling body (one not acted upon by such forces as electromagnetism and too small
to be affected by tidal gravitational forces) is independent of its internal structure and
composition. In the simplest case of dropping two different bodies in a gravitational
field, WEP states that the bodies fall with the same acceleration (this is often termed
the Universality of Free Fall).
A more powerful and far-reaching equivalence principle is known as the Einstein
equivalence principle (EEP). It states that [419]:
1. WEP is valid.
2. The outcome of any local nongravitational experiment is independent of
the velocity of the freely falling reference frame in which it is performed.
3. The outcome of any local nongravitational experiment is independent of
where and when in the Universe it is performed.
The second piece of EEP is called local Lorentz invariance (LLI), and the third piece
is called local position invariance (LPI).
For example, a measurement of the electric force between two charged bodies
is a local nongravitational experiment; a measurement of the gravitational force
between two bodies (Cavendish experiment) is not.
The Einstein equivalence principle is the heart and soul of gravitational theory,
for it is possible to argue convincingly that if EEP is valid, then gravitation must
be a curved spacetime phenomenon, in other words, the effects of gravity must be
equivalent to the effects of living in a curved spacetime. As a consequence of this
argument, the only theories of gravity that can embody EEP are those that satisfy
the postulates of metric theories of gravity, which are:
1. Spacetime is endowed with a symmetric metric.
2. The trajectories of freely falling bodies are geodesics of that metric.
3. In local freely falling reference frames, the nongravitational laws of physics
are those written in the language of special relativity.
The argument that leads to this conclusion simply notes that, if EEP is valid, then
in local freely falling frames, the laws governing experiments must be independent
of the velocity of the frame (local Lorentz invariance), with constant values for the
various atomic constants (in order to be independent of location). The only laws we
know of that fulfill this are those that are compatible with special relativity, such
as Maxwell’s equations of electromagnetism. Furthermore, in local freely falling
frames, test bodies appear to be unaccelerated, in other words they move on straight
lines; but such “locally straight” lines simply correspond to “geodesics” in a curved
spacetime.
General relativity is a metric theory of gravity, but then so are many others,
including the Brans–Dicke theory. Neither, in this narrow sense, is superstring theory,
which, while based fundamentally on a spacetime metric, introduces additional fields
(dilatons, moduli) that can couple to material stress–energy in a way that can lead to
violations, say, of WEP. Therefore, the notion of curved spacetime is a very general
and fundamental one, and therefore it is important to test the various aspects of the
Einstein Equivalence Principle thoroughly.
A direct test of WEP is the comparison of the acceleration of two laboratorysized
bodies of different composition in an external gravitational field. If the principle
were violated, then the accelerations of different bodies would differ. The simplest
way to quantify such possible violations of WEP in a form suitable for comparison
with experiment is to suppose that for a body with inertial mass mI, the passive
gravitational mass mP is no longer equal to mI, so that in a gravitational field g, the
acceleration is given by
mI a = mP g . (2.1)
Many high-precision E¨ otv¨ os-type experiments have been performed, from the
pendulum experiments of Newton, Bessel and Potter, to the classic torsion-balance
measurements of E¨ otv¨ os, Dicke, Braginsky and their collaborators. In the modern
torsion-balance experiments, two objects of different composition are connected by
a rod or placed on a tray and suspended in a horizontal orientation by a fine wire.
If the gravitational acceleration of the bodies differs, there will be a torque induced
on the suspension wire, related to the angle between the wire and the direction
of the gravitational acceleration g. If the entire apparatus is rotated about some
direction with angular velocity ω, the torque will be modulated with period 2π/ω.
In the experiments of E¨ otv¨ os and his collaborators, the wire and g were not quite
parallel because of the centripetal acceleration on the apparatus due to the Earth’s
rotation; the apparatus was rotated about the direction of the wire. In the Dicke and
Braginsky experiments, g was that of the Sun, and the rotation of the Earth provided
the modulation of the torque at a period of 24 hr. Beginning in the late 1980s,
numerous experiments were carried out primarily to search for a “fifth force,” but
their null results also constituted tests of WEP. In the “free-fall Galileo experiment”
performed at the University of Colorado, the relative free-fall acceleration of two
bodies made of uranium and copper was measured using a laser interferometric
technique. The “E¨ ot-Wash” experiments carried out at the University of Washington
used a sophisticated torsion balance tray to compare the accelerations of various
materials toward local topography on Earth, movable laboratory masses, the Sun
and the galaxy, and have recently reached levels of 4 × 10−13 .

Gravitational Redshift
The gravitational redshift is one of the most prominent consequences of EEP. Consider
two labs, a distance h apart moving with constant acceleration a. At time t0,
the first experiment emits a photon of wavelength λ0. The two experiments remain
a constant distance apart, so that the photon reaches the leading apparatus after a time
∆t = h/c in the reference frame of the experiments. In this time, the apparatus have
picked up an additional velocity ∆v = a ∆t = ah/c. Therefore, the photon reaching
the leading apparatus will be redshifted by the conventional Doppler effect
∆λ/λ=∆v/c=ah/c2. (2.4)
According to the EEP, the same situation should happen in a uniform gravitational
field. So we can imagine a tower of height z on the Earth su***ce with ag the strength
of the gravitational field. This situation is supposed to be indistinguishable from the
previous one in a lab on top of the tower, where the photon from the ground will be
detected. Therefore, a photon emitted from the ground with wavelength λ0 should
be redshifted by the amount
∆λ/λ=agh/c2. (2.5)
This is the famous gravitational redshift. This effect is a direct consequence of the
EEP, not of the details of general relativity! It has been verified for the first time by
Pound and Rebka in 1960. They used the M¨ ossbauer effect to measure the change in
frequency of gamma-rays as they travelled from the ground to the top of Jefferson
labs in Harvard.
This formula for the redshift can be stated in terms of the gravitational potential
U, where ag = ∇U . From this we obtain the redshift
∆λ/λ=1/c2 S ∂zU dz =∆U/c2. (2.6)
The principle of local position invariance, the third part of EEP, can be tested
by the gravitational redshift experiment, the first experimental test of gravitation
proposed by Einstein. Despite the fact that Einstein regarded this as a crucial test of
GR, we now realize that it does not distinguish between GR and any other metric
theory of gravity, but is only a test of EEP. A typical gravitational redshift experiment
measures the frequency or wavelength shift z = ∆ν/ν = −∆λ/λ between two
identical frequency standards (clocks) placed at rest at different heights in a static
gravitational field. If the frequency of a given type of atomic clock is the same
when measured in a local, momentarily comoving freely falling frame (Lorentz
frame), independent of the location or velocity of that frame, then the comparison of
frequencies of two clocks at rest at different locations boils down to a comparison
of the velocities of two local Lorentz frames, one at rest with respect to one clock at
the moment of emission of its signal, the other at rest with respect to the other clock
at the moment of reception of the signal. The frequency shift is then a consequence
of the first-order Doppler shift between the frames. The structure of the clock plays
no role whatsoever. The result is a shift
z = ∆U/c2 , (2.7)
where ∆U is the difference in the Newtonian gravitational potential between the
receiver and the emitter. If LPI is not valid, then it turns out that the shift can be
written
z = (1 + α) ∆U/c2 , (2.
where the parameter α may depend upon the nature of the clock whose shift is being

The Strong Equivalence Principle (SEP)
In any metric theory of gravity, matter and nongravitational fields respond only to
the spacetime metric g. In principle, however, there could exist other gravitational
fields besides the metric, such as scalar fields, vector fields, and so on. If, by our
strict definition of metric theory, matter does not couple to these fields, what can
their role in gravitation theory be? Their role must be that of mediating the manner
in which matter and nongravitational fields generate gravitational fields and produce
the metric; once determined, however, the metric alone acts back on the matter in
the manner prescribed by EEP.
What distinguishes one metric theory from another, therefore, is the number and
kind of gravitational fields it contains in addition to the metric, and the equations
that determine the structure and evolution of these fields. From this viewpoint,
one can divide all metric theories of gravity into two fundamental classes: “purely
dynamical” and “prior-geometric” [419].
By “purely dynamical metric theory” we mean any metric theory whose gravitational
fields have their structure and evolution determined by coupled partial
differential field equations. In other words, the behavior of each field is influenced
to some extent by a coupling to at least one of the other fields in the
theory. By “prior geometric” theory, we mean any metric theory that contains
“absolute elements,” fields or equations whose structure and evolution are given
a priori, and are independent of the structure and evolution of the other fields of
the theory. These “absolute elements” typically include flat background metrics η,
cosmic time coordinates t, and algebraic relationships among otherwise dynamical
fields.
General relativity is a purely dynamical theory, since it contains only one gravitational
field, the metric itself, and its structure and evolution are governed by partial
differential equations (Einstein’s equations). Brans–Dicke theory and its generalizations
are purely dynamical theories, too; the field equation for the metric involves
the scalar field (as well as the matter as source), and that for the scalar field involves
the metric.
By discussing metric theories of gravity from this broad point of view, it is possible
to draw some general conclusions about the nature of gravity in different metric
theories, conclusions that are reminiscent of the Einstein equivalence principle, but
that are subsumed under the name “strong equivalence principle.”
Consider a local, freely falling frame in any metric theory of gravity. Let this
frame be small enough that inhomogeneities in the external gravitational fields can
be neglected throughout its volume. On the other hand, let the frame be large enough
to encompass a system of gravitating matter and its associated gravitational fields.
The system could be a star, a black hole, the Solar System or a Cavendish experiment.
Call this frame a “quasilocal Lorentz frame.” To determine the behavior of
the system, we must calculate the metric. The computation proceeds in two stages.
First we determine the external behavior of the metric and gravitational fields,
thereby establishing boundary values for the fields generated by the local system,
at a boundary of the quasilocal frame “far” from the local system. Second, we
solve for the fields generated by the local system. But because the metric is coupled
directly or indirectly to the other fields of the theory, its structure and evolution
will be influenced by those fields, and in particular by the boundary values taken
on by those fields far from the local system. This will be true, even if we work
in a coordinate system in which the asymptotic form of g in the boundary region
between the local system and the external world is that of the Minkowski metric.
Thus the gravitational environment, in which the local gravitating system resides,
can influence the metric generated by the local system via the boundary values of
the auxiliary fields. Consequently, the results of local gravitational experiments may
depend on the location and velocity of the frame relative to the external environment.
Of course, local nongravitational experiments are unaffected, since the gravitational
fields they generate are assumed to be negligible, and since those experiments couple
only to the metric, whose form can always be made locally Minkowskian at
a given spacetime event. Local gravitational experiments might include Cavendish
experiments, measurement of the acceleration of massive self-gravitating bodies,
studies of the structure of stars and planets, or analyses of the periods of “gravitational
clocks.” We can now make several statements about different kinds of metric
theories.
– A theory which contains only the metric g yields local gravitational physics which
is independent of the location and velocity of the local system. This follows from
the fact that the only field coupling the local system to the environment is g, and
it is always possible to find a coordinate system in which g takes the Minkowski
form at the boundary between the local system and the external environment.
Thus the asymptotic values of g are constants independent of location, and are
asymptotically Lorentz invariant, thus independent of velocity. General relativity
is an example of such a theory.
– A theory, which contains the metric g and dynamical scalar fields, yields local
gravitational physics, which may depend on the location of the frame but which
is independent of the velocity of the frame. This follows from the asymptotic
Lorentz invariance of the Minkowski metric and of the scalar fields, but now the
asymptotic values of the scalar fields may depend on the location of the frame.
An example is Brans–Dicke theory, where the asymptotic scalar field determines
the effective value of the gravitational constant, which can thus vary as the scalar
field varies. On the other hand, a form of velocity dependence in local physics
can enter indirectly if the asymptotic values of the scalar field vary with time
cosmologically. Then the rate of variation of the gravitational constant could
depend on the velocity of the frame.
– A theory which contains the metric g and additional dynamical vector or tensor
fields or prior-geometric fields yields local gravitational physics which may have
both location and velocity-dependent effects.
These ideas can be summarized in the strong equivalence principle (SEP),
which states that
1. WEP is valid for self-gravitating bodies as well as for test bodies.
2. The outcome of any local test experiment is independent of the velocity of the
(freely falling) apparatus.
3. The outcome of any local test experiment is independent of where and when in
the Universe it is performed.
The distinction between SEP and EEP is the inclusion of bodies with selfgravitational
interactions (planets, stars) and of experiments involving gravitational
forces (Cavendish experiments, gravimeter measurements). Note that SEP
contains EEP as the special case in which local gravitational forces are ignored.

2.3 Basic Calculus on Manifolds
The above discussion suggests to replace the flat Minkowskian spacetime by means
of a curved manifold. In this section, I give a short outline about structures of
manifolds. A manifold is one of the most fundamental concepts in mathematical
physics. The notion of a manifold captures the idea of a curved space, which is
however locally just flat. In fact, the entire manifold is glued together by local
patches.
A manifold is a topological space which is locally Euclidean (i.e. around every
point, there is a neighborhood which is topologically the same as the open unit
ball in Rn ). To illustrate this idea, consider the ancient belief that the Earth was
flat as contrasted with the modern evidence that it is round. This discrepancy arises
essentially from the fact that on the small scales that we see, the Earth does indeed
look flat (although the Greeks did notice that the last part of a ship to disappear
over the horizon was the mast). In general, any object which is nearly “flat” on small
scales is a manifold, and so manifolds constitute a generalization of objects we could
live on in which we would encounter the round/flat Earth problem, as first codified
by Poincar´ e. More formally, any object that can be charted is a manifold. For the
mathematical details, see any textbook on manifolds.

宇宙学 PS 术：修正扭曲的宇宙“婴儿照”| Phys. Rev. Lett. 论文推荐

原创 2016-10-16 科研圈 科研圈
撰文 Michael Schirber
编译 金庄维

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研究人员提出并验证了消除引力透镜效应的方法，恢复宇宙微波背景图原貌。
 
CMB（宇宙微波背景）图是可观测宇宙的“婴儿照”。通过研究照片中的图案，我们可以了解宇宙的历史，但是这项任务十分艰巨：因为星系和其他天体的引力透镜效应使这张照片扭曲变形。所谓的“引力透镜效应”是指时空在大质量天体附近发生扭曲，光线经过时会像通过透镜一样发生弯曲。最新的研究展示了一种“去透镜”方法：利用红外波段的背景光消除扭曲，还原照片。研究人员将这种方法应用于观测数据，首次证明了“去透镜”的可行性。
 
过去几十年中，宇宙学家利用 CMB 图确定宇宙的几何以及密度分布。进一步的研究，特别是 CMB 的偏振图样，能够提供源自暴胀的原初引力波的信息。然而，引力透镜效应使偏振信号失真，因此“去透镜”对在 CMB 上寻找引力波信号至关重要。

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以往的“去透镜”方案建议利用 CMB 自身的特征确定透镜效应发生的位置，卡弗里宇宙学研究所的 Patricia Larsen 和合作者提出并检验了一种基于 CIB（宇宙红外背景）的新方法。宇宙红外背景辐射主要来源于布满尘埃的恒星形成星系中的漫射光，CIB 上的亮点对应星系高度集中的区域，这些区域的引力透镜效应会更强。利用普朗克卫星提供的 CIB 图，研究团队创建了“去透镜”模板，并将它应用于最精确的 CMB 全天图（同样由普朗克卫星提供）。修正后的 CMB 数据显示温度涨落谱的峰更加尖锐，与理论模型的预言一致。
 
这项研究发表于《物理评论快报》。
 
原文链接：http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.117.151102[/size]