**^**This development is traced in chapters 9 through 15 of Pais 1982 and in Janssen 2005; an accessible overview can be found in Renn 2005, p. 110ff.. An early key article is Einstein 1907, cf. Pais 1982, ch. 9. The publication featuring the field equations is Einstein 1915, cf. Pais 1982, ch. 11–15.**^**See Schwarzschild 1916a, Schwarzschild 1916b and Reissner 1916 (later complemented in Nordström 1918).**^**Einstein 1917, cf. Pais 1982, ch. 15e.**^**Hubble's original article is Hubble 1929; an accessible overview is given in Singh 2004, ch. 2-4.**^**Cf. Pais 1982, p. 253-254.**^**Cf. Kennefick 2005 and Kennefick 2007.**^**Cf. Pais 1982, ch. 16.**^**Cf. Israel 1987, ch. 7.8-7.10 and Thorne 1994, ch. 3-9.**^**Cf. the sections Orbital effects and the relativity of direction, Gravitational time dilation and frequency shift and Light deflection and gravitational time delay, and references therein.**^**Cf. the section Cosmology and references therein; the historical development is traced in Overbye 1999.**^**The following exposition re-traces that of Ehlers 1973, section 1.**^**See, for instance, Arnold 1989, chapter 1.**^**See Ehlers 1973, pp. 5f..**^**See Will 1993, section 2.4 or Will 2006, section 2.**^**Cf. Wheeler 1990, chapter 2; similar accounts can be found in most other popular-science books on general relativity.**^**See Ehlers 1973, section 1.2, Havas 1964, and Künzle 1972. The simple thought experiment in question was first described in Heckmann & Schücking 1959.**^**See Ehlers 1973, pp. 10f..**^**Good introductions are, in order of increasing presupposed knowledge of mathematics, Giulini 2005, Mermin 2005, and Rindler 1991; for accounts of precision experiments, cf. part IV of Ehlers & Lämmerzahl 2006.**^**An in-depth comparison between the two symmetry groups can be found in Giulini 2006a.**^**For instance Rindler 1991, section 22; a thorough treatment can be found in Synge 1972, ch. 1 and 2.**^**E.g. Ehlers 1973, sec. 2.3.**^**Cf. Ehlers 1973, sec. 1.4. and Schutz 1985, sec. 5.1.**^**See Ehlers 1973, p. 17ff.; a derivation can be found e.g. in Mermin 2005, ch. 12. For the experimental evidence, cf. the section Gravitational time dilation and frequency shift, below.**^**Cf. Rindler 2001, sec. 1.13; for an elementary account, see chapter 2 of Wheeler 1990; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. Norton 1985.**^**Ehlers 1973, sec. 1.4. for the experimental evidence, see once more section Gravitational time dilation and frequency shift. Choosing a different connection with non-zero torsion leads to a modified theory known as Einstein-Cartan theory.**^**Cf. Ehlers 1973, p. 16; Kenyon 1990, sec. 7.2; Weinberg 1972, sec. 2.8.**^**See Ehlers 1973, pp. 19–22; for similar derivations, see sections 1 and 2 of ch. 7 in Weinberg 1972. The Einstein tensor is the only divergence-free tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the spacetime of special relativity as a solution in the absence of sources of gravity, cf. Lovelock 1972. Both*G*and_{ab}*T*are rank-2 symmetric tensors, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations. The fact that, as a consequence of geometric relations known as Bianchi identities, the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. Schutz 1985, sec. 8.3._{ab}**^**E.g. Kenyon 1990, sec. 7.4.**^**Cf. Brans & Dicke 1961 and section 3 in ch. 7 of Weinberg 1972, Goenner 2004, sec. 7.2, and Trautman 2006, respectively.**^**E.g. Wald 1984, ch. 4, Weinberg 1972, ch. 7 or, in fact, any other text-book on general relativity.**^**At least approximately, cf. Poisson 2004.**^**E.g. p. xi in Wheeler 1990.**^**E.g. Wald 1984, sec. 4.4.**^**E.g. in Wald 1984, sec. 4.1.**^**For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see Giulini 2006b.**^**E.g. section 5 in ch. 12 of Weinberg 1972.**^**Cf. the introductory chapters of Stephani et al. 2003.**^**A review showing Einstein's equation in the broader context of other PDEs with physical significance is Geroch 1996.**^**For background information and a list of solutions, cf. Stephani et al. 2003; a more recent review can be found in MacCallum 2006.**^**E.g. chapters 3, 5, and 6 of Chandrasekhar 1983.**^**E.g. ch. 4 and sec. 3.3. in Narlikar 1993.**^**Brief descriptions of these and further interesting solutions can be found in Hawking & Ellis 1973, ch. 5.**^**See Lehner 2002 for an overview.**^**For instance Wald 1984, sec. 4.4.**^**E.g. Will 1993, sec. 4.1 and 4.2.**^**Cf. section 3.2 of Will 2006 as well as Will 1993, ch. 4.**^**Cf. Rindler 2001, pp. 24–26 vs. pp. 236–237 and Ohanian & Ruffini 1994, pp. 164–172. In fact, Einstein derived these effects using the equivalence principle as early as 1907, cf. Einstein 1907 and the description in Pais 1982, pp. 196–198.**^**Rindler 2001, pp. 24–26; Misner, Thorne & Wheeler 1973, § 38.5.**^**Pound-Rebka experiment, see Pound & Rebka 1959, Pound & Rebka 1960; Pound & Snider 1964; a list of further experiments is given in Ohanian & Ruffini 1994, table 4.1 on p. 186.**^**E.g. Greenstein, Oke & Shipman 1971; the most recent and most accurate Sirius B measurements are published in Barstow, Bond & Holberg 2005.**^**Starting with the Hafele-Keating experiment, Hafele & Keating 1972a and Hafele & Keating 1972b, and culminating in the Gravity Probe A experiment; an overview of experiments can be found in Ohanian & Ruffini 1994, table 4.1 on p. 186.**^**GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see Ashby 2002 and Ashby 2003.**^**Reviews are given in Stairs 2003 and Kramer 2004.**^**General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; Ohanian & Ruffini 1994, section 4.2.**^**Cf. Ohanian & Ruffini 1994, pp. 164–172.**^**This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell Lagrangian using a WKB approximation, cf. Ehlers 1973, section 5.**^**A brief descriptions and pointers to the literature can be found in Blanchet 2006, section 1.3.**^**See Rindler 2001, section 1.16; for the historical examples, Israel 1987, p. 202–204.; in fact, Einstein published one such derivation as Einstein 1907. Such calculations tacitly assume that the geometry of space is Euclidean, cf. Ehlers & Rindler 1997.**^**From the standpoint of Einstein's theory, these derivations take into account the effect of gravity on time, but not its consequences for the warping of space, cf. Rindler 2001, sec. 11.11.**^**Cf. Kennefick 2005 for the classic early measurements by the Eddington expeditions; for an overview of more recent measurements, see Ohanian & Ruffini 1994, chapter 4.3. For the most precise direct modern observations using quasars, cf. Shapiro et al. 2004.**^**For the Sun's gravitational field using radar signals reflected from planets such as Venus and Mercury, cf. Shapiro 1964, with a pedagogical introduction to be found in Weinberg 1972, ch. 8, sec. 7; for signals actively sent back by space probes (transponder measurements), cf. Bertotti, Iess & Tortora 2003; for an overview, see Ohanian & Ruffini 1994, table 4.4 on p. 200; for more recent measurements using signals received from a pulsar that is part of a binary system, the gravitational field causing the time delay being that of the other pulsar, cf. Stairs 2003, section 4.4.**^**Will 1993, sec. 7.1 and 7.2.**^**For an overview, see Misner, Thorne & Wheeler 1973, part VIII. Note, however that for gravitational waves, the dominant contribution is not the dipole, but the quadrupole cf. Schutz 2001.**^**Any textbook on general relativity will contain a description of these properties, e.g. Schutz 1985, ch. 9.**^**For example Jaranowski & Królak 2005.**^**Rindler 2001, ch. 13.**^**See Gowdy 1971, Gowdy 1974.**^**See Lehner 2002 for a brief introduction to the methods of numerical relativity, and Seidel 1998 for the connection with gravitational wave astronomy.**^**See Schutz 2003, pp. 48–49 and Pais 1982, pp. 253–254.**^**See Rindler 2001, sec. 11.9.**^**See Will 1993, pp. 177–181.**^**In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. Will 2006, sec. 3.5 and Will 1993, sec. 7.3.**^**The most precise measurements are VLBI measurements of planetary positions; see Will 1993, chapter 5, Will 2006, section 3.5, Anderson et al. 1992; for an overview, Ohanian & Ruffini 1994, pp. 406–407.**^**See Kramer et al. 2006.**^**A figure that includes error bars is figure 7, in section 5.1, of Will 2006.**^**See Stairs 2003 and Schutz 2003, pp. 317–321; an accessible account can be found in Bartusiak 2000, pp. 70–86.**^**An overview can be found in Weisberg & Taylor 2003; for the pulsar discovery, see Hulse & Taylor 1975; for the initial evidence for gravitational radiation, see Taylor 1994.**^**Cf. Kramer 2004.**^**See e.g. Penrose 2004, §14.5, Misner, Thorne & Wheeler 1973, sec. §11.4.**^**See Weinberg 1972, sec. 9.6, Ohanian & Ruffini 1994, sec. 7.8.**^**See Bertotti, Ciufolini & Bender 1987 and, for a more recent review, Nordtvedt 2003.**^**See Kahn 2007.**^**E.g. Townsend 1997, sec. 4.2.1, Ohanian & Ruffini 1994, pp. 469–471.**^**E.g. Ohanian & Ruffini 1994, sec. 4.7, Weinberg 1972, sec. 9.7; for a more recent review, see Schäfer 2004.**^**E.g. Ciufolini & Pavlis 2004, Ciufolini, Pavlis & Peron 2006; see the entry frame-dragging for an account of the debate.**^**A mission description can be found in Everitt et al. 2001; a first post-flight evaluation is given in Everitt et al. 2007; further updates will be available on the mission website Kahn 1996–2008.**^**For overviews of gravitational lensing and its applications, see Ehlers, Falco & Schneider 1992 and Wambsganss 1998.**^**For a simple derivation, see Schutz 2003, ch. 23; cf. Narayan & Bartelmann 1997, sec. 3.**^**See Walsh, Carswell & Weymann 1979.**^**Images of all the known lenses can be found on the pages of the CASTLES project, Kochanek et al. 2007.**^**For an overview, see Roulet & Mollerach 1997.**^**See Narayan & Bartelmann 1997, sec. 3.7.**^**For an overview, Barish 2005; accessible accounts can be found in Bartusiak 2000 and Blair & McNamara 1997.**^**An overview is given in Hough & Rowan 2000.**^**See Danzmann & Rüdiger 2003.**^**See Landgraf, Hechler & Kemble 2005.**^**Cf. Thorne 1995.**^**See Cutler & Thorne 2002, sec. 2.**^**See Cutler & Thorne 2002, sec. 3.**^**See Miller 2002, lectures 19 and 21.**^**E.g. Celotti, Miller & Sciama 1999, sec. 3.**^**Cf. Springel et al. 2005 and the accompanying summary Gnedin 2005.**^**Cf. Blandford 1987, section 8.2.4,**^**For the basic mechanism, see Carroll & Ostlie 1996, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. Robson 1996.**^**For a review, see Begelman, Blandford & Rees 1984.**^**See Rees 1966.**^**For stellar end states, cf. Oppenheimer & Snyder 1939 or, for more recent numerical work, Font 2003, sec. 4.1; for supernovae, there are still major problems to be solved, cf. Buras et al. 2003; for simulating accretion and the formation of jets, cf. Font 2003, sec. 4.2.**^**Cf. Kraus 1998.**^**See Celotti, Miller & Sciama 1999.**^**Cf. Schödel et al. 2003.**^**Examination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf. Remillard et al. 2006 and, for an overview, Narayan 2006, sec. 5.**^**Cf. Falcke, Melia & Agol 2000.**^**Cf. Seidel 1998.**^**Cf. Dalal et al. 2006.**^**E.g. Barack & Cutler 2004.**^**Originally Einstein 1917; cf. the description in Pais 1982, pp. 285–288.**^**See Carroll 2001, ch. 2.**^**See Bergström & Goobar 2003, ch. 9–11; use of these models is justified by the fact that, at large scales of around hundred million light-years and more, our own universe indeed appears to be isotropic and homogeneous, cf. Peebles et al. 1991.**^**E.g. with WMAP data, see Spergel et al. 2003.**^**See Peebles 1966; for a recent account of predictions, see Coc et al. 2004; an accessible account can be found in Weiss 2006.**^**See Olive & Skillman 2004, Bania, Rood & Balser 2002, O'Meara et al. 2001, and Charbonnel & Primas 2005.**^**A review can be found in Lahav & Suto 2004.**^**Cf. Alpher & Herman 1948 and, for a pedagogical introduction, see Bergström & Goobar 2003, ch. 11; for the initial detection, see Penzias & Wilson 1965 and, for precision measurements by satellite observatories, Mather et al. 1994 (COBE) and Bennett et al. 2003 (WMAP).**^**This additional information is contained in the background radiation's polarization, cf. Kamionkowski, Kosowsky & Stebbins 1997 and Seljak & Zaldarriaga 1997.**^**See, e.g., fig. 2 in Bridle et al. 2003.**^**For a review, see Bertschinger 1998; more recent results can be found in Springel et al. 2005.**^**These additional observations involve the dynamics of galaxies and galaxy clusters cf. chapter 18 of Peebles 1993, evidence from gravitational lensing, cf. Peacock 1999, sec. 4.6, and simulations of large-scale structure formation, see Springel et al. 2005.**^**See Peacock 1999, ch. 12, and Peskin 2007; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual elementary particles ("non-baryonic matter"), cf. Peacock 1999, ch. 12.**^**Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in Mannheim 2006, sec. 9.**^**See Carroll 2001; an accessible overview is given in Caldwell 2004.**^**Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. Mannheim 2006, sec. 10; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. Buchert 2007.**^**More precisely, these are the flatness problem, the horizon problem, and the monopole problem; a pedagogical introduction can be found in Narlikar 1993, sec. 6.4, see also Börner 1993, sec. 9.1.**^**A good introduction is Linde 1990; for a more recent review, see Linde 2005.**^**See Spergel et al. 2007, sec. 5 & 6.**^**More concretely, the potential function that is crucial to determining the dynamics of the inflaton is simply postulated, but not derived from an underlying physical theory.**^**See Brandenberger 2007, sec. 2.**^**See Frauendiener 2004, Wald 1984, section 11.1, and Hawking & Ellis 1973, section 6.8 & 6.9**^**E.g. Wald 1984, sec. 9.2–9.4 and Hawking & Ellis 1973, ch. 6.**^**See Thorne 1972; for an account of more recent numerical studies, see Berger 2002, sec. 2.1.**^**For an account of the evolution of this concept, see Israel 1987. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and apparent horizons cf. Hawking & Ellis 1973, pp. 312–320 or Wald 1984, sec. 12.2; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. Ashtekar & Krishnan 2004.**^**For first steps, cf. Israel 1971; see Hawking & Ellis 1973, sec. 9.3 or Heusler 1996, ch. 9 and 10 for a derivation, and Heusler 1998 as well as Beig & Chruściel 2006 as overviews of more recent results.**^**The laws of black hole mechanics were first described in Bardeen, Carter & Hawking 1973; a more pedagogical presentation can be found in Carter 1979; for a more recent review, see chapter 2 of Wald 2001. A thorough, book-length introduction including an introduction to the necessary mathematics Poisson 2004. For the Penrose process, see Penrose 1969.**^**See Bekenstein 1973, Bekenstein 1974.**^**The fact that black holes radiate, quantum mechanically, was first derived in Hawking 1975; a more thorough derivation can be found in Wald 1975. A review is given in chapter 3 of Wald 2001.**^**Cf. Narlikar 1993, sec. 4.4.4 and 4.4.5.**^**Cf. Rindler 2001, sec. 12.4**^**Unruh 1976, cf. Wald 2001, chapter 3.**^**See Hawking & Ellis 1973, section 8.1, Wald 1984, section 9.1.**^**See Townsend 1997, chapter 2; a more extensive treatment of this solution can be found in Chandrasekhar 1983, chapter 3.**^**See Townsend 1997, chapter 4; for a more extensive treatment, cf. Chandrasekhar 1983, chapter 6.**^**See Ellis & van Elst 1999; a closer look at the singularity itself is taken in Börner 1993, sec. 1.2**^**Namely when there are trapped null surfaces, cf. Penrose 1965.**^**See Hawking 1966.**^**The conjecture was made in Belinskii, Khalatnikov & Lifschitz 1971; for a more recent review, see Berger 2002. An accessible exposition is given by Garfinkle 2007.**^**The restriction to future singularities naturally excludes initial singularities such as the big bang singularity, which in principle be visible to observers at later cosmic time. The cosmic censorship conjecture was first presented in Penrose 1969; a text-book level account is given in Wald 1984, pp. 302-305. For numerical results, see the review Berger 2002, sec. 2.1.**^**Cf. Hawking & Ellis 1973, sec. 7.1.**^**Arnowitt, Deser & Misner 1962; for a pedagogical introduction, see Misner, Thorne & Wheeler 1973, §21.4–§21.7.**^**Fourès-Bruhat 1952 and Bruhat 1962; for a pedagogical introduction, see Wald 1984, ch. 10; an online review can be found in Reula 1998.**^**See Gourgoulhon 2007; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see Lehner 2001.**^**Cf. Misner, Thorne & Wheeler 1973, §20.4.**^**Arnowitt, Deser & Misner 1962.**^**Cf. Komar 1959; for a pedagogical introduction, see Wald 1984, sec. 11.2; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. Ashtekar & Magnon-Ashtekar 1979.**^**For a pedagogical introduction, see Wald 1984, sec. 11.2.**^**See the various references given on p. 295 of Wald 1984; this is important for questions of stability—if there were negative mass states, then flat, empty Minkowski space, which has mass zero, could evolve into these states.**^**E.g. Townsend 1997, ch. 5.**^**Such quasi-local mass-energy definitions are the Hawking energy, Geroch energy, or Penrose's quasi-local energy-momentum based on twistor methods; cf. the review article Szabados 2004.**^**An overview of quantum theory can be found in standard textbooks such as Messiah 1999; a more elementary account is given in Hey & Walters 2003.**^**Cf. textbooks such as Ramond 1990, Weinberg 1995, or Peskin & Schroeder 1995; a more accessible overview can be found in Auyang 1995.**^**Cf. Wald 1994 and Birrell & Davies 1984.**^**For Hawking radiation Hawking 1975, Wald 1975; an accessible introduction to black hole evaporation can be found in Traschen 2000.**^**Cf. chapter 3 in Wald 2001.**^**Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf. section 2 in Carlip 2001.**^**E.g. p. 407ff. in Schutz 2003.**^**A timeline and overview can be found in Rovelli 2000.**^**See Donoghue 1995.**^**In particular, a technique known as renormalization, an integral part of deriving predictions which take into account higher-energy contributions, cf. chapters 17 and 18 of Weinberg 1996, fails in this case; cf. Goroff & Sagnotti 1985.**^**An accessible introduction at the undergraduate level can be found in Zwiebach 2004; more complete overviews can be found in Polchinski 1998a and Polchinski 1998b.**^**At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different modes of oscillation of one and the same type of fundamental string appear as particles with different (electric and other) charges, e.g. Ibanez 2000. The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity, e.g. Green, Schwarz & Witten 1987, sec. 2.3 and 5.3.**^**E. g. Green, Schwarz & Witten 1987, sec. 4.2.**^**E.g. Weinberg 2000, ch. 31.**^**E.g. Townsend 1996, Duff 1996.**^**Cf. section 3 in Kuchař 1973.**^**These variables represent geometric gravity using mathematical analogues of electric and magnetic fields; cf. Ashtekar 1986, Ashtekar 1987.**^**For a review, see Thiemann 2006; more extensive accounts can be found in Rovelli 1998, Ashtekar & Lewandowski 2004 as well as in the lecture notes Thiemann 2003.**^**See e.g. the systematic expositions in Isham 1994 and Sorkin 1997.**^**See Loll 1998.**^**See Sorkin 2005.**^**See ch. 33 in Penrose 2004 and references therein.**^**Cf. Hawking 1987.**^**E.g. Ashtekar 2007, Schwarz 2007.**^**Cf. Maddox 1998, pp. 52–59 and 98–122; Penrose 2004, section 34.1 and chapter 30.**^**Cf. the section Quantum gravity, above.**^**Cf. the section Cosmology, above.**^**See Nieto 2006.**^**See Friedrich 2005.**^**A review of the various problems and the techniques being developed to overcome them, see Lehner 2002.**^**See Bartusiak 2000 for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as GEO 600 and LIGO.**^**For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see Blanchet et al. 2008, and Arun et al. 2007; for a review of work on compact binaries, see Blanchet 2006 and Futamase & Itoh 2006; for a general review of experimental tests of general relativity, see Will 2006.**^**A good starting point for a snapshot of present-day research in relativity is the electronic review journal Living Reviews in Relativity.