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Books on the topic 'Gravitational field equation'

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Published: 5 June 2025
Last updated: 24 June 2025

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1

Jefimenko, Oleg D. Causality, electromagnetic induction, and gravitation: A different approach to the theory of electromagnetic and gravitational fields. 2nd ed. Electret Scientific Co., 2000.

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2

Jefimenko, Oleg D. Causality, electromagnetic induction, and gravitation: A different approach to the theory of electromagnetic and gravitational fields. Electret Scientific Co., 1992.

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3

T, Chruściel Piotr, and Królak Andrzej, eds. Mathematics of gravitation. Polish Academy of Sciences, Institute of Mathematics, 1997.

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4

Chruściel, Piotr T., and Helmut Friedrich, eds. The Einstein Equations and the Large Scale Behavior of Gravitational Fields. Birkhäuser Basel, 2004. http://dx.doi.org/10.1007/978-3-0348-7953-8.

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5

Kh, Huleihil, and Leibowitz E, eds. Gauge fields. World Scientific, 1989.

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6

United States. National Aeronautics and Space Administration., ed. General equations for the motions of ice crystals and water drops in gravitational and electric fields. Institut d'aeronomie spatiale de Belgique, 1988.

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7

P, Hsu J., and Fine Dana, eds. 100 years of gravity and accelerated frames: The deepest insights of Einstein and Yang-Mills. World Scientific, 2005.

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8

Chruściel, Piotr T. The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity. Birkhäuser Basel, 2004.

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9

Griffiths, J. B. Colliding plane waves in general relativity. Clarendon Press, 1991.

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10

Fonseca, Carlos M. da. A panorama of mathematics: Pure and applied : Conference on Mathematics and Its Applications, November 14-17, 2014, Kuwait University, Safat, Kuwait. American Mathematical Society, 2016.

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11

Mashhoon, Bahram. Field Equation of Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0006.

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In extended general relativity (GR), Einstein’s field equation of GR can be expressed in terms of torsion and this leads to the teleparallel equivalent of GR, namely, GR||, which turns out to be the gauge theory of the Abelian group of spacetime translations. The structure of this theory resembles Maxwell’s electrodynamics. We use this analogy and the world function to develop a nonlocal GR|| via the introduction of a causal scalar constitutive kernel. It is possible to express the nonlocal gravitational field equation as modified Einstein’s equation. In this nonlocal gravity (NLG) theory, the gravitational field is local, but satisfies a partial integro-differential field equation. The field equation of NLG can be expressed as Einstein’s field equation with an extra source that has the interpretation of the effective dark matter. It is possible that the kernel of NLG, which is largely undetermined, could be derived from a more general future theory.
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12

Prussing, John E. Rocket Trajectories. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198811084.003.0003.

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rocket trajectories, including equations of motion are treated. High- and low-thrust engines are analysed, including constant-specific-impulse and variable-specific-impulse engines. The equation of motion of a spacecraft which is thrusting in a gravitational field determines its trajectory.
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13

Deruelle, Nathalie, and Jean-Philippe Uzan. The law of gravitation. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0011.

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This chapter embarks on the study of Newton’s law of gravitation. It first discusses gravitational mass and inertial mass, a measure of the ‘resistance’ of the point particle to an applied force. The numerical value of the inertial mass of a body can in principle be obtained from collision experiments by assigning to a reference body a unit inertial mass of one kilogram or, more rigorously, one ‘inertial kilogram’. Next, the chapter considers the ratio of gravitational and inertial masses. It considers that, in the absence of friction, all objects, no matter what their inertial mass, or the nature of their constituents, or the internal energy or cohesive forces of their constituents, fall in the same way in an external gravitational field. Finally, this chapter studies Newton’s gravitational force and field, as well as the Poisson equation and the gravitational Lagrangian.
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14

Deruelle, Nathalie, and Jean-Philippe Uzan. Self-gravitating fluids. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0015.

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This chapter briefly describes ‘perfect fluids’. These are characterized by their mass density ρ‎(t, xⁱ), pressure p(t, ⁱ), and velocity field v(t, ⁱ). The motion and equilibrium configurations of these fluids are determined by the equation of state, for example, p = p(ρ‎) for a barotropic fluid, and by the gravitational potential U(t, ⁱ) created at a point ⁱ by other fluid elements. The chapter shows that, given an equation of state, the equations of the problem to be solved are the continuity equation, the Euler equation, and the Poisson equation. It then considers static models with spherical symmetry, as well as polytropes and the Lane–Emden equation. Finally, the chapter studies the isothermal sphere and Maclaurin spheroids.
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15

Mashhoon, Bahram. Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.001.0001.

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A postulate of locality permeates through the special and general theories of relativity. First, Lorentz invariance is extended in a pointwise manner to actual, namely, accelerated observers in Minkowski spacetime. This hypothesis of locality is then employed crucially in Einstein’s local principle of equivalence to render observers pointwise inertial in a gravitational field. Field measurements are intrinsically nonlocal, however. To go beyond the locality postulate in Minkowski spacetime, the past history of the accelerated observer must be taken into account in accordance with the Bohr-Rosenfeld principle. The observer in general carries the memory of its past acceleration. The deep connection between inertia and gravitation suggests that gravity could be nonlocal as well and in nonlocal gravity the fading gravitational memory of past events must then be taken into account. Along this line of thought, a classical nonlocal generalization of Einstein’s theory of gravitation has recently been developed. In this nonlocal gravity (NLG) theory, the gravitational field is local, but satisfies a partial integro-differential field equation. A significant observational consequence of this theory is that the nonlocal aspect of gravity appears to simulate dark matter. The implications of NLG are explored in this book for gravitational lensing, gravitational radiation, the gravitational physics of the Solar System and the internal dynamics of nearby galaxies as well as clusters of galaxies. This approach is extended to nonlocal Newtonian cosmology, where the attraction of gravity fades with the expansion of the universe. Thus far only some of the consequences of NLG have been compared with observation.
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16

Steane, Andrew M. Relativity Made Relatively Easy Volume 2. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780192895646.001.0001.

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This is a textbook on general relativity and cosmology for a physics undergraduate or an entry-level graduate course. General relativity is the main subject; cosmology is also discussed in considerable detail (enough for a complete introductory course). Part 1 introduces concepts and deals with weak-field applications such as gravitation around ordinary stars, gravimagnetic effects and low-amplitude gravitational waves. The theory is derived in detail and the physical meaning explained. Sources, energy and detection of gravitational radiation are discussed. Part 2 develops the mathematics of differential geometry, along with physical applications, and discusses the exact treatment of curvature and the field equations. The electromagnetic field and fluid flow are treated, as well as geodesics, redshift, and so on. Part 3 then shows how the field equation is solved in standard cases such as Schwarzschild-Droste, Reissner-Nordstrom, Kerr, and internal stellar structure. Orbits and related phenomena are obtained. Black holes are described in detail, including horizons, wormholes, Penrose process and Hawking radiation. Part 4 covers cosmology, first in terms of metric, then dynamics, structure formation and observational methods. The meaning of cosmic expansion is explained at length. Recombination and last scattering are calculated, and the quantitative analysis of the CMB is sketched. Inflation is introduced briefly but quantitatively. Part 5 is a brief introduction to classical field theory, including spinors and the Dirac equation, proceeding as far as the Einstein-Hilbert action. Throughout the book the emphasis is on making the mathematics as clear as possible, and keeping in touch with physical observations.
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17

Mashhoon, Bahram. Linearized Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0007.

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The only known exact solution of the field equation of nonlocal gravity (NLG) is the trivial solution involving Minkowski spacetime that indicates the absence of a gravitational field. Therefore, this chapter is devoted to a thorough examination of NLG in the linear approximation beyond Minkowski spacetime. Moreover, the solutions of the linearized field equation of NLG are discussed in detail. We adopt the view that the kernel of the theory must be determined from observation. In the Newtonian regime of NLG, we recover the phenomenological Tohline-Kuhn approach to modified gravity. A simple generalization of the Kuhn kernel leads to a three-parameter modified Newtonian force law that is always attractive. Gravitational lensing is discussed. It is shown that nonlocal gravity (NLG), with a characteristic galactic lengthscale of order 1 kpc, simulates dark matter in the linear regime while preserving causality.
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18

Deruelle, Nathalie, and Jean-Philippe Uzan. Tests in the solar system. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0051.

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This chapter describes observable relativistic effects in the solar system. In the solar system we can, as a first approximation, neglect the gravitational field of all the stars except the Sun. In Newtonian theory, the planet trajectories are then Keplerian ellipses. Relativistic effects are weak because the dimensionless ratio characterizing them is everywhere less than GM⊙/c² R⊙≃ 2 × 10–6, and so they can be added linearly to the Newtonian perturbations due to the other planets, the non-spherical shape of celestial bodies, and so on. The chapter first describes the gravitational field of the Sun using a Schwarzschild spacetime, before moving on to look at the geodesic equation. It also discusses the bending of light, the Shapiro effect, the perihelion, post-Keplerian geodesics, and spin in a gravitational field.
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19

Keszthelyi, Bettina E. Gravitational models in 2+1 dimensions with topological terms and thermo-field dynamics of black holes. 1993.

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20

Maggiore, Michele. Gravitational Waves. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198570899.001.0001.

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A comprehensive and detailed account of the physics of gravitational waves and their role in astrophysics and cosmology. The part on astrophysical sources of gravitational waves includes chapters on GWs from supernovae, neutron stars (neutron star normal modes, CFS instability, r-modes), black-hole perturbation theory (Regge-Wheeler and Zerilli equations, Teukoslky equation for rotating BHs, quasi-normal modes) coalescing compact binaries (effective one-body formalism, numerical relativity), discovery of gravitational waves at the advanced LIGO interferometers (discoveries of GW150914, GW151226, tests of general relativity, astrophysical implications), supermassive black holes (supermassive black-hole binaries, EMRI, relevance for LISA and pulsar timing arrays). The part on gravitational waves and cosmology include discussions of FRW cosmology, cosmological perturbation theory (helicity decomposition, scalar and tensor perturbations, Bardeen variables, power spectra, transfer functions for scalar and tensor modes), the effects of GWs on the Cosmic Microwave Background (ISW effect, CMB polarization, E and B modes), inflation (amplification of vacuum fluctuations, quantum fields in curved space, generation of scalar and tensor perturbations, Mukhanov-Sasaki equation,reheating, preheating), stochastic backgrounds of cosmological origin (phase transitions, cosmic strings, alternatives to inflation, bounds on primordial GWs) and search of stochastic backgrounds with Pulsar Timing Arrays (PTA).
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21

The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity. Birkhauser, 2004.

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22

(Editor), Piotr T. Chrusciel, and Helmut Friedrich (Editor), eds. The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity. Birkhäuser Basel, 2004.

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23

Deruelle, Nathalie, and Jean-Philippe Uzan. Gravitational waves and the radiative field. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0053.

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This chapter turns to the gravitational radiation produced by a system of massive objects. The discussion is confined to the linear approximation of general relativity, which is compared with the Maxwell theory of electromagnetism. In the first part of the chapter, the properties of gravitational waves, which are the general solution of the linearized vacuum Einstein equations, are studied. Next, it relates these waves to the energy–momentum tensor of the sources creating them. The chapter then turns to the ‘first quadrupole formula’, giving the gravitational radiation field of these sources when their motion is due to forces other than the gravitational force.
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24

Deruelle, Nathalie, and Jean-Philippe Uzan. Gravitational radiation. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0054.

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This chapter attempts to calculate the radiated energy of a source in the linear approximation of general relativity to infinity in the lowest order. For this, the chapter first expands the Einstein equations to quadratic order in metric perturbations. It reveals that the radiated energy is then given by the (second) quadrupole formula, which is the gravitational analog of the dipole formula in Maxwell theory. This formula is a priori valid only if the motion of the source is due to forces other than gravity. Finally, this chapter shows that, to prove this formula for the case of self-gravitating systems, the Einstein equations to quadratic order must be solved, and the radiative field in the post-linear approximation of general relativity obtained.
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25

Huleihil, Kh, E. Leibowitz, and Moshe Carmeli. Gauge Fields: Classification and Equations of Motion. World Scientific Pub Co Inc, 1989.

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26

Huleihil, Khadra. Gauge Fields: Classification and Equations of Motion. World Scientific Publishing Co Pte Ltd, 1989.

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27

Deruelle, Nathalie, and Jean-Philippe Uzan. Matter in curved spacetime. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0043.

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This chapter is concerned with the laws of motion of matter—particles, fluids, or fields—in the presence of an external gravitational field. In accordance with the equivalence principle, this motion will be ‘free’. That is, it is constrained only by the geometry of the spacetime whose curvature represents the gravitation. The concepts of energy, momentum, and angular momentum follow from the invariance of the solutions of the equations of motion under spatio-temporal translations or rotations. The chapter shows how the action is transformed, no longer under a modification of the field configuration, but instead under a displacement or, in the ‘passive’ version, under a translation of the coordinate grid in the opposite direction.
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28

Deruelle, Nathalie, and Jean-Philippe Uzan. The Einstein equations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0044.

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This chapter deals with Einstein equations. In the absence of matter there is no gravitational field, and the spacetime which represents this empty universe is Minkowski spacetime. More precisely, if the gravitational field created by the matter can be neglected, the appropriate framework for describing the matter is that of special relativity. Einstein gravitational equations relate geometry and matter: specifically, they relate the Riemann tensor, or more precisely the Einstein tensor, to the geometrical object describing ‘inertia’, the energy content of the matter—that is, the energy–momentum tensor. These equations form a set of ten nonlinear partial differential equations. The coordinate system can be chosen arbitrarily.
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29

Baulieu, Laurent, John Iliopoulos, and Roland Sénéor. General Relativity: A Field Theory of Gravitation. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198788393.003.0004.

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General relativity. The equivalence principle and the derivation of the Einstein–Hilbert equations. The geometrical notions of curvature and affine connection are introduced. Geodesics and the bending of light by a gravitational field. General relativity as a gauge invariant classical field theory.
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30

Exact Solutions Of Einsteins Field Equations. Cambridge University Press, 2009.

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31

Hoenselaers, Cornelius, Eduard Herlt, Dietrich Kramer, Malcolm MacCallum, and Hans Stephani. Exact Solutions of Einstein's Field Equations. Cambridge University Press, 2009.

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32

Maccallum, Malcolm, Cornelius Hoenselaers, Eduard Herlt, Dietrich Kramer, and Hans Stephani. Exact Solutions of Einstein's Field Equations. Cambridge University Press, 2003.

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33

Hoenselaers, Cornelius, Eduard Herlt, Dietrich Kramer, Malcolm MacCallum, and Hans Stephani. Exact Solutions of Einstein's Field Equations. Cambridge University Press, 2003.

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34

Studies on Gravitational Field Equations and Important Results of Relativistic Cosmology. Lulu Press, Inc., 2019.

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35

Gemedjiev, Georgy. Mechanics, Electrodynamics and Gravitation of Vacuum Field Theory. Lulu Press, Inc., 2009.

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36

Chrusciel, Piotr T. Einstein Equations and the Large Scale Behavior of Gravitational Fields. Island Press, 2004.

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37

Deruelle, Nathalie, and Jean-Philippe Uzan. Interacting charges I. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0038.

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This chapter addresses the problem of radiation by a system of point charges. Owing to the fact that the electromagnetic interaction propagates at finite speed, this problem can only be solved iteratively, by assuming that all speeds are small compared to the speed of light. The chapter then derives the dipole and quadrupole formulas giving the radiation field and the energy radiated by the system in the lowest orders. Finding the field and the radiation of a system of charges beyond the dipole approximation is rather more difficult, but necessary in the absence of dipole radiation. This is also a useful exercise for studying the radiation of a mass system in theories of gravitation where the gravitational mass is equal to the inertial mass. In addition, the chapter finds the equations of motion of the charges of the system to third order in the velocities.
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38

Carlip, Steven. General Relativity. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198822158.001.0001.

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This work is a short textbook on general relativity and gravitation, aimed at readers with a broad range of interests in physics, from cosmology to gravitational radiation to high energy physics to condensed matter theory. It is an introductory text, but it has also been written as a jumping-off point for readers who plan to study more specialized topics. As a textbook, it is designed to be usable in a one-quarter course (about 25 hours of instruction), and should be suitable for both graduate students and advanced undergraduates. The pedagogical approach is “physics first”: readers move very quickly to the calculation of observational predictions, and only return to the mathematical foundations after the physics is established. The book is mathematically correct—even nonspecialists need to know some differential geometry to be able to read papers—but informal. In addition to the “standard” topics covered by most introductory textbooks, it contains short introductions to more advanced topics: for instance, why field equations are second order, how to treat gravitational energy, what is required for a Hamiltonian formulation of general relativity. A concluding chapter discusses directions for further study, from mathematical relativity to experimental tests to quantum gravity.
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39

Kennefick, Daniel. Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves. Princeton University Press, 2007.

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40

Kennefick, Daniel. Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves. Princeton University Press, 2016.

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41

Schmidt, Bernd G. Einstein's Field Equations and Their Physical Implications: Selected Essays in Honour of Jürgen Ehlers. Springer London, Limited, 2008.

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42

Schmidt, Bernd G. Einstein's Field Equations and Their Physical Implications: Selected Essays in Honour of Jürgen Ehlers. Springer Berlin / Heidelberg, 2010.

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43

9780192867414, Tom Lancaster, and Stephen Blundell. General Relativity for the Gifted Amateur. Oxford University PressOxford, 2025. https://doi.org/10.1093/oso/9780192867407.001.0001.

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Abstract General relativity is a field theory that describes gravity. It engages profoundly with the nature of space and time and is based on simple ideas from the physics of fields. It can be summarised by the Einstein equation which relates a geometrical quantity, the curvature of space and time that follows from the metric field., to a physical quantity that reflects a field that describes the matter content of the Universe. We begin in Part I with an introduction to the geometry of flat spacetime, reviewing special relativity and setting up the mathematics of the metric. Part II introduces the mathematics of curvature and sets up the physics of general relativity and finishes with the Einstein field equation. Part III applies these ideas to the Universe and studies various models used in cosmology. Part IV turns to smaller structures inside the Universe: stars, black holes and their orbits. Part V contains a more formal treatment of geometry which may be of more interest to those with more mathematical inclinations. Part VI considers general relativity as a type of field theory and examines how one might link the ideas in our best theory of gravitation to our most successful theories of quantum fields.
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44

Deruelle, Nathalie, and Jean-Philippe Uzan. The physics of black holes I. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0049.

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This chapter describes two physical processes related to the Schwarzschild and Kerr solutions which can be induced by the gravitational field of a black hole. The first is the Penrose process, which suggests that rotating black holes are large energy reservoirs. Next is superradiance, which is the first step in the study of black-hole stability. The study of the stability of black holes involves the linearization of the Einstein equations about the Schwarzschild or Kerr solution. As this chapter shows, the equations of motion for perturbations of the metric are wave equations. The problem then is to determine whether or not these solutions are bounded.
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45

Einstein's Field Equations and Their Physical Implications: Selected Essays in Honour of Jürgen Ehlers (Lecture Notes in Physics). Springer, 2000.

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46

100 years of gravity and accelerated frames: The deepest insights of Einstein and Yang-Mills. World Scientific, 2005.

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47

Zaitsev, Fedor, and Vladimir Bychkov. Mathematical modeling of electromag-netic and gravitational phenomena by the methodology of continuous media mechanics. LCC MAKS Press, 2021. http://dx.doi.org/10.29003/m2011.978-5-317-06604-8.

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The book of well-known Russian scientists systematically presents a new theoretical approach to studying nature's fundamental phenomena using the hypothesis of the physical vacuum, or the ether, as some environment in which all the processes develop. In the proposed studies, the ether is represented as some one-component continuous media that satisfies generally accepted conservation laws: of matter and momentum. From the appropriate two equations, a number of consequences are obtained to which a physical interpretation is given. For the first time, 150 years after studies of Faraday and Maxwell, it is shown that these single premises mathematically give basic physical laws established experimentally: the Maxwell equations, the Lorentz force, the Gauss theorem; the laws: Coulomb, Biot - Savard, Ampere, electromagnetic induction, Ohm, Joule - Lenz, Wiedemann - Franz, universal gravitation, and etc. Details of mechanisms of many processes, that seemed previously paradoxical, have been disclosed. A method of the model substantiation adopted in the mathematical modeling methodology allows to conclude that the presented mathematical model of the ether adequately describes electromagnetic and gravitational processes. Qualitative and quantitative analysis of hundreds of known and new experimental facts allows in the methodology of physics, as science summarizing the experiments data, to confirm a conclusion about the existence of the ether (physical vacuum). The content of the book is based on the works of authors done during the last fourteen years. Many results are published for the first time. The book is intended for specialists in the field of electrodynamics, electrical engineering, gravity and kinetics, as well as for graduate students and students, interested in the fundamental principles of these scientific directions. This book is unique in terms of the comprehensive consideration of the problem and the depth of its analysis.
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48

Griffiths, J. B. Colliding plane waves in general relativity. 2016.

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49

Saha, Prasenjit, and Paul A. Taylor. Schwarzschild’s Spacetime. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198816461.003.0003.

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The concept of a metric is motivated and introduced, along with the introduction of relativistic quantities of spacetime, proper time, and Einstein’s field equations. Geodesics are cast in basic form as a Hamiltonian dynamical problem, which readers are guided towards exploring numerically themselves. The specific case of the Schwarzschild metric is presented, which is applicable to space around non-rotating black holes, and orbital motion around such objects is contrasted with that of Newtonian systems. Some well-known formulas for black hole phenomena are derived, such as those for light deflection (also known as gravitational lensing) and the innermost stable orbit, and their consequences discussed.
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50

Deruelle, Nathalie, and Jean-Philippe Uzan. The post-Newtonian approximation. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0052.

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This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.
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