Editing Curved spacetime (section)

== Sources of spacetime curvature ==
[[File:StressEnergyTensor contravariant.svg|thumb|250px|Figure 5-5. Contravariant components of the stress–energy tensor]]
In [[Law of universal gravitation|Newton's theory of gravitation]], the only source of gravitational force is [[mass]].

In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. In the [[Einstein field equations]],
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the sources of gravity are presented on the right-hand side in <math>T_{\mu \nu},</math> the [[stress–energy tensor]].<ref name=Hobson_2006/>

Fig.&nbsp;5-5 classifies the various sources of gravity in the stress–energy tensor:
* <math>T^{00}</math> (red): The total mass–energy density, including any contributions to the potential energy from forces between the particles, as well as kinetic energy from random thermal motions.
* <math>T^{0i}</math> and <math>T^{i0}</math> (orange): These are momentum density terms. Even if there is no bulk motion, energy may be transmitted by heat conduction, and the conducted energy will carry momentum.
* <math>T^{ij}</math> are the rates of flow of the {{nowrap|1=''i''-component}} of momentum per unit area in the {{nowrap|1=''j''-direction}}. Even if there is no bulk motion, random thermal motions of the particles will give rise to momentum flow, so the {{math|1=''i'' = ''j''}} terms (green) represent isotropic pressure, and the {{math|1=''i'' ≠ ''j''}} terms (blue) represent shear stresses.<ref name=Hobson_2006>{{cite book |last1=Hobson |first1=M. P. |last2=Efstathiou |first2=G. |last3=Lasenby |first3=A. N. |title=General Relativity |date=2006 |publisher=Cambridge University Press |location=Cambridge |isbn=978-0-521-82951-9 |pages=176–179}}</ref>

One important conclusion to be derived from the equations is that, colloquially speaking, ''gravity itself creates gravity''.{{refn|group=note|More precisely, the gravitational field couples to itself. In Newtonian gravity, the potential due to two point masses is simply the sum of the potentials of the two masses, but this does not apply to GR. This can be thought of as the result of the equivalence principle: If gravitation did not couple to itself, two particles bound by their mutual gravitational attraction would not have the same inertial mass (due to negative binding energy) as their gravitational mass.<ref name="Carroll" />{{rp|112–113}} }} Energy has mass. Even in Newtonian gravity, the gravitational field is associated with an energy, {{tmath|1=E = mgh,}} called the [[gravitational potential energy]]. In general relativity, the energy of the gravitational field feeds back into creation of the gravitational field. This makes the equations nonlinear and hard to solve in anything other than weak field cases.<ref name=Schutz />{{rp|240}} [[Numerical relativity]] is a branch of general relativity using numerical methods to solve and analyze problems, often employing supercomputers to study [[black holes]], [[gravitational waves]], [[neutron stars]] and other phenomena in the strong field regime.

=== Energy-momentum ===
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 | footer            = Figure 5-6. (left) Mass-energy warps spacetime. (right) Rotating mass–energy distributions with [[angular momentum]] '''J''' generate [[Gravitoelectromagnetism|gravitomagnetic fields]] '''H'''.}}

In special relativity, mass-energy is closely connected to [[momentum]]. Just as space and time are different aspects of a more comprehensive entity called spacetime, mass–energy and momentum are merely different aspects of a unified, four-dimensional quantity called [[four-momentum]]. In consequence, if mass–energy is a source of gravity, momentum must also be a source. The inclusion of momentum as a source of gravity leads to the prediction that moving or rotating masses can generate fields analogous to the magnetic fields generated by moving charges, a phenomenon known as [[gravitomagnetism]].<ref>{{cite book |author1=Thorne, Kip S. |editor1-last=Fairbank |editor1-first=J. D. |editor2-last=Deaver |editor2-first=B. S. Jr. |editor3-last=Everitt |editor3-first=W. F. |editor4-last=Michelson |editor4-first=P. F. |title=Near zero: New Frontiers of Physics |date=1988 |publisher=W. H. Freeman and Company |pages=573–586 |s2cid=12925169 |url=https://pdfs.semanticscholar.org/f9b7/4f316437af586bc20835fe2f6fc47eeca3c2.pdf |archive-url=https://web.archive.org/web/20170728121832/https://pdfs.semanticscholar.org/f9b7/4f316437af586bc20835fe2f6fc47eeca3c2.pdf |archive-date=28 July 2017 |url-status=dead }}</ref>

[[File:Special relativistic explanation of gravitomagnetism.svg|250px|thumb|Figure 5–7. Origin of gravitomagnetism]]
It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges. (An eloquent demonstration of this was presented by Feynman in volume II, {{nowrap|1=chapter 13–6}} of his ''Lectures on Physics'', available online.)<ref>{{cite book|last1=Feynman|first1=R. P.|last2=Leighton|first2=R. B.|last3=Sands|first3=M.|title=The Feynman Lectures on Physics, vol. 2|date=1964|publisher=Basic Books|isbn=978-0-465-02416-2|pages=13–6 to 13–11|edition=New Millenium|url=https://feynmanlectures.caltech.edu/II_13.html|access-date=1 July 2017|archive-date=17 January 2023|archive-url=https://web.archive.org/web/20230117023452/https://www.feynmanlectures.caltech.edu/II_13.html|url-status=live}}</ref> Analogous logic can be used to demonstrate the origin of gravitomagnetism.<ref name=Schutz />{{rp|245–253}}

In Fig.&nbsp;5-7a, two parallel, infinitely long streams of massive particles have equal and opposite velocities −''v'' and +''v'' relative to a test particle at rest and centered between the two. Because of the symmetry of the setup, the net force on the central particle is zero. Assume {{tmath|1=v \ll c}} so that velocities are simply additive. Fig.&nbsp;5-7b shows exactly the same setup, but in the frame of the upper stream. The test particle has a velocity of +''v'', and the bottom stream has a velocity of +2''v''. Since the physical situation has not changed, only the frame in which things are observed, the test particle should not be attracted towards either stream.<ref name=Schutz />{{rp|245–253}}

It is not at all clear that the forces exerted on the test particle are equal. (1) Since the bottom stream is moving faster than the top, each particle in the bottom stream has a larger mass energy than a particle in the top. (2) Because of Lorentz contraction, there are more particles per unit length in the bottom stream than in the top stream. (3) Another contribution to the active gravitational mass of the bottom stream comes from an additional pressure term which, at this point, we do not have sufficient background to discuss. All of these effects together would seemingly demand that the test particle be drawn towards the bottom stream.<ref name=Schutz />{{rp|245–253}}

The test particle is not drawn to the bottom stream because of a velocity-dependent force that serves to repel a particle ''that is moving in the same direction as the bottom stream.'' This velocity-dependent gravitational effect is gravitomagnetism.<ref name=Schutz />{{rp|245–253}}

Matter in motion through a gravitomagnetic field is hence subject to so-called ''[[frame-dragging]]'' effects analogous to [[electromagnetic induction]]. It has been proposed that such gravitomagnetic forces underlie the generation of the [[relativistic jets]] (Fig.&nbsp;5-8) ejected by some rotating [[supermassive black hole]]s.<ref>{{Cite journal|last=Williams |first=R. K. |date=1995 |title=Extracting X rays, Ύ rays, and relativistic e<sup>−</sup>–e<sup>+</sup> pairs from supermassive Kerr black holes using the Penrose mechanism |journal=Physical Review D |volume=51 |issue=10 |pages=5387–5427 |doi=10.1103/PhysRevD.51.5387 |bibcode = 1995PhRvD..51.5387W |pmid=10018300}}</ref><ref>{{Cite journal|last=Williams |first=R. K. |date=2004 |title=Collimated escaping vortical polar e<sup>−</sup>–e<sup>+</sup> jets intrinsically produced by rotating black holes and Penrose processes |journal=The Astrophysical Journal |volume=611 |issue= 2|pages=952–963 |doi=10.1086/422304 |bibcode=2004ApJ...611..952W|arxiv = astro-ph/0404135 |s2cid=1350543 }}</ref>

=== Pressure and stress ===
Quantities that are directly related to energy and momentum should be sources of gravity as well, namely internal [[pressure]] and [[Stress (physics)|stress]]. Taken together, {{nowrap|1=mass-energy}}, momentum, pressure and stress all serve as sources of gravity: Collectively, they are what tells spacetime how to curve.

General relativity predicts that pressure acts as a gravitational source with exactly the same strength as mass–energy density. The inclusion of pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation. For example, the pressure term sets a maximum limit to the mass of a [[neutron star]]. The more massive a neutron star, the more pressure is required to support its weight against gravity. The increased pressure, however, adds to the gravity acting on the star's mass. Above a certain mass determined by the [[Tolman–Oppenheimer–Volkoff limit]], the process becomes runaway and the neutron star collapses to a [[black hole]].<ref name=Schutz />{{rp|243,280}}

The stress terms become highly significant when performing calculations such as hydrodynamic simulations of core-collapse supernovae.<ref>{{Cite journal|last1=Kuroda |first1=Takami |last2=Kotake |first2=Kei |last3=Takiwaki |first3=Tomoya |title=Fully General Relativistic Simulations of Core-Collapse Supernovae with An Approximate Neutrino Transport |journal=The Astrophysical Journal |volume=755 |issue=1 |pages=11 |arxiv=1202.2487 |year=2012 |doi=10.1088/0004-637X/755/1/11 |bibcode=2012ApJ...755...11K |s2cid=119179339 }}</ref>

These predictions for the roles of pressure, momentum and stress as sources of spacetime curvature are elegant and play an important role in theory. In regards to pressure, the early universe was radiation dominated,<ref>{{cite web |last=Wollack |first=Edward J. |title=Cosmology: The Study of the Universe |work=Universe 101: Big Bang Theory |publisher=[[NASA]] |date=10 December 2010 |url=http://map.gsfc.nasa.gov/universe/ |access-date=15 April 2017  |archive-url=https://web.archive.org/web/20110514230003/http://map.gsfc.nasa.gov/universe/ |archive-date=14 May 2011 |url-status=dead}}</ref> and it is highly unlikely that any of the relevant cosmological data (e.g. [[nucleosynthesis]] abundances, etc.) could be reproduced if pressure did not contribute to gravity, or if it did not have the same strength as a source of gravity as mass–energy. Likewise, the mathematical consistency of the Einstein field equations would be broken if the stress terms did not contribute as a source of gravity.