Editing Curved spacetime (section)

=== 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>