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Geodesics in general relativity

Geodesics describe the paths of freely falling particles in curved spacetime.

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Freely falling particles in curved spacetime follow paths described by geodesics, according to general relativity.[1][2] One way to obtain the geodesic equation of motion is through the equivalence principle, and another is via parallel transport.[1][2]

Quick Facts

Field
General relativity
Known for
Geodesic equation of motion, generalization of straight lines to curved spacetime
Concept type
Mathematical and physical concept

Facts from the source article.

Key facts and history

Physicist Steven Weinberg derived the geodesic equation directly from the equivalence principle.[1] Weinberg defined the affine connection using a freely falling coordinate system.[3] Albert Einstein believed the geodesic equation could be derived from the field equations for empty space.[4] According to Einstein, the original relativistic theory of gravitation was incomplete because it introduced the independent postulate of geodesic motion.[5]

The problem of the field and the problem of motion coincide when describing a particle without singularity.[6] The claim that the geodesic equation can be obtained from the field equations remains disputed.[6] Parallel transport provides an alternative method for deriving the geodesic equation.[2]

What the cited sources add

The book "Gravitation" by Misner, published by W. H. Freeman in 1973, states that in simply connected Minkowski space only one geodesic connects any given pair of events.[7][8] Wald provides the equation of motion for a charged particle, which includes a term involving the electromagnetic field tensor.[9][10] Misner's 1973 book "Gravitation" (pages 316, 318–319) notes that in curved spacetime, two widely separated events can be connected by more than one time-like geodesic.[7][8] For certain geodesics connecting two events, a nearby curve can have either a longer or shorter proper time than the geodesic itself, according to Misner (1973, pages 316, 318–319).[7][8]

Wald's 1984 "General Relativity" (University of Chicago Press) describes the metric tensor as symmetric and notes that in free fall it locally becomes the Minkowski tensor.[9][10]

References

  1. Wikipedia, "Geodesics in general relativity" (CC BY-SA 4.0), which cites: Weinberg, Steven. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley 1972). (original not read by us) en.wikipedia.org
  2. Wikipedia, "Geodesics in general relativity" (CC BY-SA 4.0), which cites: Plebański, Jerzy and Krasiński, Andrzej. An Introduction to General Relativity and Cosmology, p. 34 (Cambridge University Press, 2006). (original not read by us) en.wikipedia.org
  3. Wikipedia, "Geodesics in general relativity" (CC BY-SA 4.0), which cites: Weinberg, Steven. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, p. 71, equation 3.2.4 (Wiley 1972). (original not read by us) en.wikipedia.org
  4. Wikipedia, "Geodesics in general relativity" (CC BY-SA 4.0), which cites: Einstein, Albert. The Meaning of Relativity, p. 113 (Routledge 2003). (original not read by us) en.wikipedia.org
  5. Wikipedia, "Geodesics in general relativity" (CC BY-SA 4.0), which cites: Einstein, "The Particle Problem in the General Theory of Relativity", Physical Review, 1 July 1935, p. 76 (original not read by us) en.wikipedia.org
  6. Wikipedia, "Geodesics in general relativity" (CC BY-SA 4.0), which cites: Tamir, M. "Proving the principle: Taking geodesic dynamics too seriously in Einstein’s theory", Studies In History and Philosophy of Modern Physics 43(2), 137–1 (original not read by us) en.wikipedia.org
  7. Wikipedia, "Geodesics in general relativity" (CC BY-SA 4.0), which cites: Charles W. Misner, "Gravitation", W. H. Freeman, 1973, p. 316, 318–319 (we also read that source) en.wikipedia.org
  8. Open Library record, ISBN 0716703440 ("Gravitation"). CC0 (metadata); book content not read. openlibrary.org
  9. Wikipedia, "Geodesics in general relativity" (CC BY-SA 4.0), which cites: Wald, "General Relativity", University of Chicago Press, 1984 (we also read that source) en.wikipedia.org
  10. Open Library record, ISBN 9780226870335 ("General relativity"). CC0 (metadata); book content not read. openlibrary.org

Each numbered sentence above was checked against the source it cites before publication. Sentences citing Wikipedia adapt text from the English Wikipedia article Geodesics in general relativity (contributors), used under CC BY-SA 4.0; this entry is available under the same licence. Other sources were used for facts only; no text or images are copied from them.

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Einstein field equationsEnergy–momentum relationFour-momentumFour-vectorGeneral covarianceGravitational constantGravitational fieldGravitational potentialGravitational singularityGravitational time dilationGravitational waveHawking radiation

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Compiled from Wikipedia and the sources listed below. Text from Wikipedia is available under CC BY-SA 4.0; this entry is adapted from it.

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