[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/borde-guth-vilenkin-theorem-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki21\/borde-guth-vilenkin-theorem-wikipedia\/","headline":"Borde\u2013Guth\u2013Vilenkin theorem – Wikipedia","name":"Borde\u2013Guth\u2013Vilenkin theorem – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 Theorem in physical cosmology The Borde\u2013Guth\u2013Vilenkin theorem, or the BGV theorem, is a","datePublished":"2019-04-01","dateModified":"2019-04-01","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1c\/Wiki_letter_w_cropped.svg\/20px-Wiki_letter_w_cropped.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1c\/Wiki_letter_w_cropped.svg\/20px-Wiki_letter_w_cropped.svg.png","height":"14","width":"20"},"url":"https:\/\/wiki.edu.vn\/en\/wiki21\/borde-guth-vilenkin-theorem-wikipedia\/","wordCount":5189,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Theorem in physical cosmologyThe Borde\u2013Guth\u2013Vilenkin theorem, or the BGV theorem, is a theorem in physical cosmology which deduces that any universe that has, on average, been expanding throughout its history cannot be infinite in the past but must have a past spacetime boundary.[1] The theorem does not assume any specific mass content of the universe and it does not require gravity to be described by Einstein field equations. It is named after the authors Arvind Borde, Alan Guth and Alexander Vilenkin, who developed its mathematical formulation in 2003.[2][3] The BGV theorem is also popular outside physics, especially in religious and philosophical debates.[3][4][5]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsFormal definition[edit]Derivation[edit]For flat spacetime[edit]Limitations and criticism[edit]Use in theology[edit]See also[edit]References[edit]Further reading[edit]Formal definition[edit]This section needs expansion. You can help by adding to it.  (April 2023)Derivation[edit]For flat spacetime[edit]Here is an example of derivation of the BGV theorem for an expanding homogeneous isotropic flat universe (in units of speed of light c=1).[6] Which is consistent with \u039bCDM model, the current model of cosmology. However, this derivation can be generalized to an arbitrary space-time with no appeal to homogeneity or isotropy.[6]The Friedmann\u2013Lema\u00eetre\u2013Robertson\u2013Walker metric is given by       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4ds=dt2\u2212a2(t)dxidxi{displaystyle ds=dt^{2}-a^{2}(t)dx_{i}dx^{i}},where t is time, xi (i=1,2,3) are the spatial coordinates and a(t) is the scale factor. Along a timeline geodesic xi = constant, we can consider the universe to be filled with comoving particles. For an observer with proper time \u03c4 following the world line x\u03bc(\u03c4), has a 4-momentum P\u03bc=mdx\u03bc\/d\u03c4=(E,p){displaystyle P^{mu }=mdx^{mu }\/dtau =(E,mathbf {p} )}, where E=p2+m2{displaystyle E={sqrt {p^{2}+m^{2}}}} is the energy, m is the mass and p=|p| the magnitude of the 3-momentum.From the geodesic equation of motion, it follows that p(t)=pfa(tf)\/a(t){displaystyle p(t)=p_{rm {f}};a(t_{rm {f}})\/a(t)} where pf is the final momentum at time tf. Thus\u222btitfH(\u03c4)d\u03c4=\u222ba(ti)a(tf)mdam2a2+p2a(tf)=F(\u03b3f)\u2212F(\u03b3i)\u2264F(\u03b3f){displaystyle int _{t_{rm {i}}}^{t_{rm {f}}}H(tau )dtau =int _{a(t_{rm {i}})}^{a(t_{rm {f}})}{frac {mda}{sqrt {m^{2}a^{2}+p^{2}a(t_{rm {f}})}}}=F(gamma _{rm {f}})-F(gamma _{rm {i}})leq F(gamma _{rm {f}})},where ti is an initial time, H=a\u02d9\/a{displaystyle H={dot {a}}\/a} is the Hubble parameter, andF(\u03b3)=12ln\u2061(\u03b3+1\u03b3\u22121){displaystyle F(gamma )={frac {1}{2}}ln left({frac {gamma +1}{gamma -1}}right)},\u03b3 being the Lorentz factor. For any non-comoving observer \u03b3>1 and F(\u03b3)>0.The expansion rate averaged over the observer world line can be defined asHav=1\u03c4f\u2212\u03c4i\u222btitfH(\u03c4)d\u03c4{displaystyle H_{rm {av}}={frac {1}{tau _{rm {f}}-tau _{rm {i}}}}int _{t_{rm {i}}}^{t_{rm {f}}}H(tau )dtau }.Assuming 0}”\/> it is follows that\u03c4f\u2212\u03c4i\u2264F(\u03b3f)Hav{displaystyle tau _{rm {f}}-tau _{rm {i}}leq {frac {F(gamma _{rm {f}})}{H_{rm {av}}}}}.Thereby any non-comoving past-directed timelike geodesic satisfying the condition 0}”\/>, must have a finite proper length, and so must be past-incomplete.Limitations and criticism[edit]Alternative models, where the average expansion of the universe throughout its history does not hold, have been proposed under the notions of emergent spacetime, eternal inflation, and  cyclic models. Vilenkin and Audrey Mithani have argued that none of these models escape the implications of the theorem.[7] In 2017, Vilenkin stated that he does not think there are any viable cosmological models that escape the scenario.[8]Sean M. Carroll argues that the theorem only applies to classical spacetime, and may not hold under consideration of a complete theory of quantum gravity. He added that Alan Guth, one of the co-authors of the theorem, disagrees with Vilenkin and believes that the universe had no beginning.[9][10] Vilenkin argues that the Carroll-Chen model constructed by Carroll and Jennie Chen, and supported by Guth, to elude the BGV theorem\u2019s conclusions persists to indicate a singularity in the history of the universe as it has a reversal of the arrow of time in the past.[11]Use in theology[edit]Vilenkin has also written about the religious significance of the BGV theorem. In October 2015, Vilenkin responded to arguments made by theist William Lane Craig and the New Atheism movement regarding the existence of God. Vilenkin stated “What causes the universe to pop out of nothing? No cause is needed.”[6]See also[edit]References[edit]^ Perlov, Delia; Vilenkin, Alexander (7 August 2017). Cosmology for the Curious. Cham, Switzerland: Springer. pp.\u00a0330\u201331. ISBN\u00a0978-3319570402.^ Borde, Arvind; Guth, Alan H.; Vilenkin, Alexander (15 April 2003). “Inflationary space-times are incomplete in past directions”. Physical Review Letters. 90 (15): 151301. arXiv:gr-qc\/0110012. Bibcode:2003PhRvL..90o1301B. doi:10.1103\/PhysRevLett.90.151301. PMID\u00a012732026. S2CID\u00a046902994.^ a b Perlov, Delia; Vilenkin, Alexander (7 August 2017). Cosmology for the Curious. Cham, Switzerland: Springer. pp.\u00a0330\u201331. ISBN\u00a0978-3319570402.^ Copan, Paul; Craig, William Lane (2017-11-16). The Kalam Cosmological Argument, Volume 2: Scientific Evidence for the Beginning of the Universe. Bloomsbury Publishing USA. ISBN\u00a09781501335891.^ Nagasawa, Y. (2012-07-25). Scientific Approaches to the Philosophy of Religion. Springer. ISBN\u00a09781137026019.^ a b c Vilenkin, Alexander (2015-10-23). “The Beginning of the Universe”. Inference. 1 (4).^ Mithani, Audrey; Vilenkin, Alexander (20 April 2012). “Did the universe have a beginning?”. arXiv:1204.4658 [hep-th].^ Alexander Vilenkin, “The Beginning of the Universe” in The Kalam Cosmological Argument: Volume 2, Bloomsbury, 2017, pp. 150\u2013158^ Carroll, Sean (2014-02-24). “Post-Debate Reflections”. Sean Carroll Blog. Archived from the original on 2014-02-25. Retrieved 2019-11-19.^ Carroll, Sean M. (2018-06-04). “Why Is There Something, Rather Than Nothing?”. arXiv:1802.02231 [physics.hist-ph].^ Vilenkin, Alexander (2013). “Arrows of time and the beginning of the universe”. Physical Review D. 88 (4): 043516. arXiv:1305.3836. Bibcode:2013PhRvD..88d3516V. doi:10.1103\/PhysRevD.88.043516. S2CID\u00a0119213877.Further reading[edit]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/borde-guth-vilenkin-theorem-wikipedia\/#breadcrumbitem","name":"Borde\u2013Guth\u2013Vilenkin theorem – Wikipedia"}}]}]