Translation examples
Tämä johti johtopäätökseen, jonka mukaan olion massa kaareuttaa sen ympärillä olevan aika-avaruuden, niin kuin Einsteinin kenttäyhtälöt kuvaavat.
This led to the conclusion that the mass of an object warps the geometry of the space-time surrounding it, as described in Einstein's field equations.
Fridman löysi kaikkeuden laajenemista kuvaavat Fridmanin yhtälöt ratkaisuna Einsteinin kenttäyhtälöihin vuonna 1922.
Friedmann derived the expanding-universe solution to general relativity field equations in 1922.
Einsteinin kenttäyhtälöt pysyvät invariantteina, kun c ja G muuttuvat sopivalla tavalla.
The Einstein field equations remain invariant through convenient joint variations of c and G in Einstein's constant.
Bruno Bertotti on hyvin tunnettu panoksesta suhteellisuusteoriaan - erityisesti Bertotti-Robinson sähkötyhjiöstä, joka on tarkka ratkaisu Einsteinin kenttäyhtälöstä.
Bertotti was well known for his contributions to general relativity – particularly the Bertotti-Robinson electrovacuum, an exact solution of the Einstein field equation.
Friedmannin–Lemaîtren–Robertsonin–Walkerin metriikka -niminen Einsteinin kenttäyhtälöiden ratkaisu on alkuräjähdyksen teoreettinen perusta.
Together with H. P. Robertson, they devised the well-known Robertson-Walker unit for the Friedmann–Lemaître–Robertson–Walker cosmological models, which are exact solutions of the Einstein field equation.
Einsteinin kenttäyhtälöt tai Einsteinin yhtälöt ovat kymmenen Albert Einsteinin yleisen suhteellisuusteorian yhtälöä, jotka kuvaavat gravitaation massan ja energian aiheuttamana aika-avaruuden kaareutumana.
The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.
Samoissa artikkeleissa todistettiin, että universumin neljäs ulottuvuus johtuu matemaattisesta stabiliteetin vaatimuksesta, koska Einsteinin kenttäyhtälöiden lisäkomponentti määrittää rajoitetun ratkaisun materiakentille, mikä on ristiriidassa yhtä stabiilisuusehtoa vastaan.
It was also shown that the four-dimensionality of the Universe is the result of the stability requirement found in mathematics since the extra component of the Einstein field equations giving the confined solution for matter fields coincides with one of the conditions of stability.
Historiallisesti tärkeä yksityiskohta on, että yhtälöön voidaan lisätä ainoastaan metrisestä tensorista riippuva termi, jolloin kenttäyhtälöt saavat muodon G μ ν + Λ g μ ν = − 8 π G c 4 T μ ν {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=-{\frac {8\pi G}{c^{4}}}T_{\mu \nu }} ilman, että se vaikuttaa energian säilymislakiin.
Einstein modified his original field equations to include a cosmological constant term Λ proportional to the metric R μ ν − 1 2 R g μ ν + Λ g μ ν = 8 π G c 4 T μ ν . {\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }\,.} Since Λ is constant, the energy conservation law is unaffected.
Matemaattisesti tämä oletus voidaan kirjoittaa seuraavasti: ∫ Ω ′ L ( α A , α A , ν , ξ μ ) d 4 ξ − ∫ Ω L ( ϕ A , ϕ A , ν , x μ ) d 4 x = 0 {\displaystyle \int _{\Omega ^{\prime }}L\left(\alpha ^{A},{\alpha ^{A}}_{,\nu },\xi ^{\mu }\right)d^{4}\xi -\int _{\Omega }L\left(\phi ^{A},{\phi ^{A}}_{,\nu },x^{\mu }\right)d^{4}x=0} missä muuttujien jälkeen yläpuolelle kirjoitetut pilkut tarkoittavat osittaisderivaattoja niiden koordinaattien suhteen, jotka seuraavat pilkun jälkeen, toisin sanoen ϕ A , σ = ∂ ϕ A ∂ x σ . {\displaystyle {\phi ^{A}}_{,\sigma }={\frac {\partial \phi ^{A}}{\partial x^{\sigma }}}\,.} Koska ξ on pelkkä integroimisvakio ja koska rajan Ω muutos oletettiin infinitesimaaliseksi, nämä kaksi integraalia voidaan yhdistää divergenssilauseen neliulotteisen version mukaisesti seuraavaan muotoon: ∫ Ω { + ∂ ∂ x σ } d 4 x = 0 . {\displaystyle \int _{\Omega }\left\{\left+{\frac {\partial }{\partial x^{\sigma }}}\left\right\}d^{4}x=0\,.} Lagrangen funktioiden erotus voidaan kirjoittaa ensimmäisessä kertaluvuissa infinitesimaalisilla muutoksilla: = ∂ L ∂ ϕ A δ ¯ ϕ A + ∂ L ∂ ϕ A , σ δ ¯ ϕ A , σ . {\displaystyle \left={\frac {\partial L}{\partial \phi ^{A}}}{\bar {\delta }}\phi ^{A}+{\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}{\bar {\delta }}{\phi ^{A}}_{,\sigma }\,.} Koska nämä muutokset kuitenkin on määritelty samassa edellä selityssä pisteessä, muutokset ja derivoinnit voidaan suorittaa myös päinvastaisessa järjestyksessä; ne kommutoivat: δ ¯ ϕ A , σ = δ ¯ ∂ ϕ A ∂ x σ = ∂ ∂ x σ ( δ ¯ ϕ A ) . {\displaystyle {\bar {\delta }}{\phi ^{A}}_{,\sigma }={\bar {\delta }}{\frac {\partial \phi ^{A}}{\partial x^{\sigma }}}={\frac {\partial }{\partial x^{\sigma }}}\left({\bar {\delta }}\phi ^{A}\right)\,.} Käyttämällä Eulerin-Lagrangen kenttäyhtälöä ∂ ∂ x σ ( ∂ L ∂ ϕ A , σ ) = ∂ L ∂ ϕ A {\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}\right)={\frac {\partial L}{\partial \phi ^{A}}}} Lagrangen funktioiden erotus voidaan kirjoittaa yksinkertaisesti muotoon = ∂ ∂ x σ ( ∂ L ∂ ϕ A , σ ) δ ¯ ϕ A + ∂ L ∂ ϕ A , σ δ ¯ ϕ A , σ = ∂ ∂ x σ ( ∂ L ∂ ϕ A , σ δ ¯ ϕ A ) . {\displaystyle \left={\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}\right){\bar {\delta }}\phi ^{A}+{\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}{\bar {\delta }}{\phi ^{A}}_{,\sigma }={\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}{\bar {\delta }}\phi ^{A}\right)\,.} Näin ollen aktion muutokseksi saadaan ∫ Ω ∂ ∂ x σ { ∂ L ∂ ϕ A , σ δ ¯ ϕ A + L ( ϕ A , ϕ A , ν , x μ ) δ x σ } d 4 x = 0 . {\displaystyle \int _{\Omega }{\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}{\bar {\delta }}\phi ^{A}+L\left(\phi ^{A},{\phi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}d^{4}x=0\,.} Koska tämä pätee missä tahansa alueessa Ω, integrandin on oltava nolla ∂ ∂ x σ { ∂ L ∂ ϕ A , σ δ ¯ ϕ A + L ( ϕ A , ϕ A , ν , x μ ) δ x σ } = 0 . {\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}{\bar {\delta }}\phi ^{A}+L\left(\phi ^{A},{\phi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}=0\,.} .
Expressed mathematically, this assumption may be written as ∫ Ω ′ L ( α A , α A , ν , ξ μ ) d 4 ξ − ∫ Ω L ( φ A , φ A , ν , x μ ) d 4 x = 0 {\displaystyle \int _{\Omega ^{\prime }}L\left(\alpha ^{A},{\alpha ^{A}}_{,\nu },\xi ^{\mu }\right)d^{4}\xi -\int _{\Omega }L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)d^{4}x=0} where the comma subscript indicates a partial derivative with respect to the coordinate(s) that follows the comma, e.g. φ A , σ = ∂ φ A ∂ x σ . {\displaystyle {\varphi ^{A}}_{,\sigma }={\frac {\partial \varphi ^{A}}{\partial x^{\sigma }}}\,.} Since ξ is a dummy variable of integration, and since the change in the boundary Ω is infinitesimal by assumption, the two integrals may be combined using the four-dimensional version of the divergence theorem into the following form ∫ Ω { + ∂ ∂ x σ } d 4 x = 0 . {\displaystyle \int _{\Omega }\left\{\left+{\frac {\partial }{\partial x^{\sigma }}}\left\right\}d^{4}x=0\,.} The difference in Lagrangians can be written to first-order in the infinitesimal variations as = ∂ L ∂ φ A δ ¯ φ A + ∂ L ∂ φ A , σ δ ¯ φ A , σ . {\displaystyle \left={\frac {\partial L}{\partial \varphi ^{A}}}{\bar {\delta }}\varphi ^{A}+{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}{\varphi ^{A}}_{,\sigma }\,.} However, because the variations are defined at the same point as described above, the variation and the derivative can be done in reverse order; they commute δ ¯ φ A , σ = δ ¯ ∂ φ A ∂ x σ = ∂ ∂ x σ ( δ ¯ φ A ) . {\displaystyle {\bar {\delta }}{\varphi ^{A}}_{,\sigma }={\bar {\delta }}{\frac {\partial \varphi ^{A}}{\partial x^{\sigma }}}={\frac {\partial }{\partial x^{\sigma }}}({\bar {\delta }}\varphi ^{A})\,.} Using the Euler–Lagrange field equations ∂ ∂ x σ ( ∂ L ∂ φ A , σ ) = ∂ L ∂ φ A {\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}\right)={\frac {\partial L}{\partial \varphi ^{A}}}} the difference in Lagrangians can be written neatly as = ∂ ∂ x σ ( ∂ L ∂ φ A , σ ) δ ¯ φ A + ∂ L ∂ φ A , σ δ ¯ φ A , σ = ∂ ∂ x σ ( ∂ L ∂ φ A , σ δ ¯ φ A ) . {\displaystyle {\begin{aligned}&\left\\={}&{\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}\right){\bar {\delta }}\varphi ^{A}+{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}{\varphi ^{A}}_{,\sigma }={\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}\right).\end{aligned}}} Thus, the change in the action can be written as ∫ Ω ∂ ∂ x σ { ∂ L ∂ φ A , σ δ ¯ φ A + L ( φ A , φ A , ν , x μ ) δ x σ } d 4 x = 0 . {\displaystyle \int _{\Omega }{\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}+L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}d^{4}x=0\,.} Since this holds for any region Ω, the integrand must be zero ∂ ∂ x σ { ∂ L ∂ φ A , σ δ ¯ φ A + L ( φ A , φ A , ν , x μ ) δ x σ } = 0 . {\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}+L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}=0\,.} For any combination of the various symmetry transformations, the perturbation can be written δ x μ = ε X μ {\displaystyle \delta x^{\mu }=\varepsilon X^{\mu }} δ φ A = ε Ψ A = δ ¯ φ A + ε L X φ A {\displaystyle \delta \varphi ^{A}=\varepsilon \Psi ^{A}={\bar {\delta }}\varphi ^{A}+\varepsilon {\mathcal {L}}_{X}\varphi ^{A}} where L X φ A {\displaystyle {\mathcal {L}}_{X}\varphi ^{A}} is the Lie derivative of φA in the Xμ direction.
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