معادلات فريدمان

معادلات فريدمان Friedmann equations هي فئة من المعادلات في cosmology تحكم تمدد الفضاء في نماذج متجانسة وisotropic للكون داخل سياق النسبية العامة. أول من اشتقهم كان ألكسندر فريدمان عام 1922^[1] من Einstein's field equations of gravitation for the Friedmann-Lemaître-Robertson-Walker metric and a fluid with a given mass density ρ and pressure $p$ . The equations for negative spatial curvature were given by Friedmann in 1924.^[2]

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الافتراضات

المعادلات

There are two independent Friedmann equations for modelling a homogeneous, isotropic universe. The first is:

{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}={\frac {8\pi G\rho +\Lambda c^{2}}{3}}

which is derived from the 00 component of Einstein's field equations. The second is:

{\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}

which is derived from the first together with the trace of Einstein's field equations (the dimension of the two equations is time⁻²).

$a$ is the scale factor, $G$ , $Λ$ , and $c$ are universal constants ( $G$ is Newton's gravitational constant, $Λ$ is the cosmological constant with dimension length⁻², and $c$ is the speed of light in vacuum). $ρ$ and $p$ are the volumetric mass density (and not the volumetric energy density) and the pressure, respectively. $k$ is constant throughout a particular solution, but may vary from one solution to another.

In previous equations, $a$ , $ρ$ , and $p$ are functions of time. $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}k/a2$ is the spatial curvature in any time-slice of the universe; it is equal to one-sixth of the spatial Ricci curvature scalar $R$ since

R={\frac {6}{c^{2}a^{2}}}({\ddot {a}}a+{\dot {a}}^{2}+kc^{2})

in the Friedmann model. $H \equiv ȧ / a$ is the Hubble parameter.

We see that in the Friedmann equations, $a (t)$ does not depend on which coordinate system we chose for spatial slices. There are two commonly used choices for $a$ and $k$ which describe the same physics:

$k = +1, 0$ or $-1$ depending on whether the shape of the universe is a closed 3-sphere, flat (Euclidean space) or an open 3-hyperboloid, respectively.^[3] If $k = +1$ , then $a$ is the radius of curvature of the universe. If $k = 0$ , then $a$ may be fixed to any arbitrary positive number at one particular time. If $k = -1$ , then (loosely speaking) one can say that $i \cdot a$ is the radius of curvature of the universe.
$a$ is the scale factor which is taken to be 1 at the present time. $k$ is the current spatial curvature (when $a = 1$ ). If the shape of the universe is hyperspherical and $R t$ is the radius of curvature ( $R 0$ at the present), then $a = R t / R 0$ . If $k$ is positive, then the universe is hyperspherical. If $k = 0$ , then the universe is flat. If $k$ is negative, then the universe is hyperbolic.

Using the first equation, the second equation can be re-expressed as

{\dot {\rho }}=-3H\left(\rho +{\frac {p}{c^{2}}}\right),

which eliminates $Λ$ and expresses the conservation of mass–energy:

T^{\alpha \beta }{}_{;\beta }=0.

These equations are sometimes simplified by replacing

{\begin{aligned}\rho &\to \rho -{\frac {\Lambda c^{2}}{8\pi G}}\\p&\to p+{\frac {\Lambda c^{4}}{8\pi G}}\end{aligned}}

to give:

{\begin{aligned}H^{2}=\left({\frac {\dot {a}}{a}}\right)^{2}&={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}\\{\dot {H}}+H^{2}={\frac {\ddot {a}}{a}}&=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right).\end{aligned}}

The simplified form of the second equation is invariant under this transformation.

The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). Evaluating the Hubble parameter at the present time yields Hubble's constant which is the proportionality constant of Hubble's law. Applied to a fluid with a given equation of state, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density.

Some cosmologists call the second of these two equations the Friedmann acceleration equation and reserve the term Friedmann equation for only the first equation.

متغير الكثافة

An expression for the critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, k, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:

\rho _{c}={\frac {3H^{2}}{8\pi G}}.

The density parameter (useful for comparing different cosmological models) is then defined as:

\Omega \equiv {\frac {\rho }{\rho _{c}}}={\frac {8\pi G\rho }{3H^{2}}}.

حلول مفيدة

The Friedmann equations can be easily solved in presence of a perfect fluid with equation of state (ideal gas law)

p=w\rho c^{2},\!

where $p$ is the pressure, $\rho$ is the mass density of the fluid in the comoving frame and $w$ is some constant. The solution for the scale factor is

a(t)=a_{0}\,t^{\frac {2}{3(w+1)}}

where $a_{0}$ is some integration constant to be fixed by the choice of initial conditions.

هناك جدل ثان يقوم علي الدراسات حول كثافة الطاقة الكلية للكون. حيث كان معروفا نظريا ومشاهدا تيا منذ مدة,أن هذه الطاقة الكلية كثافتها تقترب من الكثافة الحرجة The critical density المطلوبة لجعل الكون مسطحا ومنبسطا . أو بعبارة أخري التقوس الكوني يصبح صفرا في الزمان والمكان كما جاء في النظرية النسبية العامة لإينشتين .و حبث كانت الطاقة تعادل الكتلة كما في النظرية النسبية الخاصة (E = mc2) .وهذا يمكن التعبير عنه بكثافة الكتلة الحرجة اللازمة لجعل الكون منبسطا . فالكتلة المضيئة من مادة الكون تعادل 2-5 % من الكتلة اللازمة لكثافة هذه الكتلة . لأن المادة المظلمة لاتشع ضوءا كافيا لرؤيته, مما يجعلها كتلة مخفية. لكن من خلال الملاحظات التي توصل اليها علماء الفلك عام 1990 ،حول المجرات وعناقيدها . قد جعلتهم يخمنون أن هذه المادة المظلمة لاتتعدي 25% من كثافة الكتلة الحرجة. ومن خلال الملاحظات للمستعر الأعظم تنبأ علماء الفلك بأن الطاقة المظلمة تشكل 70%من كثافة الطاقة الحرجة . وعندما تجمع كتلة المادة مع طاقتها ، تصبح الكثافة الكلية للطاقة تعادل تماما ما يحتاجه الكون ليكون منبسطا ومسطحا .

إعادة تحجيم معادلات فريدمان

Set a=ãa₀, ρ_c=3H₀²/8πG, ρ=ρ_cΩ, $t={\tilde {t}}/H_{0}$ , Ω_c=-kc²/H₀²a₀² where a₀ and H₀ are separately the scale factor and the Hubble parameter today. Then we can have

{\frac {1}{2}}\left({\frac {d{\tilde {a}}}{d{\tilde {t}}}}\right)^{2}+U_{\rm {eff}}({\tilde {a}})={\frac {1}{2}}\Omega _{c}

حيث U_eff(ã)=Ωã²/2. For any form of the effective potential U_eff(ã), there is an equation of state p=p(ρ) that will produce it.

انظر أيضاً

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المصادر

^ Friedman, A (1922). "Über die Krümmung des Raumes". Z. Phys. 10: 377–386. doi:10.1007/BF01332580. (بالألمانية) (English translation in: Friedman, A (1999). "On the Curvature of Space". General Relativity and Gravitation. 31: 1991–2000. doi:10.1023/A:1026751225741.)
^ Friedmann, A (1924). "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes". Z. Phys. 21: 326–332. doi:10.1007/BF01328280. (بالألمانية) (English translation in: Friedmann, A (1999). "On the Possibility of a World with Constant Negative Curvature of Space". General Relativity and Gravitation. 31: 2001–2008. doi:10.1023/A:1026755309811.)
^ Ray A d'Inverno, Introducing Einstein's Relativity, ISBN 0-19-859686-3.

الكلمات الدالة:

أحمد محمد عوف

[af1922-1] Friedman, A (1922). "Über die Krümmung des Raumes". Z. Phys. 10: 377–386. doi:10.1007/BF01332580. (بالألمانية) (English translation in: Friedman, A (1999). "On the Curvature of Space". General Relativity and Gravitation. 31: 1991–2000. doi:10.1023/A:1026751225741.)

[af1924-2] Friedmann, A (1924). "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes". Z. Phys. 21: 326–332. doi:10.1007/BF01328280. (بالألمانية) (English translation in: Friedmann, A (1999). "On the Possibility of a World with Constant Negative Curvature of Space". General Relativity and Gravitation. 31: 2001–2008. doi:10.1023/A:1026755309811.)

[3] Ray A d'Inverno, Introducing Einstein's Relativity, ISBN 0-19-859686-3.

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