Notebook

SageMath para físicos¶

Rogério T. C.

Aula IV.2¶

Equações de Friedmann¶

In [1]:

%display latex
reset()

Princípio cosmológico

Observações sugerem que nosso universo é espacialmente homogêneo e isotrópico.
Homogeneidade implica em invariância na curvatura em translações (vetores de Killing).

Isotropia significa invariância por rotações (vetores de Killing).

Em relatividade geral, isso se traduz na afirmação de que o universo pode ser folheado em fatias do tipo-espaço espaço, de modo que cada fatia tridimensional seja maximalmente simétrica (Topologicamente $\mathbb{R} \times \Sigma$ , com $\Sigma$ maximalmente simétrico).

A métrica de um espaço-tempo do tipo $\mathbb{R} \times \Sigma$ pode ser escrita como

$\begin{equation} ds^2=-dt^2+a^2(t)\left(\frac{dr^2}{1-kr^2}+r^2d\Omega^2\right). \end{equation}$

Que é a famosa métrica de Friedmann-Robertson-Walker (FRW)

Vamos usar a equação de Einstein para encontrar uma equação dinâmica para o fator de escala $a(t)$ .

Declarações iniciais¶

Declarando um espaço-tempo de 4 dimensões com estrutura Lorenziana.

In [2]:

frw = Manifold(4, 'FRW', structure='Lorentzian')
show(frw)
print(frw)

$\newcommand{\Bold}[1]{\mathbf{#1}}FRW$

4-dimensional Lorentzian manifold FRW

Declarando as cartas (coordenadas) do espaço-tempo

In [3]:

Ch.<t,r,th,ph> = frw.chart(r't r:[0,+oo) th:[0,pi]:\theta ph:[0,2*pi):\phi')
Ch

Out[3]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(FRW,(t, r, {\theta}, {\phi})\right)$

Precisaremos de variaveis simbólicas para constange gravitacional $G$ , a constante cosmológica $\Lambda$ e o parâmetro de curvatura $k$ .

In [4]:

var('G, Lambda, k', domain='real')

Out[4]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(G, \Lambda, k\right)$

Além de um campo escalar $a$ para o fator de escala.

In [5]:

a = frw.scalar_field(function('a')(t), name='a')

In [6]:

a.display()

Out[6]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} a:& FRW & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & a\left(t\right) \end{array}$

Tensores fundamentais¶

Podemos agora declarar a métrica de FRW e iniciar suas componentes.

$ds^2=-dt^2+a^2(t)\frac{dr^2}{1-kr^2}+a^2(t)r^2 d\theta^2+a^2(t)r^2\sin^2\theta d\phi^2$

In [7]:

g = frw.metric()
print(g)

Lorentzian metric g on the 4-dimensional Lorentzian manifold FRW

In [8]:

g[0,0] = -1
g[1,1] = a*a/(1 - k*r^2)
g[2,2] = a*a*r^2
g[3,3] = a*a*(r*sin(th))^2
g.display()

Out[8]:

$\newcommand{\Bold}[1]{\mathbf{#1}}g = -\mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{a\left(t\right)^{2}}{k r^{2} - 1} \right) \mathrm{d} r\otimes \mathrm{d} r + r^{2} a\left(t\right)^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + r^{2} a\left(t\right)^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

In [9]:

g[:]

Out[9]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 0 & -\frac{a\left(t\right)^{2}}{k r^{2} - 1} & 0 & 0 \\ 0 & 0 & r^{2} a\left(t\right)^{2} & 0 \\ 0 & 0 & 0 & r^{2} a\left(t\right)^{2} \sin\left({\theta}\right)^{2} \end{array}\right)$

Conexão de Levi-Civita

In [10]:

nabla = g.connection()
print(nabla)

Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional Lorentzian manifold FRW

In [11]:

nabla

Out[11]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\nabla_{g}$

Símbolos de Christoffel

In [12]:

g.christoffel_symbols_display()

Out[12]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \Gamma_{ \phantom{\, t} \, r \, r }^{ \, t \phantom{\, r} \phantom{\, r} } & = & -\frac{a\left(t\right) \frac{\partial\,a}{\partial t}}{k r^{2} - 1} \\ \Gamma_{ \phantom{\, t} \, {\theta} \, {\theta} }^{ \, t \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & r^{2} a\left(t\right) \frac{\partial\,a}{\partial t} \\ \Gamma_{ \phantom{\, t} \, {\phi} \, {\phi} }^{ \, t \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & r^{2} a\left(t\right) \sin\left({\theta}\right)^{2} \frac{\partial\,a}{\partial t} \\ \Gamma_{ \phantom{\, r} \, t \, r }^{ \, r \phantom{\, t} \phantom{\, r} } & = & \frac{\frac{\partial\,a}{\partial t}}{a\left(t\right)} \\ \Gamma_{ \phantom{\, r} \, r \, r }^{ \, r \phantom{\, r} \phantom{\, r} } & = & -\frac{k r}{k r^{2} - 1} \\ \Gamma_{ \phantom{\, r} \, {\theta} \, {\theta} }^{ \, r \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & k r^{3} - r \\ \Gamma_{ \phantom{\, r} \, {\phi} \, {\phi} }^{ \, r \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & {\left(k r^{3} - r\right)} \sin\left({\theta}\right)^{2} \\ \Gamma_{ \phantom{\, {\theta}} \, t \, {\theta} }^{ \, {\theta} \phantom{\, t} \phantom{\, {\theta}} } & = & \frac{\frac{\partial\,a}{\partial t}}{a\left(t\right)} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, {\theta} }^{ \, {\theta} \phantom{\, r} \phantom{\, {\theta}} } & = & \frac{1}{r} \\ \Gamma_{ \phantom{\, {\theta}} \, {\phi} \, {\phi} }^{ \, {\theta} \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\cos\left({\theta}\right) \sin\left({\theta}\right) \\ \Gamma_{ \phantom{\, {\phi}} \, t \, {\phi} }^{ \, {\phi} \phantom{\, t} \phantom{\, {\phi}} } & = & \frac{\frac{\partial\,a}{\partial t}}{a\left(t\right)} \\ \Gamma_{ \phantom{\, {\phi}} \, r \, {\phi} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\phi}} } & = & \frac{1}{r} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & \frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \end{array}$

Tensor de Ricci

In [13]:

Ricci = nabla.ricci()
Ricci.display()

Out[13]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{Ric}\left(g\right) = -\frac{3 \, \frac{\partial^2\,a}{\partial t ^ 2}}{a\left(t\right)} \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{2 \, \left(\frac{\partial\,a}{\partial t}\right)^{2} + a\left(t\right) \frac{\partial^2\,a}{\partial t ^ 2} + 2 \, k}{k r^{2} - 1} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( 2 \, r^{2} \left(\frac{\partial\,a}{\partial t}\right)^{2} + r^{2} a\left(t\right) \frac{\partial^2\,a}{\partial t ^ 2} + 2 \, k r^{2} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + {\left(2 \, r^{2} \left(\frac{\partial\,a}{\partial t}\right)^{2} + r^{2} a\left(t\right) \frac{\partial^2\,a}{\partial t ^ 2} + 2 \, k r^{2}\right)} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

In [14]:

Ricci.display_comp()

Out[14]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \mathrm{Ric}\left(g\right)_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & -\frac{3 \, \frac{\partial^2\,a}{\partial t ^ 2}}{a\left(t\right)} \\ \mathrm{Ric}\left(g\right)_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & -\frac{2 \, \left(\frac{\partial\,a}{\partial t}\right)^{2} + a\left(t\right) \frac{\partial^2\,a}{\partial t ^ 2} + 2 \, k}{k r^{2} - 1} \\ \mathrm{Ric}\left(g\right)_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & 2 \, r^{2} \left(\frac{\partial\,a}{\partial t}\right)^{2} + r^{2} a\left(t\right) \frac{\partial^2\,a}{\partial t ^ 2} + 2 \, k r^{2} \\ \mathrm{Ric}\left(g\right)_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & {\left(2 \, r^{2} \left(\frac{\partial\,a}{\partial t}\right)^{2} + r^{2} a\left(t\right) \frac{\partial^2\,a}{\partial t ^ 2} + 2 \, k r^{2}\right)} \sin\left({\theta}\right)^{2} \end{array}$

Escalar de Ricci

In [15]:

Ricci_scalar = g.ricci_scalar()
Ricci_scalar.display()

Out[15]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} \mathrm{r}\left(g\right):& FRW & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{6 \, {\left(\left(\frac{\partial\,a}{\partial t}\right)^{2} + a\left(t\right) \frac{\partial^2\,a}{\partial t ^ 2} + k\right)}}{a\left(t\right)^{2}} \end{array}$

Tensor energia momento¶

Introduziremos o tensor energia-momento de fluido perfeito. Para isso precisaremos de um campos escalar para a pressão $p(t)$ , um campo escalar para a densidade de energia $\rho(t)$ e um tensor covariante do tipo tempo (1-forma).

In [16]:

rho = frw.scalar_field(function('rho')(t), name='rho', latex_name=r'\rho')
p = frw.scalar_field(function('p')(t), name='p')

In [17]:

show(rho)
print(rho)

$\newcommand{\Bold}[1]{\mathbf{#1}}\rho$

Scalar field rho on the 4-dimensional Lorentzian manifold FRW

Vetor

In [18]:

u = frw.vector_field('u')
u[0] = 1
print(u)
u.display()

Vector field u on the 4-dimensional Lorentzian manifold FRW

Out[18]:

$\newcommand{\Bold}[1]{\mathbf{#1}}u = \frac{\partial}{\partial t }$

In [19]:

g(u,u).expr()

Out[19]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-1$

$g(u,u) \Leftrightarrow g_{\mu\nu}u^\mu u^\nu$

u.dot(u) é equivalente a g(u,u).

In [20]:

u.dot(u).expr()

Out[20]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-1$

In [21]:

u_form = u.down(g,0) #1-forma dual a u (baixar índice)

In [22]:

print(u_form)
u_form.display()

1-form on the 4-dimensional Lorentzian manifold FRW

Out[22]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-\mathrm{d} t$

Tensor energia momento

$T_{\mu\nu} = (p+\rho)u_\mu u_\nu+pg_{\mu\nu}$ ou

$T = (p+\rho)u \otimes u+pg$

In [23]:

T = (rho+p)*(u_form*u_form) + p*g
T.set_name('T')

In [24]:

print(T)
T.display()

Field of symmetric bilinear forms T on the 4-dimensional Lorentzian manifold FRW

Out[24]:

$\newcommand{\Bold}[1]{\mathbf{#1}}T = \rho\left(t\right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{a\left(t\right)^{2} p\left(t\right)}{k r^{2} - 1} \right) \mathrm{d} r\otimes \mathrm{d} r + r^{2} a\left(t\right)^{2} p\left(t\right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + r^{2} a\left(t\right)^{2} p\left(t\right) \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

Equação de Einstein¶

$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu}$

$\text{Ric}(g) - \frac{1}{2}\text{r}(g)g+\Lambda g = 8\pi G T$

In [25]:

Ein = Ricci - Ricci_scalar/2*g + Lambda*g - (8*pi*G)*T

In [26]:

Ein.set_name('Ein')

In [27]:

Ein.display_comp()

Out[27]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} Ein_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & -\frac{8 \, \pi G a\left(t\right)^{2} \rho\left(t\right) + \Lambda a\left(t\right)^{2} - 3 \, \left(\frac{\partial\,a}{\partial t}\right)^{2} - 3 \, k}{a\left(t\right)^{2}} \\ Ein_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & \frac{8 \, \pi G a\left(t\right)^{2} p\left(t\right) - \Lambda a\left(t\right)^{2} + \left(\frac{\partial\,a}{\partial t}\right)^{2} + 2 \, a\left(t\right) \frac{\partial^2\,a}{\partial t ^ 2} + k}{k r^{2} - 1} \\ Ein_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & -8 \, \pi G r^{2} a\left(t\right)^{2} p\left(t\right) + \Lambda r^{2} a\left(t\right)^{2} - r^{2} \left(\frac{\partial\,a}{\partial t}\right)^{2} - 2 \, r^{2} a\left(t\right) \frac{\partial^2\,a}{\partial t ^ 2} - k r^{2} \\ Ein_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & -{\left(8 \, \pi G r^{2} a\left(t\right)^{2} p\left(t\right) - \Lambda r^{2} a\left(t\right)^{2} + r^{2} \left(\frac{\partial\,a}{\partial t}\right)^{2} + 2 \, r^{2} a\left(t\right) \frac{\partial^2\,a}{\partial t ^ 2} + k r^{2}\right)} \sin\left({\theta}\right)^{2} \end{array}$

Primeira equação de Friedman

In [28]:

Ein[0,0].expr().expand()==0

Out[28]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-8 \, \pi G \rho\left(t\right) - \Lambda + \frac{3 \, \frac{\partial}{\partial t}a\left(t\right)^{2}}{a\left(t\right)^{2}} + \frac{3 \, k}{a\left(t\right)^{2}} = 0$

In [29]:

eqs = [Ein[0,0].expr().expand()==0];eqs

Out[29]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[-8 \, \pi G \rho\left(t\right) - \Lambda + \frac{3 \, \frac{\partial}{\partial t}a\left(t\right)^{2}}{a\left(t\right)^{2}} + \frac{3 \, k}{a\left(t\right)^{2}} = 0\right]$

In [30]:

eqs[0] += 8*pi*G*rho.expr()

In [31]:

eqs[0]

Out[31]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-\Lambda + \frac{3 \, \frac{\partial}{\partial t}a\left(t\right)^{2}}{a\left(t\right)^{2}} + \frac{3 \, k}{a\left(t\right)^{2}} = 8 \, \pi G \rho\left(t\right)$

Segunda Forma da equação de Einstein

$R_{\mu\nu} - \Lambda g_{\mu\nu} = 8\pi G \left(T_{\mu\nu}-\frac{1}{2}T g_{\mu\nu}\right)$ ou

$\text{Ric}(g) - \Lambda g = 8\pi G \left(T-\frac{1}{2}tr(T)g\right)$

Taço de T

In [32]:

tr_T = T.up(g,0).trace()
tr_T.set_name('tr(T)')
tr_T.display()

Out[32]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} tr(T):& FRW & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & 3 \, p\left(t\right) - \rho\left(t\right) \end{array}$

In [33]:

Ein2 = Ricci - Lambda*g - (8*pi*G)*(T - tr_T/2*g)

Segunda equação de Friedmann

In [34]:

eq2 = Ein2[0,0].expr().expand() == 0; eq2

Out[34]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-12 \, \pi G p\left(t\right) - 4 \, \pi G \rho\left(t\right) + \Lambda - \frac{3 \, \frac{\partial^{2}}{(\partial t)^{2}}a\left(t\right)}{a\left(t\right)} = 0$

In [35]:

eq2+(4*pi*G*(rho+3*p)).expr().expand()

Out[35]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\Lambda - \frac{3 \, \frac{\partial^{2}}{(\partial t)^{2}}a\left(t\right)}{a\left(t\right)} = 12 \, \pi G p\left(t\right) + 4 \, \pi G \rho\left(t\right)$

In [36]:

eqs.append(eq2+(4*pi*G*(rho+3*p)).expr().expand())

In [37]:

for eq in eqs:
    show(eq)

$\newcommand{\Bold}[1]{\mathbf{#1}}-\Lambda + \frac{3 \, \frac{\partial}{\partial t}a\left(t\right)^{2}}{a\left(t\right)^{2}} + \frac{3 \, k}{a\left(t\right)^{2}} = 8 \, \pi G \rho\left(t\right)$

$\newcommand{\Bold}[1]{\mathbf{#1}}\Lambda - \frac{3 \, \frac{\partial^{2}}{(\partial t)^{2}}a\left(t\right)}{a\left(t\right)} = 12 \, \pi G p\left(t\right) + 4 \, \pi G \rho\left(t\right)$

Conservação e equação de estado¶

In [38]:

nt0 = nabla(T)
show(nt0)
print(nt0)

$\newcommand{\Bold}[1]{\mathbf{#1}}\nabla_{g} T$

Tensor field nabla_g(T) of type (0,3) on the 4-dimensional Lorentzian manifold FRW

In [39]:

T.up(g)

Out[39]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mbox{Tensor field of type (2,0) on the 4-dimensional Lorentzian manifold FRW}$

In [40]:

nT = nabla(T.up(g))
print(nT)
nT

Tensor field of type (2,1) on the 4-dimensional Lorentzian manifold FRW

Out[40]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mbox{Tensor field of type (2,1) on the 4-dimensional Lorentzian manifold FRW}$

In [41]:

cons_T0 = nT.trace(1,2)
show(cons_T0)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mbox{Vector field on the 4-dimensional Lorentzian manifold FRW}$

In [42]:

cons_T0.display()

Out[42]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left( \frac{3 \, {\left(p\left(t\right) + \rho\left(t\right)\right)} \frac{\partial\,a}{\partial t} + a\left(t\right) \frac{\partial\,\rho}{\partial t}}{a\left(t\right)} \right) \frac{\partial}{\partial t }$

In [43]:

cons_T0[0]

Out[43]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, {\left(p\left(t\right) + \rho\left(t\right)\right)} \frac{\partial\,a}{\partial t} + a\left(t\right) \frac{\partial\,\rho}{\partial t}}{a\left(t\right)}$

In [44]:

con_T = cons_T0[0].expr()==0;con_T

Out[44]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, {\left(p\left(t\right) + \rho\left(t\right)\right)} \frac{\partial}{\partial t}a\left(t\right) + a\left(t\right) \frac{\partial}{\partial t}\rho\left(t\right)}{a\left(t\right)} = 0$

In [45]:

con_T = con_T*a.expr();con_T

Out[45]:

$\newcommand{\Bold}[1]{\mathbf{#1}}3 \, {\left(p\left(t\right) + \rho\left(t\right)\right)} \frac{\partial}{\partial t}a\left(t\right) + a\left(t\right) \frac{\partial}{\partial t}\rho\left(t\right) = 0$

Equação de estado

In [46]:

om = var('omega')

In [47]:

eq_est = p.expr() == om*rho.expr();eq_est

Out[47]:

$\newcommand{\Bold}[1]{\mathbf{#1}}p\left(t\right) = \omega \rho\left(t\right)$

In [48]:

dom_eq = con_T.subs(eq_est); dom_eq

Out[48]:

$\newcommand{\Bold}[1]{\mathbf{#1}}3 \, {\left(\omega \rho\left(t\right) + \rho\left(t\right)\right)} \frac{\partial}{\partial t}a\left(t\right) + a\left(t\right) \frac{\partial}{\partial t}\rho\left(t\right) = 0$

In [49]:

dom_eq2 = solve(dom_eq,diff(rho.expr(),t))[0]/rho.expr();dom_eq2

Out[49]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\frac{\partial}{\partial t}\rho\left(t\right)}{\rho\left(t\right)} = -\frac{3 \, {\left(\omega + 1\right)} \frac{\partial}{\partial t}a\left(t\right)}{a\left(t\right)}$

In [50]:

dom_eq3 = integral(dom_eq2,t);dom_eq3

Out[50]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\rho\left(t\right)\right) = -3 \, {\left(\omega + 1\right)} \log\left(a\left(t\right)\right) + c_{1}$

In [51]:

solve(dom_eq3,rho.expr())[0].canonicalize_radical()

Out[51]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\rho\left(t\right) = a\left(t\right)^{-3 \, \omega - 3} e^{c_{1}}$

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