Exam 2 Solutions

Part I

heenumi..

You live on a four-dimensional earth, and are arguing with a ``Flat Earth Society'' member. If both of you are honest and competent differential geometers,

1. Which of your arguments below should he find convincing that the universe is indeed curved? (CIRCLE the convincing ones.)
   
   
   

   0.0.

   1. In your coordinates, some $\Gamma^{i}_{\ jk}$ are nonzero. 0.
   2. In your coordinates, $R_{ijk}^{\ \ \ \ l}$ is nonzero. 0.0.
   3. Some geodesic paths eventually return to their starting points. 0.
   4. Some initially parallel geodesics become nonparallel. 0.
   5. Observers parallel transporting a vector from $A$ to $B$ along different paths sometimes disagree about its value at $B$. 0.0.
   6. In your coordinates, $g_{ij}$ is not diagonal. 0.0.
   7. In your coordinates, $g_{ij}$ is coordinate-dependent. 0.
   8. Covariant derivatives acting on vector fields sometimes fail to commute.
   
   
   
   

2. Which of his arguments below should you find convincing that the universe is indeed flat? (CIRCLE the convincing ones.)
   
   
   

   0.

   1. He shows you a coordinate transformation which everywhere transforms your coordinates into $w, x, y,z$ with the metric $ds^2 = dw^2 + dx^2 + dy^2 + dz^2$. 0.0.
   2. He shows that no initially parallel geodesics ever converge. 0.
   3. In his coordinates, $g_{ij}$ is constant everywhere. 0.
   4. In his coordinates, $R_{ijk}^{\ \ \ \ l}$ is zero.
   
   
   
   

3. Pick one nonconvincing statement each from parts (a) and (b). For each, either describe a counterexample or explain why the statement fails to be convincing.
   
   
   

   1. Statements (a) 1 and (a) 7

      both depend on coordinate choice and not intrinsic curvature. For example, polar or spherical coordinates on flat space give a coordinate dependent metric and nonzero Christoffel symbols, and yet the covariant characterizations of curvature $R_{ijk}^{\ \ \ \ l}, R_{ik}, R$ all vanish (since this is just a coordinate transform of flat Euclidean space).
      
      
      

      Statement (a) 6

      also just depends on coordinate choice; we can change basis to make a flat diagonal metric nondiagonal. Moreover, a nondiagonal but coordinate-independent metric will by definition have all Christoffel symbols vanish, which implies vanishing curvature $R_{ijk}^{\ \ \ \ l}$.
      
      
      

      Statement (a) 3

      This is the trickiest question; curvature measures local properties of the manifold, not global (like identifications $\vec{x+a}=\vec{x}$). If the manifold is everywhere locally flat, it is flat, even though identifications make some geodesics close. A 2-dimensional example you have seen of this is the cylinder, which has circular geodesics and zero curvature (since one radius of curvature is infinite).

      
      
      
      

   2. Statement (b) 2 precludes positive curvature, but not negative curvature, which makes geodesics diverge.
   
   
   
   

4. Some of your convincing arguments from part (a) are equivalent. (By equivalent, I mean that each implies the other; if arguments $A$ and $B$ are equivalent, $A$ is true if and only if $B$ is true.) Identify two of the equivalent arguments, and explain why they are equivalent.
   
   
   
   

   Actually all 4 of the convincing arguments from part (a) are equivalent; all are manifestations of nontriviality of the Riemann tensor $R_{ijk}^{\ \ \ \ l}$ (Statement (a) 2), which is defined as a measure of the noncommutation of covariant derivatives $\nabla_i, \nabla_j$ acting on vector (and other tensor) fields (Statement (a) 8)). Both geodesic deviation (Statement (a) 4) and path dependence of parallel transport (Statement (a) 5) can be calculated, and involve contraction of $R_{ijk}^{\ \ \ \ l}$ with various tangent and perpendicular vectors to the considered paths; thus they occur for some geodesics (or some vectors along some paths) if and only if $R_{ijk}^{\ \ \ \ l}$ is nontrivial.
   
   
   

heenumi..

NOTE: Answers to these questions should be short; you needn't rederive mathematical identities or concepts from class, just cite them as needed.

1. Write down Einstein's equations for gravity (taking the cosmological constant to be zero).
   $$R_{\mu\nu} - \frac{1}{2} \ R\ g_{\mu\nu} = 8\pi G\, T_{\mu\nu}$$

2. Explain why Einstein's equations imply energy-momentum conservation.
   
   
   

   Conservation of energy-momentum $p^\nu$ is local: current flow out of a region causes a reduction in density inside the region. This is expressed in the local conservation equation for the current $T^{\mu\nu}$ associated with $p^\nu$ conservation: $\nabla_\mu\, T^{\mu\nu} = 0$ (in curved space). This is consistent with Einstein's equations because the Einstein tensor $R_{\mu\nu} - \frac{1}{2} \ R\ g_{\mu\nu}$ obeys the identity $\nabla_\mu\ (\ R_{\mu\nu} - \frac{1}{2} \ R\ g_{\mu\nu}\ ) = 0$.
   
   
   

3. Why is adding a cosmological term $- \Lambda\, g_{\mu\nu}$ to the geometric side of Einstein's equations still consistent with energy conservation?
   
   
   

   because we constructed the covariant derivative explicitly so that $\nabla_\mu\, g_{\mu\nu} = 0$, to preserve inner products under parallel transport. (Note $\Lambda$ is a constant).
   
   
   

4. What is the relationship between Einstein's and Friedmann's equations?
   
   
   

   Friedmann's equations are obtained by applying Einstein's equations to the most general spacetime metric $g_{\mu\nu}$ and stress energy tensor $T_{\mu\nu}$ consistent with the cosmological principle (spatial homogeneity and isotropy). The two Friedmann equations are linear combinations of the only 2 nontrivial equations that result from Einstein's equations for such a universe (a Friedmann-Robertson-Walker universe).
   
   
   

5. State one way in which Friedmann's equations differ from a quasiNewtonian analysis of the Hubble expansion.
   
   
   

   Friedmann's second equation states that the quantity $\rho + 3p$ (including pressure for nonrelativistic matter), instead of just the energy density $\rho$, drives gravitational acceleration. (Friedmann's first equation is identical to the quasiNewtonian energy equation.)
   
   
   

Part II

NOTE: We use units where $c=1$ throughout Part II of this exam!

heenumi..

In this problem, we solve a Friedmann-Robertson-Walker geodesic equation for the evolution of a particle's peculiar velocity, or apparent velocity in comoving coordinates. This velocity measures the particle's motion with respect to the comoving reference frame. Thus a particle which remains at the same comoving coordinate value -- physically comoving with the universe's expansion -- has zero peculiar velocity.

For the Friedmann-Robertson-Walker metric,

$$ds^2 = dt^2 - a^2(t)\ \left(\ \frac{d\sigma^2}{1-k\sigma^2} + \sigma^2\ (d\theta^2 + \sin^2\theta\ d\phi^2\ )\ \right) \ \ ,$$

1. Show that the Christoffel symbols $\Gamma^0_{\ \mu\nu}$ are given by
   $$\Gamma^0_{\ 0\nu} = \Gamma^0_{\ \mu 0} = 0 \hspace{1in} \Gamma^0_{\ ij} = -\ \frac{\dot{a}}{a}\ g_{\ ij}\ \ ,$$
   where $i,j$ index spatial coordinates only.
   

   We must calculate the Christoffel symbols

   $$\Gamma^0_{\ \ \nu\rho} = \frac{1}{2}\ g^{0\sigma}\ \left(\ \part... ...o} + \partial_\rho\, g_{\sigma \nu} - \partial_\sigma\, g_{\nu\rho} \ \right)$$
   for the diagonal metric
   $$g_{\mu\nu} = \left(\ \begin{array}{cccc} 1&&&\\ &\frac{-a^2\, (... ...igma^2&\\ &&& -a^2\, (t) \, \sigma^2\, \sin^2\theta \end{array}\ \right)\ \ .$$
   Noting that $g^{0\sigma} = \delta^{0\sigma}$ vanishes unless $\sigma = 0$, we do the $\sigma$ sum to get
   $$\Gamma^0_{\ \ \nu\rho} \quad = \quad \frac{1}{2}\ \left(\ \parti... ...u\rho} \ \right) \quad = \quad - \ \frac{1}{2} \ \partial_0\, g_{\nu\rho} \ ,$$
   using the constancy of $g_{0\rho} = \delta_{0\rho}$. But $g_{\nu\rho}$ has time dependence only for the spatial components $g_{ij}$, which depend on $a(t)$. These give
   $$\Gamma^0_{\ \ ij} = - \ \frac{1}{2} \ \left(\ \frac{2\dot{a}}{a} \ g_{ij}\ \right) = - \ \frac{ \dot{a}}{a} \ g_{ij}\ \ ,$$
   with all other $\Gamma^0_{\ \ \nu\rho}$ vanishing. $\qedsymbol$
   

2. Write down the 0 component of the geodesic equation, using proper time $\tau$ along the geodesic as your parametrization variable. Write your result in terms of $u^o, \vert\vec{u}\vert$ where $u^\mu = (u^o, \vec{u})$ is the peculiar 4-velocity $dx^\mu/d\tau$ along the geodesic. RECALL: we define the magnitude of a spatial vector in general by $\vert\vec{a}\vert^2 = - \,g_{ij}\, a^i\, a^j$.
   

   Using $\tau$ as our parametrization variable, the 0 component of the geodesic equation is
   

   $$= \frac{d^{\, 2}\, x^o}{d\, \tau^2} + \Gamma^0_{\ \ \mu\nu}\ {\frac{d\, {x^\mu}}{d\, {\tau}}}\ {\frac{d\, {x^\nu}}{d\, {\tau}}}$$ 0

   $$= {\frac{d\, {u^o}}{d\, {\tau}}} + \Gamma^0_{\ \ \mu\nu}\ u^\mu\ u^... ...ad {\frac{d\, {u^o}}{d\, {\tau}}} - \frac{ \dot{a}}{a} \ g_{ij}\ u^i\ u^j \ \ ,$$

   
   
   where we have plugged in for the Christoffel symbols $\Gamma^0_{\ \mu\nu}$. Recognizing the spatial dot product gives
   $$0 = {\frac{d\, {u^o}}{d\, {\tau}}} + \frac{ \dot{a}}{a} \ \ \vert\vec{u}\vert^{\, 2}\ \ .$$
   $\qedsymbol$

3. For $\gamma \equiv (\ 1 - \vert\vec{v}\vert^2\ )^{-1/2}$, where $v^i = dx^i/dt$, use the following identities
   

   $$d\tau = dt \ \gamma^{-1} \$$

   $$u^\mu = \gamma\ (1, \vec{v}) \quad \Rightarrow \quad u_\mu\, u^\mu = {u^o}^2 - \vert\vec{u}\vert^2 = 1$$

   
   
   to show that

   1. $$\frac{du^o}{d\tau} = u^o\ \frac{du^o}{dt}\ \ .$$
      
      
      $$\frac{du^o}{d\tau} = {\frac{d\, {t}}{d\, {\tau}}}\ \frac{du^o}{dt} = \gamma\ \frac{du^o}{dt} = u^o\ \frac{du^o}{dt}\ \ .$$
      $\qedsymbol$

   2. $$\frac{du^o}{d\tau} = \vert\vec{u}\vert\ \frac{d\, \vert\vec{u}\vert}{dt}\ \ .$$
      
      
      $$0 = {\frac{d\, {\ }}{d\, {t}}}\ \left(\ {u^o}^2 - \vert\vec{u}\ve... ... {t}}} - \vert\vec{u}\vert\ {\frac{d\, {\vert\vec{u}\vert}}{d\, {t}}}\ \right)$$
      implies that $u^o\ d u^o/dt =\vert\vec{u}\vert\ d\vert\vec{u}\vert/dt$, so
      $$\frac{du^o}{d\tau} = u^o\ \frac{du^o}{dt} = \vert\vec{u}\vert\ \frac{d\, \vert\vec{u}\vert}{dt}\ \ .$$
      $\qedsymbol$

4. Rewrite the geodesic equation you found in part (b) as a differential equation for $\vert\vec{u}\vert\ (t)$. Show that this differential equation is obeyed by $\vert\vec{u}\vert\ \propto\ a^{-1} \ (t)$.

   NOTE: This implies that the peculiar velocities redshift (along with the particle momenta $\vert\vec{p}\vert= m\vert\vec{u}\vert$); thus any peculiar velocity a particle initially has is damped out by to the expansion, and the particle eventually settles into a comoving expansion with the universe.

   
   

   Using the result from (c), the geodesic equation from (b) can be rewritten

   $$0 = \frac{d\, \vert\vec{u}\vert}{dt} + \frac{ \dot{a}}{a} \ \ \vert\vec{u}\vert\ \ .$$
   $\qedsymbol$

   Note that $\vert\vec{u}\vert = \alpha \ a^{-1} \ (t)$, with $d\, \vert\vec{u}\vert/dt = - \alpha \ a^{-2}\ \dot{a} = - (\dot{a}/a) \ \vert\vec{u}\vert$ obeys this differential equation. $\qedsymbol$

heenumi..

In this problem, we consider evolution of a Friedmann-Robertson-Walker universe containing radiation (with a relativistic equation of state $w=1/3$) and matter (with a nonrelativistic equation of state $w=0$), only. Assume that the universe's total energy density $\rho$ is initially radiation-dominated, $\rho_{\rm rad} \gg \rho_{\rm matt}$.

1. Assume that energy is conserved independently for radiation and for matter. Find the power law decay exponent $n$, for $\rho \sim a^n$, for both $\rho_{\rm matt}$ and $\rho_{\rm rad}$. Show a quick derivation, and plot the initial energy density decay on the following log-log plot, indicating slopes.

   $\parbox{3.5in}{ For each constituent, independent energy conservatio... ...3}$\ ({with $w=0$}) and $\rho_{\rm rad} \sim a^{-4}$\ ({with $w=1/3$}). \qed }$

2. Assuming the ``early time'' approximation discussed in class remains valid, solve for the scale factor evolution $a(t)$ during the radiation-dominated era (when $\rho \approx \rho_{\rm rad}$ because $\rho_{\rm rad} \gg \rho_{\rm matt}$). Again show a quick derivation and plot the initial scale factor growth on the following log-log plot, indicating slopes.

   $\parbox{3.5in}{From Friedmann's first equation, \begin{displaymath}... ...\ for the dominant $\rho$\ and $p$\ in the universe, so $a \sim t^{1/2}$.\qed }$

3. Assume that at $t_o$, $\rho_{\rm rad} = 10^4\ \rho_{\rm matt}$. Show that eventually, $\rho_{\rm matt}$ will equal then come to dominate $\rho_{\rm rad}$. Calculate $t_{\rm mre}$, the time of ``matter-radiation equality,'' when $\rho_{\rm rad} = \rho_{\rm matt}$.

   
   

   From our graph to part (a), it should be clear that $\rho_{\rm rad}$ declines faster than $\rho_{\rm matt}$ and the two will intersect, or be equal, at $t_{\rm mre}$. Quantitatively,

   $$\rho_{\rm rad} = \rho_{\rm rad, o}\ \left(\ \frac{a}{a_o}\ \right... ...rac{\rho_{\rm matt, o}}{\rho_{\rm rad, o}}\ \left(\ \frac{a}{a_o}\ \right)\ \ .$$
   Since this catching up of $\rho_{\rm matt}$ is occurring during the RD era, $a/a_o = (t/t_o)^{1/2}$. Thus, plugging in our initial $\rho_{\rm matt, o}/\rho_{\rm rad, o} = 10^{-4}$, we get matter radiation equality when
   $$1 = 10^{-4}\ (t/t_o)^{1/2} \quad \Rightarrow \quad t_{\rm mre} = 10^8\ t_o\ \ .$$
   $\qedsymbol$

   
   

4. Assume that initially, in the matter-dominated phase following matter-radiation equality, the ``early time'' approximation remains valid. Derive the scale factor evolution $a(t)$.

   
   

   The derivation in part (b) holds, with $w=0$. Thus

   $$a \sim t^{\frac{2}{3(1+w)}} \sim t^{2/3}\ \ .$$
   $\qedsymbol$

   
   

5. Now consider the cases $k=0, -1$ for the curvature of this universe. Assume that at some $t_c > t_{\rm mre}$, the curvature term in Friedmann's first equation becomes dominant for $k = -1$. Derive the scale factor evolution in this late era, for $k=0$ and $k = -1$.

   
   

   We return to the full first Friedmann equation,

   $$\left(\ \frac{\dot{a}}{a}\ \right)^2 = \frac{8\pi G\rho}{3} - \frac{k}{a^2}\ \ .$$
   For $k=0$, there is never a curvature term, and the analysis of (b) and (d) remains valid forever, with the universe matter-dominated with $a\sim t^{2/3}$ at late times. $\qedsymbol$

   For $k = -1$, eventually $a$ becomes large and the curvature term dominates. Then

   $$\dot{a}^2 = -k = +1 \quad \Rightarrow a\sim t$$
   and the expansion goes faster.$\qedsymbol$

6. Summarize your findings on the following log-log plots for the time evolution of the universe's energy density ( $\rho_{\rm rad}$ and $\rho_{\rm matt}$) and of its scale factor $a(t)$ (in the $k=0$ and $k = -1$ cases). CAREFUL: Note we're plotting $\rho$ versus time, not scale factor; be careful to clearly label slopes and to distinguish $\rho_{\rm rad}$ and $\rho_{\rm matt}$ trajectories, as well as $k=0$ and $k = -1$ trajectories for $a(t)$.

   
   

   $a$ versus $t$ is straightforward. Before $t_{\rm mre}$, we are RD with $a \sim t^{1/2}$; between $t_{\rm mre}$ and $t_c$ we are MD with $a\sim t^{2/3}$; and after $t_c$ we are MD with any nonzero curvature term dominant, so $a\sim t^{2/3}$ for $k=0$ and $a\sim t$ for $k = -1$.

   $\rho$ versus $t$ is trickier. Independent energy conservation tells us that always, $\rho_{\rm matt} \sim a^{-3}$ and $\rho_{\rm rad} \sim a^{-4}$. $a$'s scaling with $t$ depends on the dominant component of total $\rho$ and $p$ in the universe, and varies as in the previous paragraph. Thus

   $$\rho_{\rm rad} \sim a^{-4} \sim \left\{ \begin{array}{ll} t^{-2}&... ...< t< t_c\\ t^{-2} &t>t_c, k = 0\\ t^{-3} &t>t_c, k = -1 \end{array}\right.$$

   These dependences are plotted below.

   
   

   
   

   

   
   

7. (BONUS) While we can't fully calculate the late time behavior of $a(t)$ in the $k = +1$ case, qualitatively, what happens to the scale factor for this case? Sketch it on your plot above for $a(t)$.

   In Friedmann's first equation, the curvature term grows until it just balances the $\rho$ term, at which point $\dot{a} = 0$, $a$ is at a local maximum, then begins contracting. The contraction is symmetric to the expansion, and is sketched above.

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