Exam 1 Solutions

Part I

heenumi..

1. For each theoretical paradox below, 1) state (DON'T DERIVE!) the static Newtonian prediction; and 2) say why that prediction was manifestly inconsistent. NOTE: by inconsistent, we mean that result violates the theory's own assumptions or pre-existing observations, NOT that future observations proved it wrong.
   • Olbers' Paradox
     1. The flux of starlight on earth is comparable to the daytime brightness of the sun (if we account for the opaqueness of stars).

     2. The night sky looks dark, even to theorists.

   • Gravitational Collapse
     1. Whatever point we choose as our origin, the universe wants to collapse radially inward toward it (that is, galaxies experience a radially inward force).

     2. (2 reasons)
        • Physical predictions should be independent of choice of origin.
        • Collapse is obviously nonstatic (counter to static assumption).
        • Collapse would lead to nonuniformity (counter to uniform assumption).
        
        

2. In 1929 Hubble measured the distances $R$ of galaxies to be proportional to their recessional velocities $\dot{R}$. How were observers able to measure
   1. a galaxy's recessional velocity, upon observing spectral lines due to known atomic transitions emitted by stars in the galaxy?
      

      Spectral lines are redshifted when a galaxy recedes, by an amount

      $$\Delta\, \lambda = z\, \lambda \approx v/c\ \lambda$$
      in the nonrelativistic limit. Thus by measuring $\Delta \lambda$ for a line with known $\lambda$, we obtain the recession velocity $v$.

   2. a galaxy's distance, upon observing Cepheid variable stars of known luminosity in the galaxy?
      

      The star's apparent brightness (or flux) falls as $1/R^2$,

      $$\Phi = \frac{L}{4\pi\ R^2}\ .$$
      Thus by measuring the brightness $\Phi$ of a Cepheid variable with known luminosity $L$ (known, say, by measurement of its period), we obtain the Cepheid's distance $R$.

heenumi..

1. The equivalence principle of general relativity can be stated in many ways. State the equivalence principle precisely as it pertains to each of the following physical questions:
   1. an object's coupling $m_g$ to a gravitational field $\vec{g}$
      $$m_g = m_i\ \ .$$
      (An object's inertial mass is equal to its gravitational mass.)

      
      

   2. the observability of a constant gravitational field $\vec{g}$

      
      

      A constant gravitational field $\vec{g}$ in an inertial frame cannot be distinguished from an accelerated frame with $\vec{a}= -\ \vec{g}$.

      
      

   3. the existence of inertial frames for nongravitational forces

      
      

      A freely falling frame ( $\vec{a}= \vec{g}$) is locally inertial for nongravitational forces. That is, inside such frames, all observers detect no trace of gravity, but see the same motions, which are governed by $\vec{F}= m\,\vec{a}$ for all nongravitational forces $\vec{F}$.

      
      

2. Give two consequences of the Equivalence Principle, for motion of light in a uniform gravitational field $\vec{g}$.
   
   • the path of light bends in a uniform gravitational field $\vec{g}$.
   • light redshifts when moving opposite a uniform gravitational field $\vec{g}$.

   
   

3. When do tidal forces arise, and how are they relevant to the Equivalence Principle?
   
   • whenever $\vec{g}$ is nonuniform
   • They make us able to define inertial frames only locally: whenever a frame is large enough for tidal forces to be observable, we no longer have an inertial frame obeying $\vec{F}= m\vec{a}$ for all observed motions, and the gravitational field $\vec{g}$ is distinguishable from an accelerated frame.

heenumi..

For each object below, state how it transforms under Lorentz transformations: whether as a scalar, a four-vector, a covector, or other-rank tensor; or, if none of the above, say what other physical observables it mixes with).

	Lorentz transforms as
$x^\mu$	4-vector
$t$	with $\vec{x}/c$
$\eta_{\mu\nu}\,\, dx^\mu\, dx^\nu$	scalar
$\vec{p}$	with $E/c$
$\eta_{\mu\nu} p^\mu$	covector
$\partial_\mu \,\, j^\mu$	scalar
${\boldsymbol {\nabla}}\,\cdot\vec{j}$	with $\frac{\partial \rho}{\partial t}$
$\partial_{\mu} \,\, F^{\mu\nu}$	4-vector
$\omega$	with wave vector $c\, \vec{k}$
$\eta_{\rho\nu} \,\, F^{\mu\nu}$	(1,1) tensor

Part II

heenumi..

In class we considered Newtonian expansion of galaxies obeying Hubble's law, $\dot{r} = Hr$, finding approximate solutions of the form $r \ \propto\ t^{2/n}$, for $n=3$ and $4$.

Consider other possible solutions of the form $r \ \propto\ t^p$. For such solutions,

1. How does $H$ vary as a function of time?
   $$\dot{r} \propto p\, t^{p-1} \quad \Rightarrow H = \frac{\dot{r}}{r} \propto \frac{1}{t}\ \ .$$

2. How does $1-\Omega$ vary as a function of time? (You may assume the solution obeys full conservation of energy.)

   
   

   $$E = \frac{1}{2}\ m\ \left(\ \dot{r}^2 - \frac{2G\, M_r}{r}\ \right) \quad =\quad \frac{1}{2}\ m\, r^2\ \left(\ H^2 - \frac{8\pi G\, \rho}{3}\ \right)$$

   $$= \frac{1}{2}\ m\, r^2\, H^2\ \left(\ 1 - \Omega\ \right)$$

   
   
   where $M_r$ is the gravitational mass within radius $r$, $\Omega = \rho/\rho_c$ , and $\rho_c = 3H^2/ 8\pi G$. For $E$ to be constant,
   $$(\ 1 - \Omega\ ) \propto \frac{1}{r^2\, H^2} \propto \frac{1}{t^{2p}\, t^{-2}} \propto t^{2 -2p} \ \ .$$

3. For what values of $p$ does $\vert 1-\Omega\vert$ 1) remain constant? 2) increase? 3) decrease?
   1. $p=1$
   2. $p<1$ (including the cases considered in class)
   3. $p>1$

4. In homework we found a flatness problem when positive $1-\Omega$ increases in time (since $1-\Omega = 0.9$ now requires fine-tuning of $1-\Omega$ to very small values at earlier times).

   The problem is worse when positive $1-\Omega$ decreases in time. For such solutions, the observed value of $1-\Omega = 0.9$ (at $t_o$) can only be extrapolated back to a time $t_i = \alpha \, t_o$, before which $1-\Omega$ takes on unphysical values (i.e. values greater than 1). Find $\alpha$, to first order in $\delta = \frac{1}{18(p-1)}$, which you may take to be small. How many orders of magnitude back in time may such a theory hold?

   
   

   Since $(\ 1 - \Omega\ ) \propto t^{2-2p}$, we can extrapolate back in time using

   $$\frac{(\ 1- \Omega\ )_o}{(\ 1- \Omega\ )_i} = \left(\ \frac{t_o}{t_i}\ \right)^{2-2p} = \alpha^{2p-2}\ \ .$$
   Note that $\alpha < 1$ and $p>1$. So as we extrapolate back in time $\alpha$ decreases and $(\ 1 -\Omega\ )_i$ increases, becoming unphysical when it passes through the value $(\ 1 - \Omega\ )_i = 1$. At this point
   $$\alpha^{2p-2} = (\ 1- \Omega\ )_o = 0.9 \quad \Rightarrow \quad \alpha = (0.9)^{1/2(p-1)} = (10/9)^{-1/2(p-1)}\ \ .$$
   Either of 2 Taylor expansions for $\alpha$ are reasonable:
   

   $$\alpha = (1 - 0.1)^{1/2(p-1)} \approx 1 -\frac{1}{2(p-1)}\ \frac{1}{10} + O\left(\ \left(\ \frac{1}{10}\ \right)^2\ \right)$$

   $$\alpha = (1 + 1/9\ )^{-1/2(p-1)} \approx 1 -\frac{1}{2(p-1)}\ \frac{1}{9} + O\left(\ \left(\ \frac{1}{9}\ \right)^2\ \right)\ \ .$$

   
   
   Either one tells us, since $\delta = \frac{1}{18(p-1)} << 1$, that we can extrapolate back less than 1 order of magnitude in time before our theory fails. NOTE: Extrapolating back in time 1 order of magnitude would take us to $t_i \sim 0.1\, t_o$; back p orders of magnitude to $t_i \sim 10^{-p}\, t_o$, etc.

heenumi..

This is a variation of the twin paradox, where we include a global effect of curved space. Consider a spacetime in which one of the spatial directions, say the x-direction, is a circle of circumference $L$. Observer $A$ at rest is passed by another observer $B$ moving with relative velocity $v$ in the $x$-direction and they synchronize clocks. As usual, each thinks the other's clock is running slow.

1. Explain, quantitatively (in terms of specific events and their Lorentz transformations) why each observer thinks the other's clock runs slow.

   The observers have synchronized clocks. Each observer's clock ticks, in the observer's own reference frame, first at $x^{\mu} = (0, 0, 0, 0)$, then at ${x^\mu}' = (ct, 0, 0, 0)$. That is, the observer moves along with his own clock, so it ticks at the observer's spatial origin.

   B's perception of A's clock ticks are distorted, because B has velocity $v\ \hat{x}$ relative to A. He sees A's measurement ${\Delta x^\mu}_A = (ct, 0, 0, 0)$ acted on by a Lorentz boost of $+v$ in the x-direction,

   $${\Delta x^\mu}_{A{\rm\ seen by\ }B} = \Lambda \ (+v)\ {\Delta x^\... ...array}{c} \gamma\, ct\\ -\gamma\beta\, ct \\ 0\\ 0 \end{array}\ \right)\ \ .$$
   Thus, B thinks A's clock waits a time $\gamma t$ between clock ticks; that is, that A's clock ticks slow ( $\gamma > 1$). A's perception of B's clock ticks are, similarly, a Lorentz boost by $-v \ \hat{x}$ of ${\Delta x^\mu}_B = (ct, 0, 0, 0)$, also giving a perceived time $\gamma t$.

2. Unlike the usual twin paradox, both observers meet up again at time $T = L/v$ (with neither one accelerating) and compare clocks. Whose clock shows the greater elapsed time? By what factor?

   Along A's trajectory, $d\vec{x}= 0$ so that $d\tau_A = dt$. Along B's trajectory, $dx = vdt$, so that

   $$d\tau_B = \sqrt{dt^2 - \frac{dx^2}{c^2}} = \sqrt{1 - \beta^2}\ dt\ .$$
   Thus, at the time $T$ (relative to our stationary observer A) that the observers meet up again, A has traveled a proper time $\tau_A = T$, and B a proper time $\tau_B = \sqrt{1 - \beta^2}\ T$. An observer's clock is stationary with respect to himself, and so measures $t = \tau$. A's clock thus measures the time of their meeting as $T$, while B's measures it to be $\sqrt{1 - \beta^2}\ T$. Thus A's clock shows a greater elapsed time, by the factor $\gamma = 1/ \sqrt{1 - \beta^2}$.

3. This seems paradoxical, since locally each thinks the other's clock runs slower. Describe an experiment that either observer could do alone to determine whether his clock would register the greater time under the above circumstances. (HINT: since this problem involves a globally nontrivial spacetime, your experiment should somehow probe globally around the $x$-direction. Light rays are always useful experimental probes.)

   Our nontrivial boundary condition on the x-direction is not Lorentz invariant, and so the observer can probe globally around the $x$-direction to determine whether he is the stationary observer A or the moving observer B. (This is not possible in Minkowski space.)

   $\parbox{2in}{ Ignoring $y$\ and $z$\ directions for simplicity, our ... ...rs the timelike geodesics initially passing through within the shaded region. }$

   Our experiment is pictured below. We let the observer send out two beams of light in opposite directions, at the moment when he first coincides with the other observer. For the stationary observer A, traveling along the vertical geodesic, both beams will traverse the cylinder and return at the same time and place (that is, return of the beams will be one event). B instead will receive the beam sent out opposite to his x-direction of motion first, then the other beam, since B will have moved in the x-direction since emitting the light.

   

   Thus the observer who observes his emitted light beams return at the same time will measure the longer time interval when A and B eventually meet (and the observer who doesn't, won't).

   Many of you suggested emitting a light ray and noting, when it returns, a Doppler shift. This will not work because the Doppler shift measures only the relative velocity between the emitter and receiver of the light beam. In this case the observer is both emitter and receiver, so the relative velocity is zero and there is no Doppler shift. (The observer may move between receiving subsequent crests, but he made the identical motion between emission of the two subsequent crests, so the distance traveled by subsequent crests is unchanged by the observer's motion.)

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