Expanding Universe: ordinarity, difficulties and paradoxes

... how is it possible for space,
which is utterly empty, to expand?
How can nothing expand?
The answer is: space does not expand.
Cosmologists sometimes talk about expanding space,
but they should know better.

Warm-up

Problem 1: a spider on a string

An elastic rubber cord of $1$ meter length $1$ is attached to a wall. A spider sits on it at the junction to the wall, and a man holds the other end. The man starts moving away from the wall with velocity $1\, m/s$, and at the same time the spider starts to run along the cord with velocity $1\, cm/s$. Will the spider come up with the man?

Problem 2: law of inertia

Does the law of inertia hold in an expanding Universe?

Problem 3: a particle's momentum in expanding Universe

Suppose a particle's mean free path in an expanding Universe is small enough. Show that its momentum decreases as $^*$ $p(t)\propto a(t)^{-1}$.

$^*$This is a generalization of the previous problem to relativistic case, but still a simplification of the general formulation.

Problem 4: comoving phase volume

Show that the comoving phase volume equals to the physical one.

Show that the twins' paradox is resolved in the FLRW Universe in the same way as in the Minkowski spacetime: the twin who experienced non-zero acceleration appears to be younger than his brother.

O.Gron, S. Braeck, arXiv:0909.5364

Problem 6: Hubble time

Show that the Hubble's time $H_{0}^{-1}$ gives the characteristic time scale for any stage of evolution of the Universe.

Show that in a Universe which expands with acceleration the Hubble's radius decreases.

Problem 8: velocity of Hubble sphere

Show that the surface of Hubble's sphere recedes with velocity $V=c(1+q)$, where $q=-a\ddot{a}/(\dot{a})^2$ is the deceleration parameter.

Problem 9: redshift and Hubble sphere

Show that the standard definition of redshift is valid only inside the Hubble's sphere.

The tethered galaxy problem

Problem 10: cosmological recession and peculiar velocities

Let us consider radial motion in the uniform and homogeneous Universe. For this case the FLRW metric reduces to $ds^2 =c^2 dt^2 -a^{2}(t)d\chi^{2}.$ Proper (physical) distance is defined as the distance (measured along the constant time section $dt=0$) between an observer and a galaxy with given comoving coordinate. Let us define the total velocity of a test galaxy as the time derivative of the proper distance $v_{tot}=\dot{D},\quad \dot{D}=\dot{a}\chi+a\dot{\chi},\quad v_{tot}=v_{rec}+v_{pec}.$ Here $v_{rec}$ is the recession velocity of the test galaxy and $v_{pec}$ is its peculiar velocity. What can be said of the possible values of these velocities?

Problem 11: lightspeed recession

Determine the distance to a galaxy which, due to the Hubble's expansion, recedes from us with the speed of light.

Problem 12: superluminal cosmological velocities

Is it possible for cosmological objects to recede from us with superluminal speeds?

Problem 13: observing superluminal velocities

Is it possible to observe galaxies receding with superluminal speeds?

Problem 14: tethered galaxy problem

Imagine that we separate a small test galaxy from the Hubble flow by tethering it to an observer such that the proper distance between them remains constant. We can think of the tethered galaxy as one that has received a peculiar velocity boost toward the observer that exactly matches its recession velocity. We then remove the tether (or turn off the boosting rocket) to establish the initial condition of constant proper distance $\dot{D}_{0}=0$. Determine the future fate of the test galaxy: will it approach the observer, recede from him or remain at constant distance?

Problem 15: Hubble flow

Show that, provided the Universe expands forever, the test galaxy considered in the previous problem asymptotically joins the Hubble flow.

Problem 16: Hubble's law for acceleration

Obtain the analogue of the Hubble's law for acceleration in presence of radial peculiar velocity.

Problem 17: another derivation

Derive the result of the previous problem by direct differentiation of the Hubble's law.

Problem 18: relative motion and redshifts

In the context of special relativity (Minkowski space), objects at rest with respect to an observer have zero redshift. However, in an expanding universe special relativistic concepts do not generally apply. "At rest" is defined to be "at constant proper distance" ($v_{tot}=\dot{D}= 0$), so our untethered galaxy with $\dot{D}=0$ satisfies the condition for being at rest. Will it therefore have zero redshift? That is, are $z_{tot}=0$ and $v_{tot}=0$ equivalent?

Show that, although radial recession and peculiar velocities add vectorially, their corresponding redshifts combine as $1+z_{tot}=(1+z_{rec})(1+z_{pec}).$

Cosmological redshift

Inspired by E. Bunn, D. Hogg. The kinematic origin of the cosmological redshift. Am. J. Phys. 77:688-694, (2009); arXiv:0808.1081.

Problem 20: cosmological redshift as sum of Doppler shifts

Derive the cosmological redshift as the result of addition of infinitesimal Dopper shifts due to relative velocities of galaxies along the worldline of a photon.

Problem 21: relative velocities of distant galaxies

Suppose the source galaxy $A$ and detector galaxy $B$ are moving with the Hubble flow. Imagine a family of comoving observers situated along the trajectory of the photon. Let the observer 1, closest to the source galaxy, measure his velocity $v_1$ relative to the galaxy and send this information along with the photon to the next closest to him observer 2. Observer 2 measures his velocity $u$ relative to observer 1 and calculates his velocity relative to the galaxy $v_2$ according to the special relativistic formula $v_{2}=\frac{v_1 +u}{1+v_1 u}.$ He sends this information along. What will be the velocity $v_{rel}$ of the observers relative to the galaxy, defined this way, in terms of scale factors at the moment of emission and at the moment of detection?

Problem 22: cosmological redshift and relative velocity

Show that the registered cosmological redshift corresponds to Doppler effect with this very velocity $v_{rel}$.

Problem 23: parellel-transported relative velocity

Find the relative physical velocity of two particles with $4$-velocities $u_{1}^{\mu}$ and $u_{2}^{\mu}$. Let $u_{1}^{\mu}$ be the $4$-velocity of the comoving detector at the moment of detection, and let $u_{2}^{\mu}$ be the $4$-velocity of the source at the moment of emission, parallel transported to the detector along the worldline of the photon$^*$.

$^*$A vector $a$ is parallel transported along a curve with tangent vector $u^{\mu}$, if $u^{\mu}\nabla_{\mu}a^{\nu}=0$.

Problem 24: Killing tensors

A Killing tensor $K_{\mu\nu}$ is a tensor field, which obeys the generalization of the Killing equation $\nabla_{(\mu}K_{\nu\lambda)}=0,$ where parenthesis denote symmetrization over all indices. Prove that the quantity $K_{\mu\nu}u^{\mu}u^{\nu}$ is conserved along a geodesics with tangent vector $u^{\mu}$.

Problem 25: Killing tensor in FLRW metric

Verify that the tensor $$\label{FLRWKillingTensor} K_{\mu\nu} =a^{2}\big(u_{\mu}u_{\nu}-g_{\mu\nu}\big),$$ where $u^{\mu}$ is the $4$-velocity of a comoving particle, is a Killing tensor for the FLRW metric.

Problem 26: a particle's momentum in expanding Universe revisited

Show that due to the Killing tensor (\ref{FLRWKillingTensor}), the physical momentum of a particle, measured by comoving observers, changes with time as $p\sim 1/a$.