(b) Suppose that two objects,

A

and

B

, are both comoving with the Hubble\ flow.

A

observes radiation from

B

. Use the result from part (a) to find\

\ u _()(A)/()\ u _((

)

B)

, where

\ u _((

)

A)

is the frequency of the radiation measured by the observer\ and

\ u _((

)

B)

is the frequency measured by the source. Write your answer in terms\ of

a(t_(B))

, the scale factor at the emission event, and

a(t_(A))

, the scale factor\ at the observation event.\ [10 marks]\ (c) In addition to the objects

A

and

B

introduced in part (b), suppose that\ there is a third object

C

which is also comoving with the Hubble flow. If\ objects

A,B

, and

C

have respective spatial coordinates

vec(x)_(A),vec(x)_(B)

, and

vec(x)_(C)

,\ find the apparent angle between

B

and

C

as observed by

A

. You may use\ that for a photon traveling from, e.g.,

B

to

A

, the constant

vec(c)

which appears\ in part (a) must be proportional to

vec(x)_(A)-vec(x)_(B)

. Hint: Your final answer\ should not depend on time.\ [12 marks]\ (d) Although the observed angle between

B

and

C

is constant, the proper\ distance between

B

and

C

depends on time. If

vec(x)_(C)=vec(x)_(B)+dvec(x)

for some\ small separation vector

dvec(x)

, find the proper distance between

B

and

C

, as\ measured between events on both worldlines which have the same coordinate\ time.\ [8 marks]\ (e) Suppose that an astronaut moves from

A

to

B

and then back to

A

. The\ proper time between the event at which the astronaut leaves

A

and the event\ at which they return to

A

can be measured both by

A

and by astronaut.\ Which of these two times will be longer? Explain your answer.

(b) Suppose that two objects, A and B, are both comoving with the Hubble flow. A observes radiation from B. Use the result from part (a) to find A/B, where A is the frequency of the radiation measured by the observer and B is the frequency measured by the source. Write your answer in terms of a(tB), the scale factor at the emission event, and a(tA), the scale factor at the observation event. [10 marks] (c) In addition to the objects A and B introduced in part (b), suppose that there is a third object C which is also comoving with the Hubble flow. If objects A,B, and C have respective spatial coordinates XA,XB, and XC, find the apparent angle between B and C as observed by A. You may use that for a photon traveling from, e.g., B to A, the constant c which appears in part (a) must be proportional to XAXB. Hint: Your final answer should not depend on time. [12 marks] (d) Although the observed angle between B and C is constant, the proper distance between B and C depends on time. If XC=XB+dX for some small separation vector dX, find the proper distance between B and C, as measured between events on both worldlines which have the same coordinate time. [8 marks] (e) Suppose that an astronaut moves from A to B and then back to A. The proper time between the event at which the astronaut leaves A and the event at which they return to A can be measured both by A and by astronaut. Which of these two times will be longer? Explain your answer. [8 marks]