What will Bob have observed about Alice's clock in the same scenario?
Bob will have observed Alice's clock running twice as slow as his.
So here we go.
time=0
distance=0
photons emitted (A/B = 0
photons received (A/B) = 0
Now Bob is going to travel at the speed of light to z=1.
How far is that, and how long would it take?
From my "
lab" I get:
v=HD, t=9170 d=13229.593
v=c-HD, t=9170 d=6614.623
c / (1 + H × D)^2, t=15238 d=10356.065
In an expanding universe, and in v=c-HD, you would get there in the same time, 9.17 billion years. You would be either 13.2 Bly away or 6.6.
But I'm going with the inverse square law after seeing how it fits the evidence for dark energy.
So what happens now?
Bob needs to reach of distance of 10.3 Bly.
So let's say Bob heads off at speed c.
How is Bob going to know he's gone far enough?
He can't just watch his clock for 10.3 billion years, because his clock should be moving at 0, if he's moving at c.
So that's a detail I'd like to know.
But let's say he knows, so he goes out there, now we're at:
t=15.2 billion years, d=10.3 billion light years
Alice will have transmitted 15.2 billion * seconds in a year clock pulses.
Bob will have received zero, since he's traveling with the first clock signal. Since his watch hasn't moved yet, he won't have have sent any clock pulses.
When he stops at his destination, his clock moves. He sends out 60 pulses over 60 seconds, then heads back to Alice at the speed of light.
If Bob's clock is moving slower, Alice will see those clock pulses over 120 seconds.
Alice will have sent out 120 pulses while Bob was observed to be stopped.
Bob would have observed 30 pulses while stopped. The other 90 would still be in transit when he started moving. He would encounter them all on his return trip.