The Code wrote:
Henning Makholm wrote:There is (unless I'm mistaken, and then I'd appreciate having pointed out where) something wrong with the the proposition that
IF (a) the universe is described by a Robertson-Walker metric,
THEN (b) two galaxies that recede from each other at speed >c are causally decoupled until their mutual speed drops below c.
That,s easy no information can travel faster than light at c. Distant parts of the Universe are expanding greater than c. Which is why we can't see it.
The information in my example is in the form of a photon that always travels at speed c relative to the galaxies it passes on its way towards us. Nevertheless, given enough time it eventually reaches us even though it started at a galaxy whose distance to us increases by more than one lightyear a year.
As a concrete example with invented numbers, start the experiment 10 gigayears after the Big Bang. At that time Galaxy A is 20 gigalightyears from us and receding at speed 2c. It emits a photon. Initially the photon recedes from us at speed c.
Now let's predict its trajectory roughly using 10-gigayear integration steps.
At T=20 Gy, galaxy A is 40 Gly from us. The photon, not receding quite as fast as galaxy A, is only 30 Gly away. At that point it passes galaxy B, which recedes from us at 2c*(30/40) = 1.5c. So the photon is receding from us at speed 0.5c.
At T=30 Gy, galaxy A is 60 Gly from us. The photon is now 35 Gly away. At that point it passes galaxy C which recedes from us at 2c*(35/60) = 1.166 c. So the photon is receding from us at speed 0.166c.
At T=40 Gy, galaxy A is 80 Gly from us. The photon is now 36.66 Gly away. At that point it passes a galaxy which recedes from us at 2c*(36.66/80) = 0.9166c. So now the photon is actually getting 0.0833 light years closer to us for each passing year.
At T=50 Gy, galaxy A is 100 Gly from us. The photon is 35.833 Gly away, passing a galaxy that recedes from us at 2c*(35.833/100) = 0.7166c. It is approaching us at 0.2833 lightyears per year.
At T=60 Gy, galaxy A is 120 Gly away, the photon is 32 Gly away, passing a galaxy that recedes at 2c*(32/120) and itself closing in at 0.466c.
At T=70 Gy, galaxy A is 140 Gly away, photon is 27.33 Gly away, closing in on us at 1c - 2c*(27.33/140) = 0.61c.
T=80 Gy, galaxy A 160 Gly away, photon 21.1 Gly away, closing at 1c - 2c*(21.1/160) = 0.73c.
T=90 Gy, galaxy A 180 Gly away, photon 13.8 Gly away, closing at 1c - 2c*(13.8/180) = 0.85c.
T=100 Gy, galaxy A, 200 Gly away, photon 5.3 Gly away, closing at 0.95c.
About 105 gigayears after the Big Bang, the photon finally reaches us.