Yes, we would, if it is old enough.
Just to pick up on that point, we can* and already have observed galaxies that are receding from us at greater than c.
If we could observe aliens playing basketball on planets in these galaxies they would appear slowed down**, though this is not due to time dilation, but due to redshift***. We certainly would not see them playing basketball in reverse.
*the recession velocity in an expanding Universe of the hypothetical furthest possible observable galaxy depends on the equation of state, and for any realistic equation of state, it is unavoidable that we can observe galaxies receding from us faster than c.
**a galaxy on the very edge of the observable Universe would appear frozen in time, however as the size of the observable Universe must increase quicker than galaxies recede they wouldn’t appear frozen for long.
*** yes, time dilation and redshift are linked, but in this case it is fair to say the redshift is not due to time dilation.
I feel there is a basic mistake in the Interstellar movie concept of time differential.
For there to be such a big shift in time from orbit, to surface. Then the planet itself would have to be the huge mass effect. If we are to believe that such a big time shift happens from orbit, to surface, due to a nearby massive gravity object. Then every orbit of the ship would bring it closer to the massive object, and dilate time more than for those on the surface. Also, the orbit around the planet would have to be pretty odd. Very elliptical? Then again…maybe that big elliptical orbit would even out the time differentials. Very far away from the planet and massive object. Then very close to the planet. Averaging out to the supposed time difference. But overall. I don’t think the orbiting and surface time differences could be that great, given the percentages of distance in orbit and surface, from the object.
As I did the calculations:
The formula for the recessional velocity of a galaxy at the edge of observable Universe for a model with equation of state w = pressure/energy density and assuming space is flat (k=0) is:
2c/(3w +1)
In a (ultrarelativistic) radiation dominated w =1/3 and the recessional velocity at the edge of the observable Universe is c, in a matter dominate Universe w = 0 and the recessional velocity is 2c. The above equation diverges at w = -1/3, and in fact the particle horizon only exists for models with w >-1/3, so for the de Sitter Universe (w = 1, cosmological constant, but no matter and radiation) the observable Universe covers the whole of space and the recessional velocity of the furthest “galaxy” would not be limited.
However the above formula assumes that w is constant, whereas a more realistic model it changes with time. In our Universe (ignoring inflation) w would’ve been a little less than 1/3 at early times (when the Universe was dominated by radiation) close to 0 for a signifcant portion of its history (when the Universe was dominated by matter) and for the last few billion years a bit less than -1/3 (as dark energy has come to dominate the large scale dynamics of the Universe).
This means in the very early Universe the recessional velocity of the furthest observable objects (in this would’ve been before galaxies had formed) would’ve been slightly above c, but as time has gone on this value has increased and is currently slightly more than 3c and will go on increasing without limit.
The distance of a photon emitted towards us from such an object would be really interesting; at first it would appear to be going away from us, since the space-time it is embedded in would be travelling faster in the opposite direction to its travel; but in due course it would reach regions which were not expanding so rapidly, and would stand still for a moment, before starting to accelerate towards us. All the while travelling at exactly c.
Crazy.
Asympotically fat, I don’t doubt your calculations and assessment. But, I am interested to understand where my assumption of apparent backward time is in error.
I certainly understand that the arrow of time is not literally reversed in distant galaxies, but why would it not appear that way to us if they are receding from our frame of reference at >c?
Won’t the light from those >c galaxies eventually fade from our view in the future? If yes, would it be because we’ve exceeded the visual point in time before those galaxies were created and started emitting light (in which case, I’d think we would see them as going backward in time before dimming, while still receding faster than c), or is it for another reason? IOW, I don’t understand why time dilation and red-shift are unlinked in this case.
The best way to look at it is that recessional velocity does not cause time dilation as the recessional velocity is not the same thing as the velocity in Lorentz’s formula for (kinematic) time dilation. However as the distance is growing between us and distant observers due to expansion there is still redshift and an apparent slowing effect.
Take for example special relativity, we would say that an observer moving with velocity v relative to us is slowed down by factor gamma(v). However if such an observer is moving directly towards us, what we actually see is their clock running faster than our own due to blueshift. The slowing down effect is only after we have factored out redshift/blueshift. In the cosmological case, once we have factored out the redshift, clocks in distant galaxies are running at the same rate as ours.
Call it what you may, when you see a star in the night you are actually seeing the light it gave off millions of years ago. In that since we are seeing the past in real time
right?