If we look at a clock falling into a black hole (Schwarzschild metric), we will see its time slowing down further and further as it approaches the event horizon.
What would we see by looking, from far away, at a clock that starts from the inside of a white hole horizon, and moves outwards?
What would we see by looking, from far away, at a clock that starts from the inside of a white hole horizon, and moves outwards?
A white hole is a time-reversed black hole. Your scenario may be phrased as, a distant observer receives light from a clock exiting a white hole. The time reversal of that is the situation a clock entering a black hole receives light from a distant observer. So your question is equivalent to asking: "as you fall into a black hole, how do you see time pass for the outside universe?"
The answer seems to be that nothing dramatic happens. Some parts of the sky are redshifted (slowed in time) while others are blueshifted (sped up in time), basically because of your motion as you fall. Translated into your question about the clock, this means that some observers see the clock slowed (redshifted), while others see it sped up (blueshifted), basically in accordance with the direction it is moving as it exits the white hole.
If we look at a clock falling into a black hole (Schwarzschild metric), we will see its time slowing down further and further as it approaches the event horizon.
However, it is also relevant to think about the time reversal of this scenario. The time reversal of a distant observer receives light from a clock falling into a black hole is a clock exiting a white hole receives light from a distant observer. The infalling clock appears to freeze at the horizon, remaining visible in principle to observers in the arbitrarily distant future, but in an increasingly redshifted form. This means that in the time reversed scenario, the clock exiting the white hole gets incinerated by extremely blueshifted light originating from the entire past universe.
So ultimately, there is no clock exiting the white hole, because it was destroyed as it passed the event horizon.
