# Lifespan of a black hole as measured by different observers

#### T S Bailey

I've read that a stellar-mass black hole has a lifespan on the order of 10^67 years. Does this mean that a clock which is at rest with respect to (and sufficiently far away from) a stellar-mass black hole will tick off 10^67 years before the black hole evaporates? Also, will shell observers hovering at different distances from the event horizon measure different lifespans for the black hole? Lastly, if the time dilation with respect to distant external clocks goes to infinity does that mean that, according to shell observers arbitrarily close to the horizon, the lifespan of the black hole approaches zero? If so, will an in-falling observer disagree?

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#### bcrowell

Staff Emeritus
Gold Member
I've read that a stellar-mass black hole has a lifespan on the order of 10^67 years. Does this mean that a clock which is at rest with respect to (and sufficiently far away from) a stellar-mass black hole will tick off 10^67 years before the black hole evaporates?
Yes.

Also, will shell observers hovering at different distances from the event horizon measure different lifespans for the black hole?
Yes.

Lastly, if the time dilation with respect to distant external clocks goes to infinity does that mean that, according to shell observers arbitrarily close to the horizon, the lifespan of the black hole approaches zero?
Yes.

If so, will an in-falling observer disagree?
Do you mean an observer who crosses the event horizon? Such an observer can't observe the evaporation of the black hole, because the Hawking radiation is never in their past light cone.

#### T S Bailey

I guess my intuition is that, since my clock is running so slowly as I approach the horizon, I would witness the emission process sped up by an equal amount. Or perhaps not sped up, but blue-shifted (trans-planckian problem). Could I not replace the ticks of my clock with the ticks of a Hawking radiation sensor on the ship?

#### wabbit

Gold Member
In addition to the light cone argument, I think there's another, more heuristic reason for which freefalling observers cannot see it : the Hawking radiation is equivalent (at least close to the horizon of a large black hole) to Unruh radiation, i.e. it is a property of the perspective of accelerated observers - freefallers just see empty spacetime.

#### PAllen

And one more observation: it is only for hovering observers that you can say each one's clock is slow compared to distant hovering clock according to how close the horizon they are (and there are no hovering observes at the horizon - it is impossible). A free fall observer observes hovering clocks going slower and slower as it approaches the horizon. Depending on where the free fall started from, it may see distant clocks going the same rate as its own, as it crosses the horizon. Thus, for it, the evaporation of the BH would appear equally as far in the future as for a distant observer, as it crosses the horizon.

#### T S Bailey

And one more observation: it is only for hovering observers that you can say each one's clock is slow compared to distant hovering clock according to how close the horizon they are (and there are no hovering observes at the horizon - it is impossible). A free fall observer observes hovering clocks going slower and slower as it approaches the horizon. Depending on where the free fall started from, it may see distant clocks going the same rate as its own, as it crosses the horizon. Thus, for it, the evaporation of the BH would appear equally as far in the future as for a distant observer, as it crosses the horizon.
I thought that gravitational time dilation depends only on the gravitational potential. Or is the dilation experienced by a shell observer due to its acceleration?

#### wabbit

Gold Member
Actually, for any subluminal wordline, $ds^2=(1-\frac{r_s}{r})dt^2-...$ so $\frac{ds}{dt}\leq\sqrt{1-\frac{r_s}{r}}$ with equality for a hoverer, so I would expect an infaller to have a slower clock than a hoverer at the same $r$ coordinate, in essence combining gravitational redshift with velocity redshift. Is that incorrect ?

#### PAllen

I thought that gravitational time dilation depends only on the gravitational potential. Or is the dilation experienced by a shell observer due to its acceleration?
All time dilation is a function of how an observer establishes simultaneity with some body under consideration. There are not more than one type in GR, and one can show many ways that all GR time dilation is just a generalization of SR time dilation. A static (or more generally, stationary) spacetime (a very special case) allow one to define a potential that defines relative clock rates as function of position for a particular class of observers - those for whom the stationary character of the solution is manifest (more tenchincally, those whose world lines are integral curves of the timelike killing vector field). In fact, one of the main values of defining this potential for a stationary spacetime is that time dilation relationships for other observers can be factored into difference of position * factor for relative motion compared to a stationary world line. This is a great computational convenience.

#### PAllen

Actually, for any subluminal wordline, $ds^2=(1-\frac{r_s}{r})dt^2-...$ so $\frac{ds}{dt}\leq\sqrt{1-\frac{r_s}{r}}$ with equality for a hoverer, so I would expect an infaller to have a slower clock than a hoverer at the same $r$ coordinate, in essence combining gravitational redshift with velocity redshift. Is that incorrect ?
You forget that relative motion time dilation is reciprocal. For the infaller, it is the hovering clock that is slow, and therefore they see the distant clock as comparable to theirs. The latter effect is really more of balancing Doppler effect - growing gravitational blueshift per a hovering observer, but red shift from motion relatave to that [they are moving away from the distant clock compared to the hovering clock, so red shift, growing without bound * blue shift growing without bound leading to finite shift at horizon crossing], resulting in distant clocks being seen as going similar rate to theirs. Meanwhile, for hovering observers they are passing, if they compare at closest approach (so only transverse Doppler = time dilation), they see hovering observers as very slow compared to their clocks.

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#### wabbit

Gold Member
Thanks, I can see that my reasoning was flawed.
One thing I still can't understand is the part I bolded below in your previous comment :
A free fall observer observes hovering clocks going slower and slower as it approaches the horizon. Depending on where the free fall started from, it may see distant clocks going the same rate as its own, as it crosses the horizon. Thus, for it, the evaporation of the BH would appear equally as far in the future as for a distant observer, as it crosses the horizon
Since the infaller cannot see the horizon, Hawking radiation, or the event "final flash at the end of the black hole's life"(*), saying that for him the black hole simply does not evaporate and has no defined lifetime, would seem natural. How can he instead assign to the black hole a lifetime other than "undefined" or "longer than mine" ?

(*) except for those infallers who "lose the race against evaporation" and "arrive at the horizon after it has disappeared". For those I can imagine a lifetime.

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#### PAllen

Thanks, I can see that my reasoning was flawed.
One thing I still can't understand is the part I bolded below in your previous comment :

Since the infaller cannot see the horizon, Hawking radiation, or the event "final flash at the end of the black hole's life"(*), saying that for him the black hole simply does not evaporate and has no defined lifetime, would seem natural. How can he instead assign to the black hole a lifetime other than "undefined" or "longer than mine" ?

(*) except for those infallers who "lose the race against evaporation" and "arrive at the horizon after it has disappeared". For those I can imagine a lifetime.
You are correct in a literal sense. I was allowing an inference as follows: since the external universe appears to be proceeding at the same rate as mine (per the initial conditions I chose), and since I know I heading for a BH (irrespective of local detection of the horizon, I can compute, per GR, when I should cross it, per my clock), and since I know per theory I am using what the lifetime of the BH is for a distant observer, then I infer that for me it is the same. Admittedly, this is long chain of inferences, some of which are debatable.

#### wabbit

Gold Member
Yes I see your point I think - if I agree with the guy at infinity about every time interval I measure whenever we can exchange information, perhaps I should trust him when he tells me about that lifespan, even though I have no chance of checking by myself.

#### PeterDonis

Mentor
I was allowing an inference as follows: since the external universe appears to be proceeding at the same rate as mine (per the initial conditions I chose), and since I know I heading for a BH (irrespective of local detection of the horizon, I can compute, per GR, when I should cross it, per my clock), and since I know per theory I am using what the lifetime of the BH is for a distant observer, then I infer that for me it is the same. Admittedly, this is long chain of inferences, some of which are debatable.
One way of formalizing this chain of inferences is for both observers to adopt a coordinate chart that can encompass both of their worldlines, such as the Painleve chart (or its analogue for a black hole with variable mass). In this chart, the coordinate time elapsed along the horizon from the hole's formation to its evaporation is about the same as the lifetime assigned to the hole by the distant observer (because along the distant observer's worldline, Painleve time and Schwarzschild time are almost the same). And the event of the infaller crossing the horizon can be assigned a definite Painleve coordinate time; so the infaller can simply view the difference between that Painleve coordinate time and the Painleve coordinate time of the hole's evaporation as the "lifetime" of the hole from his perspective.

#### pervect

Staff Emeritus
I guess my intuition is that, since my clock is running so slowly as I approach the horizon, I would witness the emission process sped up by an equal amount. Or perhaps not sped up, but blue-shifted (trans-planckian problem). Could I not replace the ticks of my clock with the ticks of a Hawking radiation sensor on the ship?
Your intuition probably assuming "absolute time". As other posters have mentioned, the process of comparing the rates of clocks is an inherently observer-dependent, because different observers have different notions of simultaneity, i.e. different notions of how to compare distant clocks to determine whether or not they are "at the same time".

It's a rather abstract concept but an important one. It first shows up in the context of special relativity, where twin A can measure twin B's clock as going slow (by a suitable exchange of radio signals), while twin B can measure twin A's clock as being slow. This is simply inconsistent with the notion of a single, universal concept of what it means to be simultaneous - i.e. the notion that I refer to as "absolute time". I have a short thread on this elsewhere on PF, https://www.physicsforums.com/threads/symmetrical-time-dilation-implies-relativity-of-simultaneity.805210/.

It's probably easiest (though still notoriously difficult) to try and understand the special relativity case first, then leverage the understanding of this case to to understand the GR case you are specifically interested in. But it's a bit of a digression, and probably best done in another thread.

Meanwhile, we can say that the infalling observers notion of the lifetime of the black hole is different from the hovering observers notion, though the mechanism of how and why it is different may not be intuitive.

#### bcrowell

Staff Emeritus
Gold Member
One way of formalizing this chain of inferences is for both observers to adopt a coordinate chart that can encompass both of their worldlines, such as the Painleve chart (or its analogue for a black hole with variable mass).
My default reaction to this is that in GR coordinate systems are arbitrary, and therefore any attempt to find physical meaning in a specific coordinate system is guilty until proven innocent.

#### PeterDonis

Mentor
My default reaction to this is that in GR coordinate systems are arbitrary, and therefore any attempt to find physical meaning in a specific coordinate system is guilty until proven innocent.
Yes, I agree; since the horizon is a null surface, there can't be any unique way of assigning a "length of time" to the portion of it between two given events (such as the formation and final evaporation of the black hole). The way I suggested is coordinate-dependent, like any other way would be.

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