What is the radiation pressure on Earth if the sun was replaced by a black hole?

[ryan albery]

Since black holes have a temperature, and emit (Hawking) radiation, do they also exert a radiation pressure on things? Say the sun was replaced by a gravitationally equivalent 1 solar mass black hole, what would be the radiation pressure felt by the earth?

 

[phinds]

Since black holes have a temperature, and emit (Hawking) radiation, do they also exert a radiation pressure on things? Say the sun was replaced by a gravitationally equivalent 1 solar mass black hole, what would be the radiation pressure felt by the earth?

Yes although except for tiny BHs it is so close to zero as to be pretty much indistinguishable.

Approximately zero. Maybe a rounding error in the 20th or so decimal place, something like that. "Insignificant" would be a vast overestimate.

 

[Chalnoth]

Since black holes have a temperature, and emit (Hawking) radiation, do they also exert a radiation pressure on things? Say the sun was replaced by a gravitationally equivalent 1 solar mass black hole, what would be the radiation pressure felt by the earth?

Small. To give a rough idea, the radiation pressure felt by the Earth from the Sun is about one part in $10^{13}$ of the attractive gravitational force between the Earth and Sun.

The Sun's temperature is roughly 6000K.
The temperature of a black hole with the Sun's mass would be roughly $10^{-7}$K.

Radiation pressure goes as temperature to the fourth power, so the roughly 11 orders of magnitude difference makes the radiation pressure of a black hole of the Sun's mass about $10^{44}$ times weaker than the Sun's radiation pressure, which is (very roughly) $10^{57}$ times weaker than the gravitational force between this hypothetical black hole and the Earth (at the same distance).

 

[ryan albery]

Thanks, I do appreciate the responses. I'll assume that Hawking radiation is the same as regular radiation- in that its pressure on things is a function of temperature^4.

 

[Chalnoth]

Thanks, I do appreciate the responses. I'll assume that Hawking radiation is the same as regular radiation- in that its pressure on things is a function of temperature^4... and incidence?

Yes. Black hole radiation is no different from any other thermal radiation.

 

[ryan albery]

Is there a point where/when the force of radiation pressure balances with the force of gravity?

 

[Chalnoth]

Is there a point where/when the force of radiation pressure balances with the force of gravity?

Well, as a black hole gets brighter as its mass gets lower, there must be.

Edit: Of course, this depends upon the optical properties of the object in the radiation field, as well as the surface area it presents to the black hole compared to its mass. It doesn't depend upon distance to the black hole, however, as the radiation field drops off at the same rate as gravity does.

 

[phinds]

Is there a point where/when the force of radiation pressure balances with the force of gravity?

As Chalnoth said, yes there has to be, but the BH would be pretty small by then so that point of balance would be very fleeting and not of any practical significance.

 

[Chalnoth]

I decided to plug in some numbers. If we imagine the radiation pressure between two black holes of equal mass, then, assuming my calculations are correct, the force from radiation pressure is equal to their gravitational attraction at around a mass of $10^{-10}$kg, at which point the remaining lifetime of the black hole is about $10^{-46}$ seconds.

One interesting thing I found was that in this case, the force of the radiation pressure between two black holes is actually independent of the mass if the two black holes have the same mass. It only varies with distance.

This is because:

$$T \propto {1 \over M}$$
$$R \propto M$$
$$F_r \propto {R^4 T^4 \over r^2} \propto {1 \over r^2}$$

 

[timmdeeg]

If we imagine the radiation pressure between two black holes of equal mass, then, assuming my calculations are correct, the force from radiation pressure is equal to their gravitational attraction at around a mass of $10^{-10}$kg, at which point the remaining lifetime of the black hole is about $10^{-46}$ seconds.

Hmm, Planck-time is about $10^{-44}$ seconds. This seems another hint that no longer GR or semiclassical theories but Quantum Gravity determines this epoch.

 

[Chalnoth]

Hmm, Planck-time is about $10^{-44}$ seconds. This seems another hint that no longer GR or semiclassical theories but Quantum Gravity determines this epoch.

Yes, you're probably right. Though to be fair, the radiation pressure on a classical object that is far less dense would be substantially higher and might well be in the classical regime.

 

[timmdeeg]

Yes, you're probably right. Though to be fair, the radiation pressure on a classical object that is far less dense would be substantially higher and might well be in the classical regime.

Yes, but my reasoning was whether a remaining lifetime considered as a true physical process which is much shorter than Planck-time makes any sense.

 

[ryan albery]

As I understand it then, there can exist two tiny black holes where respective radiation pressure balances with gravity, but those black holes are 'infinitely' short lived.

Could you have a stable field of such tiny black holes (that don't evaporate), as each of them would be radiating out their mass/energy at the same rate they're being radiated into by their neighbors? Like a ground state I guess, could a photon travel through such a field, or an electron... or a planet?

Appreciation, I'm humbled and thankful you're taking the time to think about this.

 

[mfb]

Such a setup would always have leakage - energy escaping to the environment. Therefore, it cannot be permanent.
And even for short-living systems, I don't see any way to get any system where Hawking radiation pressure would be relevant.

 

[timmdeeg]

Could you have a stable field of such tiny black holes (that don't evaporate), ...

No, there is common belief that black holes do evaporate. However despite of that your scenario isn't known physics in my opinion.

 

[phinds]

Yes, but my reasoning was whether a remaining lifetime considered as a true physical process which is much shorter than Planck-time makes any sense.

There seems to be a common misconception that one Plank time is the smallest time interval possible for physical phenomenon. That MAY be true but my understanding is that there is no evidence for it. The Plank time IS the smallest time that we can, with current known physics, theoretically measure. Actual measurements are something like 25 orders of magnitude larger than the Plank time, so we may never even get to that theoretical minimum of measurable time. BUT, that doesn't mean necessarily that physical phenomena are limited to that as a minimum time.

 

[timmdeeg]

There seems to be a common misconception that one Plank time is the smallest time interval possible for physical phenomenon. That MAY be true but my understanding is that there is no evidence for it. The Plank time IS the smallest time that we can, with current known physics, theoretically measure.

Well, the Planck-time is defined via the Planck constant, from this there is no smaller time interval. It's a theoretical rather than an empirical limit. If there are no smaller time units than one Planck-time, how then would you be able to define or describe a physical process which occurs within one Planck-time?

 

[phinds]

Well, the Planck-time is defined via the Planck constant, from this there is no smaller time interval. It's a theoretical rather than an empirical limit. If there are no smaller time units than one Planck-time, how then would you be able to define or describe a physical process which occurs within one Planck-time?

You have said it yourself --- it is a THEORETICAL limit, not an empirical one. I can easily define a time smaller than a Plank time. Here it is: 10 E-100 seconds

The fact that we cannot measure it, and probably never will be able to measure it, does not mean it doesn't exist, although that fact certainly gives weight in practical terms to your contention (expressed in question form) that we would not be able to describe it in any physically meaningful way. DEFINE it, yes, but actually make any use of it, other than mathematical, no.

 

[Chalnoth]

There seems to be a common misconception that one Plank time is the smallest time interval possible for physical phenomenon. That MAY be true but my understanding is that there is no evidence for it. The Plank time IS the smallest time that we can, with current known physics, theoretically measure. Actual measurements are something like 25 orders of magnitude larger than the Plank time, so we may never even get to that theoretical minimum of measurable time. BUT, that doesn't mean necessarily that physical phenomena are limited to that as a minimum time.

Once you start dealing with events that occur on the order of a Planck time, however, quantum gravity surely says quite a lot about how those sorts of events unfold.

 

[Chalnoth]

No, there is common belief that black holes do evaporate. However despite of that your scenario isn't known physics in my opinion.

The exact same phenomenon has been demonstrated physically with sound horizons.

 

[mfb]

Once you start dealing with events that occur on the order of a Planck time, however, quantum gravity surely says quite a lot about how those sorts of events unfold.

Right. In other words, we just don't know how time-evolution quicker than the Planck scale looks like. It does not mean our universe would have to evolve in steps of the Planck time.