Why Do Smaller Black Holes Have Higher Temperatures?

[Tail]

To this day I haven't been able to understand why small black holes have higher temperature (and thus radiate more) than big black holes.

I need a non-mathemathical explanation...

Anyone? Pretty please?

 

[futz]

A black hole's temperature comes from emitted radiation, which follows a blackbody spectrum. This radiation is quantum mechanical in nature.

According to quantum mechanics (and the Heisenberg uncertainty principle) the vacuum of space cannot be a true vacuum; that is, there has to be some energy there to satisfy the HUP. This comes from virtual particles pairs that spontaneously appear, then annihilate with each other almost immediately after (producing energy).

Now, if one of these virtual pairs forms near the event horizon, it is possible for one of the particles to fall into the black hole, while the other can escape, thus preventing their annihilation. The particles that escape radiate outwards -- this is Hawking radiation. To satisfy energy conservation, the "hole" left behind by this particle must be filled -- part of the mass of the black hole is converted to energy. This is why radiating black holes eventually evaporate and disappear.

As far as size, I'm not 100% sure why smaller ones are hotter, but perhaps one way of thinking of it is that the smaller the black hole, the less distance these particles have to travel, and the more likely they are to radiate away (and of course, the more particles that get radiated, the higher the temperature).

 

[marcus]

Originally posted by Tail
To this day I haven't been able to understand why small black holes have higher temperature (and thus radiate more) than big black holes.

I need a non-mathemathical explanation...

Anyone? Pretty please?

I'll start the ball rolling on this one

there is a possibly confusing sense in which small holes have more extreme or intense surface gravity than large ones

for any BH the actual gees at the event horizon is infinite
but theorists have a finite quantity they CALL the surface gravity which is bigger for smaller holes

and the temp is proportional to that surface gravity parameter

I'll get the formula for it in a moment

anyway maybe this could be intuitive
if the surface grav is more extreme then more of that
Hawking radiation happens per square centimeter
(with gravity pulling one partner in and promoting the other
from virtual to real)
so the more intense radiation means higher temp

 

[marcus]

my textbook (Frank Shu---The Physical Universe)

defines the surface gravity parameter and proves that the Hawking temp is proportional to it

The intuitive content of his proof is (put very roughly) that
the more extreme the gravity is the more of this Hawking radiation-producing process goes on

notice that with the little BH the spatial gradient of the gravity is steeper-----it would ramp up faster as you approached

for those who happen to like formulas, in natural units the H. temp
is given by the formula

kT = 1/(8pi M) where M is the mass

also kT = (surface grav parameter)/(2pi)

the actual gravitational acceleration as you get within distance R
is given by

(M/R²) divided by sqrt(1 - 2M/R)

The Schw. radius R_Schw = 2M
so the sqrt thing you divide by goes to zero as you approach
the event horizon making the fraction go to infinity.
Its the numerator (M/R²)
which they call the "surface gravity" and which the temp is proportional to.

Let's calculate the surface gravity at the event horizon (that is, where R = R_Schw = 2M
M/R² = M/R_Schw ² =
M/(2M)² = 1/(4M)

So surface gravity divided by 2pi is 1/(8piM) which is
the Hawking formula for the temperature (kT) in terms of mass.

The smaller the mass the bigger that number 1/(8piM) gets.
As the hole evaporates it gets progressively hotter etc.
until ("pop") we've all heard the story

 

[LURCH]

Or in simpler terms (and if I understand the process correctly), the energy to create the virtual particle pairs comes from the curvature of space around an object, as if the curvature were placing a strain on the fabric of space, and that strain is the energy to make vp's. So anything that curves space makes vp pairs. The sharper the curve, the more vp's it will create. Those pairs that happen to come into existence with the EH between them form the Hawking radiation. The curvature of space at the EH of a small BH is much sharper, and therefore much more active in vp production, right?

 

[marcus]

Originally posted by LURCH
Or in simpler terms (and if I understand the process correctly), the energy to create the virtual particle pairs comes from the curvature of space around an object, as if the curvature were placing a strain on the fabric of space, and that strain is the energy to make vp's. So anything that curves space makes vp pairs. The sharper the curve, the more vp's it will create. Those pairs that happen to come into existence with the EH between them form the Hawking radiation. The curvature of space at the EH of a small BH is much sharper, and therefore much more active in vp production, right?

I say right, to that. It is a different perspective----I was saying the gravity is more extreme near the event horizon of a small BH and you put it more geometrically by saying the space is more radically curved---and it makes sense to me that would breed more virtual particles. Maybe we will get yet another perspective on this. I find Hawking radiation hard to understand so could use whatever other viewpoints on it.

 

[Tail]

How comes the surface gravity of smaller black holes is bigger than that of bigger ones?

 

[marcus]

Originally posted by Tail
How comes the surface gravity of smaller black holes is bigger than that of bigger ones?

Glad you asked

well actually I was wondering if and when you might respond

(disclaimer as usual: can't give an authoritative answer etc etc)

The gravity from ANYTHING has this term M/R²
in it----proportional, that is, to the mass of the thing
and falling off as the square of the distance

and the point about the little hole-lets is that you can get in very very close and so even tho M is less
M/R² (because of dividing by the square of a small number) is larger

EXPOSITORY BRAINSTORM: maybe instead of saying that gravity "falls off as the square of the distance"
physicists should say that it
"increases with the square of the closeness"

the event horizon radius which gives an idea of the size of the BH
(also called "Schwarzschild radius") is proportional to the mass
So if you cut the mass in half you can get TWICE AS CLOSE
so that means gravity per unit mass is four times stronger
which more than compensates for having only half as much mass

 

[Tail]

I didn'e get it. I mean, aren't the mass and the radius of a black hole directly proportional?

 

[marcus]

I said:
-----------
the event horizon radius which gives an idea of the size of the BH
(also called "Schwarzschild radius") is proportional to the mass
So if you cut the mass in half you can get TWICE AS CLOSE
so that means gravity per unit mass is four times stronger
which more than compensates for having only half as much mass
---------------

In response (I think) to this, you said:

I didn'e get it. I mean, aren't the mass and the radius of a black hole directly proportional?

Indeed they are
so if you reduce the mass by half
you cut the radius down by half
so you are "twice as close" when you are at the boundary
(if you like thinking of one-over-the-distance as the closeness)

 

[Tail]

Ok, so you can ghet twice as close... but what's the point?

The gravitation just outside a big black hole should be bigger than the gravitation just outside a small black hole, no?

 

[marcus]

Originally posted by Tail
Ok, so you can ghet twice as close... but what's the point?

The gravitation just outside a big black hole should be bigger than the gravitation just outside a small black hole, no?

No. or maybe I should say Why?
gravity depends on nearness

the pull between two massive planets can be less than the pull between two small less-massive planets if the small ones are much closer together

two big things don't necessarily have the strongest attractive force between them, becausse being big they cannot get so close together----their bulk gets in the way

if you had infinitely strong strain-gauges and lab apparatus so you could measure really strong forces, you could make the strongest gravitational force in the universe only using a couple of small black holes-----they could attract each other more than two neutron stars or anything else, more than two galaxies-----because they could get closer together

the force is proportional to the two masses multiplied together and DIVIDED by the square of the separation

(so you can make the force big by making the denominator of the fraction small----making the separation, and thus its square, small)

 

[meteor]

I was wondering today: why the gobbled particle get negative energy,and can't occur the vice versa, that the attracted particle remains with possitive energy and the runaway particle get negative energy? Anyone?

 

[Tail]

Well, marcus, I hope you're patient... I know you're trying to tell me something and I'm genuinely trying to understand.

The distance to the event horizon is directly proportional to mass of the black hole. Which means that the gravitation right on the horizon is the same for every black hole. Am I wrong?

And it takes longer for the gravitation of a big black hole to lessen over distance than that of a smaller black hole, no?

 

[LURCH]

Originally posted by Tail
Well, marcus, I hope you're patient... I know you're trying to tell me something and I'm genuinely trying to understand.

The distance to the event horizon is directly proportional to mass of the black hole. Which means that the gravitation right on the horizon is the same for every black hole. Am I wrong?

And it takes longer for the gravitation of a big black hole to lessen over distance than that of a smaller black hole, no?

This one has always had me spun too, Tail. But the big idea here is that if you cut the mass in half, you can get twice as close (half the mass = half the Schwartzchild radius) and getting twice as close means FOUR TIMES the gravitational pull. That's because as you approach, the pull of gravity grows exponentially. So one-half the distance means four times the pull, one-third the distance means nine times the pull, etc.

This brings up the very counterintuitive situation once mentioned in the Astronomy Q&A game; appearently, a black hole of about 1 billion SM wuold have a gravitational pull at its event horizon of one G. I'm still having trouble reconciling this with my own preconceptions about what an event horizon is.

 

[Hurkyl]

Maybe a non-black hole example will help demonstrate.

The Earth has a mass of 5.97*10^24 kg and a radius of 6.38*10^6 m. If we compute the surface gravity as Gm/r^2, we get 9.78 m/s^2 (roundoff error. boo!)

Uranus has a mass of 8.66*10^25 kg and a radius of 2.56*10^7 m, making it over 10 times heaver and twice as big as earth. However, when we compute the surface gravity, we get 8.81 m/s^2

!

If you look at the equation for g:

g = Gm/r^2

you see that if you simultaneously double the mass and the radius, g gets cut in half. So larger black holes have less "surface" gravity. (Of course, Newton's formula is only an approximation, but it demonstrates the idea)


One thing that might help understand this phenomenon is if one considers the density of the object. Remember that density is ρ = m/V. If we simultaneously double the mass and the radius, we octuple the volume, so the density gets cut by a fourth. Large black holes are much less dense than small black holes! (taking the volume of the black hole to be everything inside the event horizon)

 

[marcus]

Originally posted by LURCH

This brings up the very counterintuitive situation once mentioned in the Astronomy Q&A game; appearently, a black hole of about 1 billion SM wuold have a gravitational pull at its event horizon of one G. I'm still having trouble reconciling this with my own preconceptions about what an event horizon is.

Lurch, I was wrong when I said that back in the Q&A game! The actual acceleration of gravity at the event horizon of any BH is infinite.
But astrophysicists have a jargon term "surface gravity" for a useful parameter describing the black hole.
In fact you can have a black hole massive enough that its "surface gravity" is one gee.

But the actual acceleration due to gravity as you approach the event horizon is equal to "surface gravity" divided by a term that goes to zero! So the actual gravity goes to infinity as you approach the horizon.

The surface gravity parameter = GM/R_Schw²

 

[marcus]

Originally posted by marcus

...the actual acceleration due to gravity as you approach the event horizon is equal to "surface gravity" divided by a term that goes to zero! So the actual gravity goes to infinity as you approach the horizon.

The surface gravity parameter = GM/R_Schw²

One handy formula is for the temperature, which is
proportional to the "surface gravity" parameter. Infinity is not
very useful for calculation purposes, so it makes sense to have
a finite parameter as a handle on gravity at the event horizon


The actual grav. accel as you approach the surface is

(GM/r²)/sqrt(1 - R_Schw/r)

As r --> R_Schw
the denominator goes to sqrt( 1 - 1) = 0

and the numerator goes to GM/R_Schw² = surface gravity parameter

so the fraction goes to "surface gravity"/0 = infinity

sorry about the confusion, at one point a while ago I was confused by the terminology in something I was reading about this

 

[jcsd]

The area inside a black hole's event horizon with the same mass our the galaxy would have the density that is a trillionth of that of water.

 

[marcus]

Originally posted by jcsd
The area inside a black hole's event horizon with the same mass our the galaxy would have the density that is a trillionth of that of water.

This seems like a good kind of calculation to make. So I will imitate you and do this for a trillion solar mass galaxy
(I'm not sure about mass of our galaxy but trillion seems in ballpark)

I believe the Schw. radius for a solarmass black hole is 3 kilometer
So the radius for a trillion solarmass hole is 3 trillion km.

that seems to be one solar mass per E26 cubic km.

2E30 kilograms per E26 cubic km
20, 000 kg per cubic km.

that is 20 billionths of the density of water

we differ by a factor of 50, which could be accounted for
by different assumptions, esp about mass of galaxy.
I guess the density goes as the inverse square of the mass of the black hole. You may have assume a galaxy some 7 times more massive which is hardly any different for rough calculation purposes

 

[jcsd]

Your right, it was from a textbook I read a long time ago, thinking about it more I don't think it specified the Milky Way it just said a balck hole the mass of a galaxy.

 

[marcus]

JCSD, I thought that was an interesting example and the difference (in rough calculation) really unimportant. I've been using Frank Shu's text "the Physical Universe" as a source. Do you, since you calculate stuff, want to reply to this guy's question? Someone probably should (unless we answered and I just didnt notice)

Originally posted by Tail

And it takes longer for the gravitation of a big black hole to lessen over distance than that of a smaller black hole, no?

Tail you are saying that gravity ramps up more steeply as you approch a small BH, yes?

Or falls off more sharply as you go away from a small one.

I think that is so. Maybe Jcsd can confirm it, if someone hasnt already.

 

[Hurkyl]

But the actual acceleration due to gravity as you approach the event horizon is equal to "surface gravity" divided by a term that goes to zero! So the actual gravity goes to infinity as you approach the horizon.

In what coordinate chart?

 

[marcus]

Originally posted by marcus

The actual grav. accel as you approach the surface is

(GM/r²)/sqrt(1 - R_Schw/r)

As r --> R_Schw
the denominator goes to sqrt( 1 - 1) = 0

and the numerator goes to GM/R_Schw² = surface gravity parameter

so the fraction goes to "surface gravity"/0 = infinity

Hi Hurkyl,

I copied this from page 138 of Frank Shu's textbook and, I should have explained, he was using a special position parameter r
which is just given by the Euclidean formula for the circumference---C = 2 pi r

In other words, to tell how far from the center of the BH you are in "r" terms, You measure the circumference of a circle around the BH at your distance from it, and then divide by 2pi = 6.28

 

[Hurkyl]

Hrm... the formula now makes me feel strongly suspicious that there is something implicit and/or misleading going on.

How does this r parameter behave? I have a sneaky suspicion it has a nonzero lower bound... and may even start increasing again once you get too close to the singularity. In any case, I think this r has some highly nonintuitive behavior, and the infinite acceleration is just an artifact of that.

 

[jcsd]

Originally posted by marcus
JCSD, I thought that was an interesting example and the difference (in rough calculation) really unimportant. I've been using Frank Shu's text "the Physical Universe" as a source. Do you, since you calculate stuff, want to reply to this guy's question? Someone probably should (unless we answered and I just didnt notice)



Tail you are saying that gravity ramps up more steeply as you approch a small BH, yes?

Or falls off more sharply as you go away from a small one.

I think that is so. Maybe Jcsd can confirm it, if someone hasnt already.

 

[jcsd]

D'oh, that post is wrong, ignore it for the minute.

a = GM_BH/r²

Let d = r - R_BH (i.e. the distance from the event horizion)=>

=> a = GM_BH/(d + R_BH)²


=> ∂a/∂d = -GM_BH/2(d + R_BH)³

The size of a Schwarzschild black hole can either be represented by it's radius (R_BH) or it's mass (M_BH), in this case I choose to represent it by it's radius as it's more convient, for a balck holemass and radius are functions of each other=>

M = c²R_BH/2G

substituting that in we get:

∂a/∂d = -c²R_BH/4(d+R_BH)³

taking d= 0 (i.e. considering the caseat the event horzion) we get:

∂a/∂d = -c²/4R_BH²

As you should be able to see the gradient of acceleration is much larger at the event horzion the smaller the value of R_BH, i.e. the smaller the black hole the quicker the acclerating force gets larger.

phew!

 

[Tail]

You're confusing me with all the maths.

Anyway, first I need to understand about the gravity at the event horizon. The event horizon is where light can no longer get out. If you're telling me that gravity at a big black hole's horizon is 4 times as small as that of a smaller hole's horizon, it seems impossible; if the gravity is even a bit smaller than that of the smaller hole's horizon, LIGHT CAN GET OUT. It's not the event horizon anymore.

Ok, WHERE am I wrong?

 

[Tail]

Hmm?

Has everyone lost patience with me...?

 

[jcsd]

No, we haven't! the escape velocity is the same for all size black holes at the event horizon (i.e V_e = c):

c² = 2GM_BH/R_BH

a = GM_BH/R_BH² =>

a = c²/2R_BH


So as you can see from the bottom equation the smaller the event horizon (R_BH) the larger the acceleration due to graviation (a) at the event horizon as c² is constant.


btw This was derived just by using Newtons laws of gravitation