Hawking radiation

[paddoboy]

I need feedback from knowledgeable folks versed in the topic, not vacuous drivel with a witch hunting agenda attached. If you qualify - by all means provide the relevant and technically useful info.

I already have.

Anyone else think there may be more than a whiff of insanity with the standard narrative? I do NOT apologize if this makes devoted followers of consensus position hurt in their heads.

If you believe there is any whiff of insanity with the standard accepted model, get out and do something about it.
Educate those that in your opinion are wrong....Pretty simple stuff actually, if you have a reasonable argument, and/or experimental or observational evidence.
I certainly also do not apologise to any alternative hypothesis pushers that have trouble accepting GR and the overwhelming evidence that supports it.

 

[paddoboy]

My quote in that thread has the similar objection, but could not find any answer anywhere in the literature...In fact I am of the contrary opinion about HR, read below, it may evaporate Universe....Disclaimer:Just an observation

Your pseudoscientific take on the subject is noted.
Professor Bennett Link also showed how your maths and thoughts were astray in another thread, but to no avail.
Professor Link obviously left in disgust at your continued pseudoscience rants.

We must remember that forums such as this, with Alternative and pseudoscience sections, among the ghosts and goblins nonsense, are the only outlet these rather eccentric posters have.
Again, as I have stated many times, if any of them had anything of substance, they would not be here...There answers to that factual claim are just ifs, buts, maybes and perhaps.

 

[Q-reeus]

I already have.



If you believe there is any whiff of insanity with the standard accepted model, get out and do something about it.
Educate those that in your opinion are wrong....Pretty simple stuff actually, if you have a reasonable argument, and/or experimental or observational evidence.
I certainly also do not apologise to any alternative hypothesis pushers that have trouble accepting GR and the overwhelming evidence that supports it.

Your predictable tirade is beneath contempt. Fact is, you and the other rat-pack members of the professional head-kicker gang that infests this forum have had ample time by now to come up with meaningful, specific responses to each point I have raised in #15 & #20. That none of you have, despite no doubt spending collectively many hours Googling for rebuttal articles, is testament to your inability to do so. Maybe tashja is busy contacting experts that can actually do so. We shall see.

And btw - this a bloody FORUM - nitwit! You should go study the dictionary definition some time. Fact is, I, as others, have every right to make just the kind of incisive points I have in this thread and many others. Your foolish challenges ignore the very purpose of having an (admittedly somewhat dysfunctional) forum such as SF.

 

[The God]

Your pseudoscientific take on the subject is noted.
Professor Bennett Link also showed how your maths and thoughts were astray in another thread, but to no avail.
Professor Link obviously left in disgust at your continued pseudoscience rants.

We must remember that forums such as this, with Alternative and pseudoscience sections, among the ghosts and goblins nonsense, are the only outlet these rather eccentric posters have.
Again, as I have stated many times, if any of them had anything of substance, they would not be here...There answers to that factual claim are just ifs, buts, maybes and perhaps.

Paddoboy, you have made yourself a laughing stock. As if the open proof of your blatant lies was not enough, now shamelessly you come up with some other funky stuff. Focus on the content, Paddoboy, not on the members. And show some intelligence to figure out what the objection is, do not start that broken tape again and again that mainstream scientists are doing this and that and they cannot be wrong and they are peer reviewed, if one could be he should be, and on and on and on...all bullshit nonsense, I am bored by your predictive posts.

SForums are not restricted to second and endorse the mainstream theories or ideas, and they are not ground for kiddy tutorials also, forums are for meaningful scientific discussions and that would certainly include voice of dissent.......You and some lolos of your ilk, start abusing others on the slight indication of dissent. Grow up and discuss the contents, I am sure you are beyond teen-adult (25+).

 

[brucep]

Thank you all. This is getting better, but where might I find a particle with negative energy?

Between uncharged Casimir plates. BTW there is no event horizon for the virtual pair. In the sense there's no universal speed limit for virtual particles.

 

[brucep]

Where does the kinetic energy come from for the positive-energy particle (whether particle or anti-particle) to escape to infinity from a black-hole event horizon? Why doesn't it instead also fall into the event horizon?

Post 10 says something about your question.

 

[paddoboy]

Your predictable tirade is beneath contempt. Fact is, you and the other rat-pack members of the professional head-kicker gang that infests this forum .

The usual rantings and ravings from an alternative nut!
Just like the rest of them.
Anfd On my tirade that you claim, I suggest you grow up and stop being an obvious hypocritical baby.

And btw - this a bloody FORUM - nitwit! You should go study the dictionary definition some time. Fact is, I, as others, have every right to make just the kind of incisive points I have in this thread and many others. Your foolish challenges ignore the very purpose of having an (admittedly somewhat dysfunctional) forum such as SF.

Like I said, you have SFA to offer, and if you did have anything, you would not be here.
The truth hurts my old friend, but then there there are the facts.
The only outlet you and your ilk have.....I bet our other nut the god, will quickly cling to your shirt tails re this...typical of him.
Let me repeat......
Educate those that in your opinion are wrong....Pretty simple stuff actually, if you have a reasonable argument, and/or experimental or observational evidence.
Take it to the appropriate quarters...get it properly peer reviewed.
Instead you waste your time debating a lay person who has shown you your short comings.
Over to you qreeus for your next hypocrisy laden tirade.

 

[paddoboy]

Paddoboy, you have made yourself a laughing stock.

We'll let your peers on this forum decide that won't we my friend.
And in most cases those that have crossed swords with your arrogant lying nature, have labeled you the laughing stock....long ago.

As if the open proof of your blatant lies was not enough, now shamelessly you come up with some other funky stuff. Focus on the content, Paddoboy, not on the members. And show some intelligence to figure out what the objection is, do not start that broken tape again and again that mainstream scientists are doing this and that and they cannot be wrong and they are peer reviewed, if one could be he should be, and on and on and on...all bullshit nonsense, I am bored by your predictive posts.

Yet my tutorial stands correct and unchanged against your arrogance and your lies, plus the fact that you havn't the guts to post under your original handle.
But if you like we'll ask the administrators and mods decide on the evidnece available as to who is really lying, and stupidly misinterpreting on other facts.
I 'll certainly stand by their judgement.....you'll probably offer some ifs and buts and maybes and what ifs. Typical of your posting style.

SForums are not restricted to second and endorse the mainstream theories or ideas, and they are not ground for kiddy tutorials also, forums are for meaningful scientific discussions and that would certainly include voice of dissent.......You and some lolos of your ilk, start abusing others on the slight indication of dissent. Grow up and discuss the contents, I am sure you are beyond teen-adult (25+).

I'll keep on refuting your nonsense to the nth degree, as long as you keep stating it as fact, and as long as you keep ignoring requests for reputable links and what credentials you have.
What are you afraid of? Being a sparky isn't the end of the world you know.
Mainstream is not perfect, but it is head and shoulders above whatever you or other would be's if they could be's can offer this forum, and I'll keep on stating it as it is, and I'll keep showing how your own interpretations are totally invalid, as long as my arse keeps pointing towards the ground.
Understand?
And as previously said, my tutorial stands unchanged and unchallenged, a testament to your flagrant manipulative style and lies.
It will be Interesting though to see how you handle my next tutorial.

 

[Q-reeus]

...The truth hurts my old friend, but then there there are the facts...

You are not my friend - old or not. Also, you have presented no facts of any kind as counterarguments. Just bald innnuendo. Which speaks volumes.

 

[paddoboy]

The Universe it appears, is a weird and wonderful place.
I agree, it sounds far fetched on face value, but isn't this the case with most of quantum mechanics?
I suppose also a couple of centuries ago, the non absolute nature of space and time would have also seemed far fetched.
I've read two books on string theory and it's derivitives, and many scientists describe it as beautifully mathematically represented.
But talking of 6, 10 or more dimensions is still hard to fathom.
And despite a hell of a lot of anti string propaganda that's going round at the moment, the problem really is one of technology or lack thereof, and our inability to observe at such levels.
Thinking about this stuff makes my head ache...
Irrespective of my head ache but, we do have evidence that it certainly takes place...The Casimir effect.

The particle pair creation scenario is not violating any conservation law, because they are only virtual, existing for less than the Planck time.
Occurring at such levels, and within the limit of the uncertainty principal, means that no law of matter/energy of which we are more familiar is violated.
Far fetched? hard to understand? agreed.....but the Casimir effect supports it.

Here's another take on the subject.......

http://physics.stackexchange.com/questions/30597/black-holes-and-positive-negative-energy-particles
Q:
I was reading Brian Greene's "Hidden Reality" and came to the part about Hawking Radiation. Quantum jitters that occur near the event horizon of a black hole, which create both positive-energy particles and negative-energy particles, produce the so-called Hawking radiation. However, I do not understand why only the negative-energy particles are being absorbed into the black hole, while the positive-energy particles shoot outward. Shouldn't there be 50/50 chance that each type of particle is being absorbed by the black hole? Also, the book mentions that a negative-energy particle would appear to an observer inside the black hole as positive. Why?

A:
There are two ways to approach your question. The first is to explain what Brian Greene means, and the second is to point out that the "particles being swallowed" explanation is a metaphor and isn't actually how the calculation is done. I'll attempt both, but I'm outside my comfort zone so if others can expand or correct what follows please jump in!

When a pair of virtual particles are produced there isn't a negative energy particle and a positive energy particle. Instead the pair form an entangled system where it's impossible to distinguish between them. This entangled system can interact with the black hole and split, and the interaction guarantees that the emerging particle will be the positive one. NB "positive" and "negative" doesn't mean "particle" and "anti-particle" (for what it does mean see below), and the black hole will radiate equal numbers of particles and anti-particles.

Now onto the second bit, and I approach this with trepidation. When you quantise a field you get positive frequency and negative frequency parts. You can sort of think of these as representing particles and anti-particles. How the positive and negative frequencies are defined depends on your choice of vacuum, and in quantum field theory the vacuum is unambiguously defined. The problem is that in a curved spacetime, like the region near a black hole, the vacuum changes. That means observers far from the black hole see the vacuum as different from observers near the black hole, and the two observers see different numbers of particles (and antiparticles). A vaccum near the event horizon looks like excess particles to observers far away, and this is the source of the radiation.

See the Wikipedia article on the Bogoliubov transformation for more information, though I must admit I found this article largely incomprehensible.

Exactly the same maths gives the Unruh effect, i.e. the production of particles in an accelerated frame. The fact that the Unruh effect also produces particles shows that a black hole is not necessary for the radiation, so it can't simply be virtual particles being swallowed.

 

[paddoboy]

You are not my friend - old or not. Also, you have presented no facts of any kind as counterarguments. Just bald innnuendo. Which speaks volumes.

Whatever my old friend!

PS: I have already told you what to do with your counterarguments.

 

[paddoboy]

Here's another take on the subject.......

http://physics.stackexchange.com/questions/30597/black-holes-and-positive-negative-energy-particles
Q:
I was reading Brian Greene's "Hidden Reality" and came to the part about Hawking Radiation. Quantum jitters that occur near the event horizon of a black hole, which create both positive-energy particles and negative-energy particles, produce the so-called Hawking radiation. However, I do not understand why only the negative-energy particles are being absorbed into the black hole, while the positive-energy particles shoot outward. Shouldn't there be 50/50 chance that each type of particle is being absorbed by the black hole? Also, the book mentions that a negative-energy particle would appear to an observer inside the black hole as positive. Why?

A:
There are two ways to approach your question. The first is to explain what Brian Greene means, and the second is to point out that the "particles being swallowed" explanation is a metaphor and isn't actually how the calculation is done. I'll attempt both, but I'm outside my comfort zone so if others can expand or correct what follows please jump in!

When a pair of virtual particles are produced there isn't a negative energy particle and a positive energy particle. Instead the pair form an entangled system where it's impossible to distinguish between them. This entangled system can interact with the black hole and split, and the interaction guarantees that the emerging particle will be the positive one. NB "positive" and "negative" doesn't mean "particle" and "anti-particle" (for what it does mean see below), and the black hole will radiate equal numbers of particles and anti-particles.

Now onto the second bit, and I approach this with trepidation. When you quantise a field you get positive frequency and negative frequency parts. You can sort of think of these as representing particles and anti-particles. How the positive and negative frequencies are defined depends on your choice of vacuum, and in quantum field theory the vacuum is unambiguously defined. The problem is that in a curved spacetime, like the region near a black hole, the vacuum changes. That means observers far from the black hole see the vacuum as different from observers near the black hole, and the two observers see different numbers of particles (and antiparticles). A vaccum near the event horizon looks like excess particles to observers far away, and this is the source of the radiation.

See the Wikipedia article on the Bogoliubov transformation for more information, though I must admit I found this article largely incomprehensible.

Exactly the same maths gives the Unruh effect, i.e. the production of particles in an accelerated frame. The fact that the Unruh effect also produces particles shows that a black hole is not necessary for the radiation, so it can't simply be virtual particles being swallowed.

Here is a video on Brian Greene's "Hidden Reality"

 

[paddoboy]

Here is a video on Brian Greene's "Hidden Reality"

No, I havn't watched it all as yet....quite lengthy.

 

[Q-reeus]

paddoboy said: ↑
...Irrespective of my head ache but, we do have evidence that it certainly takes place...The Casimir effect.

Evidently, despite my having repeatedly posted links to a recognized expert's counterargument to this popular viewpoint, you keep ignoring or forgetting it. Yet again:
http://arxiv.org/abs/hep-th/0503158

... The particle pair creation scenario is not violating any conservation law, because they are only virtual, existing for less than the Planck time...

More ignorance. Nothing to do with the Planck time. In fact, given the claim the virtual pairs have zero *net* energy, there is no reason to link such to the time-energy expression ΔtΔE ≤ ℏ, which becomes meaningless if *net* E of pair is zero. Something often touted by 'zero energy universe' proponents when confronted by arguments the nascent universe should have disappeared in an instant.

... Occurring at such levels, and within the limit of the uncertainty principal, means that no law of matter/energy of which we are more familiar is violated.
Far fetched? hard to understand? agreed.....but the Casimir effect supports it.

See above - you are ignorant of what is actually involved or rather the various disputed claims by professionals with actual knowledge of the subjects.

Here's another take on the subject.......
http://physics.stackexchange.com/questions/30597/black-holes-and-positive-negative-energy-particles
Q:
I was reading Brian Greene's "Hidden Reality" and came to the part about Hawking Radiation. Quantum jitters that occur near the event horizon of a black hole, which create both positive-energy particles and negative-energy particles, produce the so-called Hawking radiation. However, I do not understand why only the negative-energy particles are being absorbed into the black hole, while the positive-energy particles shoot outward. Shouldn't there be 50/50 chance that each type of particle is being absorbed by the black hole? Also, the book mentions that a negative-energy particle would appear to an observer inside the black hole as positive. Why?

A:
There are two ways to approach your question. The first is to explain what Brian Greene means, and the second is to point out that the "particles being swallowed" explanation is a metaphor and isn't actually how the calculation is done. I'll attempt both, but I'm outside my comfort zone so if others can expand or correct what follows please jump in!

When a pair of virtual particles are produced there isn't a negative energy particle and a positive energy particle. Instead the pair form an entangled system where it's impossible to distinguish between them. This entangled system can interact with the black hole and split, and the interaction guarantees that the emerging particle will be the positive one. NB "positive" and "negative" doesn't mean "particle" and "anti-particle" (for what it does mean see below), and the black hole will radiate equal numbers of particles and anti-particles.

Fine to a point. It flatly contradicts the statements and picture presented in #13, but I guess some don't mind 'shopping around and changing their mind to suit'. Except, at the end of it all, one still has to deal with the necessity that for any +ve energy quanta emitted, an equal amount of -ve energy quanta has to be swallowed. As I wrote in #20, how then to explain -ve energy quanta, having -ve gravitating mass (implying also tachyonic -ve inertia) falling in rather than 'falling' outwards? To actual bona fide -ve energy quanta, a BH should actually appear as a supposed 'white hole' - impossible to penetrate.
And the issue is particularly severe in the case of virtual photon pairs. Which would make up the vast bulk of HR 'precursors'. What sense can there be to the notion of a physically real negative energy/frequency photon?

Now onto the second bit, and I approach this with trepidation. When you quantise a field you get positive frequency and negative frequency parts. You can sort of think of these as representing particles and anti-particles. How the positive and negative frequencies are defined depends on your choice of vacuum, and in quantum field theory the vacuum is unambiguously defined. The problem is that in a curved spacetime, like the region near a black hole, the vacuum changes. That means observers far from the black hole see the vacuum as different from observers near the black hole, and the two observers see different numbers of particles (and antiparticles). A vaccum near the event horizon looks like excess particles to observers far away, and this is the source of the radiation.

I posed the issue in #15 of the banal nature of 'negative frequencies' as used by e.g. electrical engineers. As an intermediate computational tool, equating 'negative frequencies' to 'negative energies' might be useful and legitimate in QFT, but makes no physical sense if it's implied 'negative energy/frequency' quanta have a real independent existence. And that is unavoidable if one truly follows the logic of HR through all stages implied.

See the Wikipedia article on the Bogoliubov transformation for more information, though I must admit I found this article largely incomprehensible.

Exactly the same maths gives the Unruh effect, i.e. the production of particles in an accelerated frame. The fact that the Unruh effect also produces particles shows that a black hole is not necessary for the radiation, so it can't simply be virtual particles being swallowed.

There is exactly zero confirmed experimental evidence for Unruh effect: https://en.wikipedia.org/wiki/Unruh_effect
Even if UE/UR were confirmed eventually, it does not follow HR leading to loss of BH mass would be in any sense thus confirmed. For reasons already given.

 

[paddoboy]

Evidently, despite my having repeatedly posted links to a recognized expert's counterargument to this popular viewpoint, you keep ignoring or forgetting it. Yet again:
http://arxiv.org/abs/hep-th/0503158

First, nice to see you have calmed down some.
Second, not all scientific papers are factual [Schmelzer's paper for example] ...Quantum theory Is not fully known, and once again if you have anything concrete take it to the proper quarters for proper peer review.....
In the meantime......
http://iopscience.iop.org/article/10.1088/0264-9381/18/11/307/pdf
Abstract
The Casimir energy for the massless scalar field of two parallel conductors in a two-dimensional Schwarzschild black hole background with Dirichlet boundary conditions is calculated by making use of general properties of the renormalized stress tensor. We show that the vacuum expectation value of the stress tensor can be obtained from the Casimir effect, the trace anomaly and Hawking radiation.

More ignorance. Nothing to do with the Planck time. In fact, given the claim the virtual pairs have zero *net* energy, there is no reason to link such to the time-energy expression ΔtΔE ≤ ℏ, which becomes meaningless if *net* E of pair is zero. Something often touted by 'zero energy universe' proponents when confronted by arguments the nascent universe should have disappeared in an instant.

Whatever, I stand by my claim and that of others.
Again, If your opposition to Hawking radiation is so strong, and if you have anything of substance, you know what to do.

See above - you are ignorant of what is actually involved or rather the various disputed claims by professionals with actual knowledge of the subjects.

As previous answer.

The rest of your anti mainstream tirade is probably about as valid as your take on GR, but that's another matter.


And before I go, another.......

http://arxiv.org/abs/hep-th/0011198
ABSTRACT:
The Casimir energy for massless scalar field of two parallel conductor, in two dimensional Schwarzchild black hole background, with Dirichlet boundary conditions is calculated by making use of general properties of renormalized stress tensor. We show that vacuum expectation value of stress tensor can be obtain by Casimir effect, trace anomaly and Hawking radiation. Four-dimensional of this problem, by this method, is under progress by this author.
EXTRACT:
We have found direct relation between trace anomaly and total Casimir energy. In addition, by considering the Hawking radiation for observer far from black hole,

 

[Q-reeus]

...In the meantime......
http://iopscience.iop.org/article/10.1088/0264-9381/18/11/307/pdf
...And before I go, another.......

http://arxiv.org/abs/hep-th/0011198

Wildly casting around for articles in internet land that include both 'Casimir effect' and 'Hawking radiation' as textual terms, and then pasting quotes here, is some kind of useful counterargument? He he he. Of course there is zero chance you understand anything of technical significance in either linked to article.

 

[paddoboy]

Wildly casting around for articles in internet land that include both 'Casimir effect' and 'Hawking radiation' as textual terms, and then pasting quotes here, is some kind of useful counterargument? He he he. Of course there is zero chance you understand anything of technical significance in either linked to article.

While as an amateur at this game there is much I don't understand, what I do understand is that talk and claims against what is generally accepted as mainstream is cheap, and even cheaper on forums such as this that open to any and every Tom , Dick, and Harry.
My other good friend "the god" will probably give you a confidence boost later on today, as he is pretty apt at clinging to someone's apron strings, as he obviously is clinging to yours.
I reiterate my thoughts on Hawking Radiation and the reputable links and articles supporting it,
It is generally accepted as a viable quantum effect and remains so until evidence shows otherwise.


http://link.springer.com/article/10.1007/BF01219075#page-1

Particle Creation by a Black Hole as a Consequence of the Casimir Effect.
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

http://arxiv.org/pdf/hep-th/0011198v2.pdf

Conclusion:
We have found the renormalized energy-momentum tensor for massless scalar field on background of 1+1 dimensional Schwarzschild black hole for two parallel plates with Dirichlet boundary conditions, by making use of general properties of stress tensor only. We propose that if we know the stress tensor for a given boundary in Minkowski space-time, the Casimir effect in gravitational background can be calculated. We have found direct relation between trace anomaly and total Casimir energy. In addition, by considering the Hawking radiation for observer far from black hole, who is the same as Minkowski observer, this radiation contributes to the Casimir effect. In this paper we have derived three renormalized energy-momentum tensors for our case of study. This is due to selecting three vacuum states for calculation. If we consider Boulware vacuum, stress tensor will have two parts: boundary part and gravitational part. But using Hartle-Hawking and Unruh vacuums will result in another term added to stress tensor, which respectively corresponds to a bath of thermal radiation and Hawking radiation. It seems that similar results can be obtain for four-dimensional Schwarzschild black hole

 

[brucep]

Here's another take on the subject.......

http://physics.stackexchange.com/questions/30597/black-holes-and-positive-negative-energy-particles
Q:
I was reading Brian Greene's "Hidden Reality" and came to the part about Hawking Radiation. Quantum jitters that occur near the event horizon of a black hole, which create both positive-energy particles and negative-energy particles, produce the so-called Hawking radiation. However, I do not understand why only the negative-energy particles are being absorbed into the black hole, while the positive-energy particles shoot outward. Shouldn't there be 50/50 chance that each type of particle is being absorbed by the black hole? Also, the book mentions that a negative-energy particle would appear to an observer inside the black hole as positive. Why?

A:
There are two ways to approach your question. The first is to explain what Brian Greene means, and the second is to point out that the "particles being swallowed" explanation is a metaphor and isn't actually how the calculation is done. I'll attempt both, but I'm outside my comfort zone so if others can expand or correct what follows please jump in!

When a pair of virtual particles are produced there isn't a negative energy particle and a positive energy particle. Instead the pair form an entangled system where it's impossible to distinguish between them. This entangled system can interact with the black hole and split, and the interaction guarantees that the emerging particle will be the positive one. NB "positive" and "negative" doesn't mean "particle" and "anti-particle" (for what it does mean see below), and the black hole will radiate equal numbers of particles and anti-particles.

Now onto the second bit, and I approach this with trepidation. When you quantise a field you get positive frequency and negative frequency parts. You can sort of think of these as representing particles and anti-particles. How the positive and negative frequencies are defined depends on your choice of vacuum, and in quantum field theory the vacuum is unambiguously defined. The problem is that in a curved spacetime, like the region near a black hole, the vacuum changes. That means observers far from the black hole see the vacuum as different from observers near the black hole, and the two observers see different numbers of particles (and antiparticles). A vaccum near the event horizon looks like excess particles to observers far away, and this is the source of the radiation.

See the Wikipedia article on the Bogoliubov transformation for more information, though I must admit I found this article largely incomprehensible.

Exactly the same maths gives the Unruh effect, i.e. the production of particles in an accelerated frame. The fact that the Unruh effect also produces particles shows that a black hole is not necessary for the radiation, so it can't simply be virtual particles being swallowed.

The key is for one or both of the virtual pair becoming real as explained by Kip Thorne in the text of Black Holes and Time Warps. There's no requirement that either fall into the black hole or escape to boundary. They both can fall into the hole and the hole M remains the same or one, or both, escape to boundary and the hole mass decreases. The fact that black holes radiate a thermal spectrum became an important part of the path to quantum gravity.

 

[OnlyMe]

The fact that black holes radiate a thermal spectrum became an important part of the path to quantum gravity.

Bruce, was the use of the word fact an error above? If not could you provide a reference for the observation....

 

[Daecon]

Bruce, was the use of the word fact an error above? If not could you provide a reference for the observation....

https://en.wikipedia.org/wiki/Black-body_radiation

4th paragraph down.