There are lots of articles, calling Hawking radiation a theory, but doesn't the definition of a scientific theory state that a theory is substantiated by a repeated testing and an overwhelming amount of evidence? Are those articles using non strict definition of a word *theory*, or do they make a mistake?

# [Physics] Is Hawking radiation a theory or a hypothesis

black-holeshawking-radiation

#### Related Solutions

**Q1** - The black hole spacetime sees actual particles in the "static" frame because the diminishing curvature across the black hole spacetime continuously connects the static frame at infinity - which is really static and corresponds to freely falling, static observers - with the static observers who otherwise live in a strong gravitational field near the black hole event horizon.

So the Hamiltonian corresponding to the static time in the black hole case looks like the "static frame" of the Unruh case at infinity while it looks like the "accelerating frame" of the Unruh case near the event horizon. However, the natural Hamiltonian near the event horizon is linked to the freely falling frame - because the particles can continue to fall after they cross the event horizon.

So that's the frame in which there are no particles near the event horizon. Because the static frame is accelerating relatively to it, it will see a particle flux near the event horizon. And because the static coordinates connect this place with infinity, the same particle flux must be seen at infinity, too.

In the Unruh case, you can't turn the radiation to real particles in a flat space because every frame is either inertial or accelerating and we know very well that only the inertial frame has a flat spacetime and it sees no particles. Note that Unruh's argument is only valid locally - you may show that if one observer sees nothing, a *nearby* accelerating observer does see particles. You need to propagate the particles through a general curved spacetime to figure out how these particles will manifest themselves at a completely different location - and that's where the nontrivial curvature of the black hole spacetime kicks in.

But, as Anna suggested, it may be more pedagogic to link the creation of Hawking particles to the split virtual pairs near the event horizon. Or to the quantum tunneling from the black hole - the uncertainty principle prevents particles from being completely safely hidden inside the black hole.

**Q2** - An observer in a general state of acceleration will detect a radiation that will generalize the radiation or no radiation seen in the two frames and it will be approximately thermal, too. There is always a frame in which the radiation may be argued to vanish - and all other frames in the same place will see Unruh-like radiation corresponding to their acceleration.

**Q3** - Before the horizon gets formed, the situation is not translationally symmetric in time. So the nice Hamiltonian associated with the Killing vector field - showing that the spacetime is static - doesn't exist yet. At that moment, it's not possible to say whether some particles are "real". However, quite generally, freely falling observers define the frames in which there are no particles produced by the future black hole (there can be other "real" particles participating in the collapse, of course). That's not surprising because these freely falling particles are analogous to those that cross the event horizon of the completed black hole. All observers that accelerate relatively to them - in the same place - will see an Unruh-like radiation. It's important to realize that it's hard to detect the "exact radiation" during the formative moments of the black hole. The reason is that the creation of the black hole approximately takes time $t = R/c$ or so, and because the typical energy of the Hawking quanta is just $E = \hbar / t$ or so, we're near the saturation point of the uncertainty principle, so we can't measure the energy of the Hawking quanta too accurately. And the density of the Hawking particles is of order one per this $R^3\times t$ region of spacetime, too. Also note that once it's decided that the black hole is going to form, you can't slow the process down. A much longer time than the time of the black hole birth is needed to be sure about the precise spectrum of the Hawking radiation.

**Q4** - The static observer sitting on a heavy star doesn't see any radiation! It's because there is no event horizon. So the accounting is zero equals zero. There is not even an outgoing flux equal to an incoming flux - after all, one couldn't guarantee any permanent incoming flux from an empty space. The only sensible frame - relatively to which the actual state of the empty space is approximately the ground state - is the static frame. And there are no particles. The presence of the rock solid star's surface guarantees that the freely falling frame is no good to define the vacuum state.

As you said, the case of black holes is conceptually totally analogous to the burning books. In principle, the process is reversible, but the probability of the CPT-conjugated process (more accurate a symmetry than just time reversal) is different from the original one because $$ \frac{Prob(A\to B)}{Prob(B^{CPT}\to A^{CPT})} \approx \exp(S_B-S_A ).$$ This is true because the probabilities of evolution between ensembles are obtained by summing over final states but averaging over initial states. The averaging differs from summing by the extra factor of $1/N = \exp(-S)$, and that's why the exponential of the entropy difference quantifies the past-future asymmetry of the evolution.

At the qualitative level, a white hole is exactly as impossible in practice as a burning coal suddenly rearranging into a particular book. Quantitatively speaking, it's more impossible because the drop of entropy would be much greater: black holes have the greatest entropy among all localized or bound objects of the same total mass.

However, the Hawking radiation isn't localized or bound and it actually has an even greater entropy – by a significant factor – than the black hole from which it evaporated. That's needed and that's true because even the Hawking evaporation process agrees with the second law of thermodynamics.

At the level of classical general relativity, nothing prevents us from drawing a white hole spacetime. In fact, the spacetime for an eternal black hole is already perfectly time-reversal-symmetric. We still mostly call it a black hole but it's a "white hole" at the same moment. Such solutions don't correspond to the reality in which black holes always come from a lower-entropy initial state – because the initial state of the Universe couldn't have any black holes.

So the real issue are the realistic diagrams for a star collapsing into a black hole which later evaporates. Such a diagram is clearly time-reversal-asymmetric. The entropy increases during the star collapse as well as during the Hawking radiation. You may flip the diagram upside down and you will get a picture that solves the equations of general relativity. However, it will heavily violate the second law of thermodynamics.

Any consistent classical or quantum theory explains and guarantees the thermodynamic phenomena and laws microscopically, i.e. by statistical physics applied to its phase space or Hilbert space. That's true for burning books but that's true for theories containing black holes, too. So if one has a consistent microscopic quantum theory for this process – but the same comment would hold for a classical theory as well: your question has really nothing to do with quantum mechanics per se – then this theory must predict that the inverted processes that decrease entropy are exponentially unlikely. Whenever there is a specific model with well-defined microstates and a microscopic T or CPT symmetry, it's easy to prove the equation I started with.

A genuine microscopic theory really establishes that the inverted processes (those that lower the total entropy) are possible but very unlikely. A classical theory of macroscopic matter however "averages over many atoms". For solids, liquids, and gases, this is manifested by time-reversal-asymmetric terms in the effective equations - diffusion, heat diffusion, friction, viscosity, all these things that slow things down, heat them up, and transfer heat from warmer bodies to cooler ones.

The transfer of heat from warmer bodies to cooler ones may either occur by "direct contact" which really looks classical but it may also proceed via the black body radiation – which is a quantum process and may be found in the first semiclassical corrections to classical physics. The Hawking radiation is an example of the "transfer of heat from warmer to cooler bodies", too. The black hole has a nonzero temperature so it radiates energy away to the empty space whose temperature is zero. Again, it doesn't "realistically" occur in the opposite chronological order because the entropy would decrease and a cooler object would spontaneously transfer its heat to a warmer one.

In an approximate macroscopic effective theory that incorporates the microscopic statistical phenomena collectively, much like friction terms in mechanics, those time-reversal-violating terms appear explicitly: they are replacements/results of some statistical physics calculations. In the exact microscopic theory, however, there are no explicit time-reversal-breaking terms. And indeed, according to the full microscopic theory – e.g. a consistent theory of quantum gravity – the entropy-lowering processes aren't strictly forbidden, they may just be calculated to be exponentially unlikely.

The probability that we arrange the initial state of the black hole so that it will evolve into a star with some particular shape and composition is extremely tiny. It is hard to describe the state of the black hole microstates explicitly, but even in setups where we know them in principle, it's practically impossible to locate black hole microstates that have evolved from a recent star (or will evolve into a star soon, which is the same mathematical problem). Your $U^{-1}$ transformation undoubtedly exists in a consistent theory of quantum gravity – e.g. in AdS/CFT – but if you want the final state $U^{-1}|initial\rangle$ to have a lower entropy than the initial one, you must carefully cherry-pick the initial one and it's exponentially unlikely that you will be able to prepare such an initial state, whether it is experimental preparation or a theoretical one. For "realistically preparable" initial states, the final states will have a higher entropy. This is true everywhere in physics and has nothing specific in the context of quantum gravity with black holes.

Let me also say that the "white hole" microstates exist but they're the same thing as the "black hole microstates". The reason why these microstates almost always behave as black holes and not white holes is the second law of thermodynamics once again: it's just very unlikely for them to evolve to a lower-entropy state (at least if we expect this entropy drop to be imminent: within a long enough, Poincaré recurrence time, such thing may occur at some point). That's true for burned books, too. A "white hole" is analogous to a "burned book that will conspire its atomic vibrations and rearrange itself into a nice and healthy book again". But macroscopically, such "books waiting to be revived" don't differ from other piles of ashes; that's the analogous claim to the claim that there is no visible difference between black hole and white hole microstates, and due to their "very likely" future evolution, the whole class should better be called "black hole microstates" and not "white hole microstates" even the microstates that will drop entropy soon represent a tiny fraction of this set.

My main punch line is that at the level of general reversibility, there has never been any qualitative difference between black holes and other objects that are subject to thermodynamics and, which is related, there has never been (and there is not) any general incompatibility between the general principles of quantum mechanics, microscopic reversibility, and macroscopic irreversibility, whether black holes are present or not. The only "new" feature of black holes that sparked the decades of efforts and debates was the causality. While a burning book may still transfer the information in both ways, the material inside the black hole should no longer be able to transfer the information about itself to infinity because it's equivalent to superluminal signals forbidden in relativity. However, we know today that the laws of causality aren't this strict in the presence of black holes and the information is leaked, so the qualitative features of a collapsing star and evaporating black hole are *literally* the same as in a book that is printed by diffusing ink and then burned.

## Best Answer

In the case of Hawking radiation, the direct answer is "no, there is no direct test, nor can we imagine one with anything like current technology." But it is not some wild speculation made in vacuum. The extremely closely related Unruh effect can be derived from basic quantum field theory on a curved spacetime, and many QFT and GR texts have at least an outline of the derivation. (And some people even claim the Unruh effect is testable.) Maybe our derivation or interpretation or even some basic principle is wrong, sure, but there is still good evidentiary support for Hawking radiation.

Indirect evidence, when reasoned with correctly, is not so weak as it seems to many people. What follows below is my elaboration/rant on this last point.

The problem with the scientific method, as taught to school children and promulgated by science pundits, is that it is too narrow and linear. The story goes something like

That's all well and good, but there is no room for logical deduction in this simple picture. And inferential reasoning isn't really there either. A more accurate outline would involve several processes operating

in parallelon our collective "web of knowledge" $W$ (i.e. our set of scientific beliefs and their relationships to one another):Derive new beliefs to augment $W$:

Infer new beliefs to augment $W$:

Test the consistency of $W$ with the real world. These are the same steps as given in the beginning.

This last process of experimental verification is certainly necessary on the whole. But it is entirely reasonable for there to be beliefs $B$ that are not

directlyexperimentally verified. They could be inferred from large bodies of evidence as the only sensible thing we can come up with. Or the can be deduced from other parts of $W$. In fact, they can be deduced from very core principles of $W$ that would require monumental amounts of contrary evidence to overthrow.For example, if $W$ contains Coulomb's law and the idea that physics is invariant under the Lorentz group with fixed speed of light $c$, then the Lorentz force law and all of Maxwell's equations can be derived logically. There is no need to test these to add them to $W$, because logically you should believe them as much as you believe in Coulomb's law and Lorentz transformations. And if you do find experimental evidence against, e.g., Maxwell's equations, then the doubt it casts may very well propagate back to Coulomb's law, since you can't throw out Maxwell yet leave its logical antecedents untouched.