Sean Carroll has no clue about physics and is helping to bury the good name of 2 graduate students
Sean Carroll can sometimes give popular talks about physics, science, and atheism and most of the content is more or less OK. He wrote an OK textbook general relativity. However, when it comes to things sold as his original research, he has been a borderline crackpot for years. I think it's obvious that after his and two graduate students' latest salvo,
Recall that last summer, Boddy and Carroll argued that we have to be grateful for the unstable Higgs field (in the real world, the Higgs field cannot be unstable because it's an inconsistency) because the instability will soon destroy our world and that's a good thing because in an eternally existing approximately de Sitter world, "we" would inevitably have to become "Boltzmann Brains", thermal fluctuations that resemble the human brain, allow it to feel the same thing, and that inevitably occur after an exponentially long time.
Your humble correspondent and Jacques Distler would explain why this reasoning completely violates the rules of the probability calculus – Bayesian probability, frequentist probability, or any other well-known approach to probability – as well as causality and basic common sense, too.
We can see that we are not Boltzmann Brains and there exists no rational argument implying that "we" would have to "be" Boltzmann Brains just because there are infinitely many of them in an infinite spacetime. Claiming that "we" have to be generic in this sense is just a hypothesis, one that must be tested and one that may be easily falsified (within a split second). Our observations of our present and the past – and ourselves – clearly cannot depend on some future events happening in our Universe (like the number of "Boltzmann Brains" in a future region of the spacetime), anyway, because such an influence would be acausal.
Boddy and Carroll now generously admit that the predictions of "unavoidable Boltzmann Brains" might be physically invalid. Except that their reasons for thinking that all their previous papers have been rubbish are totally wrong, too. They are just adding another, equally obnoxious pile of crap on their previously carefully spread layers of crap.
Their new reason why the Boltzmann Brains could be unphysical is that... quantum fluctuations are unphysical, too, at least in de Sitter space. In fact, both quantum and thermal fluctuations are absent, they tell us, and they don't even have to be distinguished from one another! This "discovery" that quantum fluctuations don't exist has numerous consequences, we may "learn" from these "researchers". For example, inflation can't be eternal.
The character of the thermal fluctuations in de Sitter space may be subtle. Some issues are difficult but well understood, others are not understood, and one should be ready for some diversity in people's remarks about these matters. But what Boddy, Carroll, and Pollack are saying about pretty much anything is beyond the pale. Caltech used to be a prominent place – it's where Feynman worked and where string theory had headquarters in the decades when it was just waiting to be understood by hundreds of researchers – but some of its departments have clearly become ludicrous.
De Sitter space: basics and temperature
As we have known since the late 1990s, the expansion of our Universe initiated by the "big bang" is actually accelerating. The driver of this (soon to be) exponential acceleration is the dark energy – more precisely the cosmoloogical constant, its special type (everything is compatible with the claim that it's the cosmological constant that explains this acceleration).
The cosmological constant was initially introduced by Einstein into his equations of GR in order to make the Universe static (when Hubble found the expansion, Einstein opportunistically relabeled the constant as the greatest blunder of his life). Einstein's Static Universe wouldn't really work because the static equilibrium would be unstable; a small kick in either direction would send the Universe to an accelerating departure from the equilibrium size (either to expansion or to contraction). However, the term of the same type – also with a positive value but a slightly higher one so that it doesn't just cancel the attractive gravity at the cosmological scale but beats it to produce an accelerating expansion – has been empirically demonstrated.
Already today, the cosmological constant represents 70% of the average energy density \(T_{00}\) of the Universe we see. The percentage will approach 100% because the cosmological constant is, well, constant (in space and especially in time): that's why it's called this way. On the other hand, the density of visible and dark matter is decreasing as \(1/a(t)^3\) as the "total mass" stays the same but the volume of the Universe grows. The density of the energy carried by the radiation decreases even more quickly, as \(1/a(t)^4\), because every photon (or another particle of radiation) is seeing its wavelength increase as \(a\) and its energy goes like \(E\sim 1/ \lambda\) which adds another \(1/a(t)\) to the time dependence of the energy density carried by the radiation.
In the asymptotic future, and we're already "close to it", the linear distances \(a(t)\) e.g. between two galaxies will be growing exponentially\[
\Large a(t) = a(t_0) \times 2^{ (t-t_0) / \text{(11 billion years)} }.
\] In other words, each 11 billion years, the linear distances will double, the density of dust will decrease \(2^3=8\) times, and the energy density carried by radiation will decrease \(2^4=16\) times. Note that the spacetime geometry for the de Sitter space (our Universe in a distant future) may be written as a special case of the FRW metric:\[
- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + {a(t)}^2 \mathrm{d}\mathbf{\Sigma}^2
\] Assuming that the spatial sections are flat, and they're very close to being flat – thanks to the long enough cosmic inflation – the spatial part is simply\[
\mathrm{d}\mathbf{\Sigma}^2 = \mathrm{d}r^2 + r^2 \mathrm{d}\mathbf{\Omega}^2
\] See how it would work if the spatial slices were curved. Fine, this FRW metric is an Ansatz for a generic \(a(t)\), i.e. for a complicated dependence of the size of the Universe on time. However, the de Sitter space is very special. It is a hyperboloid of a sort.
In the far future, our Universe will be the space \(dS_4\), the four-dimensional de-Sitter space, which is the hyperboloid given by\[
A^2-B^2-C^2-D^2-E^2 = -R_0^2
\] So you may start in the 4+1-dimensional (in total, 5-dimensional) Lorentzian flat space parameterized by coordinates \(A,B,C,D,E\in\RR\) and pick all points whose proper distance from the origin is \(R_0\) and the separation is "spacelike" (like the directions of most of the axes). With the minus sign in front of \(R_0^2\), the resulting manifold has signature 3+1, like our Universe, and indeed, our Universe will be very close to the hyperboloid above i.e. for \(t\to\infty\) or, equivalently, \(A\to\infty\).
For our Universe, the asymptotic curvature radius \(R_0\) of the de Sitter space is (or will be) tens of billions of light years (or years – both spatial and temporal coordinates are involved), perhaps 20 billion light years, I am not sure about the exact figure and it is not the main goal of this blog post but if you ask, I will calculate it.
The hyperboloid is defined by an "invariant" so as a set, it is invariant under the transformations that preserve this invariant. Indeed, the whole group \(SO(4,1)\) Рthe origin-preserving group of isometries of the original parental 4+1-dimensional space Рis the symmetry group of the de Sitter space. Like \(SO(5)\), this group is \(5\times 4 / 2\times 1 = 10\)-dimensional. That's just like the \(6+4=10\) dimensions of the Poincar̩ group (Lorentz group semi-times translations) and it is no coincidence. The isometry group \(SO(4,1)\) actually "replaces" the whole Poincar̩ group in the de Sitter space.
Curved manifolds, including this de Sitter space, admit infinitely many choices of coordinates. And even the useful choices of coordinates may be numerous. De Sitter spaces are no exceptions. For example, one may choose the static coordinates on the de Sitter space. A subset of the de Sitter spacetime admits this form of the metric:\[
ds^2 = -\left(1-\frac{r^2}{\alpha^2}\right)dt^2 + \left(1-\frac{r^2}{\alpha^2}\right)^{-1}dr^2 + r^2 d\Omega_{n-2}^2.
\] You see that the components of the metric tensor \(g_{tt}\) and \(g_{rr}\) are inverse to one another, just like in the original form of the Schwarzschild metric for a black hole:\[
\eq{
c^2 {d \tau}^{2} &=
\left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \left(1-\frac{r_s}{r}\right)^{-1} dr^2 -\\
&- r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right)
}
\] This analogy has consequences. In fact, the de Sitter space is a kind of a black hole, a spherically symmetric static metric with the same Ansatz for the angular coordinates and with the event horizon. The de Sitter space has a horizon at \(r=\alpha\) – using the coordinates in the displayed formula which is the one before the last one. The gravitational acceleration at this horizon – which is called the cosmic horizon – dictates the temperature of the thermal radiation by the Bekenstein-Hawking formula just like in the case of the black hole.
But there is a funny difference. The "normal space to live" for the black hole is outside the event horizon, i.e. for \(r\gt r_s\) for the Schwarzschild metric above. On the other hand, the normal space to live in the de Sitter space is inside, \(r\lt \alpha\) in the coordinates I have used. So the analogy of "falling inside the black hole" is "leaving the observable Universe".
This "exterior of the visible patch" is infinite – while the black hole interior is finite. And in the de Sitter space, we are fully surrounded by the horizon. It means that every quantum of radiation that is emitted from the cosmic horizon in the direction inside is reabsorbed after some time. It just can't escape anywhere because the "living space" in the de Sitter space is finite. It follows that while the black hole is evaporating and shrinking as the Hawking quanta are fleeing to infinity, the radius of the de Sitter space isn't macroscopically changing because the emitted Hawking quanta are reabsorbed by the horizon.
Another, related difference is that the black hole's event horizon is agreed upon by all observers. All of them use the causal relationships with the region at infinity. And the identity of that region is shared by all of them. On the other hand, in the de Sitter case, the cosmic horizon depends on the observer because we have to think about the causal relationships with the world line of a particular observer which is not at infinity. It is in the middle of the de Sitter space (from her viewpoint). This is why every observer has her own cosmic horizon – different from the horizon of others.
But the thermal phenomena are equally real as in the case of the black hole.
The equivalence principle dictates that. One must immediately add that this whole science about the thermal behavior of the de Sitter space is just a talk as a matter of principle. The associated temperature is insanely tiny for all practical purposes – the characteristic wavelength is comparable to the radius of the Universe (40 billion light years is much longer and cooler than the fractions of a millimeter, the wavelength for the room temperature). And in fact, the temperature is so tiny and it's hard to measure things accurately in de Sitter space that these thermal phenomena might be non-existent in some sense even as a matter of principle.
Boddy, Carroll, and Pollack could have written many correct things about these matters. Unfortunately, they have only written wrong things. Let me just mention some rudimentary errors that appear both in the article and in Sean Carroll's blog post about the article:
Moreover, even if the claim "Boltzmann Brains affect physics" were implied by a rational argument – it is not – they would follow not from "quantum fluctuations" as Carroll argues but from "thermal fluctuations". After all, Ludwig Boltzmann proposed this whole thought experiment decades before quantum theory or quantum mechanics was born so it's obvious that they couldn't have been "quantum fluctuations". They were thermal fluctuations described by statistical mechanics. Statistical mechanics may also explain thermal phenomena in the quantum context but the fluctuations still remain "thermal", not "quantum", because they depend on a nonzero temperature.
You see that Carroll's understanding is fuzzy already when it comes to distinguishing two major subdisciplines of physics, quantum mechanics and statistical physics. He is mostly ignorant about both which makes it easier for him to confuse them.
Quantum fluctuations always exist.
You can't make them "disappear". They follow from the uncertainty principle. Moreover, in a more complicated setup, e.g. in de Sitter space, there are more fluctuations than otherwise. So even if quantum fluctuations could be "made disappeared", and they cannot be, they would have a chance to disappear exactly in the opposite regime than Carroll describes – in a sufficiently cool and static situation. The more processes are taking place, the more types of fluctuations are present. Quantum fluctuations exist in every physical system. But de Sitter space also has a nonzero temperature, as discussed above, so it also has thermal fluctuations on top of the omnipresent quantum fluctuations.
It's important that when we talk about the fluctuations, we actually specify (or at least implicitly have in mind) some particular observables that are fluctuating, like the position and the momentum. And those are fluctuating whether a crackpot named Carroll likes it or not. We may only confirm that they are fluctuating if we actually measure them. But even if we are not measuring them, we may know that it is not possible that \(x=0\) and \(p=0\) at the same moment because that would contradict the uncertainty principle – and, mathematically, the fact that \(xp-px=i\hbar\). The commutator is nonzero even if we don't measure anything and it's the nonzero commutator that guarantees that there have to be fluctuations in the ground state of the harmonic oscillator.
So when we talk about fluctuations, it is the observables – the physical quantities given by Hermitian operators – whose values are fluctuating, not the wave function. The wave function is not an observable; the wave function isn't observable. The wave function is a collection of complex amplitudes encoding all of our (subjective, probabilistic) knowledge about the physical system – about its observables (such as the positions and momenta). It doesn't matter that the wave function of the ground state (and any energy eigenstate) is stationary (constant in time, up to the phase). People have always known that the solution for the wave function is constant. But that has never meant that there were no quantum fluctuations. Does Carroll really believe that the people who have been talking about quantum fluctuations for almost 90 years have never noticed that the wave functions of energy eigenstates are stationary? Does he really believe that all the quantum mechanical practitioners have always been complete idiots? Is it really so hard to consider the alternative explanation, namely that it is him who is the idiot?
In the other passages, he combines his delusions about quantum mechanics and statistical physics with bizarre speculations about the many worlds that he clearly treats literally as some worlds that exist somewhere, and so on. I've read these portions of the text but I don't have the nerves to discuss all these things. I would feel like a person drowning in a pond full of feces. Just one pair of sentences that has a specific enough blunder:
\rho = C \exp(-\beta H)
\] which clearly commutes with the Hamiltonian (it's a function of the Hamiltonian, stupid), and because the time-derivative of the density matrix is given by the commutator\[
i\hbar \frac{d\rho}{dt} = [H,\rho] = \dots = 0,
\] well, the density matrix remains constant in time. Technically, we should use the term "stationary" but it's surely not "non-static" in the sense of having fluctuations in the matrix elements. Again, when we say that there are quantum fluctuations, we don't mean fluctuating (relative) amplitudes in the state vector or the density matrix. We mean fluctuating values of observables – Hermitian operators such as positions and momenta – that could be measured to see that the values are not constant but fluctuating.
There are whole paragraphs in which Carroll praises himself and sells this stinky shit as gold, for example:
At the end, he says that there shouldn't be eternal inflation – in which regions of space repeatedly see a random jump of the inflaton field to a higher level which produces an exponential expansion which may start somewhere in the newly created regions again and again – because the eternal inflation requires quantum fluctuations and they just "proved" that quantum fluctuations don't exist. Or, in other words, eternal inflation is on par with the Boltzmann Brains so because they decided that they don't like the Boltzmann Brains, they must dismiss the eternal inflation, too.
But all of this is just atrocious nonsense. There are numerous fundamental differences between eternal inflation and the Boltzmann Brains. One of them is that the eternal inflation is a credible picture in the cutting-edge cosmology discovered by some of the leading minds of the scientific community such as Andrei Linde while the Boltzmann Brains sold as a modern cosmological theory are a symptom of a brain defect of some individuals many of whom only stayed in the Academia by licking the asses of stupid leftists and mobs. I just can't believe that the arrogance of someone like Carroll could lose all the control so that he would compare his most atrocious crackpot papers to Linde's most important ones.
Technically, the difference is in the probabilities. The probability that one produces a random brain of the human size is something like \(\exp(-10^{26})\) and it may be approximated by zero. The probability that one produces a region that expands into the size of the visible Universe is much higher, perhaps at least \(\exp(-1,000)\) but it may even be comparable to one (we should talk about the probability density per unit volume and unit time). And these small but realistic probabilities are large enough so that they actually mattered in the early Universe and may matter in some future that is much closer than the Boltzmann Brain (or Poincaré recurrence) time scale. The fact that these numbers are comparable to one is the reason why we say that cosmic inflation is a natural theory.
Another difference is that most of the sane people who accept eternal inflation don't believe the idiotic arguments claimed to imply that we are predicted to be a generic, \(\infty\)-th Boltzmann Brain. Instead, the Borde-Guth-Vilenkin theorem even implies that if there were many stages of expansion in the past, their total number (or proper time of a sort) has been finite. So even if the Universe has the ability to generate infinitely many "generations" of expanding pocket Universes in the future (like the life form or the evolved mankind may have arbitrarily many generations in the future), we're still close to those "several special generations" at the beginning, with a "reasonably short pre-history", so the qualitative picture remains the same as the usual big-bang picture with the evolution tree of a reasonable number of our ancestors in evolution.
I don't plan to proofread this blog post because this arrogant junk showing dysfunctional filters in the contemporary Academia just drives me up the wall. I am utterly disgusted that Carroll is allowed to do these things under the Caltech trademark.
Sean Carroll can sometimes give popular talks about physics, science, and atheism and most of the content is more or less OK. He wrote an OK textbook general relativity. However, when it comes to things sold as his original research, he has been a borderline crackpot for years. I think it's obvious that after his and two graduate students' latest salvo,
De Sitter Space Without Quantum Fluctuations,it's time to permanently erase the adjective "borderline". I had to divide the reading of those texts to 5 sessions because the breathtaking ignorance and stupidity described in the paper has driven my adrenaline level above the healthy levels five times.
Recall that last summer, Boddy and Carroll argued that we have to be grateful for the unstable Higgs field (in the real world, the Higgs field cannot be unstable because it's an inconsistency) because the instability will soon destroy our world and that's a good thing because in an eternally existing approximately de Sitter world, "we" would inevitably have to become "Boltzmann Brains", thermal fluctuations that resemble the human brain, allow it to feel the same thing, and that inevitably occur after an exponentially long time.
Your humble correspondent and Jacques Distler would explain why this reasoning completely violates the rules of the probability calculus – Bayesian probability, frequentist probability, or any other well-known approach to probability – as well as causality and basic common sense, too.
We can see that we are not Boltzmann Brains and there exists no rational argument implying that "we" would have to "be" Boltzmann Brains just because there are infinitely many of them in an infinite spacetime. Claiming that "we" have to be generic in this sense is just a hypothesis, one that must be tested and one that may be easily falsified (within a split second). Our observations of our present and the past – and ourselves – clearly cannot depend on some future events happening in our Universe (like the number of "Boltzmann Brains" in a future region of the spacetime), anyway, because such an influence would be acausal.
Boddy and Carroll now generously admit that the predictions of "unavoidable Boltzmann Brains" might be physically invalid. Except that their reasons for thinking that all their previous papers have been rubbish are totally wrong, too. They are just adding another, equally obnoxious pile of crap on their previously carefully spread layers of crap.
Their new reason why the Boltzmann Brains could be unphysical is that... quantum fluctuations are unphysical, too, at least in de Sitter space. In fact, both quantum and thermal fluctuations are absent, they tell us, and they don't even have to be distinguished from one another! This "discovery" that quantum fluctuations don't exist has numerous consequences, we may "learn" from these "researchers". For example, inflation can't be eternal.
The character of the thermal fluctuations in de Sitter space may be subtle. Some issues are difficult but well understood, others are not understood, and one should be ready for some diversity in people's remarks about these matters. But what Boddy, Carroll, and Pollack are saying about pretty much anything is beyond the pale. Caltech used to be a prominent place – it's where Feynman worked and where string theory had headquarters in the decades when it was just waiting to be understood by hundreds of researchers – but some of its departments have clearly become ludicrous.
De Sitter space: basics and temperature
As we have known since the late 1990s, the expansion of our Universe initiated by the "big bang" is actually accelerating. The driver of this (soon to be) exponential acceleration is the dark energy – more precisely the cosmoloogical constant, its special type (everything is compatible with the claim that it's the cosmological constant that explains this acceleration).
The cosmological constant was initially introduced by Einstein into his equations of GR in order to make the Universe static (when Hubble found the expansion, Einstein opportunistically relabeled the constant as the greatest blunder of his life). Einstein's Static Universe wouldn't really work because the static equilibrium would be unstable; a small kick in either direction would send the Universe to an accelerating departure from the equilibrium size (either to expansion or to contraction). However, the term of the same type – also with a positive value but a slightly higher one so that it doesn't just cancel the attractive gravity at the cosmological scale but beats it to produce an accelerating expansion – has been empirically demonstrated.
Already today, the cosmological constant represents 70% of the average energy density \(T_{00}\) of the Universe we see. The percentage will approach 100% because the cosmological constant is, well, constant (in space and especially in time): that's why it's called this way. On the other hand, the density of visible and dark matter is decreasing as \(1/a(t)^3\) as the "total mass" stays the same but the volume of the Universe grows. The density of the energy carried by the radiation decreases even more quickly, as \(1/a(t)^4\), because every photon (or another particle of radiation) is seeing its wavelength increase as \(a\) and its energy goes like \(E\sim 1/ \lambda\) which adds another \(1/a(t)\) to the time dependence of the energy density carried by the radiation.
In the asymptotic future, and we're already "close to it", the linear distances \(a(t)\) e.g. between two galaxies will be growing exponentially\[
\Large a(t) = a(t_0) \times 2^{ (t-t_0) / \text{(11 billion years)} }.
\] In other words, each 11 billion years, the linear distances will double, the density of dust will decrease \(2^3=8\) times, and the energy density carried by radiation will decrease \(2^4=16\) times. Note that the spacetime geometry for the de Sitter space (our Universe in a distant future) may be written as a special case of the FRW metric:\[
- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + {a(t)}^2 \mathrm{d}\mathbf{\Sigma}^2
\] Assuming that the spatial sections are flat, and they're very close to being flat – thanks to the long enough cosmic inflation – the spatial part is simply\[
\mathrm{d}\mathbf{\Sigma}^2 = \mathrm{d}r^2 + r^2 \mathrm{d}\mathbf{\Omega}^2
\] See how it would work if the spatial slices were curved. Fine, this FRW metric is an Ansatz for a generic \(a(t)\), i.e. for a complicated dependence of the size of the Universe on time. However, the de Sitter space is very special. It is a hyperboloid of a sort.
In the far future, our Universe will be the space \(dS_4\), the four-dimensional de-Sitter space, which is the hyperboloid given by\[
A^2-B^2-C^2-D^2-E^2 = -R_0^2
\] So you may start in the 4+1-dimensional (in total, 5-dimensional) Lorentzian flat space parameterized by coordinates \(A,B,C,D,E\in\RR\) and pick all points whose proper distance from the origin is \(R_0\) and the separation is "spacelike" (like the directions of most of the axes). With the minus sign in front of \(R_0^2\), the resulting manifold has signature 3+1, like our Universe, and indeed, our Universe will be very close to the hyperboloid above i.e. for \(t\to\infty\) or, equivalently, \(A\to\infty\).
For our Universe, the asymptotic curvature radius \(R_0\) of the de Sitter space is (or will be) tens of billions of light years (or years – both spatial and temporal coordinates are involved), perhaps 20 billion light years, I am not sure about the exact figure and it is not the main goal of this blog post but if you ask, I will calculate it.
The hyperboloid is defined by an "invariant" so as a set, it is invariant under the transformations that preserve this invariant. Indeed, the whole group \(SO(4,1)\) Рthe origin-preserving group of isometries of the original parental 4+1-dimensional space Рis the symmetry group of the de Sitter space. Like \(SO(5)\), this group is \(5\times 4 / 2\times 1 = 10\)-dimensional. That's just like the \(6+4=10\) dimensions of the Poincar̩ group (Lorentz group semi-times translations) and it is no coincidence. The isometry group \(SO(4,1)\) actually "replaces" the whole Poincar̩ group in the de Sitter space.
Curved manifolds, including this de Sitter space, admit infinitely many choices of coordinates. And even the useful choices of coordinates may be numerous. De Sitter spaces are no exceptions. For example, one may choose the static coordinates on the de Sitter space. A subset of the de Sitter spacetime admits this form of the metric:\[
ds^2 = -\left(1-\frac{r^2}{\alpha^2}\right)dt^2 + \left(1-\frac{r^2}{\alpha^2}\right)^{-1}dr^2 + r^2 d\Omega_{n-2}^2.
\] You see that the components of the metric tensor \(g_{tt}\) and \(g_{rr}\) are inverse to one another, just like in the original form of the Schwarzschild metric for a black hole:\[
\eq{
c^2 {d \tau}^{2} &=
\left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \left(1-\frac{r_s}{r}\right)^{-1} dr^2 -\\
&- r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right)
}
\] This analogy has consequences. In fact, the de Sitter space is a kind of a black hole, a spherically symmetric static metric with the same Ansatz for the angular coordinates and with the event horizon. The de Sitter space has a horizon at \(r=\alpha\) – using the coordinates in the displayed formula which is the one before the last one. The gravitational acceleration at this horizon – which is called the cosmic horizon – dictates the temperature of the thermal radiation by the Bekenstein-Hawking formula just like in the case of the black hole.
But there is a funny difference. The "normal space to live" for the black hole is outside the event horizon, i.e. for \(r\gt r_s\) for the Schwarzschild metric above. On the other hand, the normal space to live in the de Sitter space is inside, \(r\lt \alpha\) in the coordinates I have used. So the analogy of "falling inside the black hole" is "leaving the observable Universe".
This "exterior of the visible patch" is infinite – while the black hole interior is finite. And in the de Sitter space, we are fully surrounded by the horizon. It means that every quantum of radiation that is emitted from the cosmic horizon in the direction inside is reabsorbed after some time. It just can't escape anywhere because the "living space" in the de Sitter space is finite. It follows that while the black hole is evaporating and shrinking as the Hawking quanta are fleeing to infinity, the radius of the de Sitter space isn't macroscopically changing because the emitted Hawking quanta are reabsorbed by the horizon.
Another, related difference is that the black hole's event horizon is agreed upon by all observers. All of them use the causal relationships with the region at infinity. And the identity of that region is shared by all of them. On the other hand, in the de Sitter case, the cosmic horizon depends on the observer because we have to think about the causal relationships with the world line of a particular observer which is not at infinity. It is in the middle of the de Sitter space (from her viewpoint). This is why every observer has her own cosmic horizon – different from the horizon of others.
But the thermal phenomena are equally real as in the case of the black hole.
The equivalence principle dictates that. One must immediately add that this whole science about the thermal behavior of the de Sitter space is just a talk as a matter of principle. The associated temperature is insanely tiny for all practical purposes – the characteristic wavelength is comparable to the radius of the Universe (40 billion light years is much longer and cooler than the fractions of a millimeter, the wavelength for the room temperature). And in fact, the temperature is so tiny and it's hard to measure things accurately in de Sitter space that these thermal phenomena might be non-existent in some sense even as a matter of principle.
Boddy, Carroll, and Pollack could have written many correct things about these matters. Unfortunately, they have only written wrong things. Let me just mention some rudimentary errors that appear both in the article and in Sean Carroll's blog post about the article:
Squelching Boltzmann Brains (And Maybe Eternal Inflation)He says that quantum fluctuations are important, e.g. for the observed traces of inflation in the CMB, but:
But quantum fluctuations are a bit of a mixed blessing: in addition to providing an origin for density perturbations and gravitational waves (good!), they are also supposed to give rise to Boltzmann brains (bad) and eternal inflation (good or bad, depending on taste).That's rubbish, of course. Quantum fluctuations don't imply that we should be the Boltzmann Brains. Only Carroll's irrational thinking (or the lack of thinking – and this is not meant to be an insult but a very accurate, technical problem with his story), as I and Jacques have explained in quite some detail, implies these preposterous predictions.
Moreover, even if the claim "Boltzmann Brains affect physics" were implied by a rational argument – it is not – they would follow not from "quantum fluctuations" as Carroll argues but from "thermal fluctuations". After all, Ludwig Boltzmann proposed this whole thought experiment decades before quantum theory or quantum mechanics was born so it's obvious that they couldn't have been "quantum fluctuations". They were thermal fluctuations described by statistical mechanics. Statistical mechanics may also explain thermal phenomena in the quantum context but the fluctuations still remain "thermal", not "quantum", because they depend on a nonzero temperature.
You see that Carroll's understanding is fuzzy already when it comes to distinguishing two major subdisciplines of physics, quantum mechanics and statistical physics. He is mostly ignorant about both which makes it easier for him to confuse them.
Kim Boddy, Jason Pollack and I have been re-examining how quantum fluctuations work in cosmology, and in a new paper we’ve come to a surprising conclusion: cosmologists have been getting it wrong for decades now. In an expanding universe that has nothing in it but vacuum energy, there simply aren’t any quantum fluctuations at all.What really pisses me off is that I can visualize the millions of people who can notice that Carroll has learned to emit grammatically correct English sentences with physics buzzwords so most of them will completely miss the key fact that he is an arrogant hopeless crank.
Quantum fluctuations always exist.
You can't make them "disappear". They follow from the uncertainty principle. Moreover, in a more complicated setup, e.g. in de Sitter space, there are more fluctuations than otherwise. So even if quantum fluctuations could be "made disappeared", and they cannot be, they would have a chance to disappear exactly in the opposite regime than Carroll describes – in a sufficiently cool and static situation. The more processes are taking place, the more types of fluctuations are present. Quantum fluctuations exist in every physical system. But de Sitter space also has a nonzero temperature, as discussed above, so it also has thermal fluctuations on top of the omnipresent quantum fluctuations.
Our approach shows that the conventional understanding of inflationary perturbations gets the right answer, although the perturbations aren’t due to “fluctuations”; they’re due to an effective measurement of the quantum state of the inflaton field when the universe reheats at the end of inflation.This is a downright lie – because Carroll must know that he is not saying the truth. To deny the omnipresent quantum fluctuations does mean to predict that there cannot be any signs of them, like the patterns discovered in the cosmic microwave background. He is just dishonestly spreading fog. The word "measurement" cannot change an epsilon about this story. Everything that quantum mechanics discusses is something that may be measured by a measurement. If it cannot be measured, even in principle, it isn't physical. Quantum fluctuations are omnipresent and it of course means that they may be identified by a measurement. In the case of the quantum fluctuations of the gravitational field during the cosmic inflation, we have actually measured them – you may say that the measurement was done already at the end of inflation but you may postpone the "moment of the measurement" to 2014 as well, it's up to you – and we are looking at the results of the measurement!
In contrast, less empirically-grounded ideas such as Boltzmann brains and eternal inflation both rely crucially on treating fluctuations as true dynamical events, occurring in real time — and we say that’s just wrong.Boltzmann Brains are just objects or events that appear in thermal chaos and that are exponentially unlikely. But they may occur and, with enough time and space, they surely do occur. They do occur as real events in real space and real time. It's complete rubbish to say that they don't. What is wrong about Carroll and Boddy's (and other people's) previous papers about the Boltzmann Brain is that "we" must be the Boltzmann Brains just because there are many Boltzmann Brains. It is just not true. There are 7 billion people in this world who have no idea why string theory is the correct description of the Universe which doesn't imply that I must be one of them.
All very dramatically at odds with the conventional wisdom, if we’re right.It's very dramatically at odds with the material that Carroll should have mastered before he was allowed to acquire an undergraduate degree, too. One just cannot confuse quantum mechanics with statistical physics or deny the existence of quantum fluctuations if he wants to be a physicist.
Which means, of course, that there’s always a chance we’re wrong (although we don’t think it’s a big chance).They don't think it's a big chance because they don't think at all. They are not using their brains. Not sure whether one may say there is a "chance" that the paper is a pile of crap because it is a "certainty".
This paper is pretty conceptual, which a skeptic might take as a euphemism for “hand-waving”; we’re planning on digging into some of the mathematical details in future work, but for the time being our paper should be mostly understandable to anyone who knows undergraduate quantum mechanics."Hand-waving" is quite an euphemism for this stinky pile of crap.
The basic idea is simple: what we call “quantum fluctuations” aren’t true, dynamical events that occur in isolated quantum systems.Yes, it's simple. I think that all "crackpot indices" would give him a spectacularly high score. The idea is simple, all ingenious ideas are simple, aren't they? Quantum fluctuations don't exist. Why hasn't someone discovered the same thing before us? Well, they have. Thousands of crackpots have "discovered" that the basic predictions of quantum mechanics don't exist. Evolution doesn't work and the Earth is flat, too.
Rather, they are a poetic way of describing the fact that when we observe such systems, the outcomes are randomly distributed rather than deterministically predictable.Oh, I see. So quantum fluctuations belong to poetry. Holy crap.
But when we’re not looking, a system in its ground state (like an electron in its lowest-energy orbital around an atomic nucleus) isn’t fluctuating at all; it’s just sitting there.When we're not looking, there is nothing out there at all. But even if one doesn't understand that quantum mechanics is a framework to find probabilistic relationships between measurements, without assuming (and in fact, while actively rejecting) any objective reality, Carroll's claim is more stupid at an even more elementary level. When we talk about quantum fluctuations, we talk about quantum fluctuations of particular observables. So for example, the ground state of the harmonic oscillator has the energy of \(E=\hbar\omega/2\) which may be said to result from quantum fluctuations in the position and the momentum.
It's important that when we talk about the fluctuations, we actually specify (or at least implicitly have in mind) some particular observables that are fluctuating, like the position and the momentum. And those are fluctuating whether a crackpot named Carroll likes it or not. We may only confirm that they are fluctuating if we actually measure them. But even if we are not measuring them, we may know that it is not possible that \(x=0\) and \(p=0\) at the same moment because that would contradict the uncertainty principle – and, mathematically, the fact that \(xp-px=i\hbar\). The commutator is nonzero even if we don't measure anything and it's the nonzero commutator that guarantees that there have to be fluctuations in the ground state of the harmonic oscillator.
So when we talk about fluctuations, it is the observables – the physical quantities given by Hermitian operators – whose values are fluctuating, not the wave function. The wave function is not an observable; the wave function isn't observable. The wave function is a collection of complex amplitudes encoding all of our (subjective, probabilistic) knowledge about the physical system – about its observables (such as the positions and momenta). It doesn't matter that the wave function of the ground state (and any energy eigenstate) is stationary (constant in time, up to the phase). People have always known that the solution for the wave function is constant. But that has never meant that there were no quantum fluctuations. Does Carroll really believe that the people who have been talking about quantum fluctuations for almost 90 years have never noticed that the wave functions of energy eigenstates are stationary? Does he really believe that all the quantum mechanical practitioners have always been complete idiots? Is it really so hard to consider the alternative explanation, namely that it is him who is the idiot?
In the other passages, he combines his delusions about quantum mechanics and statistical physics with bizarre speculations about the many worlds that he clearly treats literally as some worlds that exist somewhere, and so on. I've read these portions of the text but I don't have the nerves to discuss all these things. I would feel like a person drowning in a pond full of feces. Just one pair of sentences that has a specific enough blunder:
Globally (including outside the horizon), the quantum state is static. It only appears thermal to an observer because the horizon cuts them off from the rest of the world.So Carroll is also saying that "static" is the opposite of "thermal". But this ain't so. They have independent definitions. And are they positively or negatively "correlated"? Well, on the contrary, every truly thermal state is actually "static". The thermal states are given by density matrices\[
\rho = C \exp(-\beta H)
\] which clearly commutes with the Hamiltonian (it's a function of the Hamiltonian, stupid), and because the time-derivative of the density matrix is given by the commutator\[
i\hbar \frac{d\rho}{dt} = [H,\rho] = \dots = 0,
\] well, the density matrix remains constant in time. Technically, we should use the term "stationary" but it's surely not "non-static" in the sense of having fluctuations in the matrix elements. Again, when we say that there are quantum fluctuations, we don't mean fluctuating (relative) amplitudes in the state vector or the density matrix. We mean fluctuating values of observables – Hermitian operators such as positions and momenta – that could be measured to see that the values are not constant but fluctuating.
There are whole paragraphs in which Carroll praises himself and sells this stinky shit as gold, for example:
My confidence in this story about quantum fluctuations and de Sitter space is extremely high, even though it does conflict with the way many cosmologists think about the situation.It's really disgusting to read it. He then misinterprets what the "no-hair theorem" means. The no-hair theorems are theorems about classical solutions for black-hole-like geometries, theorems in classical general relativity; they completely ignore thermal statistical fluctuations as well as the quantum structure of the objects (quantum black hole microstates) by working in the \(\hbar=0\) and \(k=0\) limit. Carroll clearly thinks that the no-hair theorems may be used to invalidate both statistical physics and quantum mechanics and he is even applying this totally wrong attitude in the cosmological context.
At the end, he says that there shouldn't be eternal inflation – in which regions of space repeatedly see a random jump of the inflaton field to a higher level which produces an exponential expansion which may start somewhere in the newly created regions again and again – because the eternal inflation requires quantum fluctuations and they just "proved" that quantum fluctuations don't exist. Or, in other words, eternal inflation is on par with the Boltzmann Brains so because they decided that they don't like the Boltzmann Brains, they must dismiss the eternal inflation, too.
But all of this is just atrocious nonsense. There are numerous fundamental differences between eternal inflation and the Boltzmann Brains. One of them is that the eternal inflation is a credible picture in the cutting-edge cosmology discovered by some of the leading minds of the scientific community such as Andrei Linde while the Boltzmann Brains sold as a modern cosmological theory are a symptom of a brain defect of some individuals many of whom only stayed in the Academia by licking the asses of stupid leftists and mobs. I just can't believe that the arrogance of someone like Carroll could lose all the control so that he would compare his most atrocious crackpot papers to Linde's most important ones.
Technically, the difference is in the probabilities. The probability that one produces a random brain of the human size is something like \(\exp(-10^{26})\) and it may be approximated by zero. The probability that one produces a region that expands into the size of the visible Universe is much higher, perhaps at least \(\exp(-1,000)\) but it may even be comparable to one (we should talk about the probability density per unit volume and unit time). And these small but realistic probabilities are large enough so that they actually mattered in the early Universe and may matter in some future that is much closer than the Boltzmann Brain (or Poincaré recurrence) time scale. The fact that these numbers are comparable to one is the reason why we say that cosmic inflation is a natural theory.
Another difference is that most of the sane people who accept eternal inflation don't believe the idiotic arguments claimed to imply that we are predicted to be a generic, \(\infty\)-th Boltzmann Brain. Instead, the Borde-Guth-Vilenkin theorem even implies that if there were many stages of expansion in the past, their total number (or proper time of a sort) has been finite. So even if the Universe has the ability to generate infinitely many "generations" of expanding pocket Universes in the future (like the life form or the evolved mankind may have arbitrarily many generations in the future), we're still close to those "several special generations" at the beginning, with a "reasonably short pre-history", so the qualitative picture remains the same as the usual big-bang picture with the evolution tree of a reasonable number of our ancestors in evolution.
I don't plan to proofread this blog post because this arrogant junk showing dysfunctional filters in the contemporary Academia just drives me up the wall. I am utterly disgusted that Carroll is allowed to do these things under the Caltech trademark.
Combining Boltzmann brains with additional psychotic explosions
![Combining Boltzmann brains with additional psychotic explosions]()
Reviewed by MCH
on
May 05, 2014
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