Hawking radiation: black hole vs de Sitter

This paper argues that while the Gibbons-Hawking entropy formula ($S=A/4G$) correctly describes the Hubble volume entropy in (3+1) spacetime by matching local thermodynamic integration, this equivalence breaks down in general $d+1$ dimensions where the local entropy density yields a modified result of $S=(d-1)A/8G$.

The Gist

The Big Picture: Two Types of "Hot" Spaces

Imagine the universe is filled with a special kind of "heat" that comes from gravity itself. Physicists call this Hawking Radiation.

For a long time, we thought this heat worked the same way everywhere. But this paper argues that there are actually two different kinds of "hot" spaces, and they behave very differently:

1. The Black Hole: A finite, compact object (like a heavy stone).
2. The De Sitter Universe: An infinite, expanding space (like our current universe).

The author, G.E. Volovik, shows that while both have "horizons" (invisible boundaries you can't see past), the rules for their temperature and entropy (disorder/heat content) are not the same.

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Analogy 1: The Black Hole vs. The Infinite Ocean

The Black Hole (The Stone):
Imagine a black hole is a giant, heavy stone sitting in a calm lake. It has a specific size. If you throw a pebble (a particle) at it, it might bounce off, or if it gets too close, it falls in.

• The Rule: The "heat" of this stone is directly tied to its surface area. If you cut the stone in half, the total "heat" doesn't just add up simply; it behaves in a weird, non-linear way (like a puzzle where the pieces don't fit together perfectly).
• The Temperature: The temperature is determined by how fast the surface is "pulling" things in.

The De Sitter Universe (The Infinite Ocean):
Now imagine the universe is an infinite ocean that is expanding. There is no "edge" to the ocean, but there is a Horizon for you, the swimmer. Because the ocean is expanding so fast, there is a point far away where the water is moving away from you faster than you can swim. You can never reach it. That is your "Cosmological Horizon."

• The Rule: This isn't a finite stone; it's an infinite fluid. The "heat" here comes from the whole volume of water around you, not just the surface.

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The Big Surprise: The "Local" Temperature vs. The "Horizon" Temperature

This is the core discovery of the paper.

The Old View (The Horizon Thermometer):
Physicists used to think the temperature of this expanding universe was determined by the horizon itself. They calculated a temperature called the Gibbons-Hawking temperature ($T_{GH}$).

• Analogy: Imagine checking the temperature of the ocean by looking at the distant horizon.

The New View (The Local Thermometer):
Volovik argues that if you put a real atom (like a hydrogen atom) right here, in the middle of the ocean, it feels a different temperature.

• Analogy: Imagine you are swimming in the ocean. Even though the horizon is far away, the water around your skin is actually twice as hot as the temperature you calculated from the horizon.
• Why? The atom is getting "ionized" (losing its electron) because of the local vibration of space itself. This local vibration happens at a temperature of $T_{local} = H/\pi$. The horizon temperature is only $H/2\pi$.
• The Takeaway: The "local" temperature is the real, physical heat that matter feels. The horizon temperature is just a mathematical shadow of that local heat.

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The "Entropy" Puzzle: Counting the Disorder

Entropy is a measure of disorder or how much information is hidden in a system.

The Old Formula (The Area Law):
For a black hole, the entropy is proportional to its Surface Area ($A$).
$$Entropy = \frac{Area}{4}$$
This is the famous "Holographic Principle": the information of the 3D object is stored on its 2D surface.

The New Calculation (The Volume Law):
Volovik asks: "If the universe is infinite, and the heat is everywhere inside the volume, shouldn't the entropy be the sum of all that heat inside?"

• He calculates the entropy by adding up the "local heat" of every tiny bit of space inside the horizon.
• The Result: In our 3D universe (4 dimensions of spacetime), the math works out perfectly! The sum of the local heat inside the volume exactly equals the surface area formula.
  • Meaning: The "Surface Entropy" is actually just the "Volume Entropy" in disguise. The horizon is just a window into the heat of the whole room.

The Twist (Higher Dimensions):
Here is where it gets weird. The paper asks: "What if the universe had 4, 5, or 10 dimensions?"

• If you do the math for higher dimensions using the local temperature (the real one), the "Volume Entropy" no longer matches the old "Area/4" formula.
• The New Formula: The entropy is actually:
  $$Entropy = \frac{(d-1)}{8} \times Area$$
  (Where $d$ is the number of spatial dimensions).
• Why it matters: In our 3D world ($d=3$), the math simplifies to the old formula ($Area/4$). But in other dimensions, the old rule is wrong. The "Surface Entropy" must be modified to match the "Volume Entropy."

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The "Two-Fluid" Universe

To explain how this works, Volovik uses a clever analogy from Superfluids (like liquid helium that flows without friction).

He suggests the universe acts like a two-fluid mixture:

1. The Superfluid (Dark Energy): This is the smooth, invisible background that expands the universe. It has no friction and no heat.
2. The Normal Fluid (Gravity): This is the "rough" part that carries the heat and entropy.

The "Second Sound" Analogy:
In superfluids, there are two types of sound waves:

1. First Sound: Pressure waves (like normal sound).
2. Second Sound: Heat waves. In a superfluid, heat can actually travel as a wave!

Volovik shows that Gravitons (the particles that carry gravity) are exactly like these "Second Sound" waves in the universe. They are waves of entropy moving through the "normal fluid" part of the cosmic soup. This proves that gravity and thermodynamics are deeply linked, just like heat and sound in liquid helium.

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The "Contracting" Universe (The Negative Side)

Finally, the paper looks at a universe that is shrinking instead of expanding (a "White Hole" scenario).

• In this case, the "temperature" becomes negative.
• Consequently, the entropy becomes negative.
• This sounds scary, but it just means the rules flip. A shrinking universe is the "mirror image" of an expanding one. The math holds up, but the signs change.

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Summary: What Did We Learn?

1. Local vs. Global: The temperature an atom feels locally ($H/\pi$) is the true temperature, not the horizon temperature ($H/2\pi$).
2. Volume = Surface: The entropy of the universe inside the horizon is the same as the entropy of the horizon itself. The horizon is just a label for the heat inside the room.
3. The Dimension Rule: The famous "Area/4" rule for entropy is actually a special case that only works in our specific 3D universe. In other dimensions, the formula changes to account for the volume.
4. Gravity is Heat: Gravity behaves like the "second sound" (heat wave) in a cosmic fluid, linking the motion of the universe directly to the laws of thermodynamics.

In a nutshell: The universe isn't just a cold, empty stage. It's a hot, fluid-like substance where the "heat" of space itself creates the gravity we feel, and the "surface" of our observable universe is just a reflection of the "volume" of heat inside it.

Technical Summary

1. Problem Statement

The paper addresses a fundamental discrepancy in the thermodynamics of black holes versus de Sitter (dS) space. While both systems exhibit Hawking radiation and possess horizons, their thermodynamic interpretations differ:

• Black Holes: Finite, compact objects with a well-defined horizon position in asymptotically flat spacetime. Their entropy ($S_{BH} = A/4G$) is widely accepted as the Bekenstein-Hawking entropy.
• de Sitter Space: An infinite, homogeneous state where the cosmological horizon's position depends on the observer. The physical meaning of the Gibbons-Hawking entropy ($S_{GH} = A/4G$) is less clear.

The core problem is whether the standard Gibbons-Hawking entropy formula ($S = A/4G$) holds universally for de Sitter space in arbitrary spacetime dimensions ($d+1$), and how the "local" thermodynamics of the de Sitter vacuum (characterized by local activation processes) relates to the "global" entropy of the cosmological horizon. Previous work suggested a holographic correspondence where the bulk entropy (volume integral) equals the horizon entropy, but this relied on using the Gibbons-Hawking temperature ($T_{GH} = H/2\pi$) as the local temperature, which the author argues is incorrect.

2. Methodology

Volovik employs a multi-faceted approach combining quantum tunneling, local thermodynamics, and macroscopic quantum tunneling analogies:

• Painlevé-Gullstrand (PG) Formalism: The analysis utilizes the PG metric form ($ds^2 = -dt^2 + (dr - v(r)dt)^2$) to describe both black holes and de Sitter space. This allows for a unified treatment of particle spectra and tunneling probabilities across horizons.
• Local Activation Processes: The author argues that the true local temperature of the de Sitter vacuum is determined by local activation processes (e.g., ionization of a hydrogen atom or particle decay) occurring deep inside the horizon, rather than radiation emitted from the horizon.
• Two-Fluid Hydrodynamics Analogy: The de Sitter state is modeled using Landau's two-fluid hydrodynamics, where:
  • The superfluid component represents the dark energy (vacuum energy, $w=-1$).
  • The normal (thermal) component represents gravitational degrees of freedom (scalar curvature $R$).
• Dimensional Generalization: The thermodynamics are generalized from $3+1$ dimensions to arbitrary $(d+1)$ dimensions to test the validity of the area law.
• First Law of Thermodynamics: The author applies the first law ($TdS = dE + PdV$) to the Hubble volume, incorporating the pressure term arising from the volume dependence of the horizon.

3. Key Contributions and Results

A. The Local Temperature of de Sitter Space

The paper establishes that the local temperature governing activation processes in de Sitter space is $T_{dS} = H/\pi$.

• This is derived from the tunneling rate of an electron escaping a hydrogen atom in the de Sitter gravitational potential.
• Crucially, $T_{dS} = 2 T_{GH}$, where $T_{GH} = H/2\pi$ is the standard Gibbons-Hawking temperature associated with horizon radiation.
• The author argues that $T_{dS}$ is the universal temperature of the de Sitter medium, independent of dimension $d$.

B. Modification of Gibbons-Hawking Entropy in $d+1$ Dimensions

By integrating the local entropy density (derived from $T_{dS}$) over the Hubble volume ($V_H$), the author derives the entropy of the Hubble volume ($S_H$).

• In $3+1$ dimensions ($d=3$): The volume integral yields $S_H = A/4G$, coinciding with the standard Gibbons-Hawking entropy.
• In general $(d+1)$ dimensions: The volume integral yields a modified entropy:
  $$S_H(d) = \frac{d-1}{8G} A$$
• Conclusion: The standard formula $S = A/4G$ is only valid for $d=3$. For general dimensions, the Gibbons-Hawking entropy must be modified to maintain the holographic correspondence (bulk entropy = horizon entropy) using the correct local temperature $T_{dS} = H/\pi$.

C. Two-Fluid Thermodynamics and Second Sound

The paper demonstrates that de Sitter thermodynamics behaves like a two-fluid system:

• Dark Energy: Acts as the superfluid component ($P = -\epsilon$).
• Gravitational Degrees of Freedom: Act as the thermal normal component.
• Gravitons as Second Sound: The propagation of entropy density (second sound) in this two-fluid model yields a velocity $s_2 = c$ (speed of light). This identifies the graviton in de Sitter space as the second sound mode, providing a thermodynamic derivation of gravitational wave propagation.

D. Negative Entropy in Contracting de Sitter

The paper resolves the "sign problem" in the first law of de Sitter thermodynamics.

• For a contracting de Sitter universe ($H < 0$), the local temperature becomes negative ($T < 0$).
• Consequently, both the Hubble volume entropy and the cosmological horizon entropy become negative.
• This contracting horizon is identified as a "white horizon" (analogous to a white hole), consistent with the properties of Hawking radiation for contracting geometries.

E. Quantum Fluctuations and Uncertainty Relations

The author connects thermodynamic fluctuations of the horizon area to quantum fluctuations.

• Using the variance of entropy fluctuations, a quantum uncertainty relation between the horizon area ($A$) and the gravitational coupling ($K = 1/16\pi G$) is derived:
  $$\Delta A \Delta K \propto \hbar$$
• This suggests that $K$ and $A$ are canonically conjugate variables, linking thermal fluctuations of the horizon to quantum uncertainty.

F. Non-Extensive Entropy and Tsallis Statistics

While the de Sitter bulk entropy is extensive, black hole entropy is non-extensive.

• Black hole splitting/merging follows a composition rule consistent with Tsallis-Cirto statistics ($\delta = 2$): $\sqrt{S(M_1+M_2)} = \sqrt{S(M_1)} + \sqrt{S(M_2)}$.
• The paper suggests that for a black hole in a de Sitter environment (Schwarzschild-de Sitter), the combined system obeys a generalized composition rule involving both horizons, potentially linking to Susskind's dS Matrix Theory.

4. Significance

• Resolution of the Bulk-Horizon Correspondence: The paper provides a physical justification for the Gibbons-Hawking entropy by showing it arises from the integration of local entropy density in the Hubble volume, provided the correct local temperature ($H/\pi$) is used.
• Dimensional Dependence: It challenges the universality of the $S=A/4G$ law, showing it is a specific feature of $3+1$ dimensions. In higher dimensions, the entropy coefficient changes, which has implications for string theory and holographic dualities in higher-dimensional spacetimes.
• Thermodynamic Origin of Gravity: By modeling de Sitter space as a two-fluid system where gravitons are second sound modes, the paper reinforces the view that gravity is an emergent thermodynamic phenomenon.
• Clarification of Sign Issues: The identification of negative entropy with contracting (white) horizons resolves long-standing ambiguities regarding the sign of the first law in de Sitter thermodynamics.

In summary, Volovik argues that the thermodynamics of the de Sitter universe is governed by a local temperature $T=H/\pi$ that is distinct from the horizon radiation temperature. This distinction necessitates a modification of the Gibbons-Hawking entropy formula in dimensions other than three, unifying the bulk and horizon descriptions through a consistent thermodynamic framework.