Hawking Temperature from Bogoliubov Transformations

1. The Spacetime of a Collapsing Star

To study particle creation, we cannot use the eternal Schwarzschild black hole. An eternal black hole is in thermal equilibrium with its surroundings; it absorbs as much radiation as it emits. To see a net outward flux of particles, we must consider the non-stationary process of black hole formation.

The simplest model for this is the Vaidya spacetime. It describes a collapsing, spherically symmetric shell of null radiation (or "null dust").

• For early times, before the collapse, the spacetime is flat Minkowski space.
• For late times, after the shell has collapsed past its own Schwarzschild radius, the spacetime outside the shell is described by the Schwarzschild metric.

The key feature is that the spacetime evolves from being empty to containing a black hole. This time-dependence is what drives the particle creation.

The crucial idea from the reading is that while the details of the collapse matter, the late-time thermal radiation is universal—it depends only on the properties of the final black hole (its mass, charge, and spin), not on how it was formed. This allows us to use the highly idealized Vaidya model to derive a general result.

2. Tracing Modes from the Future to the Past

Our task is to find the Bogoliubov coefficients that relate the "out" modes detected at future null infinity (${\mathcal{J}}^{+}$) to the "in" modes that originated at past null infinity (${\mathcal{J}}^{-}$). The "out" modes, which an observer at ${\mathcal{J}}^{+}$ perceives as particles, are positive-frequency solutions with respect to the retarded time coordinate $u=t-r_{\ast }$. A basis for these modes is given by $p_{\omega }\propto e^{-i\omega u}$.

To find the $\beta$ coefficients, we need to express this outgoing mode $p_{\omega }$ in the basis of "in" modes, which are positive-frequency with respect to the advanced time coordinate $v=t+r_{\ast }$ at ${\mathcal{J}}^{-}$. This means we must trace the outgoing mode backward in time from ${\mathcal{J}}^{+}$ to ${\mathcal{J}}^{-}$ and see what it looks like.

Solving the full Klein-Gordon equation for a mode propagating through the collapsing spacetime is prohibitively difficult. Instead, we use the geometric optics approximation. This is valid for high-frequency modes (short wavelengths) and allows us to approximate the propagation of a field mode by tracing the path of a classical null geodesic (a light ray).

Consider a light ray that reaches ${\mathcal{J}}^{+}$ at a very late retarded time $u$. For this ray to have escaped the black hole, it must have passed through the collapsing matter just before the event horizon formed. This means it must have originated from ${\mathcal{J}}^{-}$ at an advanced time $v$ that is exponentially close to the time $v_H$ of the very last light ray that could have escaped.

The relationship between the retarded time $u$ of the outgoing ray at ${\mathcal{J}}^{+}$ and the advanced time $v$ of the corresponding ingoing ray at ${\mathcal{J}}^{-}$ is found by integrating the geodesic equation across the near-horizon region. The result is:
$u\approx -4M\ln \left(\frac{v_H-v}{C}\right)$
where $v_H$ is the advanced time corresponding to the horizon-forming ray, $M$ is the black hole mass, and $C$ is a constant. This logarithmic relationship is the mathematical heart of Hawking radiation. It shows that a linear, infinite stretch of outgoing time $u$ is compressed into a finite, exponentially shrinking interval of ingoing time $v$ as $v\to v_H$. This extreme "stretching" of the modes is what mixes positive and negative frequencies.

3. Calculating the Bogoliubov Coefficients

We are now ready for the central calculation. We take an outgoing positive-frequency mode at ${\mathcal{J}}^{+}$, $p_{\omega }\propto \frac{1}{r}{e}^{-i\omega u}$, and use the relation we just found to express it in terms of the "in" coordinate $v$.

Inverting the relation gives $v_H-v=C{e}^{-u/4M}$. Substituting this into the mode gives:
$p_{\omega }(v)\propto \frac{1}{r}{e}^{-i\omega [-4M\ln ((v_H-v)/C)]}=\frac{1}{r}{\left(\frac{v_H-v}{C}\right)}^{4iM\omega }$
This function of $v$ is clearly not a simple plane wave $e^{-i{\omega }^{\prime }v}$. It is a complex superposition of different frequencies. To find the Bogoliubov coefficients, we must project this mode onto the basis of "in" modes $f_{{\omega }^{\prime }}$ and their complex conjugates $f_{{\omega }^{\prime }}^{\ast }$.

where $(\cdot ,\cdot {)}_{\text{KG}}$ is the Klein-Gordon inner product.

The actual calculation of these integrals is a beautiful application of complex analysis. We will not perform it line-by-line, but instead study the key steps and the logic involved.

The critical result from this calculation is the relationship between the magnitudes of the Bogoliubov coefficients for a given mode frequency $\omega$:
$|{\alpha }_{\omega }|=e^{4\pi M\omega }|{\beta }_{\omega }|$
Since the surface gravity of a Schwarzschild black hole is $\kappa =\frac{1}{4M}$, we can write this more generally as:
$|{\alpha }_{\omega }|=e^{\pi \omega /\kappa }|{\beta }_{\omega }|$
This single equation contains all the physics of the thermal emission. It tells us that ${\beta }_{\omega }$ is non-zero, confirming that particle creation occurs. Furthermore, the exponential factor is precisely what will give rise to a thermal distribution.

4. The Planck Spectrum and Hawking Temperature

We have now assembled all the necessary pieces. The Bogoliubov coefficients must satisfy a normalization condition that arises from the preservation of the quantum field's commutation relations. This is a statement of unitarity.

So, we have a system of two equations for the two unknown coefficients, $|{\alpha }_{\omega }|$ and $|{\beta }_{\omega }|$:

1. $|{\alpha }_{\omega }{|}^2-|{\beta }_{\omega }{|}^2=1$ (Unitarity)
2. $|{\alpha }_{\omega }|=e^{\pi \omega /\kappa }|{\beta }_{\omega }|$ (From our mode calculation)

The average number of "out" particles with frequency $\omega$ detected at ${\mathcal{J}}^{+}$, assuming the "in" state was the vacuum, is given by $⟨N_{\omega }⟩=|{\beta }_{\omega }{|}^2$. Therefore, we have found:
$⟨N_{\omega }⟩=\frac{1}{e^{2\pi \omega /\kappa }-1}$
This is the Bose-Einstein distribution for a gas of bosons in thermal equilibrium!

By comparing this to the standard formula for a Planckian blackbody spectrum, $⟨N_{\omega }⟩=\frac{1}{e^{\hslash \omega /k_{B}T}-1}$, we can directly read off the temperature of the radiation. Restoring fundamental constants for a moment, the correspondence is:
$\frac{\hslash \omega }{k_{B}T}=\frac{2\pi \omega }{\kappa }$
This gives the celebrated formula for the Hawking temperature:
$T_H=\frac{\hslash \kappa }{2\pi k_B}$
Setting $\hslash =k_B=c=G=1$, and using the surface gravity for a Schwarzschild black hole $\kappa =1/(4M)$, the temperature is:
$T_H=\frac{1}{8\pi M}$
This is a stunning result. It states that a black hole radiates as a perfect blackbody with a temperature inversely proportional to its mass.

Conclusion

In this lesson, you have followed the steps of one of the most profound arguments in theoretical physics. By combining the principles of general relativity with quantum field theory, we have shown that black holes are not truly "black" but instead radiate particles with a perfect thermal spectrum.

Key Takeaways:

• The derivation relies on modeling a non-stationary collapse, which drives the particle creation process.
• The geometric optics approximation simplifies the problem to tracing null rays. The key physical input is the extreme gravitational redshift near the forming event horizon.
• This redshift leads to a logarithmic relationship between the null coordinates at past and future infinity, $u\approx -4M\ln (C(v_H-v))$.
• This coordinate stretching mixes positive and negative frequency modes, leading to non-zero Bogoliubov $\beta$ coefficients.
• By combining the calculated relation $|{\alpha }_{\omega }|=e^{\pi \omega /\kappa }|{\beta }_{\omega }|$ with the unitarity condition $|{\alpha }_{\omega }{|}^2-|{\beta }_{\omega }{|}^2=1$, we found that the spectrum of created particles is a perfect Planck distribution.
• The temperature of this distribution is the Hawking Temperature, $T_H=\frac{\kappa }{2\pi }$, which for a Schwarzschild black hole is $T_H=\frac{1}{8\pi M}$.

Preview of the Next Lesson:

We have successfully connected the laws of black hole mechanics from a previous module with thermodynamics, by giving a physical temperature to the black hole. The Hawking temperature $T_H$ is proportional to surface gravity $\kappa$, just as the First Law of Black Hole Mechanics suggested. This leads to our next topic: identifying the entropy. In the next lesson, we will relate Hawking radiation to black hole thermodynamics, identifying the Bekenstein-Hawking entropy, and discuss the profound implications of a black hole having entropy.