What is the Hawking temperature of a black hole? IPhO Problem
TLDRIn this educational video, the concept of Hawking radiation is explored, explaining how black holes emit radiation and possess a temperature. The presenter uses the first principle of thermodynamics and the definition of entropy to derive the formula for the temperature of a black hole, showing that it is inversely proportional to its mass. By plugging in values for an average-sized black hole, the video illustrates that the resulting Hawking temperature is extremely low, highlighting the significance of this theory in understanding the behavior of black holes and their potential to lose mass over time.
Takeaways
- 🌌 Stephen Hawking was the first to propose that black holes emit what is now known as Hawking radiation.
- 📚 The problem discussed is from the 2007 International Physics Olympiad, part 3.1, and involves calculating the Hawking temperature of a black hole.
- 🔄 The first principle of thermodynamics and the definition of entropy are used in conjunction with Einstein's mass-energy equivalence formula, E = mc^2, to derive the Hawking temperature.
- ⚙️ The script rearranges the thermodynamic equation dE = TdS + δW to find dQ = TdS under the assumption of no work done (δW = 0) on an isolated black hole.
- 🔢 The temperature θ is derived to be θ = dE/dS, which is then inverted to differentiate the entropy with respect to mass.
- 📉 The entropy of a black hole is differentiated with respect to its mass, yielding an expression involving the gravitational constant, Planck's constant, and Boltzmann's constant.
- 🌡 The final expression for the Hawking temperature is derived as θ = hc^3 / 8πGkBM, where h is Planck's constant, c is the speed of light, G is the gravitational constant, kB is Boltzmann's constant, and M is the mass of the black hole.
- 🔢 The script calculates the Hawking temperature for a black hole of about five solar masses, resulting in a very low temperature of approximately 9.7 × 10^−7 Kelvin.
- 🌡 The significance of the Hawking temperature is highlighted, indicating that if black holes have a temperature, they must radiate and therefore lose mass over time.
- 🎥 The video promises a follow-up to derive the rate at which a black hole's mass changes due to Hawking radiation in the next installment.
Q & A
Who first proposed the concept of black holes emitting radiation known as Hawking radiation?
-Stephen Hawking was the first scientist to argue that black holes emit radiation, now known as Hawking radiation.
What is the significance of the term 'Hawking temperature' in the context of black holes?
-Hawking temperature refers to the temperature at which a black hole emits Hawking radiation, indicating that black holes are not completely black but have a faint glow due to this radiation.
What principle from thermodynamics is used in the script to derive the Hawking temperature?
-The first principle of thermodynamics, which states that the change in internal energy (dE) is equal to the heat added to the system (dQ) plus the work done by the system (dW), is used to derive the Hawking temperature.
How is the entropy of a black hole related to its temperature in the script?
-The entropy of a black hole is related to its temperature through the equation ds = dq/T, where ds is the change in entropy, dq is the heat added, and T is the temperature.
What assumption is made regarding the work done by the black hole in the derivation?
-The assumption made is that dw (the work done by the black hole) is equal to zero, as there is no work done in the case of an isolated black hole in the middle of space.
What mathematical trick is used to differentiate the entropy of a black hole?
-The mathematical trick used is to take the inverse of the expression for temperature (dE/dS) and then differentiate the entropy expression with respect to mass (m).
What is the role of Einstein's famous equation E=mc^2 in the derivation of Hawking temperature?
-Einstein's equation E=mc^2 is applied to an infinitesimal mass (dm) to relate the change in energy (dE) to the change in mass, which is then used in the expression for the Hawking temperature.
How is the expression for the Hawking temperature derived from the entropy and mass of a black hole?
-The expression for the Hawking temperature is derived by differentiating the entropy with respect to mass, and then plugging it into the rearranged first principle of thermodynamics, considering the energy-mass equivalence.
What fundamental constants are involved in the final expression for the Hawking temperature?
-The fundamental constants involved in the final expression for the Hawking temperature are the speed of light (c), Planck's constant (h), the gravitational constant (G), and Boltzmann's constant (kB).
What is the calculated Hawking temperature for a black hole of about five solar masses?
-The calculated Hawking temperature for a black hole of about five solar masses is approximately 9.7 x 10^-7 Kelvin.
What does the Hawking temperature imply about the behavior of black holes?
-The existence of a Hawking temperature implies that black holes emit radiation and therefore must lose mass over time, a phenomenon known as black hole evaporation.
Outlines
🌌 Introduction to Hawking Radiation and Black Hole Thermodynamics
Stephen Hawking's groundbreaking theory on black holes emitting radiation, known as Hawking radiation, is introduced. The script discusses a problem from the 2007 International Physics Olympiad, aiming to calculate the Hawking temperature of a black hole. It starts by referencing the entropy formula for a black hole and then utilizes the first principle of thermodynamics alongside the definition of entropy and Einstein's mass-energy equivalence (E=mc²) to derive the temperature. The script simplifies the problem by assuming no work is done on the black hole and rearranges thermodynamic equations to find an expression for temperature in terms of entropy and energy changes.
🔢 Calculating the Hawking Temperature of a Black Hole
The script continues by applying the derived formula for the Hawking temperature to calculate the temperature of a black hole with an average mass of five solar masses. It substitutes fundamental constants such as the speed of light, Planck's constant, the gravitational constant, and Boltzmann's constant, along with the mass of the black hole, into the formula. The resulting Hawking temperature is extremely low, approximately 9.7 x 10^-7 Kelvin, indicating that black holes radiate very faintly. The significance of this result is highlighted, as it implies that black holes can lose mass through radiation. The script concludes with a teaser for the next video, which will explore the rate at which black holes lose mass.
Mindmap
Keywords
💡Hawking Radiation
💡Black Hole Entropy
💡First Law of Thermodynamics
💡Entropy
💡Thermodynamics
💡Energy-Mass Equivalence
💡Planck's Constant
💡Boltzmann's Constant
💡Gravitational Constant (G)
💡Hawking Temperature
Highlights
Stephen Hawking was the first to argue that black holes emit radiation, now known as Hawking radiation.
The problem is derived from the 2007 International Physics Olympiad, part 3.1.
The entropy of a black hole is expressed with a specific formula.
The first principle of thermodynamics and the definition of entropy are used to find Hawking radiation.
dS is equal to dq/T, where T is the temperature, rearranged to find dq.
Assuming no work is done (dw=0) for an isolated black hole in space.
Theta (temperature) is derived as dE/dS, rearranged from the first law of thermodynamics.
The expression for the Hawking temperature is inverted for mathematical differentiation.
E=mc^2 is applied to an infinitesimal mass dm to find an expression for dE.
Differentiating the entropy expression with respect to mass (m) yields a new formula for dS/dM.
The Hawking temperature formula is derived involving fundamental constants and the black hole's mass.
Hawking temperature is expressed as a function of the speed of light, Planck's constant, gravitational constant, Boltzmann's constant, and the black hole's mass.
A practical calculation of the Hawking temperature for a black hole of average size (5 solar masses) is performed.
The calculated Hawking temperature is approximately 9.7 x 10^-7 Kelvin.
The significance of black holes having a temperature implies they emit radiation and lose mass.
An upcoming video will derive the rate of mass change for a black hole.
Transcripts
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