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Arthur Suvorov
Arthur Suvorov
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This is an interesting question, and I think ultimatly relates to the black hole information paradox. The first law of black hole thermodynamics states that $$TdS = dE - dW$$ where the variables have their usual meaning. From this one can derive what is called the Bekenstein-Hawking entropy $$S_{BH} = \frac {c^3 A} {4 G \hbar}$$ where $A$ is the surface area of the black hole. As correctly pointed out by JamalS, in the Schwarzschild geometry this takes the form: $$S = 4 \pi G M^2$$ What this is in effect saying is that only the $\textit{mass}$ of an object that has the unfortunate fate of bonding with the black hole will contribute to its entropy. So, while in a classical system you are correct in saying opening the box increases the entropy of the system, the information contained in the partition is scrambled as it crosses the horizon. As such, only the combined mass content of you and the box contribute to the entropy regardless of whatever atomic structure the two of you may have contained.

For more complicated black hole models the surface area may contain terms involving the angular momentum, $J$, and the net charge, $Q$ with electric potential $\Phi$, and the term $dW$ in the first equation has the form $$dW = \Omega dJ + \Phi d Q$$ However, the classical black hole no hair theorems state that the black hole is uniquely characterised by $M$, $Q$ and $J$. This is what is called 'no-hair', since no other information contained in the infalling particles is retained. When one introduces some strange Yang-Mills fields (exotic matter) and things into the scenario the black hole may exhibit some hair (see e.g [http://cds.cern.ch/record/312731/files/9610019.pdf]). So, one could argue that there may exist exotic states of matter that when they fall into the black hole they maintain some of their internal information. In light of this, the Bekenstein-Hawking formula would need a correction term and you may indeed be able to increase the entropy (and consequentially surface area) by having some clever partition set up like you describe provided the internal atoms consist of exotic matter.

Edit: I may have messed up some of the multiplicative constants in the above, but the picture is the same.

This is an interesting question, and I think ultimatly relates to the black hole information paradox. The first law of black hole thermodynamics states that $$TdS = dE - dW$$ where the variables have their usual meaning. From this one can derive what is called the Bekenstein-Hawking entropy $$S_{BH} = \frac {c^3 A} {4 G \hbar}$$ where $A$ is the surface area of the black hole. As correctly pointed out by JamalS, in the Schwarzschild geometry this takes the form: $$S = 4 \pi G M^2$$ What this is in effect saying is that only the $\textit{mass}$ of an object that has the unfortunate fate of bonding with the black hole will contribute to its entropy. So, while in a classical system you are correct in saying opening the box increases the entropy of the system, the information contained in the partition is scrambled as it crosses the horizon. As such, only the combined mass content of you and the box contribute to the entropy regardless of whatever atomic structure the two of you may have contained.

For more complicated black hole models the surface area may contain terms involving the angular momentum, $J$, and the net charge, $Q$ with electric potential $\Phi$, and the term $dW$ in the first equation has the form $$dW = \Omega dJ + \Phi d Q$$ However, the classical black hole no hair theorems state that the black hole is uniquely characterised by $M$, $Q$ and $J$. When one introduces some strange Yang-Mills fields and things into the scenario the black hole may exhibit some hair (see e.g [http://cds.cern.ch/record/312731/files/9610019.pdf]). So, one could argue that there may exist exotic states of matter that when they fall into the black hole they maintain some of their internal information. In light of this, the Bekenstein-Hawking formula would need a correction term and you may indeed be able to increase the entropy (and consequentially surface area) by having some clever partition set up like you describe provided the internal atoms consist of exotic matter.

This is an interesting question, and I think ultimatly relates to the black hole information paradox. The first law of black hole thermodynamics states that $$TdS = dE - dW$$ where the variables have their usual meaning. From this one can derive what is called the Bekenstein-Hawking entropy $$S_{BH} = \frac {c^3 A} {4 G \hbar}$$ where $A$ is the surface area of the black hole. As correctly pointed out by JamalS, in the Schwarzschild geometry this takes the form: $$S = 4 \pi G M^2$$ What this is in effect saying is that only the $\textit{mass}$ of an object that has the unfortunate fate of bonding with the black hole will contribute to its entropy. So, while in a classical system you are correct in saying opening the box increases the entropy of the system, the information contained in the partition is scrambled as it crosses the horizon. As such, only the combined mass content of you and the box contribute to the entropy regardless of whatever atomic structure the two of you may have contained.

For more complicated black hole models the surface area may contain terms involving the angular momentum, $J$, and the net charge, $Q$ with electric potential $\Phi$, and the term $dW$ in the first equation has the form $$dW = \Omega dJ + \Phi d Q$$ However, the classical black hole no hair theorems state that the black hole is uniquely characterised by $M$, $Q$ and $J$. This is what is called 'no-hair', since no other information contained in the infalling particles is retained. When one introduces some strange Yang-Mills fields (exotic matter) and things into the scenario the black hole may exhibit some hair (see e.g [http://cds.cern.ch/record/312731/files/9610019.pdf]). So, one could argue that there may exist exotic states of matter that when they fall into the black hole they maintain some of their internal information. In light of this, the Bekenstein-Hawking formula would need a correction term and you may indeed be able to increase the entropy (and consequentially surface area) by having some clever partition set up like you describe provided the internal atoms consist of exotic matter.

Edit: I may have messed up some of the multiplicative constants in the above, but the picture is the same.

Source Link
Arthur Suvorov
Arthur Suvorov
  • 1.5k
  • 2
  • 12
  • 25

This is an interesting question, and I think ultimatly relates to the black hole information paradox. The first law of black hole thermodynamics states that $$TdS = dE - dW$$ where the variables have their usual meaning. From this one can derive what is called the Bekenstein-Hawking entropy $$S_{BH} = \frac {c^3 A} {4 G \hbar}$$ where $A$ is the surface area of the black hole. As correctly pointed out by JamalS, in the Schwarzschild geometry this takes the form: $$S = 4 \pi G M^2$$ What this is in effect saying is that only the $\textit{mass}$ of an object that has the unfortunate fate of bonding with the black hole will contribute to its entropy. So, while in a classical system you are correct in saying opening the box increases the entropy of the system, the information contained in the partition is scrambled as it crosses the horizon. As such, only the combined mass content of you and the box contribute to the entropy regardless of whatever atomic structure the two of you may have contained.

For more complicated black hole models the surface area may contain terms involving the angular momentum, $J$, and the net charge, $Q$ with electric potential $\Phi$, and the term $dW$ in the first equation has the form $$dW = \Omega dJ + \Phi d Q$$ However, the classical black hole no hair theorems state that the black hole is uniquely characterised by $M$, $Q$ and $J$. When one introduces some strange Yang-Mills fields and things into the scenario the black hole may exhibit some hair (see e.g [http://cds.cern.ch/record/312731/files/9610019.pdf]). So, one could argue that there may exist exotic states of matter that when they fall into the black hole they maintain some of their internal information. In light of this, the Bekenstein-Hawking formula would need a correction term and you may indeed be able to increase the entropy (and consequentially surface area) by having some clever partition set up like you describe provided the internal atoms consist of exotic matter.