The road to black hole thermodynamics with Λ
by Dmitry Chernyavsky and Kamal Hajian
What are volume and pressure in black hole thermodynamics? That is the question!
What do the gas in a balloon and a black hole have in common? For a regular CQG reader the answer should be obvious; both can be described within the framework of thermodynamics. However we know that the gas in balloon is characterised by volume and pressure, as well as other thermodynamic quantities. So, a natural question arises about analogues of the volume and pressure for a black hole.
Answering this question, black hole physicists have noticed that if the universe is filled with a non-zero cosmological constant Λ, this mysterious entity can be absorbed in the energy-momentum tensor of matter, and its contribution resembles a perfect fluid with a pressure proportional to Λ. Continuing with this analogy, one can also introduce a ‘thermodynamic volume’ for a black hole. For instance, the appropriate volume which satisfies the first law of thermodynamics for the Schwarzschild black hole is equal to the volume of a ball with the same radius, but in flat space! Using the notions of the black hole pressure P and volume V, it is standard to vary the cosmological constant generalising the first law of black hole thermodynamics by V δP.
“Why a ball in the flat space, while we study the curved Schwarzschild?” This is a question that an acute reader might ask. We can help the reader to pose some other relevant questions, if they have not already asked them!
- Does this geometric analogy for V work for more complicated examples, e.g. the
Kerr-AdS black hole? What could be a rigorous definition for the V?
- In the traditional approach cosmological constant is an external parameter, i.e. not a property of the black hole. Varying such an external parameter in the first law is not legal. Is it?
- The energy of black hole which appears in the first law is the conserved charge of the time translation. So, it can not be enthalpy which is not a conserved charge. Hence, should not we expect P δV (or more precisely, an intensive quantity times the variation of an extensive quantity), instead of V δP in the first law?
- In the first law temperature and all chemical potentials are associated to the black hole horizon, and can be found by the information around it. But neither V nor P has such a property. Why?
It is surprising that in the literature one might not find convincing answers to these questions. Fortunately, there is a way out of the dilemma, which nicely resolves the questions all together; cosmological constant can be presented as the on-shell value of a top-form, which itself originates from an associated gauge field. A few words more to clarify; cosmological term in Lagrangian can always be removed, and be re-introduced to the field equations by the on-shell value of an auxiliary top-form field which resembles the electrodynamic field F = dA but with more indices. A top-form is a differential-form with number of indices equal to the dimension of space-time. This formulation resolves automatically the question about the legitimacy of the variation of Λ, because Λ in this formulation is a parameter in the top-form F, not the Lagrangian. We used this formulation in our work to attack other questions raised above. Firstly, we showed that in this approach cosmological constant is a conserved charge, associated to global part of a gauge symmetry. From this point of view Λ can be put on the same footing with other extensive black hole charges, such as mass, angular momentum and electric charge. Secondly, we introduced a rigorous definition for the conjugate chemical potential of this quantity, i.e. alternative quantity to V . This quantity can be read from the behaviour of the underlying gauge field on the horizon.
Telling the story in an informal way might be illuminating: “Dear reader, if you are a black hole physicist who knows how to find the thermodynamic contribution of the electromagnetic field to the black hole thermodynamics, do similar things this time for a top-form instead of the the Maxwell 2-form. You will find a conserved charge (analogous to electric charge), and a chemical potential (analogous to horizon electric potential). The former is related to the Λ, and the latter is the alternative to V, and their contribution to the first law is what we are searching for as V δP.”
Black hole thermodynamics with cosmological constant has recently received much attention. Our analysis demonstrates that the black hole thermodynamics with cosmological constant is naturally described within the top-form framework, in which cosmological constant takes its origin from a gauge field. It is of great interest to look at black hole chemistry from this point of view.
To find out more, read our paper in CQG here.
This work is licensed under a Creative Commons Attribution 3.0 Unported License.