# Black hole accretion of dark energy

Dark energy physically can be interpreted as either a fluid with positive mass but pressure the negative of its density (pressure has units of energy/volume, and energy is mass), or a property of space. If it's a fluid, it should add to the mass of black holes like any form of energy (no hair), and the black hole should grow? However, if dark energy is a property of space, then this won't happen. Is my reasoning correct that we can differentiate (in theory) by looking at black hole's growth rate?

Edit: There are many ways to define growth quantitativly. For example, we can put test particles in orbit and look at decreases in their proper time to orbit the hole.

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Just a small point: pressure is exactly the negative of energy density (which itself is positive) for canonical dark energy (and for $c=G=1$). –  Chris White Feb 17 '13 at 5:33

If the dark energy is a true cosmological constant, there is a known exact solution of Einstein's equation where there is no accretion of dark energy, and there is a time-translation symmetry to the global spacetime. You get this by simply replacing $1-\frac{2M}{r}$ in the Schwarzschild solution with $1-\frac{2M}{r} - \frac{1}{3}\Lambda r^{2}$. Generalizations to charged and spinning black holes exist that also maintain this time-translation symmetry.

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No, it is not possible to differentiate between these two interpretations because they're ultimately physically equivalent.

First, we should separate the discussions in cases where the total energy is conserved and where it isn't. The existence of conserved energy in general relativity (which must be ADM-like) actually requires vanishing or negative cosmological constant – the spacetime is Minkowski or AdS. In de Sitter space, there's no nonzero gauge-invariant definition of energy that would be conserved because the de Sitter space has no asymptotic region at infinity.

In the de Sitter space, masses of objects may therefore change in various general ways and by measuring them, you can't deduce pretty much anything.

In Minkowski or AdS space, the energy is conserved. Let's consider an anti de Sitter space with a negative cosmological constant. This means $\rho\lt 0$, a negative energy density, with a positive pressure $p=-\rho\gt 0$. The energy of the mass that ends up as the black hole is conserved, it's the total energy in the spacetime, assuming that everything collapses. However, the value of this total mass/energy is given before the black hole is formed – it stays the same by the conservation law – which means that we can't deduce anything new if we measure the same value at the end.

What you really want to do is to "attribute" or "divide" the total mass/energy of the black hole into different regions – either the generic black hole interior or the singularity. But this "attribution" or "localization" of matter is exactly what is impossible according to general relativity. The conserved total mass/energy cannot be written as an integral of a well-defined energy density. Such a thing may only be written in the "Newtonian" limit of weak gravitational fields and the existence of black hole is exactly the opposite situation in which the "weak fields" condition is dramatically violated.

So no, your verbal descriptions of the situations are just heuristic and to see what actually happens, you need to discuss things quantitatively, using the right concepts suggested by general relativity and using the right equations. The (Einstein's) equations say a very clear thing about the impact of cosmological constant in the absence of black holes much like in their presence and any idea about "two possibilities" (the cosmological constant is a property of space or a form of energy) is a mere illusion, an artifact of non-quantitative thinking about the problem.

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Are you sure it's the same? If we have dark energy "sucked in" it can dilute the density of dark energy near the hole, but a cosmological constant by definition cannot be diluted. –  Kevin Kostlan Feb 19 '13 at 18:24
Dark energy and cosmological constant is the same thing - at most, the cosmological constant is the simplest, most well-behaved, and most likely type of dark energy (dark energy is a bit more general term). So whatever holds for dark energy holds for the cosmological constant, too. Proper dark energy can never be "diluted". One may consider generalizations such as quintessence etc. where the equation of state i.e. $p/\rho$ is postulated to be time-dependent but not dependent on the distance from a black hole. –  Luboš Motl Feb 19 '13 at 20:32
Slight quibble--"absence" should be "presence" in the last paragraph. –  Jerry Schirmer Feb 19 '13 at 21:02
@Jerry, apologies, I don't follow. The last paragraph explicitly talks about both presence and absence. –  Luboš Motl Feb 20 '13 at 8:18
@LubošMotl: misread, sorry. :) –  Jerry Schirmer Feb 20 '13 at 15:16

Here is an answer that should be more clear:

Understanding the density-volume equation of state: The relationship between density and volume for normal mass is inverse, which is intuitive because we think of mass as being conserved. We can write this as: d~x^(-3). This ignores pressure, which is accurate for normal matter (even diamond) below nuclear densities. However, if we have photons, the energy decreases as we decompress it. The number of photons is the same, but each one is red-shifted. We get: d~x^(-4). Normally, the missing energy goes into pushing the walls of the piston we expand. Energy is also conserved for photons falling into the hole. However, if space itself expands, we lose this energy permanently.

Reconciling the two interpretations Canonical dark energy has density independent of volume; it does not dilute on expanding. As it falls from a distant place into the hole, it compresses on the way in, but it does not get denser! No mass will be added to the hole. Note that only what happens outside the hole actually matters. Similarly, it won't get diluted outside the hole. The surrounding space won't get "sucked dry", and the hole won't grow. Thus dark energy permeates all space at a constant density, so it is convient mathematically to represent it as a cosmological constant, or a property of space itself.

There are other models of dark energy that do not have this property, but observations make these unlikely.

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