- Absence of an Effective Horizon for Black Holes in Gravity's Rainbow
Ahmed Farag Ali, Mir Faizal, Barun Majumder
Europhys.Lett. 109 (2015) 20001
The paper uses a modification of General Relativity known under the name of “rainbow gravity” which means that the metric and so the space-time background is energy-dependent. Dependent on which energy, you ask rightfully. I don’t know. Everyone who writes papers on this makes their own pick. Rainbow gravity is an ill-defined framework that has more problems than I can list here. In the paper the authors motivate it, amazingly enough, by string theory.The argument goes somewhat like this: rainbow gravity has something to do with deformed special relativity (DSR), some versions of which have something to do with a minimal length, which has something to do with non-commutative geometry, which has something to do with string theory. (Check paper if you don’t believe this is what they write.) This argument has more gaps than the sentence has words.
To begin with DSR was formulated in momentum space. Rainbow gravity is supposedly a formulation of DSR in position space, plus that it takes into account gravity. Except that it is known that the only ways to do DSR in position space in a mathematically consistent way either lead to violations of Lorentz-invariance (ruled out) or violations of locality (also ruled out).
This was once a nice idea that caused some excitement, but that was 15 years ago. For what I am concerned, papers on the topic shouldn’t be accepted for publication any more unless these problems are solved or at least attempted to be solved. At the very least the problems should be mentioned in an article on the topic. The paper in question doesn’t list any of these issues. Rainbow gravity isn’t only not new, it is also not a theory. It once may have been an idea from which a theory might have been developed, but this never happened. Now it’s a zombie idea that doesn’t die because journal editors think it must be okay if others have published papers on it too.
There is one way to make sense of rainbow gravity which is in the context of running coupling constants. Coupling constants, including Newton’s constant, aren’t actually constant, but depend on the energy scale that the physics is probed with. This is a well-known effect which can be measured for the interactions in the standard model and it is plausible that it should also exist for gravity. Since the curvature of spacetime depends on the strength of the gravitational coupling, the metric then becomes a function of the energy that it is probed with. This is to my knowledge also the only way to make sense of deformed special relativity. (I wrote a paper on this with Xavier and Roberto some years ago.) Alas, to see any effect from this you’d need to do measurements at Planckian energies (com), and the energy-dependent metric would only apply directly in the collision region.
In their paper the authors allude to some “measurement” that supposedly sets the energy in their metric. Unfortunately, there is never any observer doing any measurement, so one doesn’t know which energy it is. It’s just a word that they appeal to. What they do instead is making use of a known relation in some versions of DSR that prevents one from measuring distances below the Planck length. They then argue that if one cannot resolve structures below the Planck length then the horizon of a black hole cannot be strictly speaking defined. That quantum gravity effects should blur out the horizon to finite width is correct in principle.
Generally, all surfaces of zero width, like the horizon, are mathematical constructs. This is hardly a new insight, but it’s also not very meaningful. The “surface of the Earth” for example doesn’t strictly speaking exist either. You will still smash to pieces if you jump out of a window, you just can’t tell exactly where you will die. Similarly, that the exact location of the horizon cannot be measured doesn’t mean that the space-time does no longer have a causally disconnected region. You just can’t tell exactly when you enter it. The authors’ statement that:
“The absence of an effective horizon means there is nothing absolutely stopping information from going out of the black hole.”is therefore logically equivalent to the statement that there is nothing absolutely stopping you at the surface of the Earth when you jump out the window.
The paper also contains a calculation. The authors first point out that in the normal metric of the Schwarzschild black hole an infalling observer needs a finite time to cross the horizon, but for a faraway observer it looks like it takes an infinite time. This is correct. If one calculates the time in the faraway observer’s coordinates it diverges if the infalling observer approaches the horizon. The authors then find out that it takes only a finite time to reach a surface that is still a Planck length away from the horizon. This is also correct. It’s also a calculation that normally is assigned to undergrad students.
They try to conclude from this that the faraway observer sees a crossing of the horizon in finite time, which doesn’t make sense because they’ve previously argued that one cannot measure exactly where the horizon is, though they never say who is measuring what and how. What it really means is that the faraway observer cannot exactly tell when the horizon is crossed. This is correct too, but since it takes an infinite time anyway, the uncertainty is also infinite. The authors then argue: “Divergence in time is actually an signal of breakdown of spacetime description of quantum theory of gravity, which occurs because of specifying a point in spacetime beyond the Planck scale.” The authors, in short, conclude that if an observer cannot tell exactly when he reaches a certain distance, he can never cross it. Thus the position at which the asymptotic time diverges is never reached. And the observer is never causally connected.
In their paper, this reads as follows:
“Even though there is a Horizon, as we can never know when a string cross it, so effectively, it appears as if there is no Horizon.”Talking about strings here is just cosmetics, the relevant point is that they believe if you cannot tell exactly when you cross the horizon, you will never become causally disconnected, which just isn’t so.
The rest of the paper is devoted to trying to explain what this means, and the authors keep talking about some measurements which are never done by anybody. If you would indeed make a measurement that reaches the Planck energy (com) at the horizon, you could indeed locally induce a strong perturbation, thereby denting away the horizon a bit, temporarily. But this isn’t what the authors are after. They are trying to convince the reader that the impossibility of resolving distances arbitrarily well, though without actually making any measurement, bears some relevance for the causal structure of spacetime.
A polite way to summarize this finding is that the calculation doesn’t support the conclusion.
This paper is a nice example though to demonstrate what is going wrong in theoretical physics. It isn’t actually that the calculation is wrong, in the sense that the mathematical manipulations are most likely correct (I didn’t check in detail, but it looks good). The problem is that not only is the framework that they use ill-defined (in their version it is plainly lacking necessary definitions, notably the transformation behavior under a change of coordinate frame and the meaning of the energy scale that they use), but that they moreover misinterpret their results.
The authors do not only not mention the shortcomings of the framework that they use but also oversell it by trying to connect it to string theory. Even though they should know that the type of uncertainty that results from their framework is known to NOT be valid in string theory. And the author of the phys.org article totally bought into this. The tragedy is of course that for the authors their overselling has worked out just fine and they’ll most likely do it again. I’m writing this in the hope to prevent it, though on the risk that they’ll now hate me and never again cite any of my papers. This is how academia works these days, or rather, doesn’t work. Now I’m depressed. And this is all your fault for pointing out this article to me.
I can only hope that Lisa Zyga, who wrote the piece at phys.org, will learn from this that solely relying on the author’s own statements is never good journalistic practice. Anybody working on black hole physics could have told her that this isn’t a newsworthy paper.