Tryggth’s Blog

DARPA Mathematical Challenge Twenty-two:

Settle the Smooth Poincare Conjecture in Dimension 4

• What are the implications for space-time and cosmology? And might the answer unlock the secret of “dark energy”?

OK, where was I?

Wile E. Coyote

You know how this is going to end. The coyote is not going to catch the roadrunner. But what if the coyote’s rocket were to travel at the speed of light? And what if the roadrunner constantly accelerates but never attains the speed of light?

It turns out that if  roadrunner has a sufficient head-start (and we know coyote isn’t going to light the fuse until after roadrunner swooshes by) and roadrunner has ‘constant proper acceleration’ (though the bird will never reach the speed of light!) coyote can never catch roadrunner.  Which is a little surprising. After all, its a scenario where the coyote is always traveling at a velocity greater (namely c)  than the roadrunner.

The phenomenon is known as the “rindler horizon” and a great description of this Zeno-like ‘paradox’ is provided by the technically adept science (and science fiction) writer Greg Egan here.

In the roadrunner/coyote scenario above, the coyote cannot catch and consequently cannot influence (or cook and eat) the roadrunner. You might say the coyote is ‘causally disconnected’ from the roadrunner. In fact, the roadrunner’s constant acceleration defines a rindler horizon which is also a sort of ‘causal horizon’. Another causal horizon most people are aware of is the event horizon of a black hole. Are these horizon’s similar at a more fundamental level? Remember that Unruh wrote in 1976, when writing about his acceleration temperature/radiation:

The similarity of this case with the behavior of a detector near the black hole is brought out, and it is shown that a geodesic detector near the horizon will not see the Hawking flux of particle

Ted Jacobson and Renaud Parentani wrote a wonderful survey article which also suggests there is a common underlying framework for causal horizons. In their article they write:

Any causal horizon is endowed with a surface entropy density of 1/4.

The realization that horizon entropy is an intrinsically observer dependent notion raises the obvious question of what are the states that the horizon entropy counts? The notion that it counts the number of internal configuations, i.e. configurations behind the horizon, was argued against in [57] on various grounds. It seems only even possibly viable if the holographic conjecture[58] holds, i.e. if the entire description of the world behind any horizon can be fully described on its bounding surface. It was argued in [57] that the holographic conjecture is at best valid when there is no trapped surface behind the horizon, but it may otherwise in some sense be true. Whether or not it is true, the fact remains that, for the observers who remain confined to the “outside” of the horizon, the horizon entropy somehow captures the number of ways that the world beyond the horizon can affect the world outside.

[57] T. Jacobson, “On the nature of black hole entropy,” in General Relativity and Relativistic Astrophysics: Eighth Canadian Conference, AIP Conference Proceedings 493, C. Burgess and R.C. Myers, eds. (AIP Press, 1999), pp. 85-97.

[58] R. Bousso,“The holographic principle,” Rev. Mod. Phys. 74, 825 (2002).

They conclude their 2003 article with these sentences:

While no definitive answer as to the ultimate nature of horizon entropy seems immediately at hand, an abundance of insight has been gleaned from the three decades of work. Perhaps the time is ripe to synthesize this insight and make the leap to a new conception.

In the next post I will return to Verlinde’s paper. To be continued…